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Open billiards; Lyapunov exponents; Non-wandering set; Billiard deformation +Mathematics Subject Classification (2010). 37D50, 37B10, 37D20, 34D08 +1 +Introduction +The stability and instability of a dynamical system can be studied by means of Lyapunov expo- +nents. A dynamical system is considered chaotic if it has a positive Lyapunov exponent. Examples +of chaotic systems are the dispersing billiards or so-called Sinai billiards (see [15], [16]). Billiards +are dynamical systems in which a particle moves with constant speed and hits the billiard’s wall +(boundary of the billiard’s domain) according to the law of geometrical optics,“the angle of inci- +dence equals the angle of reflection”. Open billiards are a particular case of billiards in unbounded +domains. The domain is the exterior of finitely many strictly convex compact obstacles satisfying +the no-eclipse condition (H) of Ikawa [6]: the convex hull of any two obstacles does not intersect +with another obstacle; in other words, there does not exist a straight line that intersects more +than two obstacles. It follows from Sinai [15], [16] (see also [14]) that the non-wandering set of +the open billiard map is hyperbolic (i.e. there exist positive and negative Lyapunov exponents). +Many studies have investigated Lyapunov exponents for billiards (see [22], [1], [4], [9], [10]). In +this paper, we estimate the largest Lyapunov exponent for open billiard in R2. We demonstrate +that the Lyapunov exponent depends continuously on a parameter α related to a deformation of +the billiard as defined in [21]. Moreover, we prove that the Lyapunov exponent is differentiable +with respect to the deformation parameter α. +Here we state the main results: +In the following theorems, we denote the billiard deformation by K(α) where α ∈ [0, b]. +See +Section 4 for the precise definition. +aDepartment of Mathematics and Statistics, School of Physics, Mathematics and Computing, University of +Western Australia, Perth, WA 6009, Australia +Email address: amal.aldowais@research.uwa.edu.au +bDepartment of Mathematics, College of Science and Arts, Najran University, Najran, Saudi Arabia +Email address: amalduas@nu.edu.sa +1 + +Theorem 1.1. (Continuity) Let K(α) be a C4,2 billiard deformation in R2. Let λ1(α) be the +largest Lyapunov exponent for K(α). Then the largest Lyapunov exponent is continuous as a +function of α. +Theorem 1.2. (Differentiability) Let K(α) be a C5,3 billiard deformation in R2. Let λ1(α) be +the largest Lyapunov exponent for K(α). Then λ1(α) is C1 with respect to α. +There are many works studying continuity properties of Lyapunov exponents (see e.g. [20], [3]). +However, to our knowledge, in all these continuity is established generically, i.e. with respect to +“most” (typical) values of the parameters/perturbations involved. In the case of the open billiard +considered in the present paper we establish continuity and even differentiability for all values +of the parameter that appears in the perturbation, which is a truly remarkable property of this +physical system. +2 +Preliminaries +This section provides some preliminary concepts for open billiards, billiard flow, symbolic coding +and stable/unstable manifolds. We also describe some notations related to curvatures, distances, +and collision angles. In the last part of this section, we state the Oseledets multiplicative ergodic +theorem and its consequence for open billiards. +2.1 +Open billiard +Let Ki, where i = 1, 2, 3, ..., z0, be strictly convex compact domains with smooth boundaries ∂Ki +in R2. In this paper, we assume that K = � +i Ki satisfies the following condition(H) of Ikawa [6]: +for any i ̸= j ̸= k the convex hull of Ki ∪Kk does not have any common points with Kj. Let Ω be +the exterior of K (i.e., Ω = R2\K). Let Φt, t ∈ R, be the billiard flow such that for any particle +x = (q, v), where q ∈ Ω represents the position of x and v is the unit velocity of the particle x, +then Φt(x) = (qt, vt) = (q + tv, v). When the particle hits the boundary, then the velocity follows +the collision law vnew = vold − 2 < vold, n > n, where n is the outwards unit normal vector to ∂K +at q, and φ the angle between n = n(q) and v. +We denote the time of the j-th reflection of x by tj(x) ∈ (−∞, ∞) for j ∈ Z. We say tj(q) = ∞ +(tj(q) = −∞) if the forwards (backwards) trajectory of x has less than j reflections. We denote +the non-wandering set of the flow Φt by Λ = {x ∈ �Ω, |tj(x)| < ∞, for all j ∈ Z}, where +�Ω = {(q, v) ∈ int Ω × S1 or (q, v) ∈ ∂Ω × S1 : ⟨n(q), v⟩ ≥ 0}. +Now let M = {x = (q, v) ∈ ∂K × S1 : ⟨n(q), v⟩ ≥ 0} and let π : M → ∂K be the canonical +projection map defined by π(q, v) = q. Let t1(x) be the time of the first reflection of x and let +M1 = {x ∈ M : t1(x) < ∞}. Define the billiard ball map B : M1 → M by B(x) = Φt1(x)(x), (e.g. +if y = (p0, w0), where p0 lies on ∂Ki then B(y) = B(p0, w0) = (p1, w1) where p1 = p0+t1w0 ∈ ∂Kj +and w1 = w0 − 2 < w0, n > n). The non-wandering set of the open billiard map is M0 = {x ∈ +M : |tj(x)| < ∞} which is a subset of Λ. Finally, let B : M0 → M0 be the restriction of the +open billiard map on the non-wandering set M0. It is obvious that the non-wandering set is an +invariant set. See [15], [16], [4], [5], [14], for general information about billiard dynamical systems. +2 + +2.2 +Symbolic coding for open billiards +Each particular x ∈ M0 can be coded by a bi-infinite sequence +ξ(x) = (..., ξ−1, ξ0, ξ1, ...) ∈ {1, 2, ..., z0}Z, +in which ξi ̸= ξi+1, for all i ∈ Z, and ξj indicates the obstacle Kξj such that πBj(x) ∈ ∂Kξj. For +example, if there are three obstacles K1, K2 and K3 as above and a particular q repeatedly hits +K1, K3, K2, K1, K3, K2, then the bi-infinite sequence is (..., 1, 3, 2, 1, 3, 2, ...). Let Σ be the symbol +space which is defined as: +Σ = {ξ = (..., ξ−1, ξ0, ξ1, ...) ∈ {1, 2, ..., z0}Z : ξi ̸= ξi+1, ∀i ∈ Z}. +Define the representation map R : M0 → Σ by R(x) = ξ(x). Let σ : Σ → Σ be the two-sided +subshift map defined by σ(ξi) = ξi+1. Given θ ∈ (0, 1) define the metric dθ on Σ by: +dθ(ξ, η) = +� 0 +if +ξi = ηi for all i ∈ Z +θn +if +n = max{j ≥ 0 : ξi = ηi for all |i| < j} +Then σ is continuous with respect to dθ ([2]). +It is also known that the representation map +R : M0 → Σ is a homeomorphism (see e.g. [14]). See [6], [8], [11], [14], [17], for topics related to +symbolic dynamics for open billiards. +2.3 +Lyapunov exponents +Here we state a consequence of Oseledets Multiplicative Ergodic Theorem for billiards (see e.g. +Ch. 2 in [13], also see [12], [20], [7]). +For the open billiard map B : M0 −→ M0 in R2 we will use the coding R : M0 −→ Σ from Section +2.2, which conjugates B with the shift map σ : Σ −→ Σ, to define Lyapunov exponents. It is +well known that there are ergodic σ-invariant measures µ on Σ. Let µ be an ergodic σ-invariant +probability measure on Σ. The following is a consequence of Oseledets Multiplicative Ergodic +Theorem: +Theorem 2.1 (A Consequence of Oseledets Multiplicative Ergodic Theorem). There exist real +numbers λ1 > 0 > −λ1 and one-dimensional vector subspaces Eu(x) and Es(x) of Tx(∂K), +x ∈ M0, depending measurably on R(x) ∈ Σ such that: +(i) Eu(x) and Es(x) for almost all x ∈ M0; +(ii) DxB(Eu(x)) = Eu(B(x)) and DxB(Es(x)) = Es(B(x)) for almost all x ∈ M0, and +(iii) For almost all x ∈ M0 there exists +lim +n→∞ +1 +n log ∥DxBn(w)∥ = λ1 +whenever 0 ̸= w ∈ Eu(x). +Here ”for almost all x” means ”for almost all R(x)” with respect to µ. The numbers λ1 > 0 > −λ1 +are called Lyapunov exponents, while the invariant subspaces Eu(x) and Es(x) are called Oseledets +subspaces. +3 + +2.4 +Propagation of unstable manifolds for open billiards +We describe a formula which is useful in getting estimates for +lim +m→∞ +1 +m log ∥DxBmw∥, (0 ̸= w ∈ Eu(x), x ∈ M0). +Let M0 be the non-wandering set of the billiard ball map B of an open billiard. Then +Λ = {Φt(x) : x ∈ M0 , t ∈ R}, +is the non-wandering set for the billiard flow Φt. For x ∈ Λ and a sufficiently small ǫ > 0 let +� +W s +ǫ (x) = {y ∈ Λ : d(Φt(x), Φt(y)) ≤ ǫ for all t ≥ 0 , d(Φt(x), Φt(y)) →t→∞ 0 }, +� +W u +ǫ (x) = {y ∈ Λ : d(Φt(x), Φt(y)) ≤ ǫ for all t ≤ 0 , d(Φt(x), Φt(y)) →t→−∞ 0 } +be the (strong) stable and unstable manifolds of size ǫ for the billiard flow. Then �Eu(x) = Tx� +W u +ǫ (x) +and �Es(x) = Tx� +W s +ǫ (x). In a similar way one defines stable/unstable manifolds for the billiard +ball map B. For any x = (q, v) ∈ M0 define +W s +ǫ (x) = {y ∈ M0 : d(Bn(x), Bn(y)) ≤ ǫ for all n ∈ N , d(Bn(x), Bn(y)) →n→∞ 0 }, +W u +ǫ (x) = {y ∈ M0 : d(B−n(x), B−n(y)) ≤ ǫ for all n ∈ N , d(B−n(x), B−n(y)) →n→∞ 0 }. +In what follows we will just write W u(x) and W s(x) for W u +ǫ (x) and W s +ǫ (x), assuming some +appropriately chosen sufficiently small ǫ > 0 is involved. Similarly for � +W u and � +W s. +It is well-known that there is an one-to-one correspondence between the stable/unstable man- +ifolds for the billiard ball map and these for the flow. +Geometrically the easiest (and most +convenient way) to describe this is as follows. +Given x = (q, v) ∈ M0 (so q ∈ ∂K and v ∈ S1), and a small 0 < r < t1(x), set y = (q + rv, v). +Then there is a 1-1 correspondence +ϕ : W u(x) −→ � +W u(y) +such that ϕ(z, w) = (z + t w, w) for all (z, w) ∈ W u(x), where t = t(z, w) > 0. Similarly, there is +a correspondence between W s(x) and � +W s(y). Moreover +Dϕ(x) : TxM0 −→ TyΛ +is so that Dϕ(x)(Eu(x)) = �Eu(y) and Dϕ(x)(Es(x)) = �Es(y). +It is known that � +W u(y) has the form � +W u(y) = �Y , where +�Y = {(p, νY (p)) : p ∈ Y } +for some smooth curve Y in R2 containing the point y such that Y is strictly convex with respect +to the unit normal field νY , i.e. the curvature of Y is strictly positive. +4 + +Next, let x and y be as above and let x1 = (q1, v1) = B(x). Then q1 = q + t1 v. Define +y1 = (q1 + r′v1, v1) for some small 0 < r′ < t2(x) − t1(x), where 0 = t0(x) < t1(x) < t2(x). Then +there is a 1-1 correspondence +ϕ1 : W u(x1) −→ ˜W u(y1) +defined as above. Again, we can write � +W u(y1) = �Y1, where +�Y1 = {(p1, νY (p1)) : p1 ∈ Y1} +for some smooth curve Y1 in R2 containing the point y1 such that Y1 is strictly convex with respect +to the unit normal field νY1. Moreover the following diagram is commutative, where t = t1 + r′: +W u(x) +B +−→ +W u(x1) +�ϕ +�ϕ1 +� +W u(y) = �Y +Φt +−→ +� +W u(y1) = �Y1 +Similarly, the following diagram is commutative: +Eu(x) +DB(x) +−→ +Eu(x1) +�Dϕ +�Dϕ1 +�Eu(y) +DΦt(y) +−→ +�Eu(y1) +Since the derivatives Dϕ and Dϕ1 are uniformly bounded, the above conjugacy can be used later +to calculate the Lyapunov exponents of the billiard ball map using propagation of appropriate +convex curves Y which we describe as follows. +Let x0 = (q0, v0) ∈ M0 and let W u +ǫ (x0) be the local unstable manifold for x0 for sufficiently +small ǫ > 0. Let t1(x0) be the time of the first reflection of x0. Then � +X = W u +ǫ (x0) = {(q, nX(q)) : +q ∈ X} for some C3 curve X in Ω such that q0 ∈ X and X is strictly convex curve with respect to +the outer unit normal field nX(q). Let X be parametrized by q(s), s ∈ [0, a], such that q(0) = q0, +and has unit normal field nX(q(s)). +Set q0(s) = q(s). +Let qj(s), j ≥ 1 be the jth-reflection +points of the forward billiard trajectory γ(s) generated by x(s) = (q(s), nX(q(s)). We assume +that a > 0 is sufficiently small so that the jth-reflection points qj(s) belong to the same boundary +component ∂Kξj for every s ∈ [0, a]. Let 0 = t0(x(s)) < t1(x(s)) < ... < tm+1(x(s)) be the times +of the reflections of the ray γ(s) at ∂K. Let κj(s) be the curvature of ∂Kξj at qj(s) and φj(s) +be the collision angle between the outward unit normal to ∂K and the reflection ray of γ(s) at +qj(s). Also, let dj(s) be the distance between two reflection points i.e. dj(s) = ∥qj+1(s) − qj(s)∥, +j = 0, 1, . . . , m. +Given a large m ≥ 1, let tm(x(s)) < t < tm+1(x(s)). Set Φt( � +X) = � +Xt. Let π(Φt(x(s))) = p(s). +Then p(s), s ∈ [0, a], is a parametrization of the C3 curve Xt = π(Φt( � +X). +Next, let k0(s) > 0 be the curvature of X at q(s). +Let tj(x(s)) < τ < tj+1(x(s)), j = +1, 2, . . . , m. Denote by uτ(s) be the shift of (q(s), n(q(s))) along the forward billiard trajectory +γ(s) after time τ > 0. Then Xτ = {uτ(s) : s ∈ [0, a]} is a C3 convex curve with respect to the +outward normal field n(uτ(s)). Let kj(s) > 0 be the curvature of Xtj = limτցtj(s) Xτ at qj(s). It +follows from Sinai [15] that +kj+1(s) = +kj(s) +1 + dj(s)kj(s) + 2 +κj+1(s) +cos φj+1(s) +, +0 ≤ j ≤ m − 1 . +(2.1) +5 + +Moreover, the curvature of Xτ at uτ(s) is +kτ(s) = +kj(s) +1 + (τ − tj(s))kj(s). +(2.2) +Set +δj(s) = +1 +1 + dj(s)kj(s) +, +1 ≤ j ≤ m . +(2.3) +Theorem 2.2. [18] For all s ∈ [0, a] we have +∥ ˙q(s)∥ = ∥ ˙p(s)∥δ1(s)δ2(s) . . . δm(s) . +(2.4) +This was proved in [18] in the 2D case and in [19] in the general case. +Finally, we want to introduce some notation related to the maximum and minimum of previous +billiard characteristies dj(s),κj(s), φj(s) and kj(s). +For all j, we have dmin ≤ dj(s) ≤ dmax, +where dmax and dmin are constants independent of j such that dmax = max{d(Ki, Kk)} and +dmin = min{d(Ki, Kk)} for i ̸= k. Also, since ∂K is strictly convex, we have constants κmin > 0 +and κmax > 0 independent of j such that κmin ≤ κj(s) ≤ κmax. And it follows from the condition +(H) that there exists a constant φmax ∈ (0, π +2 ) such that 0 ≤ φj(s) ≤ φmax < π +2 , (see e.g. [17]). +Let kj(s) be as in equation (2.1). It follows easily that kmin ≤ kj(s) ≤ kmax, where kmin = 2κmin +and kmax = +1 +dmin + +2κmax +cos φmax . +3 +Estimation of the largest Lyapunov exponent for open billiards +A formula for the largest Lyapunov exponents for a rather general class of billiards can be found +in [5], see Theorem 3.41 there. In our case we derive this formula again (see (3.1) below) and then +we use Theorem 2.2 to derive important regularity properties of the largest Lyapunov exponent. +Assume that µ is an ergodic σ-invariant measure on Σ, and let x0 = (q0, v0) ∈ M0 correspond +to a typical point in Σ with respect to µ via the representation map R. That is as in Theorem +2.1, we have +λ1 = lim +m→∞ +1 +m log ∥Dx0Bm(w)∥, +with 0 ̸= w ∈ Eu(x0). As in Sect. 2.4, let X be a (small) C3 strictly convex curve containing q0 and +having a unit normal field nX so that nX(q0) = v0. As in Sect. 2.4 again, let X be parametrised +by arc length via q(s), s ∈ [0, a], such that q(0) = q0. Let again qj(s), j = 1, 2, . . . , m + 1, be the +consecutive reflection points of the billiard trajectory γ(s) determined by x(s) = (q(s), nX(q(s)). +Given an integer m > 0 and assuming the interval [0, a] is sufficiently small, the jth reflection +points qj(s) belong to the same boundary component ∂Kξj for all s ∈ [0, a]. Next, define dj(s), +tj(x(s)), etc. as in Sect. 2.4, let tm(x(0)) < t < tm+1(x(0)), and let p(s) be the parametrisation +of � +Xt corresponding to q(s). Then the formula (2.4) in Theorem 2.2 (holds with ∥ ˙q(s)∥ = 1 from +our assumptions). Now the discussion in Sect. 2.4 implies that there exist some global constants +c1 > c2 > 0, independent of x0, X, m, etc. such that +c2∥ ˙p(s)∥ ≤ ∥Dx0Bm(w)∥ ≤ c1∥ ˙p(s)∥ +6 + +for all s ∈ [0, a]. So, by (2.4), +c2 +δ1(0)δ2(0) . . . δm(0) ≤ ∥Dx0Bm(w)∥ ≤ +c1 +δ1(0)δ2(0) . . . δm(0) +for all s ∈ [0, a]. Using this for s = 0, taking logarithms and limits as m → ∞, we obtain +− lim +m→∞ +1 +m log (δ1(0)δ2(0) . . . δm(0)) ≤ lim +m→∞ +1 +m log ∥Dx0Bm(w)∥ +≤ − lim +m→∞ +1 +m log (δ1(0)δ2(0) . . . δm(0)) . +Hence, +λ1 = lim +m→∞ − 1 +m +m +� +i=1 +log δi(0). +This implies that the largest Lyapunov exponent at the initial point x0, so at almost every point +wilt respect to the given measure µ, is given by +λ1 = lim +m→∞ +1 +m +m +� +i=1 +log +� +1 + di(0)ki(0) +� +. +(3.1) +From equation (3.1), we can estimate the largest Lyapunov exponent from below and above as +log (1 + dminkmin) ≤ λ1 ≤ log (1 + dmaxkmax). +4 +Billiard deformations +In this section, we consider some changes to the billiards in the plane, such as moving, rotating, +and changing the shape of one or multiple obstacles. This kind of billiard transformation is called a +billiard deformation as defined in [21]. We describe this deformation by adding an extra parameter +α ∈ [0, b] for some b ∈ R+, which is called the deformation parameter, to the parametrization of +the boundary of obstacles i.e., if the boundary of an obstacle parametrized by ϕ(u), it will become +ϕ(u, α). In this section, we provide the definition a billiard deformation as defined in [21]. In +addition, we describe the propagation of unstable manifolds for billiard deformations. We also +estimate the higher derivatives of some of the billiard characteristics such as distance, collision +angle and curvature, with respect to deformation parameter α. +Let α ∈ I = [0, b], for some b ∈ R+, be a deformation parameter and let ∂Ki(α) be +parametrized counterclockwise by ϕi(ui, α) and parametrized by arc-length ui. Let qi = ϕi(ui, α) +be a point that lies on ∂Ki(α). Denote the perimeter of ∂Ki(α) by Li(α), and let Pi = {(ui, α) : +α ∈ I, ui ∈ [0, Li(α)]}. +Definition 4.1. [21] For any α ∈ I = [0, b], let K(α) be a subset of R2. For integers r ≥ 4, r′ ≥ 2, +we call K(α) a Cr,r′-billiard deformation (i.e. Cr with respect to u and Cr′ with respect to α) if +the following conditions hold for all α ∈ I: +1. K(α) = �z0 +i=1 Ki(α) satisfies the no-eclipse condition (H). +7 + +2. Each Ki(α) is a compact, strictly convex set with Cr boundary and total arc length Li(α). +3. Each Ki is parametrized counterclockwise by arc-length with Cr,r′ functions ϕi : Pi → R2. +4. For all integers 0 ≤ l ≤ r, 0 ≤ l′ ≤ r′ (apart from l = l′ = 0), there exist constants C(l,l′) +ϕ +depending only on the choice of the billiard deformation and the parametrizations ϕi, such +that for all integers i = 1, 2, 3, ..., z0, +��� ∂l+l′ϕi +∂ul +i∂αl′ +��� ≤ C(l,l′) +ϕ +. +Let Bα be the open billiard map on a non-wandering set Mα for K(α). Let Σ defined in Sec. +2.2, we defined Rα : Mα → Σ by Rα(x(α)) = ξ(x(α)). We can write the points that correspond +to the billiard trajectories according to the parameterization in previous definition as follows, +π(Bj(x(α))) = qξj(α) = ϕξj(uξj(α), α) ∈ ∂Kξj(α), where uξj(α) ∈ [0, Lξj(α)]. For brevity, we will +write qj(α) = ϕj(uj(α), α). +The next corollary shows that uj(α) = uξj(α) for a fixed ξ ∈ Σ, is differentiable with respect +to α. This corollary is proved in [21]. +Theorem 4.2. [21] Let K(α) be a Cr,r′ billiard deformation with r, r′ ≥ 2. +Then uj(α) is +Cmin{r−1,r′−1} with respect to α, and there exist constants C(n) +u +> 0 such that +���dnuj(α) +dαn +��� ≤ C(n) +u . +The next corollary follows from Definition 4.1 and Theorem 4.2. +Corollary 4.3. Let K(α) be a Cr,r′ billiard deformation with r, r′ ≥ 2. Let qj(α) belongs to ∂Kξj. +Then qj(α) is Cn, where n = min{r − 1, r′ − 1}, with respect to α, and there exist constants +C(n) +q +> 0 such that +���dnqj(α) +dαn +��� ≤ C(n) +q +. +4.1 +Propagation of unstable manifolds for billiard deformations +We described the unstable manifolds propagation in Section 2.4 for open billiards. Here in this +section, we describe it for billiard deformations. +Let K(α), α ∈ [0, b] be a Cr,r′ billiard deformation as in Definition 4.1 with r ≥ 3, r′ ≥ 1. +x0(α) = (q0(α), v0(α)) ∈ Mα and let W u +ǫ (x0(α)) be the local unstable manifold for x0(α) for +sufficiently small ǫ > 0. Take a curve Xα containing q0(α) such that Xα = {q0(s, α) : s ∈ [0, a]} +is a convex curve with outer unit normal field nX(q0(s, α)) = v0(α) and C3 with respect to s. +It follows from Sinai [15], [16] that W u +ǫ (x0(α)) = {(q0(s, α), nX(q0)) : s ∈ [0, a]}. +Set � +Xα = +W u +ǫ (x0(α)). Let a ∈ R+ be small enough such that all reflection points qj(s, α), j = 1, 2, ..., m, +that are generated by x0(s, α) = (q0(s, α), nX(q0(s, α))) belong to the same boundary ∂Kξj(α). +Let dj(s, α) = ∥qj+1(s, α) − qj(s, α)∥ be the distance between two reflection points qj+1(s, α) and +qj(s, α). Denote the curvature of ∂K(α) at qj(s, α) by κj(s, α), the collision angle between the +unit normal to ∂K(α) and the reflection vector at qj(s, α) by φj(s, α), and the curvature of X at +q0(s, α) by k0(s, α) . +8 + +Let tj(x(s, α)) = tj(s, α) be the time of the j-th reflection. Given t with tj < t < tj+1 for +some j = 1, 2, ..., m, set π(Φt( � +Xα)) = Xαt. Then Xαt = {uαt(s, α) : s ∈ [0, a]} is C3 with respect +to s and a convex curve with outer unit normal field nXαt(uαt(s, α)). Denote the curvature of +Xαtj(s,α) at qj(s, α) by kj(s, α), where Xαtj(s,α) = limtցtj(s,α) Xαt. As in equation (2.1), we can +define kj(s, α) as follows: +kj+1(s, α) = +kj(s, α) +1 + dj(s, α)kj(s, α) + 2 +κj+1(s, α) +cos φj+1(s, α) +, +0 ≤ j ≤ m − 1 . +(4.1) +From now on, we will need to use previous characteristics in the case s = 0, so for brevity, +we will write dj(α) = dj(0, α), etc. Also, we denote the billiard deformation by K(α), so all of +its characteristics will be denoted dj(α), kj(α), etc. The initial open billiard is K(0) so all of its +characteristics will be denoted dj(0), etc. +4.2 +The higher derivatives of billiard characteristics +Let K(α) be a Cr,r′ billiard deformation as in +Definition 4.1 with r ≥ 4, r′ ≥ 2. Recall that +∂Kξj(α) is parametrized by arc-length ujand qj(α) = ϕj(uj(α), α) ∈ ∂Kξj(α). Here, we state +some corollaries related to bounds of the higher derivatives of curvature, distance and collision +angle of a billiard deformation. These corollaries are forthright consequences of condition 4 in +Definition 4.1. +Corollary 4.4. Let K(α) be a Cr,r′ billiard deformation with r ≥ 4, r′ ≥ 2. Then the curvature +κj(α) at qj(α) is Cn, where n = min{r − 3, r′ − 1} with respect to α and there exist constants +C(n) +κ +> 0 depending only on n such that +���dnκ +dαn +��� ≤ C(n) +κ . +Proof. Suppose K(α) is a a Cr,r′ billiard deformation with r ≥ 3, r′ ≥ 1. +Since ∂Kj(α) is +paramitrized by arc-length uj, then the curvature of ∂Kj(α) at qj(α) = ϕj(uj(α), α) is κj = ∂2ϕj +∂u2 +j , +for j = 0, 1, ..., m. Then κj(α) is Cmin{r−3,r′−1} with respect to α. +For the first derivative, we have +���dκj +dα +��� = +���∂3ϕj +∂u3 +j +∂uj +∂α + ∂3ϕj +∂u2 +j∂α +��� ≤ C(1) +κ , +this estimate was obtained in [21]. Next, we continue to estimate the second derivative, so we +have +���d2κj +dα2 +��� = +���∂4ϕj +∂u4 +j +�∂uj +∂α +�2 + ∂3ϕj +∂u3 +j +∂u2 +j +∂α2 + 2 ∂4ϕj +∂u3 +j∂α +∂uj +∂α + +∂4ϕj +∂u2 +j∂α2 +���. +By using condition 4 in Definition 4.1 and Theorem 4.2, there exists a constant C(2) +κ +> 0 such +that +9 + +���d2κj +dα2 +��� ≤ C(4,0) +ϕ +(C(1) +u )2 + C(3,0) +ϕ +C(2) +u ++ 2Cϕ(3,1)C(1) +u ++ C(2,2) +ϕ += C(2) +κ . +Continuing by induction we see that the n-th derivative, where n = min{r −3, r′ −1}, is bounded +by a constant C(n) +κ +> 0 which depends only on n such that +���dnκ +dαn +��� ≤ C(n) +κ . +Corollary 4.5. Let K(α) be a Cr,r′ billiard deformation with r ≥ 3, r′ ≥ 1. Then the distance +dj(α) between two points qj+1(α) and qj(α) is Cn, where n = min{r − 1, r′ − 1} with respect to α +and there exist constants C(n) +d +> 0 depending only on n such that +���dndj +dαn +��� ≤ C(n) +d . +Proof. Since dj = ∥qj+1(α) − qj(α)∥ = ∥ϕj+1(uj+1(α), α) − ϕj(uj(α), α)∥ for j = 0, 1, ..., m, then +dj is Cmin{r−1,r′−1}. The first derivative is +ddj +dα = +� ϕj+1(uj+1(α), α) − ϕj(uj(α), α) +∥ϕj+1(uj+1(α), α) − ϕj(uj(α), α)∥, ∂ϕj+1 +∂uj+1 +∂uj+1 +∂α ++ ∂ϕj+1 +∂α ++ ∂ϕj +∂uj +∂uj +∂α + ∂ϕj +∂α +� +. +And then +���ddj +dα +��� = +���∂ϕj+1 +∂uj+1 +∂uj+1 +∂α ++ ∂ϕj+1 +∂α ++ ∂ϕj +∂uj +∂uj +∂α + ∂ϕj +∂α +��� ≤ C(1) +d , +which was estimated in [21]. For the second derivative, using condition 4 in Definition 4.1 and +Theorem 4.2 it follows that +���d2dj +dα2 +��� = +���∂2ϕj+1 +∂u2 +j+1 +�∂uj+1 +∂α +�2 ++ ∂ϕj+1 +∂uj+1 +∂2uj+1 +∂α2 ++ 2 ∂2ϕj+1 +∂uj+1∂α +∂uj+1 +∂α ++ ∂2ϕj+1 +∂α2 ++ ∂2ϕj +∂u2 +j +�∂uj +∂α +�2 ++ ∂ϕj +∂uj +∂2uj +∂α2 + 2 ∂2ϕj +∂uj∂α +∂uj +∂α + ∂2ϕj +∂α2 +���. +By using condition 4 in Definition 4.1 and Theorem 4.2, there exists a constant C(2) +κ +> 0 such +that +���d2dj +dα2 +��� ≤ 2C(2,0) +ϕ +(C(1) +u )2 + 2C(2) +u ++ 4C(1,1) +ϕ +(C(1) +u )2 + 2C(0,2) +ϕ += C(2) +d . +Continuing by induction, we can see that there exists a constant C(n) +d +> 0 depends only on n such +that +���dndj +dαn +��� ≤ C(n) +d +. +Corollary 4.6. Let K(α) be a Cr,r′ billiard deformation with r ≥ 4, r′ ≥ 2. Then cos φj(α) is +Cmin{r−1,r′−1} and there exists a constant C(n) +φ +> 0 depending only on n such that +���dn cos φj +dαn +��� ≤ C(n) +φ . +10 + +Proof. We can write +cos 2φj = +� +qj+1(α) − qj(α) +� +· +� +qj(α) − qj−1(α) +� +|qj+1(α) − qj(α)||qj(α) − qj−1(α)| += +� +ϕj+1(, uj+1, α) − ϕj(uj, α) +� +· +� +ϕj(uj, α) − ϕj−1(uj−1α) +� +|ϕj+1(uj+1, α) − ϕj(uj, α)||ϕj(uj, α) − ϕj−1(uj−1, α)| +. +And then, cos φj(α) = +� +cos 2φj(α)+1 +2 +. Therefore, the statement follows from condition 4 in Defi- +nition 4.1 and Corollary 4.2. +The next corollary follows from Corollaries 4.4, 4.6. +Corollary 4.7. Let K(α) be a Cr,r′ billiard deformation with r ≥ 4, r′ ≥ 2. Then the expression +gj(α) = +2κj +cos φj is Cmin{r−3,r′−1} and there exist constants C(n) +g +> 0 depending only on n such that +���dngj +dαn +��� ≤ C(n) +g +. +The next corollary concerning the curvature kj, defined in (4.1), follows from Corollaries 4.5 and +4.7. +Corollary 4.8. Let K(α) be a Cr,r′ billiard deformation with r ≥ 4, r′ ≥ 2. Then the curvature +kj(α) is Cn, where n = min{r − 3, r′ − 1} and here exist constants C(n) +k +depending only on n such +that +���dnkj +dαn +��� ≤ C(n) +k . +Proof. First, we recall +kj+1(α) = +kj(α) +1 + dj(α)kj(α) + 2 +κj+1(α) +cosφj+1(α) +, +0 ≤ j ≤ m − 1 . +We will write kj+1(α) simply as follows +kj+1(α) = +kj(α) +1 + dj(α)kj(α) + gj+1(α), +where gj+1(α) = +2κj+1 +cos φj+1 . [21] contains an estimate that the first derivative of kj(α) with respect +to α is bounded by a constant C(1) +k . Here, we use the same argument in [21] and show that the +second derivative of kj(α) with respect to α is also bounded. These estimates are useful and will +be used later in Section 5. +Next, we start with the first derivative of kj+1 with respect to α and we will use the notation +˙k, ¨k,...etc. to simplify equations. So, we have +˙kj+1 = +˙kj +(1 + djkj)2 − +˙djk2 +j +(1 + djkj)2 + ˙gj+1. +And for the second derivative, we have +¨kj+1 = +¨kj +(1 + djkj)2 − +k2 +j ( ¨dj + ¨djdjkj − 2 ˙d2 +jkj) + 2˙kj(˙kjdj + 2 ˙djkj) +(1 + djkj)3 ++ ¨gj+1. +11 + +Let +βj = +1 +(1 + djkj)2 , +ηj = − +k2 +j( ¨dj + ¨djdjkj − 2 ˙d2 +jkj) + 2˙kj(˙kjdj + 2 ˙djkj) +(1 + djkj)3 ++ ¨gj+1 +, +0 ≤ j ≤ m − 1 . +From Corollaries 4.5 and 4.7, and the estimate of ˙kj, we have +|βj| ≤ βmax = +1 +(1 + dminkmin)2 , +|ηj| ≤ ηmax = k2 +max(C(2) +d ++ C(2) +d dmaxkmax + 2(C(1) +d )2kmax) +(1 + dminkmin)3 ++ 2C(1) +k (C(1) +k dmax + 2C(1) +d kmax) +(1 + dminkmin)3 ++ C(2) +g . +Then, we have +¨km(α) = ηm−1 + βm−1¨km−1(α) += ηm−1 + βm−1 ηm−2 + .... + βm−1....β1 η0 + βm−1....β0 ¨k0(α). +To solve this equation, we assume that (q(α), v(α)) is periodic such that Bm +α (q(α), v(α)) = +(q(α), v(α)). Then km(α) = k0(α). From this, we can solve the previous equation as follows +¨km(α) − βm−1....β0 ¨k(α) = ηm−1 + βm−1 ηm−2 + .... + βm−1....β1 η0 +¨km(α) = +1 +1 − βm−1....β0 +� +ηm−1 + βm−1 ηm−2 + .... + βm−1....β1 +� +By the maximum value of ηj and βi, we have +|¨km(α)| ≤ +ηmax +1 − βm +max +� +1 + βmax + .... + βm−1 +max +� += +ηmax +1 − βm +max +�1 − βm +max +1 − βmax +� += +ηmax +1 − βmax +. +This means there exists a constant C(2) +k +> 0 does not depend on m or α such that |¨kj(α)| ≤ C(2) +k , +for every j = 0, 1, ..., m. Continuing by induction we can see that the n-th derivative of kj(α) +with respect to α is bounded by constant C(n) +k +> 0 that depending only on n. +5 +Continuity of the largest Lyapunov exponent +In this section, we show that the largest Lyapunov exponent λ1 depends continuously on a planar +billiard deformation. Let K(α) be a billiard deformation as defined in Definition 4.1 and let K(0) +be the initial open billiard. Let kj(α), kj(0) and dj(α), di(0) be the curvatures and the distances +that are described in section 4.1. +12 + +For every α ∈ [0, b], let Mα be the non-wandering set for the billiard map and let Rα : +Mα −→ Σ be the analogue of the conjugacy map R : M0 −→ Σ, so that the following diagram is +commutative: +Mα +Bα +−→ +Mα +�Rα +�Rα +Σ +σ +−→ +Σ +where Bα is the billiard ball map on Mα. By Theorem 2.1 there exists a subset Aα of Σ with +µ(Aα) = 1 so that +λ1(α) = lim +m→∞ +1 +m log ∥Dx0Bm +α (w)∥ +(5.1) +for all x ∈ Mα with Rα(x) ∈ Aα. Similarly, let A0 be the set with µ(A0) = 1 which we get from +Theorem 2.1 for α = 0. +Lemma 5.1. Given an arbitrary sequence +α1, α2, . . . , αp, . . . +of elements of [0, b], for µ-almost all ξ ∈ Σ the formula (5.1) is valid for α = αp and x = R−1 +α (ξ) +for all p = 1, 2, . . . and also for α = 0 and x = R−1(ξ). +Proof. The set A = A0 ∩ ∩∞ +p=1Aαp has µ(A) = 1 since +Σ \ A = (Σ \ A0) ∪ ∪∞ +p (Σ \ Aαp) +has measure zero as a countable union of sets of measure zero. If α = αp for some p and Rα(x) ∈ A, +then Rα(x) ∈ Aαp so formula (5.1) holds. Similarly (5.1) holds for α = 0 as well. +Thus, using the notation x(0, α) ∈ Mα, we can choose ξ ∈ Σ so that formula (5.1) applies for +α = αp and x = x(0, αp) for all p = 1, 2, . . ., and also for α = 0 and x = (0, 0). +From the formula for the largest Lyapunov exponent (3.1), we can write the Lyapunov expo- +nents for K(α) and K(0) as follows: +λ1(α) = lim +m→∞ +1 +m +m +� +j=1 +log +� +1 + dj(α)kj(α) +� += lim +m→∞ λ(m) +1 +(α), +λ1(0) = lim +m→∞ +1 +m +m +� +j=1 +log +� +1 + dj(0)kj(0) +� += lim +m→∞ λ(m) +1 +(0), +where +λ(m) +1 +(α) = 1 +m +m +� +j=1 +log +� +1 + dj(α)kj(α) +� +and +λ(m) +1 +(0) = 1 +m +m +� +j=1 +log +� +1 + dj(0)kj(0) +� +. +(5.2) +Now, we prove Theorem 1.1 +13 + +Proof of Theorem 1.1: Let K(α) be a C4,2 billiard deformation in R2, and let +α ∈ [0, b]. Assume that λ1(α) is not continuous at α = 0. Then there exists ε > 0 and a sequence +α1 > α2 > ... > αp > ... → 0 in [0, b] with αp → 0 such that |λm +1 (αk) − λm +1 (0)| ≥ ε for all p ≥ 1. +By using Lemma 5.1 and the previous expressions of λm +1 (α) for α = αp and λm +1 (0) in (5.2), we +have +�����λm +1 (αp) − λm +1 (0) +����� = +����� +1 +m +m +� +j=1 +(log δj(αp) − log δj(0)) +����� += +����� +−1 +m +m +� +j=1 +(log(1 + dj(αp)kj(αp)) − log(1 + dj(0)kj(0))) +����� +≤ 1 +m +m +� +j=1 +����� log(1 + dj(αp)kj(αp)) − log(1 + dj(0)kj(0)) +����� +≤ 1 +m +m +� +j=1 +����� +1 + dj(αp)kj(αp) − (1 + dj(0)kj(0)) +1 + min{dj(αp)kj(αp), dj(0)kj(0)} +����� += 1 +m +m +� +j=1 +����� +dj(αp)kj(αp) − dj(0)kj(0) +1 + dminkmin +����� += 1 +m C0 +m +� +j=1 +�����dj(αp)kj(αp) − dj(0)kj(0) +����� += 1 +m C0 +m +� +j=1 +�����(dj(αp) − dj(0))kj(αp) + dj(0)(kj(αp) − kj(0)) +�����, +where C0 = +1 +1+dminkmin > 0 is a global constant independent of αp. +Fix a small δ > 0; we will state later how small δ > 0 should be. Next consider p sufficiently large so +that αp < δ. For all p, we have |kj(αp)−kj(0)| = αp|˙kj(s(αp))| and |dj(αp)−dj(0)| = αp| ˙dj(r(αp))|, +for some s(αp), r(αp) ∈ [0, αp]. From Corollaries 4.5 and 4.8 , there exist constants Ck and Cd +such that |˙kj(s(αp))| ≤ Ck and | ˙dj(s(αp))| ≤ Cd. Therefore for all j, +|kj(αp) − kj(0)| ≤ αpCk < δCk, and |dj(αp) − dj(0)| ≤ αpCd < δCd. Then +���λm +1 (αp) − λm +1 (0) +��� ≤ 1 +m C0 +m +� +j=1 +����dj(αp) − dj(0) +���kj(αp) + dj(0) +���kj(αp) − kj(0) +��� +� +< 1 +m C0 +m +� +j=1 +δ(Cdkmax + Ckdmax) += C0δ(Cdkmax + Ckdmax) < ε, +if we take δ < +ε +Cdkmax+Ckdmax . We now have a contradiction because with the choice of the sequence +α1 > α2 > ... > αp > ... → 0 in [0, b]. Therefore the statement is proved. +14 + +6 +Differentiability of the largest Lyapunov exponent +Here we prove Theorem 1.2 +Proof of Theorem 1.2: We will prove differentiability at α = 0. From this differentiability at any +α ∈ [0, b] follows. To prove the differentiability at α = 0, we have to show that there exists +lim +α→0 +λ1(α) − λ1(0) +α +. +Equivalently, there exists a number F such that +lim +p→∞ +λ1(αp) − λ1(0) +αp += F, +for any sequence α1 > α2 > ... > αp > ... → 0 as p → ∞ in [0, b]. +Let K(α) ⊂ R2 be a C5,3 billiard deformation and α ∈ [0, b] for a positive number b. Let λ1(α) be +the largest Lyapunov exponent for K(α) and λ1(0) be the largest Lyapunov exponent for K(0). +By using Lemma 5.1 and the expressions of λm +1 (α) for α = αp and λm +1 (0) in (5.2), we have +λ(m) +1 +(αp) → λ1(αp) and λ(m) +1 +(0) → λ1(0) when m → ∞. Also, +λ(m) +1 +(αp) − λ(m) +1 +(0) +αp += − 1 +m +m +� +j=1 +log δj(αp) − log δj(0) +αp += − 1 +m +m +� +j=1 +log +� +1 + dj(αp)kj(αp) +� +− log +� +1 + dj(0)kj(0) +� +αp +. +Set fj(αp) = log +� +1 + dj(αp)kj(αp) +� +and fj(0) = log +� +1 + dj(0)kj(0) +� +. Then +λ(m) +1 +(αp) − λ(m) +1 +(0) +αp += − 1 +m +m +� +j=1 +fj(αp) − fj(0) +αp +. +Taylor’s formula gives +fj(αp) = fj(0) + αp ˙fj(0) + α2 +p +2 +¨fj(rj(αp)) +for some rj(αp) ∈ [0, αp]. Then +fj(αp) − fj(0) +αp +− ˙fj(0) = αp +2 +¨fj(rj(αp)). +Let +Fm = 1 +m +m +� +j=1 +˙fj(0). +Summing up the above for j = 1, 2, ..., m, we get +λ(m) +1 +(αp) − λ(m) +1 +(0) +αp +− Fm = − 1 +m +m +� +j=1 +�fj(αp) − fj(0) +αp +− ˙fj(0) +� +. +15 + +From the definition of fj(αp), +˙fj(αp) = +˙dj(αp)kj(αp) + dj(αp)˙kj(αp) +1 + dj(αp)kj(αp) +, +and therefore, +¨fj(αp) = +� ¨dj(αp)kj(αp) + 2 ˙dj(αp)˙kj(αp) + dj(αp)¨kj(αp) +�� +1 + dj(αp)kj(αp) +� +� +1 + dj(αp)kj(αp) +�2 +− +� ˙dj(αp)k(αp) + dj(αp)˙kj(αp) +�2 +� +1 + dj(αp)kj(αp) +�2 +. +Then from Corollaries 4.5 and 4.8, we get +��� ˙fj(αp) +��� ≤ C(1) +d kmax + dmaxC(1) +k +1 + dminkmin += C1, +��� ¨fj(αp) +��� ≤ +� +C(2) +d kmax + 2C(1) +d C(1) +k ++ dmaxC(2) +k +�� +1 + dmaxkmax +� +� +1 + dminkmin +�2 ++ +� +C(1) +d kmax + dmaxC(1) +k +�2 +� +1 + dminkmin +�2 += C2. +Therefore +| ¨fj(rj(αp))| ≤ C2, +for some constant C2 > 0 independent of rj(αp) and j. This implies +���λ(m) +1 +(αp) − λ(m) +1 +(0) +αp +− Fm +��� ≤ 1 +m +m +� +j=1 +αp +2 +��� ¨fj(tj(αp)) +��� +≤ C2 +2 αp. +Since | ˙fj(αp)| ≤ C1, we have |Fm| ≤ +1 +m +�m +j=1 | ˙fj(0)| ≤ C1, for all m. Therefore, the sequence +{Fm} has convergent subsequences. Let for example Fmh → F, for some sub-sequence {mh}. +Then +���λ(mh) +1 +(αp) − λ(mh) +1 +(0) +αp +− Fmh +��� ≤ C2 +2 αp, +for all h ≥ 1. So, letting h → ∞, we get +���λ1(αp) − λ1(0) +αp +− F +��� ≤ C2 +2 αp, +and letting αp → 0 as p → ∞ we get that there exists +lim +p→∞ +λ1(αp) − λ1(0) +αp += F. +for every sequence α1 > α2 > ... > αp > ... → 0 as p → ∞ in [0, b]. +Thus, there exists +F = limm→∞ 1 +m +�m +j=1 ˙fj(0). This is true for every subsequence {mh}, so for any subsequence we +have Fmh → F. Hence, Fm converges to F as well. This implies that there exists +16 + +lim +α→0 +λ1(α) − λ1(0) +α += F, +so λ1 is differentiable at α = 0 and ˙λ1(0) = F. +Corollary 6.1. Let K(α) be a C5,3 billiard deformation. Then there exists a constant Cλ1 > 0 +such that +���dλ1(α) +dα +��� ≤ Cλ1, +for all α ∈ [0, b]. +Proof. We have +λ1(α) = lim +m→∞ +1 +m +m +� +j=1 +log(1 + dj(α)kj(α)). +By Theorem 1.2, λ1(α) is C1. So, from the formula in the previous proof that +˙λ1(0) = limm→∞ 1 +m +�m +j=1 ˙fj(0), we have +dλ1 +dα = lim +m→∞ +1 +m +m +� +j=1 +ddj +dα kj(α) + dj(α)dkj +dα +1 + dj(α)kj(α) +. +From Corollaries 4.5 and 4.8, there exist constants C(1) +d , C(1) +k +> 0 such that +���ddj +dα +��� ≤ C(1) +d +and +���dkj +dα +��� ≤ C(1) +k . Then, we have +���dλ1 +dα +��� ≤ lim +m→∞ +1 +m +m +� +j=1 +C(1) +d kmax + dmaxC(1) +k +1 + dminkmin += C(1) +d kmax + dmaxC(1) +k +1 + dminkmin += Cλ1. +This proves the statement. +Acknowledgment +The author would like to thank Prof. Luchezar Stoyanov for his suggestions, comments, and help. +This work was supported by a scholarship from Najran University, Saudi Arabia. +References +[1] L. Barreira and Ya. Pesin, Lyapunov exponents and smooth ergodic theory. Univ. Lect. Series 23, +American Mathematical Society, Providence, RI, 2001. +[2] R. Bowen, Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429-460. +[3] P. Duarte, S. Klein and M. Poletti, H¨older continuity of the Lyapunov exponents of linear cocycles over +hyperbolic maps. Math. Z. 302 (2022), 2285–2325. +17 + +[4] N. Chernov, Entropy, Lyapunov exponents, and mean free path for billiards. Journal of Statistical +Physics, 88 (1997), 1-29. +[5] N. Chernov and R. Markarian, Chaotic Billiards. Math. Surveys and Monographs Vol. 127, Amer. +Math. Soc. 2006. +[6] M. Ikawa, Decay of solutions of the wave equation in the exterior of several strictly convex bodies. Ann. +Inst. Fourier 38 (1988), 113-146. +[7] A. Katok and J. M. Strelcyn, Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singular- +ities. Lecture Notes in Mathematics 1222, Springer, 1986. +[8] A. Lopes and R. Markarian, Open billiards: invariant and conditionally invariant probabilities on +Cantor sets. SIAM J. Appl. Math. 56 (1996), 651-680. +[9] R. Markarian, Billiards with Pesin Region of Measure one. Comm. in Math Phys. 118 (1988), 87-97. +[10] R. Markarian, New ergodic Billiards: exact results. Nonlinearity 6. (1993), 819-841 +[11] T. Morita, The symbolic representation of billiards without boundary condition. Trans. Amer. Math. +Soc. 325 (1991), 819-828. +[12] V. I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical +systems. Trans. Moscow Math. Soc. 19 (1968), 197-221. +[13] M. Pollicott, Lectures on ergodic theory and Pesin theory on compact manifolds. Cambridge Univ. +Press, Cambridge 1993. +[14] V. Petkov and L. Stoyanov, Geometry of Reflecting Rays and Inverse Spectral Problems. Wiley, Chich- +ester, (1992). +[15] Ya. Sinai, Dynamical systems with elastic reflections. Russian Math. Surveys 25 (1970), 137-190. +[16] Ya. Sinai, Development of Krylov’s ideas, An addendum to: N.S.Krylov ”Works on the foundations +of statistical physics”. Princeton Univ. Press, Princeton 1979, 239-281. +[17] L. Stoyanov, Exponential instability and entropy for a class of dispersing billiards. Ergod. Th. & +Dynam. Sys. 19 (1999), 201-226. +[18] L. Stoyanov, Spectrum of the Ruelle operator and exponential decay of correlation for open billiard +flows. Amer. J. Math. 123 (2001), 715-759. +[19] L. Stoyanov, Non-integrability of open billiard flows and Dolgopyat-type estimates. Ergodic Th. & Dyn. +Systems 32 (2012), 295-313. +[20] M. Viana, Lectures on Lyapunov exponents, Cambridge Studies in Adv. Math. vol.145, Cambridge +Univ. Press 2014. +[21] P. Wright, Differentiability of the Hausdorff dimension of the non-wandering set in a planar open +billiard, Discrete & Continuous Dynamical Systems 36(7) (2016), 3993-4014. +[22] M, P. Wojtkowski, Principles for the design of billiards with nonvanishing Lyapunov exponents. Com- +mun. Math. Phys. 105 (1986), 391-414. +18 + diff --git a/1dAzT4oBgHgl3EQf8v7L/content/tmp_files/load_file.txt b/1dAzT4oBgHgl3EQf8v7L/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b93317bc23dc73e32e77eadd39e0dfb2314f5899 --- /dev/null +++ b/1dAzT4oBgHgl3EQf8v7L/content/tmp_files/load_file.txt @@ -0,0 +1,680 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf,len=679 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='01910v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='DS] 5 Jan 2023 Differentiability of the largest Lyapunov exponent for planar open billiards Amal Al Dowais a,b Abstract In this paper, we estimate the largest Lyapunov exponent for open billiards in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' We show that the largest Lyapunov exponent is differentiable with respect to a billiard defor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Open billiards;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Lyapunov exponents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Non-wandering set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Billiard deformation Mathematics Subject Classification (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 37D50, 37B10, 37D20, 34D08 1 Introduction The stability and instability of a dynamical system can be studied by means of Lyapunov expo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' A dynamical system is considered chaotic if it has a positive Lyapunov exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Examples of chaotic systems are the dispersing billiards or so-called Sinai billiards (see [15], [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Billiards are dynamical systems in which a particle moves with constant speed and hits the billiard’s wall (boundary of the billiard’s domain) according to the law of geometrical optics,“the angle of inci- dence equals the angle of reflection”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Open billiards are a particular case of billiards in unbounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' The domain is the exterior of finitely many strictly convex compact obstacles satisfying the no-eclipse condition (H) of Ikawa [6]: the convex hull of any two obstacles does not intersect with another obstacle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' in other words, there does not exist a straight line that intersects more than two obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' It follows from Sinai [15], [16] (see also [14]) that the non-wandering set of the open billiard map is hyperbolic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' there exist positive and negative Lyapunov exponents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Many studies have investigated Lyapunov exponents for billiards (see [22], [1], [4], [9], [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' In this paper, we estimate the largest Lyapunov exponent for open billiard in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' We demonstrate that the Lyapunov exponent depends continuously on a parameter α related to a deformation of the billiard as defined in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Moreover, we prove that the Lyapunov exponent is differentiable with respect to the deformation parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Here we state the main results: In the following theorems, we denote the billiard deformation by K(α) where α ∈ [0, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' See Section 4 for the precise definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' aDepartment of Mathematics and Statistics, School of Physics, Mathematics and Computing, University of Western Australia, Perth, WA 6009, Australia Email address: amal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='aldowais@research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='uwa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='au bDepartment of Mathematics, College of Science and Arts, Najran University, Najran, Saudi Arabia Email address: amalduas@nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='sa 1 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' (Continuity) Let K(α) be a C4,2 billiard deformation in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let λ1(α) be the largest Lyapunov exponent for K(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then the largest Lyapunov exponent is continuous as a function of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' (Differentiability) Let K(α) be a C5,3 billiard deformation in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let λ1(α) be the largest Lyapunov exponent for K(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then λ1(α) is C1 with respect to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' There are many works studying continuity properties of Lyapunov exponents (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' [20], [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' However, to our knowledge, in all these continuity is established generically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' with respect to “most” (typical) values of the parameters/perturbations involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' In the case of the open billiard considered in the present paper we establish continuity and even differentiability for all values of the parameter that appears in the perturbation, which is a truly remarkable property of this physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 2 Preliminaries This section provides some preliminary concepts for open billiards, billiard flow, symbolic coding and stable/unstable manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' We also describe some notations related to curvatures, distances, and collision angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' In the last part of this section, we state the Oseledets multiplicative ergodic theorem and its consequence for open billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1 Open billiard Let Ki, where i = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=', z0, be strictly convex compact domains with smooth boundaries ∂Ki in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' In this paper, we assume that K = � i Ki satisfies the following condition(H) of Ikawa [6]: for any i ̸= j ̸= k the convex hull of Ki ∪Kk does not have any common points with Kj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let Ω be the exterior of K (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=', Ω = R2\\K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let Φt, t ∈ R, be the billiard flow such that for any particle x = (q, v), where q ∈ Ω represents the position of x and v is the unit velocity of the particle x, then Φt(x) = (qt, vt) = (q + tv, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' When the particle hits the boundary, then the velocity follows the collision law vnew = vold − 2 < vold, n > n, where n is the outwards unit normal vector to ∂K at q, and φ the angle between n = n(q) and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' We denote the time of the j-th reflection of x by tj(x) ∈ (−∞, ∞) for j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' We say tj(q) = ∞ (tj(q) = −∞) if the forwards (backwards) trajectory of x has less than j reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' We denote the non-wandering set of the flow Φt by Λ = {x ∈ �Ω, |tj(x)| < ∞, for all j ∈ Z}, where �Ω = {(q, v) ∈ int Ω × S1 or (q, v) ∈ ∂Ω × S1 : ⟨n(q), v⟩ ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Now let M = {x = (q, v) ∈ ∂K × S1 : ⟨n(q), v⟩ ≥ 0} and let π : M → ∂K be the canonical projection map defined by π(q, v) = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let t1(x) be the time of the first reflection of x and let M1 = {x ∈ M : t1(x) < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Define the billiard ball map B : M1 → M by B(x) = Φt1(x)(x), (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' if y = (p0, w0), where p0 lies on ∂Ki then B(y) = B(p0, w0) = (p1, w1) where p1 = p0+t1w0 ∈ ∂Kj and w1 = w0 − 2 < w0, n > n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' The non-wandering set of the open billiard map is M0 = {x ∈ M : |tj(x)| < ∞} which is a subset of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Finally, let B : M0 → M0 be the restriction of the open billiard map on the non-wandering set M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' It is obvious that the non-wandering set is an invariant set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' See [15], [16], [4], [5], [14], for general information about billiard dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2 Symbolic coding for open billiards Each particular x ∈ M0 can be coded by a bi-infinite sequence ξ(x) = (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=', ξ−1, ξ0, ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=') ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=', z0}Z, in which ξi ̸= ξi+1, for all i ∈ Z, and ξj indicates the obstacle Kξj such that πBj(x) ∈ ∂Kξj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' For example, if there are three obstacles K1, K2 and K3 as above and a particular q repeatedly hits K1, K3, K2, K1, K3, K2, then the bi-infinite sequence is (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=', 1, 3, 2, 1, 3, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let Σ be the symbol space which is defined as: Σ = {ξ = (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=', ξ−1, ξ0, ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=') ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=', z0}Z : ξi ̸= ξi+1, ∀i ∈ Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Define the representation map R : M0 → Σ by R(x) = ξ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let σ : Σ → Σ be the two-sided subshift map defined by σ(ξi) = ξi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Given θ ∈ (0, 1) define the metric dθ on Σ by: dθ(ξ, η) = � 0 if ξi = ηi for all i ∈ Z θn if n = max{j ≥ 0 : ξi = ηi for all |i| < j} Then σ is continuous with respect to dθ ([2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' It is also known that the representation map R : M0 → Σ is a homeomorphism (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' See [6], [8], [11], [14], [17], for topics related to symbolic dynamics for open billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='3 Lyapunov exponents Here we state a consequence of Oseledets Multiplicative Ergodic Theorem for billiards (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 2 in [13], also see [12], [20], [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' For the open billiard map B : M0 −→ M0 in R2 we will use the coding R : M0 −→ Σ from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2, which conjugates B with the shift map σ : Σ −→ Σ, to define Lyapunov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' It is well known that there are ergodic σ-invariant measures µ on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let µ be an ergodic σ-invariant probability measure on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' The following is a consequence of Oseledets Multiplicative Ergodic Theorem: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1 (A Consequence of Oseledets Multiplicative Ergodic Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' There exist real numbers λ1 > 0 > −λ1 and one-dimensional vector subspaces Eu(x) and Es(x) of Tx(∂K), x ∈ M0, depending measurably on R(x) ∈ Σ such that: (i) Eu(x) and Es(x) for almost all x ∈ M0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' (ii) DxB(Eu(x)) = Eu(B(x)) and DxB(Es(x)) = Es(B(x)) for almost all x ∈ M0, and (iii) For almost all x ∈ M0 there exists lim n→∞ 1 n log ∥DxBn(w)∥ = λ1 whenever 0 ̸= w ∈ Eu(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Here ”for almost all x” means ”for almost all R(x)” with respect to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' The numbers λ1 > 0 > −λ1 are called Lyapunov exponents, while the invariant subspaces Eu(x) and Es(x) are called Oseledets subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='4 Propagation of unstable manifolds for open billiards We describe a formula which is useful in getting estimates for lim m→∞ 1 m log ∥DxBmw∥, (0 ̸= w ∈ Eu(x), x ∈ M0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let M0 be the non-wandering set of the billiard ball map B of an open billiard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then Λ = {Φt(x) : x ∈ M0 , t ∈ R}, is the non-wandering set for the billiard flow Φt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' For x ∈ Λ and a sufficiently small ǫ > 0 let � W s ǫ (x) = {y ∈ Λ : d(Φt(x), Φt(y)) ≤ ǫ for all t ≥ 0 , d(Φt(x), Φt(y)) →t→∞ 0 }, � W u ǫ (x) = {y ∈ Λ : d(Φt(x), Φt(y)) ≤ ǫ for all t ≤ 0 , d(Φt(x), Φt(y)) →t→−∞ 0 } be the (strong) stable and unstable manifolds of size ǫ for the billiard flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then �Eu(x) = Tx� W u ǫ (x) and �Es(x) = Tx� W s ǫ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' In a similar way one defines stable/unstable manifolds for the billiard ball map B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' For any x = (q, v) ∈ M0 define W s ǫ (x) = {y ∈ M0 : d(Bn(x), Bn(y)) ≤ ǫ for all n ∈ N , d(Bn(x), Bn(y)) →n→∞ 0 }, W u ǫ (x) = {y ∈ M0 : d(B−n(x), B−n(y)) ≤ ǫ for all n ∈ N , d(B−n(x), B−n(y)) →n→∞ 0 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' In what follows we will just write W u(x) and W s(x) for W u ǫ (x) and W s ǫ (x), assuming some appropriately chosen sufficiently small ǫ > 0 is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Similarly for � W u and � W s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' It is well-known that there is an one-to-one correspondence between the stable/unstable man- ifolds for the billiard ball map and these for the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Geometrically the easiest (and most convenient way) to describe this is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Given x = (q, v) ∈ M0 (so q ∈ ∂K and v ∈ S1), and a small 0 < r < t1(x), set y = (q + rv, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then there is a 1-1 correspondence ϕ : W u(x) −→ � W u(y) such that ϕ(z, w) = (z + t w, w) for all (z, w) ∈ W u(x), where t = t(z, w) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Similarly, there is a correspondence between W s(x) and � W s(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Moreover Dϕ(x) : TxM0 −→ TyΛ is so that Dϕ(x)(Eu(x)) = �Eu(y) and Dϕ(x)(Es(x)) = �Es(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' It is known that � W u(y) has the form � W u(y) = �Y , where �Y = {(p, νY (p)) : p ∈ Y } for some smooth curve Y in R2 containing the point y such that Y is strictly convex with respect to the unit normal field νY , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' the curvature of Y is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 4 Next, let x and y be as above and let x1 = (q1, v1) = B(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then q1 = q + t1 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Define y1 = (q1 + r′v1, v1) for some small 0 < r′ < t2(x) − t1(x), where 0 = t0(x) < t1(x) < t2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then there is a 1-1 correspondence ϕ1 : W u(x1) −→ ˜W u(y1) defined as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Again, we can write � W u(y1) = �Y1, where �Y1 = {(p1, νY (p1)) : p1 ∈ Y1} for some smooth curve Y1 in R2 containing the point y1 such that Y1 is strictly convex with respect to the unit normal field νY1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Moreover the following diagram is commutative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' where t = t1 + r′: W u(x) B −→ W u(x1) \uf8e6\uf8e6�ϕ \uf8e6\uf8e6�ϕ1 � W u(y) = �Y Φt −→ � W u(y1) = �Y1 Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' the following diagram is commutative: Eu(x) DB(x) −→ Eu(x1) \uf8e6\uf8e6�Dϕ \uf8e6\uf8e6�Dϕ1 �Eu(y) DΦt(y) −→ �Eu(y1) Since the derivatives Dϕ and Dϕ1 are uniformly bounded,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' the above conjugacy can be used later to calculate the Lyapunov exponents of the billiard ball map using propagation of appropriate convex curves Y which we describe as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let x0 = (q0, v0) ∈ M0 and let W u ǫ (x0) be the local unstable manifold for x0 for sufficiently small ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let t1(x0) be the time of the first reflection of x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then � X = W u ǫ (x0) = {(q, nX(q)) : q ∈ X} for some C3 curve X in Ω such that q0 ∈ X and X is strictly convex curve with respect to the outer unit normal field nX(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let X be parametrized by q(s), s ∈ [0, a], such that q(0) = q0, and has unit normal field nX(q(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Set q0(s) = q(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let qj(s), j ≥ 1 be the jth-reflection points of the forward billiard trajectory γ(s) generated by x(s) = (q(s), nX(q(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' We assume that a > 0 is sufficiently small so that the jth-reflection points qj(s) belong to the same boundary component ∂Kξj for every s ∈ [0, a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let 0 = t0(x(s)) < t1(x(s)) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' < tm+1(x(s)) be the times of the reflections of the ray γ(s) at ∂K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let κj(s) be the curvature of ∂Kξj at qj(s) and φj(s) be the collision angle between the outward unit normal to ∂K and the reflection ray of γ(s) at qj(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Also, let dj(s) be the distance between two reflection points i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' dj(s) = ∥qj+1(s) − qj(s)∥, j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Given a large m ≥ 1, let tm(x(s)) < t < tm+1(x(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Set Φt( � X) = � Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let π(Φt(x(s))) = p(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then p(s), s ∈ [0, a], is a parametrization of the C3 curve Xt = π(Φt( � X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Next, let k0(s) > 0 be the curvature of X at q(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let tj(x(s)) < τ < tj+1(x(s)), j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Denote by uτ(s) be the shift of (q(s), n(q(s))) along the forward billiard trajectory γ(s) after time τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then Xτ = {uτ(s) : s ∈ [0, a]} is a C3 convex curve with respect to the outward normal field n(uτ(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let kj(s) > 0 be the curvature of Xtj = limτցtj(s) Xτ at qj(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' It follows from Sinai [15] that kj+1(s) = kj(s) 1 + dj(s)kj(s) + 2 κj+1(s) cos φj+1(s) , 0 ≤ j ≤ m − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1) 5 Moreover, the curvature of Xτ at uτ(s) is kτ(s) = kj(s) 1 + (τ − tj(s))kj(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2) Set δj(s) = 1 1 + dj(s)kj(s) , 1 ≤ j ≤ m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='3) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' [18] For all s ∈ [0, a] we have ∥ ˙q(s)∥ = ∥ ˙p(s)∥δ1(s)δ2(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' δm(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='4) This was proved in [18] in the 2D case and in [19] in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Finally, we want to introduce some notation related to the maximum and minimum of previous billiard characteristies dj(s),κj(s), φj(s) and kj(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' For all j, we have dmin ≤ dj(s) ≤ dmax, where dmax and dmin are constants independent of j such that dmax = max{d(Ki, Kk)} and dmin = min{d(Ki, Kk)} for i ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Also, since ∂K is strictly convex, we have constants κmin > 0 and κmax > 0 independent of j such that κmin ≤ κj(s) ≤ κmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' And it follows from the condition (H) that there exists a constant φmax ∈ (0, π 2 ) such that 0 ≤ φj(s) ≤ φmax < π 2 , (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let kj(s) be as in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' It follows easily that kmin ≤ kj(s) ≤ kmax, where kmin = 2κmin and kmax = 1 dmin + 2κmax cos φmax .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 3 Estimation of the largest Lyapunov exponent for open billiards A formula for the largest Lyapunov exponents for a rather general class of billiards can be found in [5], see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='41 there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' In our case we derive this formula again (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1) below) and then we use Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2 to derive important regularity properties of the largest Lyapunov exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Assume that µ is an ergodic σ-invariant measure on Σ, and let x0 = (q0, v0) ∈ M0 correspond to a typical point in Σ with respect to µ via the representation map R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' That is as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1, we have λ1 = lim m→∞ 1 m log ∥Dx0Bm(w)∥, with 0 ̸= w ∈ Eu(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' As in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='4, let X be a (small) C3 strictly convex curve containing q0 and having a unit normal field nX so that nX(q0) = v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' As in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='4 again, let X be parametrised by arc length via q(s), s ∈ [0, a], such that q(0) = q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let again qj(s), j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' , m + 1, be the consecutive reflection points of the billiard trajectory γ(s) determined by x(s) = (q(s), nX(q(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Given an integer m > 0 and assuming the interval [0, a] is sufficiently small, the jth reflection points qj(s) belong to the same boundary component ∂Kξj for all s ∈ [0, a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Next, define dj(s), tj(x(s)), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' as in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='4, let tm(x(0)) < t < tm+1(x(0)), and let p(s) be the parametrisation of � Xt corresponding to q(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='4) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2 (holds with ∥ ˙q(s)∥ = 1 from our assumptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Now the discussion in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='4 implies that there exist some global constants c1 > c2 > 0, independent of x0, X, m, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' such that c2∥ ˙p(s)∥ ≤ ∥Dx0Bm(w)∥ ≤ c1∥ ˙p(s)∥ 6 for all s ∈ [0, a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' So, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='4), c2 δ1(0)δ2(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' δm(0) ≤ ∥Dx0Bm(w)∥ ≤ c1 δ1(0)δ2(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' δm(0) for all s ∈ [0, a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Using this for s = 0, taking logarithms and limits as m → ∞, we obtain − lim m→∞ 1 m log (δ1(0)δ2(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' δm(0)) ≤ lim m→∞ 1 m log ∥Dx0Bm(w)∥ ≤ − lim m→∞ 1 m log (δ1(0)δ2(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' δm(0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Hence, λ1 = lim m→∞ − 1 m m � i=1 log δi(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' This implies that the largest Lyapunov exponent at the initial point x0, so at almost every point wilt respect to the given measure µ, is given by λ1 = lim m→∞ 1 m m � i=1 log � 1 + di(0)ki(0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1) From equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1), we can estimate the largest Lyapunov exponent from below and above as log (1 + dminkmin) ≤ λ1 ≤ log (1 + dmaxkmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 4 Billiard deformations In this section, we consider some changes to the billiards in the plane, such as moving, rotating, and changing the shape of one or multiple obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' This kind of billiard transformation is called a billiard deformation as defined in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' We describe this deformation by adding an extra parameter α ∈ [0, b] for some b ∈ R+, which is called the deformation parameter, to the parametrization of the boundary of obstacles i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=', if the boundary of an obstacle parametrized by ϕ(u), it will become ϕ(u, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' In this section, we provide the definition a billiard deformation as defined in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' In addition, we describe the propagation of unstable manifolds for billiard deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' We also estimate the higher derivatives of some of the billiard characteristics such as distance, collision angle and curvature, with respect to deformation parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let α ∈ I = [0, b], for some b ∈ R+, be a deformation parameter and let ∂Ki(α) be parametrized counterclockwise by ϕi(ui, α) and parametrized by arc-length ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let qi = ϕi(ui, α) be a point that lies on ∂Ki(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Denote the perimeter of ∂Ki(α) by Li(α), and let Pi = {(ui, α) : α ∈ I, ui ∈ [0, Li(α)]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' [21] For any α ∈ I = [0, b], let K(α) be a subset of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' For integers r ≥ 4, r′ ≥ 2, we call K(α) a Cr,r′-billiard deformation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Cr with respect to u and Cr′ with respect to α) if the following conditions hold for all α ∈ I: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' K(α) = �z0 i=1 Ki(α) satisfies the no-eclipse condition (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Each Ki(α) is a compact, strictly convex set with Cr boundary and total arc length Li(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Each Ki is parametrized counterclockwise by arc-length with Cr,r′ functions ϕi : Pi → R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' For all integers 0 ≤ l ≤ r, 0 ≤ l′ ≤ r′ (apart from l = l′ = 0), there exist constants C(l,l′) ϕ depending only on the choice of the billiard deformation and the parametrizations ϕi, such that for all integers i = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=', z0, ��� ∂l+l′ϕi ∂ul i∂αl′ ��� ≤ C(l,l′) ϕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let Bα be the open billiard map on a non-wandering set Mα for K(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let Σ defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2, we defined Rα : Mα → Σ by Rα(x(α)) = ξ(x(α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' We can write the points that correspond to the billiard trajectories according to the parameterization in previous definition as follows, π(Bj(x(α))) = qξj(α) = ϕξj(uξj(α), α) ∈ ∂Kξj(α), where uξj(α) ∈ [0, Lξj(α)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' For brevity, we will write qj(α) = ϕj(uj(α), α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' The next corollary shows that uj(α) = uξj(α) for a fixed ξ ∈ Σ, is differentiable with respect to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' This corollary is proved in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' [21] Let K(α) be a Cr,r′ billiard deformation with r, r′ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then uj(α) is Cmin{r−1,r′−1} with respect to α, and there exist constants C(n) u > 0 such that ���dnuj(α) dαn ��� ≤ C(n) u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' The next corollary follows from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let K(α) be a Cr,r′ billiard deformation with r, r′ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let qj(α) belongs to ∂Kξj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then qj(α) is Cn, where n = min{r − 1, r′ − 1}, with respect to α, and there exist constants C(n) q > 0 such that ���dnqj(α) dαn ��� ≤ C(n) q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1 Propagation of unstable manifolds for billiard deformations We described the unstable manifolds propagation in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='4 for open billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Here in this section, we describe it for billiard deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let K(α), α ∈ [0, b] be a Cr,r′ billiard deformation as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1 with r ≥ 3, r′ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' x0(α) = (q0(α), v0(α)) ∈ Mα and let W u ǫ (x0(α)) be the local unstable manifold for x0(α) for sufficiently small ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Take a curve Xα containing q0(α) such that Xα = {q0(s, α) : s ∈ [0, a]} is a convex curve with outer unit normal field nX(q0(s, α)) = v0(α) and C3 with respect to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' It follows from Sinai [15], [16] that W u ǫ (x0(α)) = {(q0(s, α), nX(q0)) : s ∈ [0, a]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Set � Xα = W u ǫ (x0(α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let a ∈ R+ be small enough such that all reflection points qj(s, α), j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=', m, that are generated by x0(s, α) = (q0(s, α), nX(q0(s, α))) belong to the same boundary ∂Kξj(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let dj(s, α) = ∥qj+1(s, α) − qj(s, α)∥ be the distance between two reflection points qj+1(s, α) and qj(s, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Denote the curvature of ∂K(α) at qj(s, α) by κj(s, α), the collision angle between the unit normal to ∂K(α) and the reflection vector at qj(s, α) by φj(s, α), and the curvature of X at q0(s, α) by k0(s, α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 8 Let tj(x(s, α)) = tj(s, α) be the time of the j-th reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Given t with tj < t < tj+1 for some j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=', m, set π(Φt( � Xα)) = Xαt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then Xαt = {uαt(s, α) : s ∈ [0, a]} is C3 with respect to s and a convex curve with outer unit normal field nXαt(uαt(s, α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Denote the curvature of Xαtj(s,α) at qj(s, α) by kj(s, α), where Xαtj(s,α) = limtցtj(s,α) Xαt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' As in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1), we can define kj(s, α) as follows: kj+1(s, α) = kj(s, α) 1 + dj(s, α)kj(s, α) + 2 κj+1(s, α) cos φj+1(s, α) , 0 ≤ j ≤ m − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1) From now on, we will need to use previous characteristics in the case s = 0, so for brevity, we will write dj(α) = dj(0, α), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Also, we denote the billiard deformation by K(α), so all of its characteristics will be denoted dj(α), kj(α), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' The initial open billiard is K(0) so all of its characteristics will be denoted dj(0), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2 The higher derivatives of billiard characteristics Let K(α) be a Cr,r′ billiard deformation as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1 with r ≥ 4, r′ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Recall that ∂Kξj(α) is parametrized by arc-length ujand qj(α) = ϕj(uj(α), α) ∈ ∂Kξj(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Here, we state some corollaries related to bounds of the higher derivatives of curvature, distance and collision angle of a billiard deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' These corollaries are forthright consequences of condition 4 in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let K(α) be a Cr,r′ billiard deformation with r ≥ 4, r′ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then the curvature κj(α) at qj(α) is Cn, where n = min{r − 3, r′ − 1} with respect to α and there exist constants C(n) κ > 0 depending only on n such that ���dnκ dαn ��� ≤ C(n) κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Suppose K(α) is a a Cr,r′ billiard deformation with r ≥ 3, r′ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Since ∂Kj(α) is paramitrized by arc-length uj, then the curvature of ∂Kj(α) at qj(α) = ϕj(uj(α), α) is κj = ∂2ϕj ∂u2 j , for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then κj(α) is Cmin{r−3,r′−1} with respect to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' For the first derivative, we have ���dκj dα ��� = ���∂3ϕj ∂u3 j ∂uj ∂α + ∂3ϕj ∂u2 j∂α ��� ≤ C(1) κ , this estimate was obtained in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Next, we continue to estimate the second derivative, so we have ���d2κj dα2 ��� = ���∂4ϕj ∂u4 j �∂uj ∂α �2 + ∂3ϕj ∂u3 j ∂u2 j ∂α2 + 2 ∂4ϕj ∂u3 j∂α ∂uj ∂α + ∂4ϕj ∂u2 j∂α2 ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' By using condition 4 in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2, there exists a constant C(2) κ > 0 such that 9 ���d2κj dα2 ��� ≤ C(4,0) ϕ (C(1) u )2 + C(3,0) ϕ C(2) u + 2Cϕ(3,1)C(1) u + C(2,2) ϕ = C(2) κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Continuing by induction we see that the n-th derivative, where n = min{r −3, r′ −1}, is bounded by a constant C(n) κ > 0 which depends only on n such that ���dnκ dαn ��� ≤ C(n) κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let K(α) be a Cr,r′ billiard deformation with r ≥ 3, r′ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then the distance dj(α) between two points qj+1(α) and qj(α) is Cn, where n = min{r − 1, r′ − 1} with respect to α and there exist constants C(n) d > 0 depending only on n such that ���dndj dαn ��� ≤ C(n) d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Since dj = ∥qj+1(α) − qj(α)∥ = ∥ϕj+1(uj+1(α), α) − ϕj(uj(α), α)∥ for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=', m, then dj is Cmin{r−1,r′−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' The first derivative is ddj dα = � ϕj+1(uj+1(α), α) − ϕj(uj(α), α) ∥ϕj+1(uj+1(α), α) − ϕj(uj(α), α)∥, ∂ϕj+1 ∂uj+1 ∂uj+1 ∂α + ∂ϕj+1 ∂α + ∂ϕj ∂uj ∂uj ∂α + ∂ϕj ∂α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' And then ���ddj dα ��� = ���∂ϕj+1 ∂uj+1 ∂uj+1 ∂α + ∂ϕj+1 ∂α + ∂ϕj ∂uj ∂uj ∂α + ∂ϕj ∂α ��� ≤ C(1) d , which was estimated in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' For the second derivative, using condition 4 in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2 it follows that ���d2dj dα2 ��� = ���∂2ϕj+1 ∂u2 j+1 �∂uj+1 ∂α �2 + ∂ϕj+1 ∂uj+1 ∂2uj+1 ∂α2 + 2 ∂2ϕj+1 ∂uj+1∂α ∂uj+1 ∂α + ∂2ϕj+1 ∂α2 + ∂2ϕj ∂u2 j �∂uj ∂α �2 + ∂ϕj ∂uj ∂2uj ∂α2 + 2 ∂2ϕj ∂uj∂α ∂uj ∂α + ∂2ϕj ∂α2 ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' By using condition 4 in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2, there exists a constant C(2) κ > 0 such that ���d2dj dα2 ��� ≤ 2C(2,0) ϕ (C(1) u )2 + 2C(2) u + 4C(1,1) ϕ (C(1) u )2 + 2C(0,2) ϕ = C(2) d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Continuing by induction, we can see that there exists a constant C(n) d > 0 depends only on n such that ���dndj dαn ��� ≤ C(n) d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let K(α) be a Cr,r′ billiard deformation with r ≥ 4, r′ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then cos φj(α) is Cmin{r−1,r′−1} and there exists a constant C(n) φ > 0 depending only on n such that ���dn cos φj dαn ��� ≤ C(n) φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' We can write cos 2φj = � qj+1(α) − qj(α) � � qj(α) − qj−1(α) � |qj+1(α) − qj(α)||qj(α) − qj−1(α)| = � ϕj+1(, uj+1, α) − ϕj(uj, α) � � ϕj(uj, α) − ϕj−1(uj−1α) � |ϕj+1(uj+1, α) − ϕj(uj, α)||ϕj(uj, α) − ϕj−1(uj−1, α)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' And then, cos φj(α) = � cos 2φj(α)+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Therefore, the statement follows from condition 4 in Defi- nition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' The next corollary follows from Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let K(α) be a Cr,r′ billiard deformation with r ≥ 4, r′ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then the expression gj(α) = 2κj cos φj is Cmin{r−3,r′−1} and there exist constants C(n) g > 0 depending only on n such that ���dngj dαn ��� ≤ C(n) g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' The next corollary concerning the curvature kj, defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1), follows from Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let K(α) be a Cr,r′ billiard deformation with r ≥ 4, r′ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then the curvature kj(α) is Cn, where n = min{r − 3, r′ − 1} and here exist constants C(n) k depending only on n such that ���dnkj dαn ��� ≤ C(n) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' First, we recall kj+1(α) = kj(α) 1 + dj(α)kj(α) + 2 κj+1(α) cosφj+1(α) , 0 ≤ j ≤ m − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' We will write kj+1(α) simply as follows kj+1(α) = kj(α) 1 + dj(α)kj(α) + gj+1(α), where gj+1(α) = 2κj+1 cos φj+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' [21] contains an estimate that the first derivative of kj(α) with respect to α is bounded by a constant C(1) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Here, we use the same argument in [21] and show that the second derivative of kj(α) with respect to α is also bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' These estimates are useful and will be used later in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Next, we start with the first derivative of kj+1 with respect to α and we will use the notation ˙k, ¨k,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' to simplify equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' So, we have ˙kj+1 = ˙kj (1 + djkj)2 − ˙djk2 j (1 + djkj)2 + ˙gj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' And for the second derivative, we have ¨kj+1 = ¨kj (1 + djkj)2 − k2 j ( ¨dj + ¨djdjkj − 2 ˙d2 jkj) + 2˙kj(˙kjdj + 2 ˙djkj) (1 + djkj)3 + ¨gj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 11 Let βj = 1 (1 + djkj)2 , ηj = − k2 j( ¨dj + ¨djdjkj − 2 ˙d2 jkj) + 2˙kj(˙kjdj + 2 ˙djkj) (1 + djkj)3 + ¨gj+1 , 0 ≤ j ≤ m − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' From Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='7, and the estimate of ˙kj, we have |βj| ≤ βmax = 1 (1 + dminkmin)2 , |ηj| ≤ ηmax = k2 max(C(2) d + C(2) d dmaxkmax + 2(C(1) d )2kmax) (1 + dminkmin)3 + 2C(1) k (C(1) k dmax + 2C(1) d kmax) (1 + dminkmin)3 + C(2) g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then, we have ¨km(α) = ηm−1 + βm−1¨km−1(α) = ηm−1 + βm−1 ηm−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='. + βm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='.β1 η0 + βm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='.β0 ¨k0(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' To solve this equation, we assume that (q(α), v(α)) is periodic such that Bm α (q(α), v(α)) = (q(α), v(α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then km(α) = k0(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' From this, we can solve the previous equation as follows ¨km(α) − βm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='.β0 ¨k(α) = ηm−1 + βm−1 ηm−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='. + βm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='.β1 η0 ¨km(α) = 1 1 − βm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='.β0 � ηm−1 + βm−1 ηm−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='. + βm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='.β1 � By the maximum value of ηj and βi, we have |¨km(α)| ≤ ηmax 1 − βm max � 1 + βmax + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='. + βm−1 max � = ηmax 1 − βm max �1 − βm max 1 − βmax � = ηmax 1 − βmax .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' This means there exists a constant C(2) k > 0 does not depend on m or α such that |¨kj(α)| ≤ C(2) k , for every j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Continuing by induction we can see that the n-th derivative of kj(α) with respect to α is bounded by constant C(n) k > 0 that depending only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 5 Continuity of the largest Lyapunov exponent In this section, we show that the largest Lyapunov exponent λ1 depends continuously on a planar billiard deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let K(α) be a billiard deformation as defined in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1 and let K(0) be the initial open billiard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let kj(α), kj(0) and dj(α), di(0) be the curvatures and the distances that are described in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 12 For every α ∈ [0, b], let Mα be the non-wandering set for the billiard map and let Rα : Mα −→ Σ be the analogue of the conjugacy map R : M0 −→ Σ, so that the following diagram is commutative: Mα Bα −→ Mα \uf8e6\uf8e6�Rα \uf8e6\uf8e6�Rα Σ σ −→ Σ where Bα is the billiard ball map on Mα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1 there exists a subset Aα of Σ with µ(Aα) = 1 so that λ1(α) = lim m→∞ 1 m log ∥Dx0Bm α (w)∥ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1) for all x ∈ Mα with Rα(x) ∈ Aα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Similarly, let A0 be the set with µ(A0) = 1 which we get from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1 for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Given an arbitrary sequence α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' , αp, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' of elements of [0, b], for µ-almost all ξ ∈ Σ the formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1) is valid for α = αp and x = R−1 α (ξ) for all p = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' and also for α = 0 and x = R−1(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' The set A = A0 ∩ ∩∞ p=1Aαp has µ(A) = 1 since Σ \\ A = (Σ \\ A0) ∪ ∪∞ p (Σ \\ Aαp) has measure zero as a countable union of sets of measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' If α = αp for some p and Rα(x) ∈ A, then Rα(x) ∈ Aαp so formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Similarly (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1) holds for α = 0 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Thus, using the notation x(0, α) ∈ Mα, we can choose ξ ∈ Σ so that formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1) applies for α = αp and x = x(0, αp) for all p = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=', and also for α = 0 and x = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' From the formula for the largest Lyapunov exponent (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1), we can write the Lyapunov expo- nents for K(α) and K(0) as follows: λ1(α) = lim m→∞ 1 m m � j=1 log � 1 + dj(α)kj(α) � = lim m→∞ λ(m) 1 (α), λ1(0) = lim m→∞ 1 m m � j=1 log � 1 + dj(0)kj(0) � = lim m→∞ λ(m) 1 (0), where λ(m) 1 (α) = 1 m m � j=1 log � 1 + dj(α)kj(α) � and λ(m) 1 (0) = 1 m m � j=1 log � 1 + dj(0)kj(0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2) Now, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1 13 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1: Let K(α) be a C4,2 billiard deformation in R2, and let α ∈ [0, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Assume that λ1(α) is not continuous at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then there exists ε > 0 and a sequence α1 > α2 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' > αp > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' → 0 in [0, b] with αp → 0 such that |λm 1 (αk) − λm 1 (0)| ≥ ε for all p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' By using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1 and the previous expressions of λm 1 (α) for α = αp and λm 1 (0) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' we have �����λm 1 (αp) − λm 1 (0) ����� = ����� 1 m m � j=1 (log δj(αp) − log δj(0)) ����� = ����� −1 m m � j=1 (log(1 + dj(αp)kj(αp)) − log(1 + dj(0)kj(0))) ����� ≤ 1 m m � j=1 ����� log(1 + dj(αp)kj(αp)) − log(1 + dj(0)kj(0)) ����� ≤ 1 m m � j=1 ����� 1 + dj(αp)kj(αp) − (1 + dj(0)kj(0)) 1 + min{dj(αp)kj(αp),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' dj(0)kj(0)} ����� = 1 m m � j=1 ����� dj(αp)kj(αp) − dj(0)kj(0) 1 + dminkmin ����� = 1 m C0 m � j=1 �����dj(αp)kj(αp) − dj(0)kj(0) ����� = 1 m C0 m � j=1 �����(dj(αp) − dj(0))kj(αp) + dj(0)(kj(αp) − kj(0)) �����,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' where C0 = 1 1+dminkmin > 0 is a global constant independent of αp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Fix a small δ > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' we will state later how small δ > 0 should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Next consider p sufficiently large so that αp < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' For all p, we have |kj(αp)−kj(0)| = αp|˙kj(s(αp))| and |dj(αp)−dj(0)| = αp| ˙dj(r(αp))|, for some s(αp), r(αp) ∈ [0, αp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' From Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='8 , there exist constants Ck and Cd such that |˙kj(s(αp))| ≤ Ck and | ˙dj(s(αp))| ≤ Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Therefore for all j, |kj(αp) − kj(0)| ≤ αpCk < δCk, and |dj(αp) − dj(0)| ≤ αpCd < δCd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then ���λm 1 (αp) − λm 1 (0) ��� ≤ 1 m C0 m � j=1 ����dj(αp) − dj(0) ���kj(αp) + dj(0) ���kj(αp) − kj(0) ��� � < 1 m C0 m � j=1 δ(Cdkmax + Ckdmax) = C0δ(Cdkmax + Ckdmax) < ε, if we take δ < ε Cdkmax+Ckdmax .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' We now have a contradiction because with the choice of the sequence α1 > α2 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' > αp > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' → 0 in [0, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Therefore the statement is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 14 6 Differentiability of the largest Lyapunov exponent Here we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2: We will prove differentiability at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' From this differentiability at any α ∈ [0, b] follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' To prove the differentiability at α = 0, we have to show that there exists lim α→0 λ1(α) − λ1(0) α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Equivalently, there exists a number F such that lim p→∞ λ1(αp) − λ1(0) αp = F, for any sequence α1 > α2 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' > αp > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' → 0 as p → ∞ in [0, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let K(α) ⊂ R2 be a C5,3 billiard deformation and α ∈ [0, b] for a positive number b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let λ1(α) be the largest Lyapunov exponent for K(α) and λ1(0) be the largest Lyapunov exponent for K(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' By using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1 and the expressions of λm 1 (α) for α = αp and λm 1 (0) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2), we have λ(m) 1 (αp) → λ1(αp) and λ(m) 1 (0) → λ1(0) when m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Also, λ(m) 1 (αp) − λ(m) 1 (0) αp = − 1 m m � j=1 log δj(αp) − log δj(0) αp = − 1 m m � j=1 log � 1 + dj(αp)kj(αp) � − log � 1 + dj(0)kj(0) � αp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Set fj(αp) = log � 1 + dj(αp)kj(αp) � and fj(0) = log � 1 + dj(0)kj(0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then λ(m) 1 (αp) − λ(m) 1 (0) αp = − 1 m m � j=1 fj(αp) − fj(0) αp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Taylor’s formula gives fj(αp) = fj(0) + αp ˙fj(0) + α2 p 2 ¨fj(rj(αp)) for some rj(αp) ∈ [0, αp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then fj(αp) − fj(0) αp − ˙fj(0) = αp 2 ¨fj(rj(αp)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let Fm = 1 m m � j=1 ˙fj(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Summing up the above for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=', m, we get λ(m) 1 (αp) − λ(m) 1 (0) αp − Fm = − 1 m m � j=1 �fj(αp) − fj(0) αp − ˙fj(0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 15 From the definition of fj(αp), ˙fj(αp) = ˙dj(αp)kj(αp) + dj(αp)˙kj(αp) 1 + dj(αp)kj(αp) , and therefore, ¨fj(αp) = � ¨dj(αp)kj(αp) + 2 ˙dj(αp)˙kj(αp) + dj(αp)¨kj(αp) �� 1 + dj(αp)kj(αp) � � 1 + dj(αp)kj(αp) �2 − � ˙dj(αp)k(αp) + dj(αp)˙kj(αp) �2 � 1 + dj(αp)kj(αp) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then from Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='8, we get ��� ˙fj(αp) ��� ≤ C(1) d kmax + dmaxC(1) k 1 + dminkmin = C1, ��� ¨fj(αp) ��� ≤ � C(2) d kmax + 2C(1) d C(1) k + dmaxC(2) k �� 1 + dmaxkmax � � 1 + dminkmin �2 + � C(1) d kmax + dmaxC(1) k �2 � 1 + dminkmin �2 = C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Therefore | ¨fj(rj(αp))| ≤ C2, for some constant C2 > 0 independent of rj(αp) and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' This implies ���λ(m) 1 (αp) − λ(m) 1 (0) αp − Fm ��� ≤ 1 m m � j=1 αp 2 ��� ¨fj(tj(αp)) ��� ≤ C2 2 αp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Since | ˙fj(αp)| ≤ C1, we have |Fm| ≤ 1 m �m j=1 | ˙fj(0)| ≤ C1, for all m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Therefore, the sequence {Fm} has convergent subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let for example Fmh → F, for some sub-sequence {mh}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then ���λ(mh) 1 (αp) − λ(mh) 1 (0) αp − Fmh ��� ≤ C2 2 αp, for all h ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' So, letting h → ∞, we get ���λ1(αp) − λ1(0) αp − F ��� ≤ C2 2 αp, and letting αp → 0 as p → ∞ we get that there exists lim p→∞ λ1(αp) − λ1(0) αp = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' for every sequence α1 > α2 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' > αp > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' → 0 as p → ∞ in [0, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Thus, there exists F = limm→∞ 1 m �m j=1 ˙fj(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' This is true for every subsequence {mh}, so for any subsequence we have Fmh → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Hence, Fm converges to F as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' This implies that there exists 16 lim α→0 λ1(α) − λ1(0) α = F, so λ1 is differentiable at α = 0 and ˙λ1(0) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Let K(α) be a C5,3 billiard deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then there exists a constant Cλ1 > 0 such that ���dλ1(α) dα ��� ≤ Cλ1, for all α ∈ [0, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' We have λ1(α) = lim m→∞ 1 m m � j=1 log(1 + dj(α)kj(α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='2, λ1(α) is C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' So, from the formula in the previous proof that ˙λ1(0) = limm→∞ 1 m �m j=1 ˙fj(0), we have dλ1 dα = lim m→∞ 1 m m � j=1 ddj dα kj(α) + dj(α)dkj dα 1 + dj(α)kj(α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' From Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content='8, there exist constants C(1) d , C(1) k > 0 such that ���ddj dα ��� ≤ C(1) d and ���dkj dα ��� ≤ C(1) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Then, we have ���dλ1 dα ��� ≤ lim m→∞ 1 m m � j=1 C(1) d kmax + dmaxC(1) k 1 + dminkmin = C(1) d kmax + dmaxC(1) k 1 + dminkmin = Cλ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' This proves the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Acknowledgment The author would like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Luchezar Stoyanov for his suggestions, comments, and help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' This work was supported by a scholarship from Najran University, Saudi Arabia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' References [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Barreira and Ya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Pesin, Lyapunov exponents and smooth ergodic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Series 23, American Mathematical Society, Providence, RI, 2001.' 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[22] M, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Wojtkowski, Principles for the design of billiards with nonvanishing Lyapunov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Com- mun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 105 (1986), 391-414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQf8v7L/content/2301.01910v1.pdf'} diff --git a/4tFKT4oBgHgl3EQfRy39/content/tmp_files/2301.11773v1.pdf.txt b/4tFKT4oBgHgl3EQfRy39/content/tmp_files/2301.11773v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..adb57b2a69c423cef056bb57852b575588ef485a --- /dev/null +++ b/4tFKT4oBgHgl3EQfRy39/content/tmp_files/2301.11773v1.pdf.txt @@ -0,0 +1,3963 @@ +1 +Automatic Modulation Classification with Deep +Neural Networks +Clayton A. Harper, Mitchell A. Thornton, and Eric C. Larson +Darwin Deason Institute for Cyber Security +{caharper, mitch, eclarson}@smu.edu +Abstract—Automatic modulation classification is a desired +feature in many modern software-defined radios. In recent years, +a number of convolutional deep learning architectures have +been proposed for automatically classifying the modulation used +on observed signal bursts. However, a comprehensive analysis +of these differing architectures and importance of each design +element has not been carried out. Thus it is unclear what +tradeoffs the differing designs of these convolutional neural +networks might have. In this research, we investigate numerous +architectures for automatic modulation classification and perform +a comprehensive ablation study to investigate the impacts of +varying hyperparameters and design elements on automatic +modulation classification performance. We show that a new +state of the art in performance can be achieved using a subset +of the studied design elements. In particular, we show that +a combination of dilated convolutions, statistics pooling, and +squeeze-and-excitation units results in the strongest performing +classifier. We further investigate this best performer according +to various other criteria, including short signal bursts, common +misclassifications, and performance across differing modulation +categories and modes. +Index Terms—Automatic modulation classification, deep learn- +ing, convolutional neural network. +I. INTRODUCTION +A +UTOMATIC modulation classification (AMC) is of par- +ticular interest for radio frequency (RF) analysis and in +modern software-defined radios to perform numerous tasks +including “spectrum interference monitoring, radio fault detec- +tion, dynamic spectrum access, opportunistic mesh network- +ing, and numerous regulatory and defense applications” [1]. +Upon detection of an RF signal with unknown characteristics, +AMC is a crucial initial procedure in order to demodulate the +signal. Efficient AMC allows for maximal usage of transmis- +sion mediums and can provide resilience in modern cognitive +radios. Systems capable of adaptive modulation schemes can +monitor current channel conditions with AMC and adjust +exercised modulation schemes to maximize usage across the +transmission medium. +Moreover, for receivers that have a versatile demodulation +capability, AMC is a requisite task. The correct demodulation +scheme must be applied to recover the modulated message +within a detected signal. In systems where the modulation +scheme is not known a priori, AMC allows for efficient predic- +tion of the employed modulation scheme. Higher performing +AMC can increase the throughput and accuracy of these +systems; therefore, AMC is currently an important research +topic in the fields of machine learning and communication +systems, specifically for software-defined radios. +Typical benchmarks are constructed on the premise that the +AMC model must classify not only the mode of modulation +(e.g., QAM), but the exact variant of that mode of modulation +(e.g., 32QAM). While many architectures have proven to be +effective at high signal to noise ratios (SNRs), performance +degrades significantly at lower SNRs that often occur in real- +world applications. Other works have investigated increasing +classification performance at lower SNR levels through the +use of SNR-specific modulation classifiers [2] and clustering +based on SNR ranges [3]. To perform classification, a variety +of signal features have been investigated. Historically, AMC +has relied upon statistical moments and higher order cumulants +[4]–[6] derived from the received signal. Recent approaches +[1], [7]–[9] use raw time-domain in-phase (I) and quadrature +(Q) components as features to predict the modulation variant +of a signal. Further works have investigated additional features +including I/Q constellation plots [10]–[12]. +After selecting the signal input features, machine learning +models are used to determine statistical patterns in the data +for the classification task. Support vector machines, decision +trees, and neural networks are commonly used classifiers for +this application [1], [3], [7]–[10], [13], [14]. Residual neural +networks (ResNets), along with convolutional neural networks +(CNNs), have been shown to achieve high classification perfor- +mance for AMC [1], [3], [7]–[10]. Thus, deep learning based +methods in AMC have become more prevalent due to their +promising performance and their ability to generalize to large, +complex datasets. +While other works have contributed to increased AMC +performance, the importance of many design elements for +AMC remains unclear and a number of architectural elements +have yet to be investigated. Therefore, in this work, we aim +to formalize the impact of a variety of architectural changes +and model design decisions on AMC performance. Numerous +modifications to architectures from previous works, including +our own [7], and novel combinations of elements applied to +AMC are considered. After an initial investigation, we provide +a comprehensive ablation study in this work to investigate +the performance impact of various architectural modifications. +Additionally, we achieve new state-of-the-art classification +performance on the RadioML 2018.01A dataset [15]. Using +the best performing model, we provide additional analyses +that characterize its performance across modulation modes and +arXiv:2301.11773v1 [cs.LG] 27 Jan 2023 + +2 +Fig. 1. ResNet architecture used in [1]. Each block represents a unit in the network, which may be comprised of several layers and connections as shown +on the right of the figure. Dimensions of the tensors on the output of each block are also shown where appropriate. +signal burst duration. +II. RELATED WORK +The area of AMC has been investigated by several research +groups. We provide a summary of results in AMC to provide +context and motivation for our contributions to AMC and the +corresponding ablation study described in this paper. +Corgan et al. [8] illustrate that deep convolutional neural +networks are able to achieve high classification performance +particularly at low SNRs on a dataset comprising 11 different +types of modulation. It was found that CNNs exceeded perfor- +mance over expertly crafted features. Comparing results with +architectures in [8] and [1], [16] improved AMC performance +utilizing self-supervised contrastive learning. First, an encoder +is pre-trained in a self-supervised manner through creating +contrastive pairs with data augmentation. By creating different +views of the input data through augmentation, contrastive loss +is used to maximize the cosine similarity between positive +pairs (augmented views of the same input). Once converged, +the encoder is frozen (i.e., the weights are set to fixed +values) and two fully-connected layers are added following the +encoder to form the classifier. The classifier is trained using +supervised learning to predict the 11 different modulation +schemes. Chen et al. applied a novel architecture to the +same dataset where the input signal is sliced and transformed +into a square matrix and apply a residual network to predict +the modulation schemes [17]. Other work has investigated +empirical and variational mode decomposition to improve few- +shot learning for AMC [18]. In our work, we utilize a larger, +more complex dataset consisting of 24 modulation schemes, +as well as modeling improvements. +Spectrograms and I/Q constellation plots in [19] were found +to be effective input features to a traditional CNN achieving +nearly equivalent performance as the baseline CNN network +in [1] which used raw I/Q signals. +Further, [10]–[12] also used I/Q constellations as an input +feature in their machine learning models on a smaller scale +of four or eight modulation types. Other features have been +used in AMC— [20], [21] utilized statistical features and +support vector machines while [22], [23] used fusion methods +in CNN classifiers. Mao et al. utilized various constellation +diagrams at varying symbol timings alleviating symbol timing +synchronization concerns [24]. A squeeze-and-excitation [25] +inspired architecture was used as an attention mechanism to +focus on the most important diagrams. +Although spectrograms and constellation plots have shown +promise, they require additional processing overhead and have +had comparable performance to raw I/Q signals. In addition, +models that use raw I/Q signals could be more adept at +handling varying-length signals than constellation plots be- +cause they are not limited by periodicity constraints for short +duration signals (i.e., burst transmissions). Consequently, we +utilize raw I/Q signals in our work. +Tridgell, in his dissertation [26], builds upon these works by +investigating these architectures when deployed on resource- +limited Field Programmable Gate Arrays (FGPAs). His work +stresses the importance of reducing the number of parameters +for modulation classifiers because they are typically deployed +in resource-constrained embedded systems. +Fig. 2. X-Vector architecture overview. The convolutional activations imme- +diately before pooling are shown. These activations are fed into two statistical +pooling layers that collapse the activations over time, creating a fixed-length +tensor that can be further processed by fully connected dense layers. + +ResNet Architecture +Residual Stack +Residual Unit +Input ↓ Batch size ×1024 ×2 +Batch size x 128 x 32 +Batch size x 512 +Residual Stack +Residual Stack +Dense + SeLU (128) +Input +Conv1D + Linear (32, 1) +Conv1D + ReLU (32, 3) + Batch size × 512 × 32 +↓ Batch size x× 64 × 32 +Batch size x 128 +Residual Stack +Residual Stack +Dense + SeLU (128) +Residual Unit +Conv1D + Linear (32, 3) +↓ Batch size × 256 × 32 +Batch size x 32 x 32 +Batch size x 128 +Dense + Softmax (24) +Residual Unit +Residual Stack +Residual Stack +Batch size x 16 × 32 +Batch size x 24 +Max Pooling (stride=2) +ndno +Flatten +Prediction +↑ andno +Conv1D +(number of filters, filter size)Time +μ +Mean +Statistics Pooling +Dense +Channels +Across Channels +Layers +0 +Variance +Fixed-length +Convolutional Activations +X-Vector +Pooled +Statistics3 +Fig. 3. Proposed CNN Architecture in [7]. This is the first work to employ an X-Vector inspired architecture for AMC showing strong performance. This +architecture is used as a baseline for the modifications investigated in this paper. The f and k variables shown designate the number of kernels and size of +each kernel, respectively, in each layer. These parameters are investigated for optimal sizing in our initial investigation. +In [1], Oshea et al. created a dataset with 24 different +types of modulation, known as RadioML 2018.01A, and +achieved high classification performance using convolutional +neural networks—specifically using residual connections (see +Figure 1) within the network (ResNet). A total of 6 residual +stacks were used in the architecture. A residual stack is defined +as a series of a convolutional layers, residual units, and a max +pooling operation as shown in Figure 1. The ResNet employed +by [1] attained approximately 95% classification accuracy at +high SNR values. +Harper et al. proposed the use of X-Vectors [27] to increase +classification performance using CNNs [7]. X-Vectors are tra- +ditionally used in speaker recognition and verification systems +making use of aggregate statistics. X-Vectors employ statistical +moments, specifically mean and variance, across convolutional +filter outputs. It can be theorized that taking the mean and +variance of the embedding layer helps to eliminate signal- +specific information, leaving global, modulation-specific char- +acteristics. Figure 2 illustrates the X-Vector architecture where +statistics are computed over the activations from a convolu- +tional layer producing a fixed-length vector. +Additionally, +this +architecture +maintains +a +fully- +convolutional structure enabling variable size inputs into +the network. Using statistical aggregations allows for this +property to be exploited. When using statistical aggregations, +the input to the first dense layer is dependent upon the +number of filters in the final convolutional layer. The number +of filters is a hyperparameter, independent of the length in +time of the input signal into the neural network. +Without the statistical aggregations, the input signals into +a traditional CNN or ResNet would need to be resampled, +cropped or padded to a fixed-length in time such that there is +not a size mismatch with the final convolutional output and +the first dense layer. While the dataset used in this work has +uniformly sized signals in terms of duration, (1024 × 2), this +is an architectural advantage in our deployment as received +signals may vary in duration. Instead of modifying the inputs +to the network via sampling, cropping, padding, etc., the X- +Vector architecture can directly operate with variable-length +inputs without modifications to the network or input signal. +Figure 3 outlines the employed X-Vector architecture in [7] +where F = [f1, f2, ..., f7] = 64 and K = [k1, k2, ..., k7] = 3. +Mean and variance pooling are performed on the final con- +volutional outputs, concatenated, and fed through a series of +dense layers creating the fixed-length X-Vector. A maximum +of 98% accuracy was achieved at high SNR levels. +Fig. 4. +Accuracy comparison of the reproduced ResNet in [1] and the X- +Vector inspired model from [7] over varying SNRs. This accuracy comparison +shows the superior performance of the X-Vector architecture, especially at +higher SNRs, and supports using this architecture as a baseline for the +improvements investigated in this paper. +The work of [7] replicated the ResNet architecture from +[1] and compared the results with the X-Vector architectures +as seen in Figure 4. Harper et al. [7] were able to reproduce +this architecture achieving a maximum of 93.7% accuracy. The +authors attribute the difference in performance to differences in +the train and test set separation they used since these parame- +ters were unavailable. As expected, the classifiers perform with +a higher accuracy as the SNR value increases. In signals with a +low SNR value, noise becomes more dominant and the signal +is harder to distinguish. In modern software-defined radio + +Input +Batch size x 1024 x 2 + Batch size x 1024 × f4 +Batch size x 1024 x fz +Conv1D + ReLU (f5, k5, 1) +Statistics Pooling +Conv1D + ReLU (f1, k1, 1) +Batch size x 1024 x f1 +I Batch size x 1024 x fs +Batch size x (f*2) +Conv1D + ReLU (f2, k2, 1) +Conv1D (f6, k6, 1) +Dense + SeLU (128) +Conv1D (number of filters, +filter size, dilation rate) +Batch size x 1024 x f2 ++ Batch size x 1024 x fe +Batch size x 128 +Conv1D + ReLU (f3, k3, 1) +Conv1D + ReLU (f7, k7, 1) +Dense + SeLU (128) +Batch size x 1024 x f3 +Batch size x 128 +Conv1D + ReLU (f4, k4, 1) +Dense + Softmax (24) +Batch size x 24 +Prediction1 +0.8 +Accuracy +0.6 +0.4 +0.2 +0 +-20 +-10 +10 +20 +30 +0 +SNR (dB)4 +applications, a high SNR value is not always a given. However, +there is still significant improvement compared to random +chance, even at low SNR values. Moreover, in systems where +the modulation type must be classified quickly, this could +become crucially important as fewer demodulation schemes +would need to be applied in a trial and error manner to discover +the correct scheme. +One challenge of AMC is that performance is desired to +work well across a large range of SNRs. For instance, Figure 4 +illustrates modulation classification performance plateaued in +peak performance beyond +8dB SNR and approached chance +classification performance below −8dB SNR on the RadioML +2018.01A dataset. This range is denoted by the shaded region. +Harper et al. investigated methods to improve classification +performance in this range by employing an SNR regression +model to aid separate modulation classifiers (MCs). While +other works have trained models to be as resilient as possible +under varying SNR conditions, Harper et al. employed SNR- +specific MCs [2]. +TABLE I +SNR GROUPINGS FOR TRAINING SNR-SPECIFIC CLASSIFIERS AND +DEMULTIPLEXED CLASSIFICATION RANGES FOR EACH PREDICTED SNR. +Training Range (dB) +Demultiplexed Classification Range (dB) +[-20, -8] +(−∞, -8) +[-8, -4] +[-8, -4) +[-4, 0] +[-4, 0) +[0, 4] +[0, 4) +[4, 8] +[4, 8) +[8, 30] +[8, ∞) +Six MCs were created by discretizing the SNR range to +ameliorate performance between −8dB to +8dB SNR (see +Figure 5). These groupings were chosen in order to provide +sufficient training data to avoid overfitting the MCs and +provide enough resolution so that combining MCs provided +more value than a single classifier. +By first predicting the SNR of the received signal with +a regression model, an SNR-specific MC that was trained +on signals with the predicted SNR is applied to make the +final prediction. Although the SNR values in the dataset +are discrete, SNR is measured on a continuous scale in a +deployment scenario and can vary over time. As a result, +regression is used over classification to model SNR. Using this +approach, different classifiers can tune their feature processing +for differing SNR ranges. Each MC in this approach uses the +same architecture as that proposed in [7]; however, each MC +is trained with signals within each MC’s SNR training range +(see Table I). +Highlighting improvements across varying SNR values, Fig- +ure 6 shows the overall performance improvement (in percent- +age accuracy) using the SNR-assisted architecture compared to +the baseline classification architecture described in [7]. While +a slight decrease in performance was observed for −8dB and +a larger decrease for −2dB, improvement is shown under most +SNR conditions—particularly in the target range of −8dB to ++8dB. A possible explanation for the decrease in performance +at particular SNRs is that the optimization for a particular +MC helped overall performance for a grouping at the expense +of a single value in the group. That is, the MC for [−4, 0) +Fig. 5. The architecture using SNR regression and SNR-specific classifiers +from [2]. Each MC block shown employs the same architecture as the baseline +from [7], but specifically trained to perform AMC within a more narrow range +of SNRs (denoted as dB ranges in each block). +boosted the overall performance by performing well at −4 +and 0dB at the expense of −2dB. Due to the large size of +the testing set, these small percentage gains are impactful +because thousands more classifications are correct. All results +are statistically significant based on a McNemar’s test [28], +therefore achieving new state-of-the-art performance at the +time. +Soltani et al. [3] found SNR regions of [−10, −2]dB, +[0, 8]dB, and [10, 30]dB having similar classification patterns. +Instead of predicting exact modulation variants, the authors +group commonly confused variants into a more generic, +coarse-grained label. This grouping increases performance of +AMC by combining modulation variants that are commonly +confused. However, it also decreases the sensitivity of the +model to the numerous possible variants. +Cai et al. utilized a transformer based architecture to aid +performance at low SNR levels with relatively few training pa- +rameters (approximately 265,0000 parameters) [29]. A multi- +scale network along with center loss [30] was used in [31]. +It was found that larger kernel sizes improved AMC perfor- +mance. We further explore kernel size performance impacts +in this work. Zhang et al. proposed a high-order attention +mechanism using the covariance matrix achieving a maximum +accuracy of 95.49% [32]. +Although many discussed works use the same RadioML +2018.01A dataset, there is a lack of a uniform dataset split +to establish a benchmark for papers to report performance. +In an effort to make AMC work more reproducible and +comparable across publications, we have made our dataset split +and accompanying code available on GitHub.1 +While numerous works have investigated architectural im- +provements, we aim to improve upon these works by intro- +ducing additional modifications as well as a comprehensive +ablation study that illustrates the improvement of each mod- +ification. With the new modifications, we achieve new state- +of-the-art AMC performance. +III. DATASET +To evaluate different machine learning architectures, we +use the RadioML 2018.01A dataset that is comprised of 24 +1https://github.com/caharper/Automatic-Modulation-Classification-with- +Deep-Neural-Networks + +SNR Regression +Model +DEMUX +MC +MC +MC +MC +MC +MC +(-8, -8) +[-8, -4) +[-4, 0] +[0, 4] +[4, 8] +(8, 8)5 +Fig. 6. Summary of residual improvement in accuracy over [7] that was first +published in [2]. This work showed how the baseline architecture could be +tuned to specific SNR ranges. Positive improvement is observed for most SNR +ranges. +different modulation types [1], [15]. Due to the complexity +and variety of modulation schemes in the dataset, it is fairly +representative of typically encountered modulation schemes. +Moreover, this variety increases the likelihood that AMC +models will generalize to more exotic or non-existing modu- +lation schemes in the training data that are derived from these +traditional variants. +There are a total of 2.56 million labeled signals, S(T), +each consisting of 1024 time domain digitized intermediate +frequency (IF) samples of in-phase (I) and quadrature (Q) +signal components where S(T) = I(T) + jQ(T). The data +was collected at a 900MHz IF with an assumed sampling +rate of 1MS/sec such that each 1024 time domain digitized +I/Q sample is 1.024 ms [33]. The 24 modulation types and +the representative groups that we chose for each are listed as +follows: +• Amplitude: OOK, 4ASK, 8ASK, AM-SSB-SC, AM- +SSB-WC, AM-DSB-WC, and AM-DSB-SC +• Phase: BPSK, QPSK, 8PSK, 16PSK, 32PSK, and +OQPSK +• Amplitude and Phase: 16APSK, 32APSK, 64APSK, +128APSK, 16QAM, 32QAM, 64QAM, 128QAM, and +256QAM +• Frequency: FM and GMSK +Each modulation type includes a total of 106, 496 obser- +vations ranging from −20dB to +30dB SNR in 2dB steps +for a total of 26 different SNR values. SNR is assumed +to be consistent over the same window length as the I/Q +sample window. For evaluation, we divided the dataset into 1 +million different training observations and 1.5 million testing +observations under a random shuffle split, stratified across +modulation type and SNR. Because of this balance, the +expected performance for a random chance classifier is 1/24 +or 4.2%. With varying SNR levels across the dataset, it is +expected that the classifier would perform with a higher degree +of accuracy as the SNR value is increased. For consistency, +each model investigated in this work was trained and evaluated +on the same train and test set splits. +IV. INITIAL INVESTIGATION +In this work, we use the architecture described in [7] as +the baseline architecture. We note that [2] improved upon the +baseline; however, each individual MC used the baseline archi- +tecture except trained on specific SNR ranges. Therefore, the +base architectural elements were similar to [7], but separated +for different SNRs. In this work, our focus is to improve upon +the employed CNN architecture for an individual MC rather +than the use of several MCs. Therefore, we use the architecture +from [7] as our baseline. +Before exploring an ablation study, we make a few notable +changes from the baseline architecture in an effort to increase +AMC performance. This initial exploration is for clarity as +it reserves the ablation study that follows from requiring an +inordinate number of models. It also introduces the general +training procedures that assist and orient the reader in fol- +lowing the ablation study—the ablation study mirrors these +procedures. We first provide an initial investigation exploring +these notable changes. +We train each model using the Adam optimizer [34] with +an initial learning rate lr = 0.0001, a decay factor of 0.1 if +the validation loss does not decrease for 12 epochs, and a +minimum learning rate of 1e-7. If the validation loss does not +decrease after 20 epochs, training is terminated and the models +are deemed converged. For all experiments, mini-batches of +size 32 are used. As has been established in most programming +packages for neural networks, we refer to fully connected +neural network layers as dense layers, which are typically +followed by an activation function. +A. Architectural Changes +A common property of neural networks is using fewer but +larger kernels in the early layers of the network, and an +increase of smaller kernels are used in the later layers than +the baseline architecture. This is commonly referred to as the +information distillation pipeline [35]. By utilizing a smaller +number of large kernels in early layers, we are able to increase +the temporal context of the convolutional features without +dramatically increasing the number of trainable parameters. +Numerous, but smaller kernels are used in later convolu- +tional layers to create more abstract features. Configuring +the network in this manner is especially popular in image +classification where later layers represent more abstract, class- +specific features. +We investigate this modification in three stages, using the +baseline architecture described in Figure 3 [7]. We denote +number of filters in the network and the filter sizes as +F = [f1, f2, ..., f7] and K = [k1, k2, ...k7] in Figure 3. The +baseline architecture used f = 64 (for all layers) and k = 3 +(consistent kernel size for all layers). Our first modification to +the baseline architecture is F = [32, 48, 64, 72, 84, 96, 108], +but keeping k = 3 for all layers. Second, we use the baseline +architecture, but change the size of filters in the network where +f = 64 (same as baseline) and K = [7, 5, 7, 5, 3, 3, 3]. Third, +we make both modifications and compare the result to the +baseline model where F = [32, 48, 64, 72, 84, 96, 108] and +K = [7, 5, 7, 5, 3, 3, 3]. These modifications are not exhaustive + +0.318 +0.306 +0.3 - +0.2600.259 +Residual Improvement (0-100%) +0.235 +0.229 +0.222 +0.216 +0.2 +0.192 +0.172 +0.170 +0.165 +0.1540.157 +0.142 +0.149 +0.134 +0.124 +0.1 +0.020 +0.008 +0.0 +-0.012 -0.008 +-0.008 +-0.060 +-0.1 +0.094 +-0.111 +20-18-16-14-12-10 +-8 +-6 +-4 +0 +2 +4 +6 +8 +10 +12 +16 +20 +22 +24 +26 +2830 +SNR (dB)6 +searches; rather, these modifications are meant to guide future +changes to the network by understanding the influence of filter +quantity and filter size in a limited context. +TABLE II +INITIAL INVESTIGATION PERFORMANCE OVERVIEW. ALL +ARCHITECTURES EMPLOY THE BASELINE WITH VARYING NUMBERS OF +KERNELS AND KERNEL SIZES. +Notes +# Params +Avg. +Accuracy +Max +Accuracy +Reproduced ResNet [1] +165,144 +59.2% +93.7% +X-Vector in [7] +110,680 +61.3% +98.0% +More Filters +(Same Filter Sizes) +149,168 +61.0% +96.1% +Larger Filter Sizes +(Same # Filters) +143,960 +62.6% +98.2% +Combined +174,000 +62.9% +98.6% +B. Initial Investigation Results +As shown in Table II, increasing the size of the filters +in earlier layers increases both average and maximum test +accuracy over [7]; but, at the cost of additional parameters. +A possible explanation for the increase in performance is the +increase in temporal context due to the larger kernel sizes. +Increasing the number of filters without increasing temporal +context decreases performance. This is possibly because it in- +creases the complexity of the model without adding additional +signal context. +Fig. 7. +SNR vs. accuracy comparison of the initial investigation using the +baseline architecture. Noticeable improvements can be observed across all +SNRs. +Figure 7 illustrates the change in accuracy with varying +SNR. The combined model, utilizing various kernel sizes +and numbers of filters, consistently outperforms the other +architectures across changing SNR conditions. +Although increasing the number of filters decreases per- +formance alone, combining the approach with larger kernel +sizes yields the best performance in our initial investigation. +Increasing the temporal context may have allowed additional +filters to better characterize the input signal. +Because increased temporal context improves AMC perfor- +mance, we are inspired to investigate additional methods such +as squeeze-and-excitation blocks and dilated convolutions that +can increase global and local context [25], [36]. +V. ABLATION STUDY ARCHITECTURE BACKGROUND +Building upon our findings from our initial investigation, +we make additional modifications to the baseline architecture. +For the MCs, we introduce dilated convolutions, squeeze- +and-excitation blocks, self-attention, and other architectural +changes. We also investigate various kernel sizes and the +quantity of kernels employed from the initial investigation. +Our goal is to improve upon existing architectures while +investigating the impact of each modification on classification +accuracy through an ablation study. In this section, we describe +each modification performed. +A. Squeeze-and-Excitation Networks +Fig. 8. +Squeeze-and-Excitation block proposed in [25]. One SE block is +shown applied to a single layer convolutional output activation. Two paths +are shown, a scaling path and an identity path. The scaling vector is applied +across channels to the identity path of the activations. +Squeeze-and-Excitation (SE) blocks introduce a channel- +wise attention mechanism first proposed in [25]. Due to the +limited receptive field of each convolutional filter, SE blocks +propose a recalibration step based on global statistics across +channels (average pooling) to provide global context. Although +initially utilized for image classification tasks [25], [37], [38], +we argue the use of SE blocks can provide meaningful global +context to the convolutional network used for AMC over the +time domain. +Figure 8 depicts an SE block. The squeeze operation is de- +fined as temporal global average pooling across convolutional +filters. For an individual channel, c, the squeeze operation is +defined as: +zc = Fsq(xc) = 1 +T +T +� +i=1 +xi,c +(1) +where X +∈ +RT ×C += +[x1, x2, ..., xC], Z +∈ +R1×C += +[z1, z2, ..., zC], T is the number of samples in time, and C is +the total number of channels. To model nonlinear interactions +between channel-wise statistics, Z is fed into a series of dense +layers followed by nonlinear activation functions: +s = Fex(z, W) = σ(g(z, W)) = σ(W2δ(W1z)) +(2) +where δ is the rectified linear (ReLU) activation function, +W1 ∈ R +C +r ×C, W2 ∈ RC× C +r , r is a dimensionality reduction +ratio, and σ is the sigmoid activation function. The sigmoid +function is chosen as opposed to the softmax function so that +multiple channels can be accentuated and are not mutually- +exclusive. That is, the normalization term in the softmax +can cause dependencies among channels, so the sigmoid +activation is preferred. W1 imposes a bottleneck to improve +generalization performance and reduce parameter counts while +W2 increases the dimensionality back to the original number +of channels for the recalibration operation. In our work, we + +0.8 +2 +0.6 +Accurac +0.4 +0.2 +0 +-20 +-10 +0 +10 +20 +30 +SNR (dB) +Model - X-Vector Model from [7] - More Filters (Same Filter Sizes) +- Larger Filter Sizes (Same # Filters) → Combined - - Random ChanceX +Fex(·, W) +Time (T) +Fsq(·) +1 ×C +1 × C +Channels (C) +Fscale(·, ·) +T×C +T×C7 +Fig. 9. Proposed architecture with modifications including SENets, dilated convolutions, optional ReLU activation before statistics pooling, and self-attention. +The output tensor sizes are also shown for each unit in the diagram. An * denotes where the sizes differ from the baseline architecture. +use r = 2 for all SE blocks to ensure a reasonable number +of trainable parameters without over-squashing the embedding +size. +The final operation in the SE block, scaling or recalibration, +is obtained by scaling the the input X by s: +ˆxc = Fscale(xc, sc) = scxc +(3) +where ˆX ∈ RT ×C = [ ˆx1, ˆx2, ..., ˆ +xC]. +B. Dilated Convolutions +Fig. 10. +Dilated convolutions diagram. The top shows a traditional kernel +applied to sequential time series points. The middle and bottom diagram +illustrate dilation rates of two and three, respectively. These dilations serve +to increase the receptive field of the filter without increasing the number of +trainable variables in the kernel. +Proposed in [36], Figure 10 depicts dilated convolutions +where the convolutional kernels are denoted by the colored +components. In a traditional convolution, the dilation rate +is equal to 1. Dilated convolutions build temporal context +by increasing the receptive field of the convolutional kernels +without increasing parameter counts as the number of entries +in the kernel remains the same. +Dilated convolutions also do not downsample the signals +like strided convolutions. Instead, the output of a dilated +convolution can be the exact size of the input after properly +handling edge effects at the beginning and end of the signal. +C. Final Convolutional Activation +We also investigate the impact of using an activation func- +tion (ReLU) after the last convolutional layer, just before +statistics pooling. Because ReLU transforms the input se- +quence to be non-negative, the distribution characterized by +the pooling statistics may become skewed. In [7] and [2], +no activation was applied after the final convolutional layer +as shown in Figure 3. We investigate if this transformation +impacts classification performance. +D. Self-Attention +Self-attention allows the convolutional outputs to interact +with one another enabling the network to learn to focus on +important outputs. Self-attention before statistics pooling es- +sentially creates a weighted summation over the convolutional +outputs weighting their importance similarly to [39]–[41]. +We use the attention mechanism described by Vaswani et +al. in [42] where each output element is a weighted sum of +the linearly transformed input where the dimensionality of K +is dk as seen in Equation (4). +Attention(Q, K, V ) = softmax +� QKT +|√dk| +� +V +(4) +In the case of self-attention, Q, K, and V are equal. A scaling +factor of +1 +|√dk| is applied to counteract vanishing gradients in +the softmax output when dk is large. +VI. ABLATION STUDY ARCHITECTURE +Applying the specified modifications to the architecture in +[7], Figure 9 illustrates the proposed architecture with every +modification included in the graphic. Each colored block +represents an optional change to the architecture that will be +investigated in the ablation study. That is, each combination +of network modifications are analyzed to aid understanding of +each modification’s impact on the network. +Each convolutional layer has the following parameters: +number of filters, kernel size, and dilation rate. The asterisk +next to each dilation rate represents the changing of dilation +rates in the ablation study. If dilated convolutions are used, + +Time +Dilation rate = 1 +Dilation rate = 2 +Dilation rate = 3Input + +Batch size × 1024 x 2 +1 Batch size × 1024 × 64 +1 Batch size x 1024 x 84 +IBatch size x 1024 x 108 +Conv1D + ReLU (32, 7, 1*) +SE Block +Conv1D + ReLU (96, 3, 2*) +Statistics Pooling ++ Batch size × 1024 x 32 +I Batch size x 1024 × 64 +↓ Batch size × 1024 x 96 +Batch size x 216 +Conv1D (number of filters, +SE Block +Conv1D + ReLU (72, 5, 2*) +SE Block +Dense + SeLU (128) +filter size, dilation rate) + Batch size × 1024 × 32 +I Batch size x 1024 ×72 +I Batch size x 1024 × 96 +Batch size × 128 +Equals 1 for the +initial investigation +Conv1D + ReLU (48, 5, 2*) +SE Block +Conv1D (108, 3, 1*) +Dense + SeLU (128) +Not included in the +initial investigation +Batch size × 1024 × 48 +I Batch size x 1024 x 72 + Batch size x 1024 × 108 +Batch size x 128 +SE Block +Conv1D + ReLU (84, 3, 2*) +ReLU +Dense + Softmax (24) ++ Batch size x 1024 × 48 +I Batch size x 1024 × 84 +I Batch size × 1024 × 108 +Batch size x 24 +Conv1D + ReLU (64, 7, 3*) +SE Block +Self-Attention +Prediction8 +TABLE III +ABLATION STUDY PERFORMANCE OVERVIEW. +Model Name +Notes +SENet +Dilated +Convolutions +Final +Activation +Attention +# Params +Avg. +Accuracy +Max +Accuracy +— +Reproduced ResNet [1] +— +— +— +— +165,144 +59.2% +93.7% +— +X-Vector in [7] +— +— +— +— +110,680 +61.3% +98.0% +0000 +Best performing model from +the initial investigation +— +— +— +— +174,000 +62.9% +98.6% +0001 +— +— +— + +221,088 +62.3% +97.6% +0010 +— +— + +— +174,000 +62.8% +98.6% +0011 +— +— + + +221,088 +62.3% +97.5% +0100 +— + +— +— +174,000 +63.2% +98.9% +0101 +— + +— + +221,088 +63.1% +97.9% +0110 +— + + +— +174,000 +63.2% +98.9% +0111 +— + + + +221,088 +63.0% +98.0% +1000 + +— +— +— +202,880 +62.9% +98.5% +1001 + +— +— + +249,968 +62.6% +98.2% +1010 + +— + +— +202,880 +62.6% +98.3% +1011 + +— + + +249,968 +62.8% +98.1% +1100 + + +— +— +202,880 +62.8% +98.2% +1101 + + +— + +249,968 +63.0% +97.7% +1110 +Overall best performing model + + + +— +202,880 +63.7% +98.9% +1111 + + + + +249,968 +63.0% +97.8% +then the dilation rate value in the graphic is used. If dilated +convolutions are not used, each dilation rate is set to 1. That +is, a traditional convolution is applied. All convolutions use a +stride of 1, and the same training procedure from the initial +investigation is used. +VII. EVALUATION METRICS +We present several evaluation metrics to compare the dif- +ferent architectures considered in the ablation study. In this +section, we will discuss each evaluation technique used in the +results section. +Due to the varying levels of SNRs in the employed dataset, +we plot classification accuracy over each true SNR value. This +allows for a visualization of the tradeoff in performance as +noise becomes more or less dominant in the received signals. +Additionally, we report average accuracy and maximum ac- +curacy across the entire test set for each model. While we +note that average accuracy is not indicative of the model’s +performance, as accuracy is highly correlated to the SNR of +the input signal, we share this result to give other researchers +the ability to reproduce and compare works. +As discussed in [26], AMC is often implemented on +resource-constrained devices. In these systems, using larger +models in terms of parameter counts may not be feasible. We +report the number of parameters for each model in the ablation +study to examine the tradeoff in AMC performance and model +size. +Additional analyses are also carried out. However, due to +the large number of models investigated in this study, we +will select the best performing model from the ablation study +for brevity and analyze the performance of this model in +greater detail. For example, confusion matrices for the best +performing model from the ablation study are provided to +show common misclassifications for each modulation type. +Additionally, there exist several use-cases where relatively +short signal bursts are received. For example, a wide-band +scanning receiver may only detect a short signal burst. There- +fore, signal duration in the time domain versus AMC perfor- +mance is investigated to determine the robustness of the best +performing model when short signal bursts are received. +VIII. ABLATION RESULTS +A. Overall Performance +Table III lists the maximum and average accuracy perfor- +mance for each model in the ablation study. A binary naming +convention is used to indicate the various methods used for +each architecture. Similarly to the result found in Section IV, +increasing the temporal context typically results in increased +performance. Models that incorporate dilated convolutions +tended to have higher average accuracies than models without +dilated convolutions. +The best performing model, in terms of average accuracy +across all SNR conditions included SE blocks, dilated convolu- +tions, and a ReLU activation prior to statistics pooling (model +1110) with an average accuracy of approximately 63.7%. This +model also achieved the highest maximum accuracy of about +98.9% at a 22dB level. +SE blocks did not increase performance compared to model +0000 with the exception of models 1110 and 1111. However, +SE blocks were incorporated in the best performing model, +1110. Self-attention was not found to aid classification perfor- +mance in general with the proposed architecture. Self-attention +introduces a large number of trainable parameters possibly +forming a complex loss space. +Table IV lists the performances of single modification (from +baseline) architectures. Each component of the ablation study, +with the exception of dilated convolutions, decreased perfor- +mance when applied individually. When combined, however, +the best performing model was found. Therefore, we conclude +that each component could possibly aid the optimization of + +9 +Fig. 11. Ablation study results in terms of classification accuracy across SNR ranges. The best performing model is in the second to last row and displays +strong performance across SNR values. +TABLE IV +INDIVIDUAL NETWORK MODIFICATION PERFORMANCE OVERVIEW. +ENTRIES ARE REPEATED FROM TABLE III FOR CLARITY. +Model Name +Notes +SENet +Dilated +Convolutions +Final +Activation +Attention +# Params +Avg. +Accuracy +Max +Accuracy +— +X-Vector in [7] +— +— +— +— +110,680 +61.3% +98.0% +0000 +— +— +— +— +174,000 +62.9% +98.6% +0001 +— +— +— + +221,088 +62.3% +97.6% +0010 +— +— + +— +174,000 +62.8% +98.6% +0100 +— + +— +— +174,000 +63.2% +98.9% +1000 + +— +— +— +202,880 +62.9% +98.5% +1110 +Best performer + + + +— +202,880 +63.7% +98.9% +each other—and, in general, dilated convolutions tend to have +the most dramatic performance increases. +B. Accuracy Over Varying SNR +Figure 11 summarizes the ablation study in terms of classi- +fication accuracy over varying SNR levels. We add this figure +for completeness and reproducibility for other researchers. +The accuracy within each SNR band is shown along with the +modifications used, similar to Table III. The coloring in the +figure denotes the accuracy in each SNR band. Performance +follows a trend similar to that of a sigmoid function, where +the rate at which peak classification accuracy is achieved is +the most distinguishing feature between the different models. +With the improved architectures, a maximum of 99% accuracy +is achieved at high SNR levels (starting around 12dB SNR). +While the proposed changes to the architectures gener- +ally improve performance at higher SNR levels, the largest +improvements occur between −12dB and 12dB compared +to the baseline model in [7]. For example, at 4dB, the +performance increases from 75% up to 82%. Incorporating +these modifications to the network may prove to be critical +in real-world situations where noisy signals are likely to be +obtained. Improving AMC performance at lower SNR ranges +(< −12dB) is still an open research topic, with accuracies +near chance level. +One observation is the best performing model can vary with +SNR. In systems that have available memory and processing +power, an approach similar to [2] may be used to utilize several +models and intelligently chose predictions based on estimated +SNR conditions. That is, if the SNR of the signal of interest is +known, a model can be tuned to increase performance slightly, +as shown in [2]. Using the results presented here, researchers +could also choose the architecture differences that perform best +for a given SNR range (although performance differences are +subtle). +C. Parameter Count Tradeoff +Fig. 12. +Ablation study parameter count tradeoff. The x-axis shows the +number of trainable variables in each model and the y-axis shows max or +average accuracy. The callout for each point denotes the model name as shown +in Table III. +An overview of each model’s complexity and overall per- +formance across the entire testing set is shown in Table III. +This information is also shown graphically in Figure 12 for +the maximum accuracy over SNR and the average accuracy +across all SNRs. Whether looking at the maximum or the +average measures of performance, the conclusions are similar. +The previously described binary model name also appears in +the figure. We found a slight correlation between the number +of model parameters and overall model performance; however, +with the architectures explored, there was a general parameter +count where performance peaked. Models with parameter +counts between approximately 170k to 205k generally per- +formed better than smaller and larger models. We note that the + +Dilated +Final + Model Name +Notes +SENet +Activation +Attention + Modulation Classification Results +Convolutions +一 +Reproduced ResNet [1] +- +- +一 +0.04 +15 +0.93 +0.94 +0.94 +0.94 +0.93 +- +X-Vector in [7] +- +- +- +- +0.04 +0.91 +0.94 +0.97 +0.98 +0.98 +0.98 +0.98 +0.98 +0.98 +0.98 +0.98 +000 +- +- +- +- +一 +0.04 +0.95 +0.97 +0.98 +0.98 +86°0 +66°0 +66°0 +66°0 +66°0 +0.99 +0.99 +0001 +- +一 +- +- +0.0 +0.94 +0.97 +0.97 +0.97 +0.97 +0.98 +0.98 +0.98 +0.98 +0010 +一 +- +V +0.98 +0.98 +0.99 +0.99 +0011 +- +V +0.89 +0.94 +0.96 +0.97 +0.97 +0.97 +0.97 +0.97 +0.97 +0.97 +0100 +一 +V +- +一 +0.0 +0.92 +0.97 +0.98 +0.99 +0.99 +0.99 +0.99 +0.99 +0.99 +0.99 +0.99 +0.99 +0.99 +0101 +一 +- +一 +0.04 +0.92 +0.97 +0.98 +0.98 +0.98 +0.98 +0.98 +0110 +- +- +0.04 +0.97 +0.99 +66:0 +0.99 +0.99 +0.99 +0.99 +0111 +- +一 +V +0.04 +0.97 +0.98 +0.98 +0.98 +0.98 +0.98 +0.98 +0.98 +1000 +一 +v +- +- +一 +0.04 +0.98 +0.98 +0.98 +0.98 +0.98 +1001 +- +v +- +- +0.04 +0.98 +1010 +- +- +v +- +0.98 +0.98 +0.98 +0.98 +0.98 +0.98 +1011 +- +v +- +v +v +0.98 +0.98 +0.96 +0.98 +0.98 +0.98 +0.98 +0.98 +0.98 +0.98 +1100 +- +v +- +一 +0.04 +0.91 +0.96 +0.97 +0.98 +0.98 +0.98 +86°0 +0.98 +0.98 +0.98 +0.98 +0.98 +0.98 +1101 +- +- +0.04 +0.81 +0.91 +0.95 +0.97 +0.97 +0.98 +86°0 +0.98 +86°0 +0.98 +0.98 +0.98 +0.98 +0.98 +1110 +Overall best performing model +V +0.28 +0.36 +0.46 +0.58 +0.69 +0.82 +0.92 +0.97 +0.98 +0.99 +66°0 +66°0 +66°0 +66°0 +66°0 +66°0 +0.99 +0.99 +66:0 +1111 +- +0.81 +0.91 +0.96 +0.97 +0.97 +0.98 +0.98 +0.98 +86°0 +0.98 +0.98 +0.98 +0.98 +20 +-18 +-16 +-14 +-12 +-10 +-6 +-2 +10 +12 +14 +16 +18 +20 +22 +24 +4 +6 +8 +28 +30 +SNR (dB)1.00 +Overall Accuracy +_0110 +0100- +←1110 +1000→ +Max Accuracy +0.98 +LLO +0101 +X-Vector Model from [7] +0001 +0011 +0.96 +0.94 +%) +Reproduced ResNet from [1] +acy +0.92 +ra +0.64 +iccur +^—1110 +0100- +→-0110 +0101 +1101. +1000+1100 +toiii +0.63 +1111 +A +0000- +—0010 +←1011 +-i010 +1001→ +0.62 +X-Vector Model from [7] +0.61 +0.60 +Reproduced ResNet from [1] +0.59 +# Params10 +(a) +(b) +(c) +Fig. 13. Accuracy over varying SNR conditions for model 1110 with (a), (b), and (c) showing the top-1, top-2, and top-5 accuracy respectively. Random +chance for each is defined as 1/24, 2/24, and 5/24. +models with more than 205k parameters included self-attention +which was found to decrease model performance with the +proposed architectures. This implies that one possible reason +self-attention did not perform as well as other modifications +is because of the increase in parameters, resulting in a more +difficult loss space from which to optimize. +IX. BEST PERFORMING MODEL INVESTIGATION +Due to the large volume of models, we focus upon the +best performing model, (model 1110), for the remainder of +this work. As previously mentioned, this model employs all +modifications except self-attention. +A. Top-K Accuracy +As discussed, in systems where the modulation schemes +must be classified quickly, it is advantageous to apply fewer +demodulation schemes in a trial and error fashion. This is +particularly significant at lower SNR values where accuracy is +mediocre. Top-k accuracy allows an in-depth view on the ex- +pected number of trials before finding the correct modulation +scheme. Although traditional accuracy (top-1 accuracy) char- +acterizes the performance of the model in terms of classifying +the exact variant, top-k accuracy characterizes the percentage +of the classifier predicting the correct variant among the top- +k predictions (sorted by descending class probabilities). We +plot the top-1, top-2, and top-5 classification accuracy over +varying SNR conditions for each modulation grouping defined +in Section III in Figure 13. +Although performance decays to approximately random +chance for the overall (all modulation schemes) performance +curves for each top-k accuracy, it is notable that some modu- +lation group performances drop below random chance. The +models are trained to maximize the overall model perfor- +mance. This could explain why certain modulation groups dip +below random chance but the overall performance and other +modulation groups remain at or above random chance. +Using the proposed method greatly reduces the correct +modulation scheme search space. While high performance in +top-1 accuracy is increasingly difficult to achieve with low +SNR signals, top-2 and top-5 accuracy converge to higher +values at a much faster rate. This indicates our proposed +method greatly reduces the search space from 24 modulation +candidates to fewer candidate types when employing trial and +error methods to determine the correct modulation scheme. +Further, if the group of modulation is known (e.g., FM), one +can view a more specific tradeoff curve in terms of SNR and +top-k accuracy given in Figure 13. +B. Short Duration Signal Bursts +Due to the rapid scanning characteristic of some modern +software-defined radios, we investigate the performance trade- +off of varying signal duration and AMC performance. This +analysis is meant to emulate the situation wherein a receiver +only detects a short RF signal burst. We investigate signal +burst durations of 1.024 ms (full length signal from original +dataset), 512 µs, 256 µs, 128 µs, 64 µs, 32 µs, and 16 µs. +We assume the same 1MS/sec sampling rate as in the previous +analyses such that 16 µs burst is captured in 16 I/Q samples. +Fig. 14. Tradeoff in accuracy for various signal lengths across SNR, grouped +by modulation category for the best performing model 1110. The top plot +shows the baseline performance using the full sequence. Subsequent plots +show the same information using increasingly smaller signal lengths for +classification. +In this section, we use the same test set as our other +investigations; however, a uniformly random starting point is + +0.8 +0.6 +0.4 +Overall +一 + Amplitude +←Phase +- Amplitude and Phase +0.2 + Frequency +- - Random Chance +0 +-20 +-10 +0 +10 +20 +30 +SNR (dB)0.8 +0.6 +0.4 +0.2 +0 +-20 +-10 +0 +10 +20 +30 +SNR (dB)0.8 +0.6 +0.4 +0.2 +0 +-20 +-10 +0 +10 +20 +30 +SNR (dB)1.024 ms (n=1024) +1 +0.8 +Overall +0.6 +Amplitude +Phase +0.4 +Amplitude and Phase +Frequency +0.2 +Random Chance +0 +-20 +-10 +0 +10 +20 +30 +512 μs (n=512) +256 μs (n=256) +1 +1 +Accuracy +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0 +0 +-20 +-10 +0 +10 +20 +30 +-20 +-10 +0 +10 +20 +30 +128 μs (n=128) +64 μs (n=64) +1 +1 +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0 +0 +-20 +-10 +0 +10 +20 +30 +-20 +-10 +0 +10 +20 +30 +32 μs (n=32) +16 μs (n=16) +1 +1 +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0 +0 +20 +-10 +0 +10 +20 +30 +-20 +0 +10 +20 +-10 +30 +SNR11 +(a) +(b) +(c) +Fig. 15. Confusion matrices for (a) model 1110 (best performing model from this work), (b) the reproduced ResNet model from [1], and (c) the X-Vector +inspired model from [19] with SNR ≥ 0dB. +determined for each signal such that a contiguous sample of +the desired duration, starting at the random point, is chosen. +Thus, the chosen segment from a test set sample is randomly +assigned. +We also note that, although the sample length for the evalu- +ation is changed, the best performing model is the same archi- +tecture with the exact same trained weights because this model +uses statistics pooling from the X-Vector inspired modification. +A significant benefit to the X-Vector inspired architecture is +its ability to handle variable-length inputs without the need +of padding, retraining, or other network modifications. This +is achieved by taking global statistics across convolutional +channels producing a fixed-length vector, regardless of signal +duration. Due to this flexibility, the same model (model 1110) +weights are used for each duration experiment. This fact also +emphasizes the desirability of using X-vector inspired AMC +architectures for receivers that are deployed in an environment +where short-burst and variable duration signals are anticipated +to be present. +For each signal duration in the time domain, we plot the +overall classification accuracy over varying SNR conditions +as well as the accuracy for each modulation grouping de- +fined in Section III. Figure 14 demonstrates the tradeoff for +various signal durations where n is the number of samples +from the time domain I/Q signal. The first observation is, +as we would expect, that classification performance degrades +with decreased signal duration. For example, the maximum +accuracy begins to degrade at 256 µs and is more noticeable +at 128 µs. This is likely a result of using sample statistics +that result in unstable or biased estimates for short signal +lengths since the number of received signal data points are +insufficient to characterize the sample statistics used during +training. Random classification accuracy is approximately 4% +and is shown in the black dotted line in Figure 14. Although +classification performance decreases with decreased duration, +we are still able to achieve significantly higher classification +accuracy than random chance down to 16 µs of signal capture. +FM (frequency modulation) signals were typically more +resilient to noise interference than AM (amplitude modulation) +and AM-PM (amplitude and phase modulation) signals in our +AMC. This was observed across all signal burst durations and +our top-k accuracy analysis. This behavior indicates that the +performance of our AMC for short bursts, in the presence +of increasing amounts of noise, is more robust for signals +modulated by changes in the carrier frequency and is more +sensitive to signals modulated by varying the carrier amplitude. +We attribute this behavior to our AMC architecture, the +architecture of the receiver, or a combination of both of the +AMC and receiver. +C. Confusion Matrices +While classification accuracy provides a holistic view of +model performance, it lacks the granularity to investigate +where misclassifications are occurring. Confusion matrices are +used to analyze the distribution of classifications for each given +class. For each true label, the proportion of correctly classified +samples is calculated along with the proportion of incorrect +predictions for each opposing class. In this way, we can see +which classes the model is struggling to distinguish from +one another. A perfect classifier would be the identity matrix +where the diagonal values represent the true class matches the +predicted class. Each matrix value represents the percentage +of classifications for the true label and each row sums to 1 +(100%). +Figure 15 illustrates the class confusion matrices for SNR +levels greater than or equal to 0dB for models 1110, the +reproduced ResNet architecture from [1], and the baseline X- +Vector architecture from [7] respectively. Shown in [7], the +X-Vector architecture was able to distinguish PSK and AM- +SSB variants to a higher degree and performed better overall +than [1]. Both architectures struggled to differentiate QAM +variants. +Model 1110 improved upon these prior results for QAM +signals and in general has higher diagonal components than +the other architectures. This again supports a conclusion that +model 1110 achieves a new state-of-the-art in AMC perfor- +mance. +X. CONCLUSION +A comprehensive ablation study was carried out with regard +to AMC architectural features using the extensive RadioML +2018.01A dataset. This ablation study built upon a strong +performance of a new baseline model that was also intro- +duced in the initial investigation of this study. This initial +investigation informed the design of a number of AMC ar- +chitecture modifications—specifically, the use of X-Vectors, +dilated convolutions, and SE blocks. 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Among these modifications, dilated convolutions +were found to be the most critical architectural feature for +model performance. Self-attention was also investigated but +was not found to increase performance—although increased +temporal context improved upon prior works. +REFERENCES +[1] T. O’Shea, T. Roy, and T. C. Clancy, “Over-the-air deep learning based +radio signal classification,” in IEEE Journal of Selected Topics in Signal +Processing, vol. 12:1. +IEEE, 2018, pp. 168–179. +[2] C. A. Harper, A. Sinha, M. A. Thornton, and E. C. Larson, “SNR- +boosted automatic modulation classification,” in 2021 55th Asilomar +Conference on Signals, Systems, and Computers. +IEEE, 2021, pp. +372–375. +[3] N. Soltani, K. Sankhe, S. Ioannidis, D. Jaisinghani, and K. Chowdhury, +“Spectrum awareness at the edge: Modulation classification using smart- +phones,” in 2019 IEEE International Symposium on Dynamic Spectrum +Access Networks (DySPAN). +IEEE, 2019, pp. 1–10. +[4] S. Soliman and S.-Z. 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Chollet, Deep learning with Python. +Simon and Schuster, 2021. +[36] F. Yu and V. Koltun, “Multi-scale context aggregation by dilated +convolutions,” in International Conference on Learning Representations +(ICLR), May 2016. +[37] X. Zhang, X. Zhou, M. Lin, and J. Sun, “Shufflenet: An extremely effi- +cient convolutional neural network for mobile devices,” in Proceedings +of the IEEE conference on computer vision and pattern recognition, +2018, pp. 6848–6856. +[38] M. Tan and Q. Le, “Efficientnet: Rethinking model scaling for con- +volutional neural networks,” in International conference on machine +learning. +PMLR, 2019, pp. 6105–6114. +[39] K. Okabe, T. Koshinaka, and K. Shinoda, “Attentive Statistics Pooling +for Deep Speaker Embedding,” in Proc. Interspeech 2018, 2018, pp. +2252–2256. +[40] P. Safari, M. India, and J. Hernando, “Self-attention encoding and +pooling for speaker recognition,” in Proc. Interspeech 2020, 2020, pp. +941–945. +[41] G. Sammit, Z. Wu, Y. Wang, Z. Wu, A. Kamata, J. Nese, and E. C. +Larson, “Automated prosody classification for oral reading fluency +with quadratic kappa loss and attentive x-vectors,” in ICASSP 2022- +2022 IEEE International Conference on Acoustics, Speech and Signal +Processing (ICASSP). +IEEE, 2022. + +13 +[42] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, +Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” Advances in +neural information processing systems, vol. 30, 2017. +Clayton A. Harper received his B.S. in mathematics +and computer engineering in 2019 and M.S. in +data engineering in 2021 from Southern Methodist +University in Dallas, TX, where he specialized in +machine learning and audio signal processing. His +main research area is the analysis of time series +signal processing in computer systems, especially +pertaining to security and privacy. He is a student +member of IEEE and is currently pursuing his Ph.D. +in computer science at Southern Methodist Univer- +sity with the co-advisors Dr. Eric C. Larson and Dr. +Mitchell A. Thornton. +Mitchell (Mitch) A. Thornton is currently the Cecil +H. Green Chair of Engineering and Professor in +the Department of Electrical and Computer Engi- +neering at Southern Methodist University in Dallas, +Texas. He also serves as the Executive Director of +the Darwin Deason Institute for Cyber Security, a +research-only unit, and as Program Director for the +interdisciplinary M.S. in Data Engineering degree +program within the Lyle School of Engineering at +SMU. His main research interests are in the areas +of cyber security and quantum informatics. His past +industrial experience includes full-time employment at the Amoco Research +Center, E-Systems, Inc. (now L3Harris Technologies Inc.), and the Cyrix +Corporation. Dr. Thornton is a member of several professional and honor +societies including the IEEE and the ACM where he is a senior member in +each organization. He was elected as Chair of the IEEE Technical Community +on Multiple-Valued Logic (TCMVL, 2010-11) and has served in various roles +for other IEEE/ACM committees. He is an author or co-author of five books +and more than 300 technical articles. He is a named inventor on over 20 +US/PCT/WIPO patents and patents pending. He holds P.E. licenses in the +states of Texas, Mississippi and Arkansas. He received the Ph.D. in computer +engineering from SMU in 1995, M.S. in computer science from SMU in 1993, +M.S. in electrical engineering from the University of Texas at Arlington in +1990, and B.S. in electrical engineering from Oklahoma State University in +1985. +Eric C. Larson is an Associate Professor in the de- +partment of Computer Science in the Bobby B. Lyle +School of Engineering, Southern Methodist Univer- +sity. His main research interests are in machine +learning, sensing, and signal / image processing +for various applications, in particular, for healthcare +and security applications. His work in both areas +has been commercialized and he holds a variety of +patents for sustainability sensing and mobile phone- +based health sensing. Dr. Larson has authored one +textbook and over 70 technical articles. He is active +in signal processing education for computer scientists and is an active member +of IEEE and the ACM. He received his Ph.D. in 2013 from the University +of Washington, where he was co-advised by Shwetak N. Patel and Les Atlas. +He received his B.S. and M.S. in Electrical Engineering in 2006 and 2008, +respectively, at Oklahoma State University, where he was advised by Damon +Chandler. + diff --git a/4tFKT4oBgHgl3EQfRy39/content/tmp_files/load_file.txt b/4tFKT4oBgHgl3EQfRy39/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1a755afe46a5e963b5c08f41efbbacf6d8d390fe --- /dev/null +++ b/4tFKT4oBgHgl3EQfRy39/content/tmp_files/load_file.txt @@ -0,0 +1,2950 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf,len=2949 +page_content='1 Automatic Modulation Classification with Deep Neural Networks Clayton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Harper, Mitchell A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Thornton, and Eric C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Larson Darwin Deason Institute for Cyber Security {caharper, mitch, eclarson}@smu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='edu Abstract—Automatic modulation classification is a desired feature in many modern software-defined radios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In recent years, a number of convolutional deep learning architectures have been proposed for automatically classifying the modulation used on observed signal bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' However, a comprehensive analysis of these differing architectures and importance of each design element has not been carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Thus it is unclear what tradeoffs the differing designs of these convolutional neural networks might have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In this research, we investigate numerous architectures for automatic modulation classification and perform a comprehensive ablation study to investigate the impacts of varying hyperparameters and design elements on automatic modulation classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We show that a new state of the art in performance can be achieved using a subset of the studied design elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In particular, we show that a combination of dilated convolutions, statistics pooling, and squeeze-and-excitation units results in the strongest performing classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We further investigate this best performer according to various other criteria, including short signal bursts, common misclassifications, and performance across differing modulation categories and modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Index Terms—Automatic modulation classification, deep learn- ing, convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' INTRODUCTION A UTOMATIC modulation classification (AMC) is of par- ticular interest for radio frequency (RF) analysis and in modern software-defined radios to perform numerous tasks including “spectrum interference monitoring, radio fault detec- tion, dynamic spectrum access, opportunistic mesh network- ing, and numerous regulatory and defense applications” [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Upon detection of an RF signal with unknown characteristics, AMC is a crucial initial procedure in order to demodulate the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Efficient AMC allows for maximal usage of transmis- sion mediums and can provide resilience in modern cognitive radios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Systems capable of adaptive modulation schemes can monitor current channel conditions with AMC and adjust exercised modulation schemes to maximize usage across the transmission medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Moreover, for receivers that have a versatile demodulation capability, AMC is a requisite task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The correct demodulation scheme must be applied to recover the modulated message within a detected signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In systems where the modulation scheme is not known a priori, AMC allows for efficient predic- tion of the employed modulation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Higher performing AMC can increase the throughput and accuracy of these systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' therefore, AMC is currently an important research topic in the fields of machine learning and communication systems, specifically for software-defined radios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Typical benchmarks are constructed on the premise that the AMC model must classify not only the mode of modulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=', QAM), but the exact variant of that mode of modulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=', 32QAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' While many architectures have proven to be effective at high signal to noise ratios (SNRs), performance degrades significantly at lower SNRs that often occur in real- world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Other works have investigated increasing classification performance at lower SNR levels through the use of SNR-specific modulation classifiers [2] and clustering based on SNR ranges [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' To perform classification, a variety of signal features have been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Historically, AMC has relied upon statistical moments and higher order cumulants [4]–[6] derived from the received signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Recent approaches [1], [7]–[9] use raw time-domain in-phase (I) and quadrature (Q) components as features to predict the modulation variant of a signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Further works have investigated additional features including I/Q constellation plots [10]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' After selecting the signal input features, machine learning models are used to determine statistical patterns in the data for the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Support vector machines, decision trees, and neural networks are commonly used classifiers for this application [1], [3], [7]–[10], [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Residual neural networks (ResNets), along with convolutional neural networks (CNNs), have been shown to achieve high classification perfor- mance for AMC [1], [3], [7]–[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Thus, deep learning based methods in AMC have become more prevalent due to their promising performance and their ability to generalize to large, complex datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' While other works have contributed to increased AMC performance, the importance of many design elements for AMC remains unclear and a number of architectural elements have yet to be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Therefore, in this work, we aim to formalize the impact of a variety of architectural changes and model design decisions on AMC performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Numerous modifications to architectures from previous works, including our own [7], and novel combinations of elements applied to AMC are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' After an initial investigation, we provide a comprehensive ablation study in this work to investigate the performance impact of various architectural modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Additionally, we achieve new state-of-the-art classification performance on the RadioML 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='01A dataset [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Using the best performing model, we provide additional analyses that characterize its performance across modulation modes and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='11773v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='LG] 27 Jan 2023 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' ResNet architecture used in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Each block represents a unit in the network, which may be comprised of several layers and connections as shown on the right of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Dimensions of the tensors on the output of each block are also shown where appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' signal burst duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' RELATED WORK The area of AMC has been investigated by several research groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We provide a summary of results in AMC to provide context and motivation for our contributions to AMC and the corresponding ablation study described in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Corgan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' [8] illustrate that deep convolutional neural networks are able to achieve high classification performance particularly at low SNRs on a dataset comprising 11 different types of modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' It was found that CNNs exceeded perfor- mance over expertly crafted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Comparing results with architectures in [8] and [1], [16] improved AMC performance utilizing self-supervised contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' First, an encoder is pre-trained in a self-supervised manner through creating contrastive pairs with data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' By creating different views of the input data through augmentation, contrastive loss is used to maximize the cosine similarity between positive pairs (augmented views of the same input).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Once converged, the encoder is frozen (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=', the weights are set to fixed values) and two fully-connected layers are added following the encoder to form the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The classifier is trained using supervised learning to predict the 11 different modulation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' applied a novel architecture to the same dataset where the input signal is sliced and transformed into a square matrix and apply a residual network to predict the modulation schemes [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Other work has investigated empirical and variational mode decomposition to improve few- shot learning for AMC [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In our work, we utilize a larger, more complex dataset consisting of 24 modulation schemes, as well as modeling improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Spectrograms and I/Q constellation plots in [19] were found to be effective input features to a traditional CNN achieving nearly equivalent performance as the baseline CNN network in [1] which used raw I/Q signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Further, [10]–[12] also used I/Q constellations as an input feature in their machine learning models on a smaller scale of four or eight modulation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Other features have been used in AMC— [20], [21] utilized statistical features and support vector machines while [22], [23] used fusion methods in CNN classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' utilized various constellation diagrams at varying symbol timings alleviating symbol timing synchronization concerns [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' A squeeze-and-excitation [25] inspired architecture was used as an attention mechanism to focus on the most important diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Although spectrograms and constellation plots have shown promise, they require additional processing overhead and have had comparable performance to raw I/Q signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In addition, models that use raw I/Q signals could be more adept at handling varying-length signals than constellation plots be- cause they are not limited by periodicity constraints for short duration signals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=', burst transmissions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Consequently, we utilize raw I/Q signals in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Tridgell, in his dissertation [26], builds upon these works by investigating these architectures when deployed on resource- limited Field Programmable Gate Arrays (FGPAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' His work stresses the importance of reducing the number of parameters for modulation classifiers because they are typically deployed in resource-constrained embedded systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' X-Vector architecture overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The convolutional activations imme- diately before pooling are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' These activations are fed into two statistical pooling layers that collapse the activations over time, creating a fixed-length tensor that can be further processed by fully connected dense layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' ResNet Architecture Residual Stack Residual Unit Input ↓ Batch size ×1024 ×2 Batch size x 128 x 32 Batch size x 512 Residual Stack Residual Stack Dense + SeLU (128) Input Conv1D + Linear (32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 1) Conv1D + ReLU (32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 3) Batch size × 512 × 32 ↓ Batch size x× 64 × 32 Batch size x 128 Residual Stack Residual Stack Dense + SeLU (128) Residual Unit Conv1D + Linear (32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 3) ↓ Batch size × 256 × 32 Batch size x 32 x 32 Batch size x 128 Dense + Softmax (24) Residual Unit Residual Stack Residual Stack Batch size x 16 × 32 Batch size x 24 Max Pooling (stride=2) ndno Flatten Prediction ↑ andno Conv1D (number of filters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' filter size)Time μ Mean Statistics Pooling Dense Channels Across Channels Layers 0 Variance Fixed-length Convolutional Activations X-Vector Pooled Statistics3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Proposed CNN Architecture in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This is the first work to employ an X-Vector inspired architecture for AMC showing strong performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This architecture is used as a baseline for the modifications investigated in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The f and k variables shown designate the number of kernels and size of each kernel, respectively, in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' These parameters are investigated for optimal sizing in our initial investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In [1], Oshea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' created a dataset with 24 different types of modulation, known as RadioML 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='01A, and achieved high classification performance using convolutional neural networks—specifically using residual connections (see Figure 1) within the network (ResNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' A total of 6 residual stacks were used in the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' A residual stack is defined as a series of a convolutional layers, residual units, and a max pooling operation as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The ResNet employed by [1] attained approximately 95% classification accuracy at high SNR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Harper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' proposed the use of X-Vectors [27] to increase classification performance using CNNs [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' X-Vectors are tra- ditionally used in speaker recognition and verification systems making use of aggregate statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' X-Vectors employ statistical moments, specifically mean and variance, across convolutional filter outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' It can be theorized that taking the mean and variance of the embedding layer helps to eliminate signal- specific information, leaving global, modulation-specific char- acteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Figure 2 illustrates the X-Vector architecture where statistics are computed over the activations from a convolu- tional layer producing a fixed-length vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Additionally, this architecture maintains a fully- convolutional structure enabling variable size inputs into the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Using statistical aggregations allows for this property to be exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' When using statistical aggregations, the input to the first dense layer is dependent upon the number of filters in the final convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The number of filters is a hyperparameter, independent of the length in time of the input signal into the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Without the statistical aggregations, the input signals into a traditional CNN or ResNet would need to be resampled, cropped or padded to a fixed-length in time such that there is not a size mismatch with the final convolutional output and the first dense layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' While the dataset used in this work has uniformly sized signals in terms of duration, (1024 × 2), this is an architectural advantage in our deployment as received signals may vary in duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Instead of modifying the inputs to the network via sampling, cropping, padding, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=', the X- Vector architecture can directly operate with variable-length inputs without modifications to the network or input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Figure 3 outlines the employed X-Vector architecture in [7] where F = [f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=', f7] = 64 and K = [k1, k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=', k7] = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Mean and variance pooling are performed on the final con- volutional outputs, concatenated, and fed through a series of dense layers creating the fixed-length X-Vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' A maximum of 98% accuracy was achieved at high SNR levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Accuracy comparison of the reproduced ResNet in [1] and the X- Vector inspired model from [7] over varying SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This accuracy comparison shows the superior performance of the X-Vector architecture, especially at higher SNRs, and supports using this architecture as a baseline for the improvements investigated in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The work of [7] replicated the ResNet architecture from [1] and compared the results with the X-Vector architectures as seen in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Harper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' [7] were able to reproduce this architecture achieving a maximum of 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='7% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The authors attribute the difference in performance to differences in the train and test set separation they used since these parame- ters were unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' As expected, the classifiers perform with a higher accuracy as the SNR value increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In signals with a low SNR value, noise becomes more dominant and the signal is harder to distinguish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In modern software-defined radio Input Batch size x 1024 x 2 Batch size x 1024 × f4 Batch size x 1024 x fz Conv1D + ReLU (f5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' k5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 1) Statistics Pooling Conv1D + ReLU (f1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 1) Batch size x 1024 x f1 I Batch size x 1024 x fs Batch size x (f*2) Conv1D + ReLU (f2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 1) Conv1D (f6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' k6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 1) Dense + SeLU (128) Conv1D (number of filters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' filter size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' dilation rate) Batch size x 1024 x f2 + Batch size x 1024 x fe Batch size x 128 Conv1D + ReLU (f3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' k3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 1) Conv1D + ReLU (f7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' k7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 1) Dense + SeLU (128) Batch size x 1024 x f3 Batch size x 128 Conv1D + ReLU (f4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' k4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 1) Dense + Softmax (24) Batch size x 24 Prediction1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2 0 20 10 10 20 30 0 SNR (dB)4 applications, a high SNR value is not always a given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' However, there is still significant improvement compared to random chance, even at low SNR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Moreover, in systems where the modulation type must be classified quickly, this could become crucially important as fewer demodulation schemes would need to be applied in a trial and error manner to discover the correct scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' One challenge of AMC is that performance is desired to work well across a large range of SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' For instance, Figure 4 illustrates modulation classification performance plateaued in peak performance beyond +8dB SNR and approached chance classification performance below −8dB SNR on the RadioML 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='01A dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This range is denoted by the shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Harper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' investigated methods to improve classification performance in this range by employing an SNR regression model to aid separate modulation classifiers (MCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' While other works have trained models to be as resilient as possible under varying SNR conditions, Harper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' employed SNR- specific MCs [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' TABLE I SNR GROUPINGS FOR TRAINING SNR-SPECIFIC CLASSIFIERS AND DEMULTIPLEXED CLASSIFICATION RANGES FOR EACH PREDICTED SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Training Range (dB) Demultiplexed Classification Range (dB) [-20, -8] (−∞, -8) [-8, -4] [-8, -4) [-4, 0] [-4, 0) [0, 4] [0, 4) [4, 8] [4, 8) [8, 30] [8, ∞) Six MCs were created by discretizing the SNR range to ameliorate performance between −8dB to +8dB SNR (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' These groupings were chosen in order to provide sufficient training data to avoid overfitting the MCs and provide enough resolution so that combining MCs provided more value than a single classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' By first predicting the SNR of the received signal with a regression model, an SNR-specific MC that was trained on signals with the predicted SNR is applied to make the final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Although the SNR values in the dataset are discrete, SNR is measured on a continuous scale in a deployment scenario and can vary over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' As a result, regression is used over classification to model SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Using this approach, different classifiers can tune their feature processing for differing SNR ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Each MC in this approach uses the same architecture as that proposed in [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' however, each MC is trained with signals within each MC’s SNR training range (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Highlighting improvements across varying SNR values, Fig- ure 6 shows the overall performance improvement (in percent- age accuracy) using the SNR-assisted architecture compared to the baseline classification architecture described in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' While a slight decrease in performance was observed for −8dB and a larger decrease for −2dB, improvement is shown under most SNR conditions—particularly in the target range of −8dB to +8dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' A possible explanation for the decrease in performance at particular SNRs is that the optimization for a particular MC helped overall performance for a grouping at the expense of a single value in the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' That is, the MC for [−4, 0) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The architecture using SNR regression and SNR-specific classifiers from [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Each MC block shown employs the same architecture as the baseline from [7], but specifically trained to perform AMC within a more narrow range of SNRs (denoted as dB ranges in each block).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' boosted the overall performance by performing well at −4 and 0dB at the expense of −2dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Due to the large size of the testing set, these small percentage gains are impactful because thousands more classifications are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' All results are statistically significant based on a McNemar’s test [28], therefore achieving new state-of-the-art performance at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Soltani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' [3] found SNR regions of [−10, −2]dB, [0, 8]dB, and [10, 30]dB having similar classification patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Instead of predicting exact modulation variants, the authors group commonly confused variants into a more generic, coarse-grained label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This grouping increases performance of AMC by combining modulation variants that are commonly confused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' However, it also decreases the sensitivity of the model to the numerous possible variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' utilized a transformer based architecture to aid performance at low SNR levels with relatively few training pa- rameters (approximately 265,0000 parameters) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' A multi- scale network along with center loss [30] was used in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' It was found that larger kernel sizes improved AMC perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We further explore kernel size performance impacts in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' proposed a high-order attention mechanism using the covariance matrix achieving a maximum accuracy of 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='49% [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Although many discussed works use the same RadioML 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='01A dataset, there is a lack of a uniform dataset split to establish a benchmark for papers to report performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In an effort to make AMC work more reproducible and comparable across publications, we have made our dataset split and accompanying code available on GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='1 While numerous works have investigated architectural im- provements, we aim to improve upon these works by intro- ducing additional modifications as well as a comprehensive ablation study that illustrates the improvement of each mod- ification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' With the new modifications, we achieve new state- of-the-art AMC performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' DATASET To evaluate different machine learning architectures, we use the RadioML 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='01A dataset that is comprised of 24 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='com/caharper/Automatic-Modulation-Classification-with- Deep-Neural-Networks SNR Regression Model DEMUX MC MC MC MC MC MC (-8, -8) [-8, -4) [-4, 0] [0, 4] [4, 8] (8, 8)5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Summary of residual improvement in accuracy over [7] that was first published in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This work showed how the baseline architecture could be tuned to specific SNR ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Positive improvement is observed for most SNR ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' different modulation types [1], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Due to the complexity and variety of modulation schemes in the dataset, it is fairly representative of typically encountered modulation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Moreover, this variety increases the likelihood that AMC models will generalize to more exotic or non-existing modu- lation schemes in the training data that are derived from these traditional variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' There are a total of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='56 million labeled signals, S(T), each consisting of 1024 time domain digitized intermediate frequency (IF) samples of in-phase (I) and quadrature (Q) signal components where S(T) = I(T) + jQ(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The data was collected at a 900MHz IF with an assumed sampling rate of 1MS/sec such that each 1024 time domain digitized I/Q sample is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='024 ms [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The 24 modulation types and the representative groups that we chose for each are listed as follows: Amplitude: OOK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 4ASK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 8ASK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' AM-SSB-SC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' AM- SSB-WC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' AM-DSB-WC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' and AM-DSB-SC Phase: BPSK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' QPSK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 8PSK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 16PSK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 32PSK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' and OQPSK Amplitude and Phase: 16APSK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 32APSK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 64APSK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 128APSK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 16QAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 32QAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 64QAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 128QAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' and 256QAM Frequency: FM and GMSK Each modulation type includes a total of 106,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 496 obser- vations ranging from −20dB to +30dB SNR in 2dB steps for a total of 26 different SNR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' SNR is assumed to be consistent over the same window length as the I/Q sample window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' For evaluation, we divided the dataset into 1 million different training observations and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='5 million testing observations under a random shuffle split, stratified across modulation type and SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Because of this balance, the expected performance for a random chance classifier is 1/24 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' With varying SNR levels across the dataset, it is expected that the classifier would perform with a higher degree of accuracy as the SNR value is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' For consistency, each model investigated in this work was trained and evaluated on the same train and test set splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' INITIAL INVESTIGATION In this work, we use the architecture described in [7] as the baseline architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We note that [2] improved upon the baseline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' however, each individual MC used the baseline archi- tecture except trained on specific SNR ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Therefore, the base architectural elements were similar to [7], but separated for different SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In this work, our focus is to improve upon the employed CNN architecture for an individual MC rather than the use of several MCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Therefore, we use the architecture from [7] as our baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Before exploring an ablation study, we make a few notable changes from the baseline architecture in an effort to increase AMC performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This initial exploration is for clarity as it reserves the ablation study that follows from requiring an inordinate number of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' It also introduces the general training procedures that assist and orient the reader in fol- lowing the ablation study—the ablation study mirrors these procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We first provide an initial investigation exploring these notable changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We train each model using the Adam optimizer [34] with an initial learning rate lr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='0001, a decay factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='1 if the validation loss does not decrease for 12 epochs, and a minimum learning rate of 1e-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' If the validation loss does not decrease after 20 epochs, training is terminated and the models are deemed converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' For all experiments, mini-batches of size 32 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' As has been established in most programming packages for neural networks, we refer to fully connected neural network layers as dense layers, which are typically followed by an activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Architectural Changes A common property of neural networks is using fewer but larger kernels in the early layers of the network, and an increase of smaller kernels are used in the later layers than the baseline architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This is commonly referred to as the information distillation pipeline [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' By utilizing a smaller number of large kernels in early layers, we are able to increase the temporal context of the convolutional features without dramatically increasing the number of trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Numerous, but smaller kernels are used in later convolu- tional layers to create more abstract features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Configuring the network in this manner is especially popular in image classification where later layers represent more abstract, class- specific features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We investigate this modification in three stages, using the baseline architecture described in Figure 3 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We denote number of filters in the network and the filter sizes as F = [f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=', f7] and K = [k1, k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='k7] in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The baseline architecture used f = 64 (for all layers) and k = 3 (consistent kernel size for all layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Our first modification to the baseline architecture is F = [32, 48, 64, 72, 84, 96, 108], but keeping k = 3 for all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Second, we use the baseline architecture, but change the size of filters in the network where f = 64 (same as baseline) and K = [7, 5, 7, 5, 3, 3, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Third, we make both modifications and compare the result to the baseline model where F = [32, 48, 64, 72, 84, 96, 108] and K = [7, 5, 7, 5, 3, 3, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' These modifications are not exhaustive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='3 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='259 Residual Improvement (0-100%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='235 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='1540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='157 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='012 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='111 20-18-16-14-12-10 8 6 4 0 2 4 6 8 10 12 16 20 22 24 26 2830 SNR (dB)6 searches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' rather, these modifications are meant to guide future changes to the network by understanding the influence of filter quantity and filter size in a limited context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' TABLE II INITIAL INVESTIGATION PERFORMANCE OVERVIEW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' ALL ARCHITECTURES EMPLOY THE BASELINE WITH VARYING NUMBERS OF KERNELS AND KERNEL SIZES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Notes # Params Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Accuracy Max Accuracy Reproduced ResNet [1] 165,144 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='7% X-Vector in [7] 110,680 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='3% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='0% More Filters (Same Filter Sizes) 149,168 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='0% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='1% Larger Filter Sizes (Same # Filters) 143,960 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2% Combined 174,000 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='9% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6% B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Initial Investigation Results As shown in Table II, increasing the size of the filters in earlier layers increases both average and maximum test accuracy over [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' but, at the cost of additional parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' A possible explanation for the increase in performance is the increase in temporal context due to the larger kernel sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Increasing the number of filters without increasing temporal context decreases performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This is possibly because it in- creases the complexity of the model without adding additional signal context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' SNR vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' accuracy comparison of the initial investigation using the baseline architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Noticeable improvements can be observed across all SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Figure 7 illustrates the change in accuracy with varying SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The combined model, utilizing various kernel sizes and numbers of filters, consistently outperforms the other architectures across changing SNR conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Although increasing the number of filters decreases per- formance alone, combining the approach with larger kernel sizes yields the best performance in our initial investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Increasing the temporal context may have allowed additional filters to better characterize the input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Because increased temporal context improves AMC perfor- mance, we are inspired to investigate additional methods such as squeeze-and-excitation blocks and dilated convolutions that can increase global and local context [25], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' ABLATION STUDY ARCHITECTURE BACKGROUND Building upon our findings from our initial investigation, we make additional modifications to the baseline architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' For the MCs, we introduce dilated convolutions, squeeze- and-excitation blocks, self-attention, and other architectural changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We also investigate various kernel sizes and the quantity of kernels employed from the initial investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Our goal is to improve upon existing architectures while investigating the impact of each modification on classification accuracy through an ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In this section, we describe each modification performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Squeeze-and-Excitation Networks Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Squeeze-and-Excitation block proposed in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' One SE block is shown applied to a single layer convolutional output activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Two paths are shown, a scaling path and an identity path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The scaling vector is applied across channels to the identity path of the activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Squeeze-and-Excitation (SE) blocks introduce a channel- wise attention mechanism first proposed in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Due to the limited receptive field of each convolutional filter, SE blocks propose a recalibration step based on global statistics across channels (average pooling) to provide global context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Although initially utilized for image classification tasks [25], [37], [38], we argue the use of SE blocks can provide meaningful global context to the convolutional network used for AMC over the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Figure 8 depicts an SE block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The squeeze operation is de- fined as temporal global average pooling across convolutional filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' For an individual channel, c, the squeeze operation is defined as: zc = Fsq(xc) = 1 T T � i=1 xi,c (1) where X ∈ RT ×C = [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=', xC], Z ∈ R1×C = [z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=', zC], T is the number of samples in time, and C is the total number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' To model nonlinear interactions between channel-wise statistics, Z is fed into a series of dense layers followed by nonlinear activation functions: s = Fex(z, W) = σ(g(z, W)) = σ(W2δ(W1z)) (2) where δ is the rectified linear (ReLU) activation function, W1 ∈ R C r ×C, W2 ∈ RC× C r , r is a dimensionality reduction ratio, and σ is the sigmoid activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The sigmoid function is chosen as opposed to the softmax function so that multiple channels can be accentuated and are not mutually- exclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' That is, the normalization term in the softmax can cause dependencies among channels, so the sigmoid activation is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' W1 imposes a bottleneck to improve generalization performance and reduce parameter counts while W2 increases the dimensionality back to the original number of channels for the recalibration operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In our work, we 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6 Accurac 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2 0 20 10 0 10 20 30 SNR (dB) Model - X-Vector Model from [7] - More Filters (Same Filter Sizes) Larger Filter Sizes (Same # Filters) → Combined - - Random ChanceX Fex(·, W) Time (T) Fsq(·) 1 ×C 1 × C Channels (C) Fscale(·, ·) T×C T×C7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Proposed architecture with modifications including SENets, dilated convolutions, optional ReLU activation before statistics pooling, and self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The output tensor sizes are also shown for each unit in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' An * denotes where the sizes differ from the baseline architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' use r = 2 for all SE blocks to ensure a reasonable number of trainable parameters without over-squashing the embedding size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The final operation in the SE block, scaling or recalibration, is obtained by scaling the the input X by s: ˆxc = Fscale(xc, sc) = scxc (3) where ˆX ∈ RT ×C = [ ˆx1, ˆx2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=', ˆ xC].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Dilated Convolutions Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Dilated convolutions diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The top shows a traditional kernel applied to sequential time series points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The middle and bottom diagram illustrate dilation rates of two and three, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' These dilations serve to increase the receptive field of the filter without increasing the number of trainable variables in the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Proposed in [36], Figure 10 depicts dilated convolutions where the convolutional kernels are denoted by the colored components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In a traditional convolution, the dilation rate is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Dilated convolutions build temporal context by increasing the receptive field of the convolutional kernels without increasing parameter counts as the number of entries in the kernel remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Dilated convolutions also do not downsample the signals like strided convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Instead, the output of a dilated convolution can be the exact size of the input after properly handling edge effects at the beginning and end of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Final Convolutional Activation We also investigate the impact of using an activation func- tion (ReLU) after the last convolutional layer, just before statistics pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Because ReLU transforms the input se- quence to be non-negative, the distribution characterized by the pooling statistics may become skewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In [7] and [2], no activation was applied after the final convolutional layer as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We investigate if this transformation impacts classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Self-Attention Self-attention allows the convolutional outputs to interact with one another enabling the network to learn to focus on important outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Self-attention before statistics pooling es- sentially creates a weighted summation over the convolutional outputs weighting their importance similarly to [39]–[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We use the attention mechanism described by Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' in [42] where each output element is a weighted sum of the linearly transformed input where the dimensionality of K is dk as seen in Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Attention(Q, K, V ) = softmax � QKT |√dk| � V (4) In the case of self-attention, Q, K, and V are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' A scaling factor of 1 |√dk| is applied to counteract vanishing gradients in the softmax output when dk is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' ABLATION STUDY ARCHITECTURE Applying the specified modifications to the architecture in [7], Figure 9 illustrates the proposed architecture with every modification included in the graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Each colored block represents an optional change to the architecture that will be investigated in the ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' That is, each combination of network modifications are analyzed to aid understanding of each modification’s impact on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Each convolutional layer has the following parameters: number of filters, kernel size, and dilation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The asterisk next to each dilation rate represents the changing of dilation rates in the ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' If dilated convolutions are used,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Time Dilation rate = 1 Dilation rate = 2 Dilation rate = 3Input + Batch size × 1024 x 2 1 Batch size × 1024 × 64 1 Batch size x 1024 x 84 IBatch size x 1024 x 108 Conv1D + ReLU (32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 1*) SE Block Conv1D + ReLU (96,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 2*) Statistics Pooling + Batch size × 1024 x 32 I Batch size x 1024 × 64 ↓ Batch size × 1024 x 96 Batch size x 216 Conv1D (number of filters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' SE Block Conv1D + ReLU (72,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 2*) SE Block Dense + SeLU (128) filter size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' dilation rate) Batch size × 1024 × 32 I Batch size x 1024 ×72 I Batch size x 1024 × 96 Batch size × 128 Equals 1 for the initial investigation Conv1D + ReLU (48,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 2*) SE Block Conv1D (108,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 1*) Dense + SeLU (128) Not included in the initial investigation Batch size × 1024 × 48 I Batch size x 1024 x 72 Batch size x 1024 × 108 Batch size x 128 SE Block Conv1D + ReLU (84,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 2*) ReLU Dense + Softmax (24) + Batch size x 1024 × 48 I Batch size x 1024 × 84 I Batch size × 1024 × 108 Batch size x 24 Conv1D + ReLU (64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 3*) SE Block Self-Attention Prediction8 TABLE III ABLATION STUDY PERFORMANCE OVERVIEW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Model Name Notes SENet Dilated Convolutions Final Activation Attention # Params Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Accuracy Max Accuracy — Reproduced ResNet [1] — — — — 165,144 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='7% — X-Vector in [7] — — — — 110,680 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='3% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='0% 0000 Best performing model from the initial investigation — — — — 174,000 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='9% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6% 0001 — — — \x13 221,088 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='3% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6% 0010 — — \x13 — 174,000 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6% 0011 — — \x13 \x13 221,088 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='3% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='5% 0100 — \x13 — — 174,000 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='9% 0101 — \x13 — \x13 221,088 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='1% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='9% 0110 — \x13 \x13 — 174,000 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='9% 0111 — \x13 \x13 \x13 221,088 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='0% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='0% 1000 \x13 — — — 202,880 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='9% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='5% 1001 \x13 — — \x13 249,968 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2% 1010 \x13 — \x13 — 202,880 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='3% 1011 \x13 — \x13 \x13 249,968 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='1% 1100 \x13 \x13 — — 202,880 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2% 1101 \x13 \x13 — \x13 249,968 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='0% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='7% 1110 Overall best performing model \x13 \x13 \x13 — 202,880 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='7% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='9% 1111 \x13 \x13 \x13 \x13 249,968 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='0% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8% then the dilation rate value in the graphic is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' If dilated convolutions are not used, each dilation rate is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' That is, a traditional convolution is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' All convolutions use a stride of 1, and the same training procedure from the initial investigation is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' EVALUATION METRICS We present several evaluation metrics to compare the dif- ferent architectures considered in the ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In this section, we will discuss each evaluation technique used in the results section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Due to the varying levels of SNRs in the employed dataset, we plot classification accuracy over each true SNR value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This allows for a visualization of the tradeoff in performance as noise becomes more or less dominant in the received signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Additionally, we report average accuracy and maximum ac- curacy across the entire test set for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' While we note that average accuracy is not indicative of the model’s performance, as accuracy is highly correlated to the SNR of the input signal, we share this result to give other researchers the ability to reproduce and compare works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' As discussed in [26], AMC is often implemented on resource-constrained devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In these systems, using larger models in terms of parameter counts may not be feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We report the number of parameters for each model in the ablation study to examine the tradeoff in AMC performance and model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Additional analyses are also carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' However, due to the large number of models investigated in this study, we will select the best performing model from the ablation study for brevity and analyze the performance of this model in greater detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' For example, confusion matrices for the best performing model from the ablation study are provided to show common misclassifications for each modulation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Additionally, there exist several use-cases where relatively short signal bursts are received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' For example, a wide-band scanning receiver may only detect a short signal burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' There- fore, signal duration in the time domain versus AMC perfor- mance is investigated to determine the robustness of the best performing model when short signal bursts are received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' ABLATION RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Overall Performance Table III lists the maximum and average accuracy perfor- mance for each model in the ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' A binary naming convention is used to indicate the various methods used for each architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Similarly to the result found in Section IV, increasing the temporal context typically results in increased performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Models that incorporate dilated convolutions tended to have higher average accuracies than models without dilated convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The best performing model, in terms of average accuracy across all SNR conditions included SE blocks, dilated convolu- tions, and a ReLU activation prior to statistics pooling (model 1110) with an average accuracy of approximately 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This model also achieved the highest maximum accuracy of about 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='9% at a 22dB level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' SE blocks did not increase performance compared to model 0000 with the exception of models 1110 and 1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' However, SE blocks were incorporated in the best performing model, 1110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Self-attention was not found to aid classification perfor- mance in general with the proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Self-attention introduces a large number of trainable parameters possibly forming a complex loss space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Table IV lists the performances of single modification (from baseline) architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Each component of the ablation study, with the exception of dilated convolutions, decreased perfor- mance when applied individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' When combined, however, the best performing model was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Therefore, we conclude that each component could possibly aid the optimization of 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Ablation study results in terms of classification accuracy across SNR ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The best performing model is in the second to last row and displays strong performance across SNR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' TABLE IV INDIVIDUAL NETWORK MODIFICATION PERFORMANCE OVERVIEW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' ENTRIES ARE REPEATED FROM TABLE III FOR CLARITY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Model Name Notes SENet Dilated Convolutions Final Activation Attention # Params Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Accuracy Max Accuracy — X-Vector in [7] — — — — 110,680 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='3% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='0% 0000 — — — — 174,000 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='9% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6% 0001 — — — \x13 221,088 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='3% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6% 0010 — — \x13 — 174,000 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6% 0100 — \x13 — — 174,000 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='9% 1000 \x13 — — — 202,880 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='9% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='5% 1110 Best performer \x13 \x13 \x13 — 202,880 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='7% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='9% each other—and, in general, dilated convolutions tend to have the most dramatic performance increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Accuracy Over Varying SNR Figure 11 summarizes the ablation study in terms of classi- fication accuracy over varying SNR levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We add this figure for completeness and reproducibility for other researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The accuracy within each SNR band is shown along with the modifications used, similar to Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The coloring in the figure denotes the accuracy in each SNR band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Performance follows a trend similar to that of a sigmoid function, where the rate at which peak classification accuracy is achieved is the most distinguishing feature between the different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' With the improved architectures, a maximum of 99% accuracy is achieved at high SNR levels (starting around 12dB SNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' While the proposed changes to the architectures gener- ally improve performance at higher SNR levels, the largest improvements occur between −12dB and 12dB compared to the baseline model in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' For example, at 4dB, the performance increases from 75% up to 82%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Incorporating these modifications to the network may prove to be critical in real-world situations where noisy signals are likely to be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Improving AMC performance at lower SNR ranges (< −12dB) is still an open research topic, with accuracies near chance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' One observation is the best performing model can vary with SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In systems that have available memory and processing power, an approach similar to [2] may be used to utilize several models and intelligently chose predictions based on estimated SNR conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' That is, if the SNR of the signal of interest is known, a model can be tuned to increase performance slightly, as shown in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Using the results presented here, researchers could also choose the architecture differences that perform best for a given SNR range (although performance differences are subtle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Parameter Count Tradeoff Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Ablation study parameter count tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The x-axis shows the number of trainable variables in each model and the y-axis shows max or average accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The callout for each point denotes the model name as shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' An overview of each model’s complexity and overall per- formance across the entire testing set is shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This information is also shown graphically in Figure 12 for the maximum accuracy over SNR and the average accuracy across all SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Whether looking at the maximum or the average measures of performance, the conclusions are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The previously described binary model name also appears in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We found a slight correlation between the number of model parameters and overall model performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' however, with the architectures explored, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='98 LLO 0101 X-Vector Model from [7] 0001 0011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='94 %) Reproduced ResNet from [1] acy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='92 ra 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='64 iccur ^—1110 0100- →-0110 0101 1101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 1000+1100 toiii 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='63 1111 A 0000- —0010 ←1011 i010 1001→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='62 X-Vector Model from [7] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='60 Reproduced ResNet from [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='59 # Params10 (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Accuracy over varying SNR conditions for model 1110 with (a), (b), and (c) showing the top-1, top-2, and top-5 accuracy respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Random chance for each is defined as 1/24, 2/24, and 5/24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' models with more than 205k parameters included self-attention which was found to decrease model performance with the proposed architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This implies that one possible reason self-attention did not perform as well as other modifications is because of the increase in parameters, resulting in a more difficult loss space from which to optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' BEST PERFORMING MODEL INVESTIGATION Due to the large volume of models, we focus upon the best performing model, (model 1110), for the remainder of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' As previously mentioned, this model employs all modifications except self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Top-K Accuracy As discussed, in systems where the modulation schemes must be classified quickly, it is advantageous to apply fewer demodulation schemes in a trial and error fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This is particularly significant at lower SNR values where accuracy is mediocre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Top-k accuracy allows an in-depth view on the ex- pected number of trials before finding the correct modulation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Although traditional accuracy (top-1 accuracy) char- acterizes the performance of the model in terms of classifying the exact variant, top-k accuracy characterizes the percentage of the classifier predicting the correct variant among the top- k predictions (sorted by descending class probabilities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We plot the top-1, top-2, and top-5 classification accuracy over varying SNR conditions for each modulation grouping defined in Section III in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Although performance decays to approximately random chance for the overall (all modulation schemes) performance curves for each top-k accuracy, it is notable that some modu- lation group performances drop below random chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The models are trained to maximize the overall model perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This could explain why certain modulation groups dip below random chance but the overall performance and other modulation groups remain at or above random chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Using the proposed method greatly reduces the correct modulation scheme search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' While high performance in top-1 accuracy is increasingly difficult to achieve with low SNR signals, top-2 and top-5 accuracy converge to higher values at a much faster rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This indicates our proposed method greatly reduces the search space from 24 modulation candidates to fewer candidate types when employing trial and error methods to determine the correct modulation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Further, if the group of modulation is known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=', FM), one can view a more specific tradeoff curve in terms of SNR and top-k accuracy given in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Short Duration Signal Bursts Due to the rapid scanning characteristic of some modern software-defined radios, we investigate the performance trade- off of varying signal duration and AMC performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This analysis is meant to emulate the situation wherein a receiver only detects a short RF signal burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We investigate signal burst durations of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='024 ms (full length signal from original dataset), 512 µs, 256 µs, 128 µs, 64 µs, 32 µs, and 16 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We assume the same 1MS/sec sampling rate as in the previous analyses such that 16 µs burst is captured in 16 I/Q samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Tradeoff in accuracy for various signal lengths across SNR, grouped by modulation category for the best performing model 1110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The top plot shows the baseline performance using the full sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Subsequent plots show the same information using increasingly smaller signal lengths for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In this section, we use the same test set as our other investigations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' however, a uniformly random starting point is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='4 Overall 一 Amplitude ←Phase Amplitude and Phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2 Frequency - Random Chance 0 20 10 0 10 20 30 SNR (dB)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2 0 20 10 0 10 20 30 SNR (dB)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2 0 20 10 0 10 20 30 SNR (dB)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='024 ms (n=1024) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8 Overall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6 Amplitude Phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='4 Amplitude and Phase Frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2 Random Chance 0 20 10 0 10 20 30 512 μs (n=512) 256 μs (n=256) 1 1 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2 0 0 20 10 0 10 20 30 20 10 0 10 20 30 128 μs (n=128) 64 μs (n=64) 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2 0 0 20 10 0 10 20 30 20 10 0 10 20 30 32 μs (n=32) 16 μs (n=16) 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='2 0 0 20 10 0 10 20 30 20 0 10 20 10 30 SNR11 (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Confusion matrices for (a) model 1110 (best performing model from this work), (b) the reproduced ResNet model from [1], and (c) the X-Vector inspired model from [19] with SNR ≥ 0dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' determined for each signal such that a contiguous sample of the desired duration, starting at the random point, is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Thus, the chosen segment from a test set sample is randomly assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We also note that, although the sample length for the evalu- ation is changed, the best performing model is the same archi- tecture with the exact same trained weights because this model uses statistics pooling from the X-Vector inspired modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' A significant benefit to the X-Vector inspired architecture is its ability to handle variable-length inputs without the need of padding, retraining, or other network modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This is achieved by taking global statistics across convolutional channels producing a fixed-length vector, regardless of signal duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Due to this flexibility, the same model (model 1110) weights are used for each duration experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This fact also emphasizes the desirability of using X-vector inspired AMC architectures for receivers that are deployed in an environment where short-burst and variable duration signals are anticipated to be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' For each signal duration in the time domain, we plot the overall classification accuracy over varying SNR conditions as well as the accuracy for each modulation grouping de- fined in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Figure 14 demonstrates the tradeoff for various signal durations where n is the number of samples from the time domain I/Q signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' The first observation is, as we would expect, that classification performance degrades with decreased signal duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' For example, the maximum accuracy begins to degrade at 256 µs and is more noticeable at 128 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This is likely a result of using sample statistics that result in unstable or biased estimates for short signal lengths since the number of received signal data points are insufficient to characterize the sample statistics used during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Random classification accuracy is approximately 4% and is shown in the black dotted line in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Although classification performance decreases with decreased duration, we are still able to achieve significantly higher classification accuracy than random chance down to 16 µs of signal capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' FM (frequency modulation) signals were typically more resilient to noise interference than AM (amplitude modulation) and AM-PM (amplitude and phase modulation) signals in our AMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This was observed across all signal burst durations and our top-k accuracy analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This behavior indicates that the performance of our AMC for short bursts, in the presence of increasing amounts of noise, is more robust for signals modulated by changes in the carrier frequency and is more sensitive to signals modulated by varying the carrier amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' We attribute this behavior to our AMC architecture, the architecture of the receiver, or a combination of both of the AMC and receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Confusion Matrices While classification accuracy provides a holistic view of model performance, it lacks the granularity to investigate where misclassifications are occurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Confusion matrices are used to analyze the distribution of classifications for each given class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' For each true label, the proportion of correctly classified samples is calculated along with the proportion of incorrect predictions for each opposing class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' In this way, we can see which classes the model is struggling to distinguish from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' A perfect classifier would be the identity matrix where the diagonal values represent the true class matches the predicted class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Each matrix value represents the percentage of classifications for the true label and each row sums to 1 (100%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Figure 15 illustrates the class confusion matrices for SNR levels greater than or equal to 0dB for models 1110, the reproduced ResNet architecture from [1], and the baseline X- Vector architecture from [7] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Shown in [7], the X-Vector architecture was able to distinguish PSK and AM- SSB variants to a higher degree and performed better overall than [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Both architectures struggled to differentiate QAM variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Model 1110 improved upon these prior results for QAM signals and in general has higher diagonal components than the other architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This again supports a conclusion that model 1110 achieves a new state-of-the-art in AMC perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' CONCLUSION A comprehensive ablation study was carried out with regard to AMC architectural features using the extensive RadioML 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='01A dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This ablation study built upon a strong performance of a new baseline model that was also intro- duced in the initial investigation of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' This initial investigation informed the design of a number of AMC ar- chitecture modifications—specifically, the use of X-Vectors, dilated convolutions, and SE blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' With the 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classification for oral reading fluency with quadratic kappa loss and attentive x-vectors,” in ICASSP 2022- 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' IEEE, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 13 [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Vaswani, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Shazeer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Parmar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Uszkoreit, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Jones, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Gomez, Ł.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Kaiser, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Polosukhin, “Attention is all you need,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Clayton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Harper received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' in mathematics and computer engineering in 2019 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' in data engineering in 2021 from Southern Methodist University in Dallas, TX, where he specialized in machine learning and audio signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' His main research area is the analysis of time series signal processing in computer systems, especially pertaining to security and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' He is a student member of IEEE and is currently pursuing his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' in computer science at Southern Methodist Univer- sity with the co-advisors Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Eric C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Larson and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Mitchell A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Thornton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Mitchell (Mitch) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Thornton is currently the Cecil H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Green Chair of Engineering and Professor in the Department of Electrical and Computer Engi- neering at Southern Methodist University in Dallas, Texas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' He also serves as the Executive Director of the Darwin Deason Institute for Cyber Security, a research-only unit, and as Program Director for the interdisciplinary M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' in Data Engineering degree program within the Lyle School of Engineering at SMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' His main research interests are in the areas of cyber security and quantum informatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' His past industrial experience includes full-time employment at the Amoco Research Center, E-Systems, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' (now L3Harris Technologies Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='), and the Cyrix Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Thornton is a member of several professional and honor societies including the IEEE and the ACM where he is a senior member in each organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' He was elected as Chair of the IEEE Technical Community on Multiple-Valued Logic (TCMVL, 2010-11) and has served in various roles for other IEEE/ACM committees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' He is an author or co-author of five books and more than 300 technical articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' He is a named inventor on over 20 US/PCT/WIPO patents and patents pending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' He holds P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' licenses in the states of Texas, Mississippi and Arkansas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' He received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' in computer engineering from SMU in 1995, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' in computer science from SMU in 1993, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' in electrical engineering from the University of Texas at Arlington in 1990, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' in electrical engineering from Oklahoma State University in 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Eric C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Larson is an Associate Professor in the de- partment of Computer Science in the Bobby B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Lyle School of Engineering, Southern Methodist Univer- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' His main research interests are in machine learning, sensing, and signal / image processing for various applications, in particular, for healthcare and security applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' His work in both areas has been commercialized and he holds a variety of patents for sustainability sensing and mobile phone- based health sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Larson has authored one textbook and over 70 technical articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' He is active in signal processing education for computer scientists and is an active member of IEEE and the ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' He received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' in 2013 from the University of Washington, where he was co-advised by Shwetak N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' Patel and Les Atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' He received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} +page_content=' in Electrical Engineering in 2006 and 2008, respectively, at Oklahoma State University, where he was advised by Damon Chandler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQfRy39/content/2301.11773v1.pdf'} diff --git a/79FLT4oBgHgl3EQfAy4h/content/tmp_files/2301.11967v1.pdf.txt b/79FLT4oBgHgl3EQfAy4h/content/tmp_files/2301.11967v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..01aaa20a354758bcb9a12263613b48046099c13b --- /dev/null +++ b/79FLT4oBgHgl3EQfAy4h/content/tmp_files/2301.11967v1.pdf.txt @@ -0,0 +1,1315 @@ +Mapi-Pro: An Energy Efficient Memory Mapping Technique +for Intermittent Computing +SATYAJASWANTH BADRI, MUKESH SAINI, and NEERAJ GOEL, Indian Institute of Technology, +Ropar +Battery-less technology evolved to replace battery usage in space, deep mines, and other environments to +reduce cost and pollution. Non-volatile memory (NVM) based processors were explored for saving the system +state during a power failure. Such devices have a small SRAM and large non-volatile memory. To make the +system energy efficient, we need to use SRAM efficiently. So we must select some portions of the application +and map them to either SRAM or FRAM. This paper proposes an ILP-based memory mapping technique for +Intermittently powered IoT devices. Our proposed technique gives an optimal mapping choice that reduces +the system’s Energy-Delay Product (EDP). We validated our system using a TI-based MSP430FR6989 and +MSP430F5529 development boards. Our proposed memory configuration consumes 38.10% less EDP than +the baseline configuration and 9.30% less EDP than the existing work under stable power. Our proposed +configuration achieves 15.97% less EDP than the baseline configuration and 21.99% less EDP than the existing +work under unstable power. This work supports intermittent computing and works efficiently during frequent +power failures. +Additional Key Words and Phrases: NVM, MSP430FR6989, ILP, Intermittent power, Memory-Mapping +1 +INTRODUCTION +The Internet of Things (IoT) is a network of sensors and nodes that allows nearby objects to +communicate and collaborate easily. Batteries are the most common source of power for IoT devices. +Because of the battery’s limited capacity and short lifespan [15], replacement is costly. IoT may +consist of billions of sensors and systems by the end of 2050 [9]. Replacing and disposing billions +of battery-operated devices is expensive and hazardous to the environment. As a result, we need +battery-free IoT devices. +Energy harvesters are a promising alternative to battery-powered devices. The energy harvester +collects energy from the environment and stores energy in capacitors. Energy harvesting is un- +reliable, power failures are unavoidable, and the application’s execution is irregular. This type of +computing is known as intermittent computing [14, 27, 34]. +For intermittently powered IoT devices, energy harvesting is the primary energy source. Energy +harvesting sources like piezo-electric materials and radio-frequency devices extract a small amount +of energy from their surroundings. We must use energy efficiently in both stable and unstable +power supply scenarios. +In order to utilize energy efficiently and to make the system energy efficient, we primarily have +two choices. The first choice is to reduce energy consumption by proposing new techniques that +use energy efficiently. The second choice is to increase the number of different energy harvesters, +which will accumulate more energy while increasing maintenance costs. We need to maintain +these many energy harvesters, which is not a feasible solution. Thus, our main concern is to reduce +energy consumption by proposing new techniques which help to design an energy-efficient system. +Gonzalez et al. [10] mentioned energy as not an ideal metric for evaluating system efficiency. By +simply reducing supply voltage or load capacitance, energy can be reduced. Instead of using energy +as a metric, they suggested using the Energy-Delay Product (EDP) as the energy-efficient design +Authors’ address: SatyaJaswanth Badri, 2018CSZ0002@iitrpr.ac.in; Mukesh Saini, mukesh@iitrpr.ac.in; Neeraj Goel, neeraj@ +iitrpr.ac.in, Indian Institute of Technology, Ropar, S.Ramanujan Block, IIT Ropar Main Campus, Ropar, Punjab, India, 140001. +arXiv:2301.11967v1 [cs.AR] 27 Jan 2023 + +2 +S.J Badri, et al. +metric. The EDP considers both performance and energy simultaneously in a design. If a design +minimizes the EDP, we can call such a design energy-efficient. We define EDP in the equation 1. +𝐸𝐷𝑃 = 𝐸𝑠𝑦𝑠𝑡𝑒𝑚 × 𝑁𝑢𝑚_𝑐𝑦𝑐𝑙𝑒𝑠 +(1) +Where 𝐸𝑠𝑦𝑠𝑡𝑒𝑚 is the system’s energy consumption, 𝑁𝑢𝑚_𝑐𝑦𝑐𝑙𝑒𝑠 is the number of CPU cycles. +During these frequent power failures, executing IoT applications becomes more difficult because +all computed data may be lost, and the application’s execution must restart from the beginning. +During power failures, we need an additional procedure to backup/checkpoint the volatile memory +contents to non-volatile memory (NVM). +Flash memory was the prior NVM technology used by modern microcontrollers at the main +memory level, such as MSP430F5529 [24]. Flash is ineffective for frequent backups and checkpointing +because its erase/write operations require a lot of energy. Emerging NVMs outperform flash, +including spin-transfer-torque RAM (STT-RAM) [4, 28], phase-change memory (PCM) [25], resistive +RAM (ReRAM), and ferroelectric RAM (FRAM) [16]. Previous works have been demonstrated by +incorporating these emerging NVMs into low-power-based microcontrollers (MCUs) [16, 18, 24]. +Recent non-volatile processors (NVPs), such as the flash-based MSP430F5529 and the FRAM-based +MSP430FR6989, encourage the use of hybrid main memory. The flash-based NVP, MSP430F5529, is +made up of SRAM and flash, while the FRAM-based NVP, MSP430FR6989, is made up of SRAM +and FRAM at the main memory level. The challenges associated with hybrid main memory-based +NVPs, such as MSP430FR6989, are as follows. +(1) FRAM consumes 2x times more energy and latency than SRAM. This design degrades system +performance and consumes extra energy even during normal operations. +(2) SRAM loses contents during a power failure and needs to execute the application from the +beginning, which consumes extra energy and time. For large-size applications, this design +will not be helpful. Anyway, using only SRAM performs better during regular operations. +(3) We can design a hybrid main memory to get the benefits from both SRAM and FRAM. The +following questions need to be answered and analyzed to use the hybrid main memory design. +(a) How do we choose the appropriate sections of a program and map them to either SRAM +or FRAM regions? A significant challenge is mapping a program’s stack, code, and data +sections to either SRAM or FRAM. +(b) How and where should volatile contents be backed up to the NVM region during frequent +power failures? +The main question is which section of an application should be placed in which memory region; +this is essentially a memory mapping problem. Concerning all of the challenges mentioned earlier, +this article makes the following contributions: +• To the best of our knowledge, this is the first work on the Integer-Linear Programming (ILP) +based memory mapping technique for intermittently powered IoT devices. +• We formulated the memory mapping problem to cover all the possible design choices. We +also formulated our problem in such a way that it supports large-size applications. +• We proposed a framework that efficiently consumes low energy during regular operation +and frequent power failures. Our proposed framework supports intermittent computing. +• We evaluated the proposed techniques and frameworks in actual hardware boards. +Our proposed ILP model recommends placing each section in either SRAM or FRAM. We com- +pared the proposed memory configuration and techniques with the baseline memory configurations +under both stable and unstable power scenarios. Our proposed memory configuration consumes +38.10% less EDP than baseline-1 and 9.30% less EDP than the existing work under stable power. + +Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing +3 +Our proposed configuration achieves 15.97% less EDP than baseline-1 and 21.99% less EDP than the +existing work under unstable power. +Paper organization: Section 2 discusses the background and related works. Section 3 explains +the motivation behind the proposed framework. Section 4 explains the system model and gives an +overview of the problem definition. Section 5 explains about proposed ILP-based memory mapping +technique and framework that supports during frequent power failures. The experimental setup +and results are described in section 6. We conclude this work in section 7. +2 +BACKGROUND AND RELATED WORKS +SRAM and DRAM are used to design registers, caches, and main memory in traditional processors. +For an intermittently aware design, we replace a regular processor’s volatile memory model +with an NVM. STT-RAM, PCM, flash, and FRAM are all relatively new NVM technologies [4– +6, 11, 17, 25, 28, 29, 31, 37]. FRAM consumes less energy than other NVM technologies, such as flash. +FRAM can be helpful for IoT devices that are operating at low power. These NVM technologies +motivated researchers because of their appealing characteristics, such as non-volatility, low cost, +and high density [2, 3]. +Researchers started using real-time NVPs for intermittent computing [16, 24, 30, 32]. Researchers +observed that using only NVMs at the cache or main memory level degrades the system’s perfor- +mance and consumes more energy, which gives an idea to explore hybrid memories. Recent NVPs +such as MSP430FR6989 [16] consists of both SRAM and FRAM. We need to utilize the SRAM and +FRAM efficiently and correctly; otherwise, we may degrade system performance and consume +extra energy. To make the system more efficient, we need to map the application contents to either +SRAM or FRAM. This is actually a memory mapping problem, similar to scratch-pad memories. +Researchers explored a similar mapping problem in scratch-pad memories (SPMs) [12, 26, 33]. +Chakraborty et al. [1] documented the existing and standard memory mapping techniques on SPMs. +In earlier works, memory mapping was done mainly between SPMs and main memory. Memory +mapping can be done statically and dynamically [21, 22]. In static memory mapping, either ILP +or the compiler can assist in determining the best placement [12, 26, 33]. ILP-solver takes inputs +obtained from profilers and memory sizes as constraints in ILP-based memory mapping works. The +ILP-solver provides the best placement option based on the objective function. In dynamic memory +allocation [7, 8, 35, 36], either the user-defined program or the compiler will decide on an optimal +placement choice at run time. +However, our problem differs from the memory mapping techniques in SPMs because intermittent +computing brings new constraints. During intermittent computation, the challenges were the +forward progress of an application, data consistency, environmental consistency, and concurrency +between the tasks. Due to these challenges, the execution model and development environment +differ from the SPM-based memory mapping techniques. As a result, we require a memory mapping +technique that supports intermittent computation. +Researchers have explored memory mapping techniques and analysis for the MSP430FR6989 +MCU. In FRAM-based MCUs, Jayakumar et al. [18] implement a checkpointing policy. They save the +system state to FRAM during a power failure. Jayakumar et al. [19, 20] propose an energy-efficient +memory mapping technique for TI-based applications in FRAM-based MCUs. Kim et al. [23] present +a detailed analysis of energy consumption for all memory sections in FRAM-based MCUs under +different memory mappings. +Earlier works investigated this problem by analyzing the possibilities to make the system efficient. +The authors [19, 20, 23] have not covered all the design choices and possibilities. In addition, there +is significantly less contribution towards memory mappings in FRAM-based MCUs that supports + +4 +S.J Badri, et al. +intermittent computation. Our work proposes an energy-efficient memory mapping technique for +intermittently powered IoT devices that experience frequent power failures. +3 +MOTIVATION +This section discusses the advantages of using hybrid SRAM and FRAM for these MSP430-based +MCUs over unified SRAM or unified FRAM designs, as well as the importance of an efficient +memory allocation. +SRAM is 2KB, and FRAM is 128KB in a FRAM-based MCU, MSP430FR6989. The first naive +approach is to use the entire 128KB of FRAM in both stable and unstable power scenarios, resulting +in longer execution cycles and higher energy consumption. Similarly, we have a second naive +approach to use the entire 2KB SRAM for small applications (whichever fits within the SRAM size), +which has advantages during regular operation. Unfortunately, it loses all 2KB SRAM data during +a power failure and takes more time to backup 2KB contents to FRAM during a power failure. +These two approaches are treated as baselines 1 and 2 for this work. As shown in figure 1, for +the baseline-1 design, we map all three sections to FRAM and all three sections to SRAM for the +baseline-2 design. +int glob1, glob2,..., globn; +func_1(){ +local_variables +} +func_2(){ +local_variables +} +func_n(){ +local_variables +} +Text +Data +Stack +For func_1 () +Text +Data +Stack +For func_2 () +For func_n () +Program +Global_Variables +Functions +Consists of +Local Variables +For global_vars +Data +.bss +Text +Data +Stack +SRAM +SRAM (2 KB) +FRAM (128 KB) +Memory +Stack(func_1) +Stack(func_n) +Text(func_1) +Data(func_1) +Data(func_n) +Text(func_n) + Map to SRAM +FRAM +SRAM (2 KB) +FRAM (128 KB) +Memory +Stack(func_1) +Stack(func_n) +Text(func_1) +Data(func_1) +Data(func_n) +Text(func_n) + Map to FRAM +Baseline-1 Design +Baseline-2 Design +global_vars +global_vars +Fig. 1. Overview of the Baseline-1 and Baseline-2 memory mappings in MSP430FR6989 + +Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing +5 +We compared baseline-1 and baseline-2 in both stable and unstable power scenarios. Baseline-1 +performs better during frequent power failures, while baseline-2 performs better during regular +operations (without any power failures), as shown in figure 2. On average, baseline-1 consumes +47.9% more energy than baseline-2 during a stable power, as shown in figure 2 (a). On average, +baseline-2 consumes 32.7% more energy than baseline-1 during an unstable power, as shown in +figure 2 (b). We also observed that MCU would pitch an error to either increase the SRAM space or +use FRAM space for any computations. For large-size applications will not run using only SRAM, it +requires FRAM as well. Thus, large applications consume more energy in baseline-2 during a stable +power scenario. +These two designs motivate us to propose a hybrid memory design that effectively uses both +SRAM and FRAM. We also encountered that baseline-2 is ineffective for larger applications. As a +result, we had to use a hybrid memory and figure out how and where to place the sections. To the +best of our knowledge, only one work explored the memory mapping issue for these MCUs [20]. +We analyzed the mapping decisions using their empirical model. Jayakumar et al. [20] calculated +the energy consumption values for each configuration. The authors suggested that allocate the +sections to either SRAM or FRAM based on the energy values. +0 +0.2 +0.4 +0.6 +0.8 +1 +16bit_2dim +aes +basicmath_small +basicmath_large +bf +crc +dhrystone +dijkstra +fft +fir +matrix_mult +patricia +qsort_small +qsort_large +sha +susan +Normalized Energy Consumption +(Normalized with Baseline-1) +Benchmarks +Baseline-1 +Baseline-2 +(a) Under Stable Power +0 +0.2 +0.4 +0.6 +0.8 +1 +16bit_2dim +aes +basicmath_small +basicmath_large +bf +crc +dhrystone +dijkstra +fft +fir +matrix_mult +patricia +qsort_small +qsort_large +sha +susan +Normalized Energy Consumption +(Normalized with Baseline-1) +Benchmarks +Baseline-1 +Baseline-2 +(b) Under Unstable Power +Fig. 2. Comparison between Baseline-1 and 2 configurations under Stable and Unstable Power Scenarios +Table 1. Analysis of the Empirical Methods Used by Jayakumar et al. [20] for qsort_small under stable and +unstable power supply scenarios +Configuration +Text +Data +Stack +𝐸𝑛𝑒𝑟𝑔𝑦𝑠𝑡𝑎𝑏𝑙𝑒 (𝑚𝐽) +𝐸𝑛𝑒𝑟𝑔𝑦𝑢𝑛𝑠𝑡𝑎𝑏𝑙𝑒 (𝑚𝐽) +1 {SSS} +SRAM +SRAM +SRAM +16.70 +79.56 +2 {SSF} +SRAM +SRAM +FRAM +21.08 +66.34 +3 {SFS} +SRAM +FRAM +SRAM +28.75 +33.79 +4 {SFF} +SRAM +FRAM +FRAM +35.97 +52.10 +5 {FSS} +FRAM +SRAM +SRAM +39.48 +68.24 +6 {FSF} +FRAM +SRAM +FRAM +57.64 +54.75 +7 {FFS} +FRAM +FRAM +SRAM +64.14 +45.83 +8 {FFF} +FRAM +FRAM +FRAM +92.09 +36.07 +The empirical method used by the authors is as follows. The authors considered functions as the +basic unit. They explored all configurations and calculated the energy values, as shown in table 1. +The authors have eight configurations because they have two memory regions (SRAM or FRAM) +and need to map three sections (stack, data, text). Using the author’s model, we calculated the + +6 +S.J Badri, et al. +energy values for the qsort_small application. For instance, the SSS configuration performs better +during a stable power supply, and during a power failure, SFS consumes less energy than all other +configurations. As a result, authors allocate text and stack sections to SRAM and data sections to +FRAM. +We observed that this empirical method becomes ineffective as the number of configurations +increases. The authors considered all global variables, arrays, and constants as data sections. Instead, +why can’t we map each global variable or array to either SRAM or FRAM? This increases the +number of configurations, and calculating/tracking energy values is challenging. Our design space +grows enormously and makes our mapping problem challenging. +This new set of challenges motivated us to propose an energy-efficient memory mapping tech- +nique. Our proposed memory mapping framework supports large-size applications and covers all +possible configurations. +4 +SYSTEM MODEL AND PROBLEM DEFINITION +This section discusses the system model for embedded MCUs and defines the mapping problem for +these MCUs. +4.1 +System Model +We consider a simple, customized RISC instruction set with a Von-Neumann architecture, where +the instructions and data share the same address space that supports at least 16-bit addressing. Base +architecture doesn’t have a cache to avoid uncertainty. To make things simple, we assume single +cycle execution of the processor. Base architecture has a small SRAM memory and a larger NVM. +The MSP430 is an example of such a processor. Non-volatile memory sizes range from 1 kilobyte +(KB) to 256 KB, while volatile RAM sizes range from 256 bytes to 2KB. Both SRAM and NVM can +be accessed by instructions using a compiler/linker script. We can modify the linker script to map +memory according to the memory ranges specified by the user. MSP430 doesn’t have any operating +system. +4.2 +Problem Definition +Definition 4.1: Optimal Memory Mapping Problem: Given a program that consists of various +functions and global variables, sizes of SRAM and FRAM, the number of reads and writes for each +function/variable, and the energy required per read/write to the SRAM/FRAM. What is the optimal +memory mapping for these functions/variables in order to reduce the system’s EDP? +The inputs are : Number of functions; number of global variables; energy per write to SRAM +and FRAM; energy per read to SRAM and FRAM; SRAM and FRAM sizes; Number of CPU cycles +per each function; the number of reads; the number of writes. +The output is: Mapping information for all functions and global variables, under which the +system’s EDP is minimized. +Definition 4.2: Support for Intermittent Computing: During power failures, we must safely +backup the volatile contents to NVM. As previously stated, we must use SRAM efficiently for +energy savings; but again, how can we save the contents of SRAM? There are two significant issues +with intermittent computation. First, during a power failure, all SRAM’s mapping information +and register contents are lost, causing the system to become inconsistent. Second, how do we +backup/restore the mapping information and register contents to ensure system consistency? + +Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing +7 +5 +MAPI-PRO: AN ENERGY EFFICIENT MEMORY MAPPING FOR INTERMITTENT +COMPUTING +In this section, we discuss the details of the proposed mapping technique. Our main objective is to +pick the optimal mapping choice among all the design choices, which reduces the system’s EDP. To +achieve this, we proposed an ILP-based mapping technique. The overview of the proposed mapping +technique is shown in figure 3. We also discuss how we support intermittent computing for these +MCUs. +int glob_1, glob_2,..., glob_n; +func_1(){ +local_variables +} +func_2(){ +local_variables +} +func_n(){ +local_variables +} +Text +Data +Stack +For func_1 () +Text +Data +Stack +For func_2 () +For func_n () +Program +Global_Variables +Functions +Consists of +Local Variables +Proposed ILP based +Mapping Technique +For global_vars +Data +.bss +Text +Data +Stack +Placement +Decision +for SRAM +Placement +Decision +for FRAM +Stack(func_2) +Stack(func_n) +Text(func_1) +Data(func_2) +Text(func_n) +Data(func_n) +Data(func_1) +Data(func_2) +Stack(func_1) +glob_3,....,glob_n +Text(func_1) +Text(func_2) +Data(func_n) +SRAM (2 KB) +FRAMn (125 KB) +Memory +Backup Region +FRAMb (3 KB) +glob_1, glob_2 +Fig. 3. Overview of the proposed memory mappings in MSP430FR6989 +5.1 +ILP Formulation for Data Mapping +We present the ILP formulation for the memory mapping problem mentioned in definition 4.1. We +divide this ILP formulation into two parts, one is for global variables, and the second is for the +functions. We have shown the overview block diagram of the proposed ILP framework in figure 4. +Application +Profiling and one-time +characterization +Assembly +Code +Number of reads & writes to +each variable +Number of reads & writes to +each function +Energy per read/write to +SRAM +Energy per read/write to +FRAM +ILP Solver +Number of CPU cycles +required for eachfunctions +and variable +Number of Functions +Number of Global variables +SRAM and FRAM sizes +Mapping Information for each +Variable and Function +MSP430FR6989 +Fig. 4. Overview of the Proposed ILP Framework + +8 +S.J Badri, et al. +For Global Variables: Let the number of global variables in a program be ‘G’. Let the number +of reads and writes to variable ‘i’ are 𝑟𝑖 and 𝑤𝑖. We divided FRAM’s 128 KB into two regions, i.e., +𝐹𝑅𝐴𝑀𝑛 and 𝐹𝑅𝐴𝑀𝑏, 𝐹𝑅𝐴𝑀𝑛 memory region has 125 KB, and the 𝐹𝑅𝐴𝑀𝑏 memory region has 3 KB. +We have two memory regions represented as 𝑀𝑒𝑚𝑗 as shown in the equation 2; when j=1, we +select the memory region as SRAM, and we use 𝐹𝑅𝐴𝑀𝑛 for j=2. +𝑀𝑒𝑚𝑗 = +� +𝑗 = 1 +; SRAM +𝑗 = 2 +; 𝐹𝑅𝐴𝑀𝑛 +(2) +Let the sizes of SRAM/FRAM as 𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗) as shown in equation 3, when j=1, we refer as +SRAM memory size in bytes, and when j=2, we refer as 𝐹𝑅𝐴𝑀𝑛 memory size in bytes. +𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗) = +� +𝑗 = 1 +; SRAM +𝑗 = 2 +; 𝐹𝑅𝐴𝑀𝑛 +(3) +Let the energy required for each read/write to 𝑀𝑒𝑚𝑗 is 𝐸𝑟_𝑗 and 𝐸𝑤_𝑗. Let the number of CPU +cycles required to execute a global variable 𝑣𝑖 be 𝑁𝐶𝑣𝑖, where ∀𝑖 ∈ [1,𝐺]). Using one-time charac- +terization and static profiling, we gathered data such as per read/write energy to SRAM/FRAM and +the number of cycles. +We define a binary variable (BV); 𝐼𝑗 (𝑣𝑖), which refers to a variable 𝑣𝑖 is allocated to memory +region 𝑗. If 𝐼𝑗 (𝑣𝑖)=1 then the variable 𝑣𝑖 is allocated and 𝐼𝑗 (𝑣𝑖)=0 indicates that the variable 𝑣𝑖 is +not allocated. 𝐼𝑗 (𝑣𝑖), where (∀𝑗 ∈ [1, 𝑀𝑒𝑚𝑗], ∀𝑖 ∈ [1,𝐺]) is defined as shown in the equation 4. +𝐼𝑗 (𝑣𝑖) = +� +1 +𝑣𝑖 is allocated to memory region 𝑗 +0 +otherwise +(4) +Constraints: There are two constraints, one is for BV; 𝐼𝑗 (𝑣𝑖) and one is a memory size constraint. +In any case, a variable 𝑣𝑖 is allocated to only one memory region, which means 𝑣𝑖 is allocated to +either SRAM or FRAM but not both. This constraint is defined in the equation 5. +𝑀𝑒𝑚𝑗 +∑︁ +𝑗=1 +𝐼𝑗 (𝑣𝑖) = 1 +(∀𝑖 ∈ [1,𝐺]) +(5) +The other constraint is related to memory sizes. The allocated variables 𝑣𝑖 and its 𝑆𝑖𝑧𝑒(𝑣𝑖); +∀𝑖 ∈ [1,𝐺]) should not be greater than the 𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗). This constraint is defined in the equation +6. +𝐺 +∑︁ +𝑖=1 +𝐼𝑗 (𝑣𝑖) ∗ 𝑆𝑖𝑧𝑒(𝑣𝑖) ≤ 𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗) +(∀𝑗 ∈ [1, 𝑀𝑒𝑚𝑗]) +(6) +Objective 4.1: The challenge of mapping global variables in a program to either SRAM or FRAM +is to reduce EDP and improve system performance. 𝐸𝑔𝑙𝑜𝑏𝑎𝑙 is defined in the equation 7. Where +𝐸𝑔𝑙𝑜𝑏𝑎𝑙 is the energy required to allocate global variables to either SRAM or FRAM. +𝐸𝑔𝑙𝑜𝑏𝑎𝑙 = +𝑀𝑒𝑚𝑗 +∑︁ +𝑗=1 +𝐺 +∑︁ +𝑖=1 +[𝐸𝑟_𝑗 × 𝑟𝑖 + 𝐸𝑤_𝑗 × 𝑤𝑖] +(7) +𝐸𝐷𝑃𝑔𝑙𝑜𝑏𝑎𝑙 is defined in the equation 8. Where 𝐸𝐷𝑃𝑔𝑙𝑜𝑏𝑎𝑙 is the energy-delay product required to +allocate global variables to either SRAM or FRAM. + +Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing +9 +𝐸𝐷𝑃𝑔𝑙𝑜𝑏𝑎𝑙 = +𝑀𝑒𝑚𝑗 +∑︁ +𝑗=1 +𝐺 +∑︁ +𝑖=1 +𝐼𝑗 (𝑣𝑖) [𝐸𝑔𝑙𝑜𝑏𝑎𝑙 × 𝑁𝐶𝑣𝑖] +(8) +For Functions: Let the number of functions in a program be ‘𝑁 ′ +𝑓 . Let the number of reads and +writes to 𝑖𝑡ℎ function are 𝑟 (𝐹𝑖) and 𝑤(𝐹𝑖), where ∀𝑖 ∈ [1, 𝑁𝑓 ]. Functions consist of procedural +parameters, local variables, and return variables. Internally the code/data of functions are divided +into the text, data, and stack sections. We map at least one section among these three sections to +either SRAM or FRAM regions, i.e., 𝑀𝑒𝑚𝑗 and 𝑆𝑒𝑐𝑘 (𝑖) defines section ‘k’ of 𝑖𝑡ℎ function as shown +in the equation 9, when k=1, we refer to the text section of 𝑖𝑡ℎ function, when k=2, we refer to the +data section of 𝑖𝑡ℎ function, and when k=3, we refer to the stack section of 𝑖𝑡ℎ function. +𝑆𝑒𝑐𝑘 (𝑖) = + + +𝑘 = 1 +; Text +𝑘 = 2 +; Data +𝑘 = 3 +; Stack +;∀𝑖 ∈ [1, 𝑁𝑓 ] +(9) +We define a BV; 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)), which refers to a section 𝑆𝑒𝑐𝑘 of 𝑖𝑡ℎ function is allocated to only +one memory region 𝑗. If 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖))=1 then the section 𝑆𝑒𝑐𝑖 is allocated and 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖))=0 that +indicates the section 𝑆𝑒𝑐𝑖 is not allocated. 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)), where (∀𝑗 ∈ [1, 𝑀𝑒𝑚𝑗], ∀𝑖 ∈ [1, 𝑁𝑓 ]), +∀𝑘 ∈ [1,𝑆𝑒𝑐𝑘 (𝑖)]) is defined as shown in the equation 10. +𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)) = +� +1 +𝑆𝑒𝑐𝑘 of 𝑖𝑡ℎ function is allocated to 𝑗 +0 +otherwise +(10) +Constraints: There are two constraints, one is for BV; 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)) and one is a memory size +constraint. In any case, a 𝑆𝑒𝑐𝑘 of 𝑖𝑡ℎ function is allocated to only one memory region, which means +𝑆𝑒𝑐𝑘 of 𝑖𝑡ℎ function is either allocated to either SRAM or FRAM but not both. This constraint is +defined in the equation 11. +3 +∑︁ +𝑘=1 +𝑀𝑒𝑚𝑗 +∑︁ +𝑗=1 +𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖))) = 1 +(∀𝑖 ∈ [1, 𝑁𝑓 ]) +(11) +The other constraint is related to memory sizes. The allocated sections 𝑆𝑒𝑐𝑘 (𝑖) and its 𝑆𝑖𝑧𝑒(𝐹𝑖); +∀𝑘 ∈ [1,𝑆𝑒𝑐𝑘 (𝑖)]), ∀𝑗 ∈ [1, 𝑀𝑒𝑚𝑗], ∀𝑖 ∈ [1, 𝑁𝑓 ] should not be greater than the 𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗). This +constraint is defined in the equation 12. +𝐺 +∑︁ +𝑖=1 +𝐼𝑗 (𝑣𝑖) ∗ 𝑆𝑖𝑧𝑒(𝑣𝑖) + +3 +∑︁ +𝑘=1 +𝑁𝑓 +∑︁ +𝑖=1 +𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)) ∗ 𝑆𝑖𝑧𝑒(𝐹𝑖) ≤ 𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗) +(12) +Objective 4.2: The challenge of mapping sections of these functions in a program to either +SRAM or FRAM is to minimize EDP and improve system performance. 𝐸𝑓 𝑢𝑛𝑐 is defined in the +equation 13, where 𝑀𝑐𝑖 is the number of the times 𝑖𝑡ℎ functions called. +𝐸𝑓 𝑢𝑛𝑐 = +𝑀𝑒𝑚𝑗 +∑︁ +𝑗=1 +𝑁𝑓 +∑︁ +𝑖=1 +[𝐸𝑟_𝑗 × 𝑟 (𝐹𝑖) + 𝐸𝑤_𝑗 × 𝑤(𝐹𝑖)] × 𝑀𝑐𝑖 +(13) +𝐸𝐷𝑃𝑓 𝑢𝑛𝑐 is defined in the equation 14. Where 𝐸𝐷𝑃𝑓 𝑢𝑛𝑐 is the energy-delay product required to +allocate all functions to either SRAM or FRAM. Where 𝐸𝑓 𝑢𝑛𝑐 is the energy required to allocate + +10 +S.J Badri, et al. +functions to either SRAM or FRAM. Where 𝑁𝐶𝐹𝑖 is the number of CPU cycles required to execute +a function 𝐹𝑖. +𝐸𝐷𝑃𝑓 𝑢𝑛𝑐 = +3 +∑︁ +𝑘=1 +𝑀𝑒𝑚𝑗 +∑︁ +𝑗=1 +𝑁𝑓 +∑︁ +𝑖=1 +𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)) [𝐸𝑓 𝑢𝑛𝑐 × 𝑁𝐶𝐹𝑖] +(14) +The overall system EDP, 𝐸𝐷𝑃𝑠𝑦𝑠𝑡𝑒𝑚, is the sum of both 𝐸𝐷𝑃𝑔𝑙𝑜𝑏𝑎𝑙 and 𝐸𝐷𝑃𝑓 𝑢𝑛𝑐 as shown in the +equation 15. +𝐸𝐷𝑃𝑠𝑦𝑠𝑡𝑒𝑚 = 𝐸𝐷𝑃𝑔𝑙𝑜𝑏𝑎𝑙 + 𝐸𝐷𝑃𝑓 𝑢𝑛𝑐 +(15) +Our objective function is shown in the equation 16. Our main objective is to minimize the +system’s EDP by choosing the optimal placement choice. +Objective Function: Minimize 𝐸𝐷𝑃𝑠𝑦𝑠𝑡𝑒𝑚 +(16) +5.2 +Implementing Mapping Technique in MSP430FR6989 +Once we obtain the placement information from the 𝐼𝐿𝑃_𝑠𝑜𝑙𝑣𝑒𝑟, we map the respective variables +and the sections of a function to either SRAM or FRAM. We modify the linker script accordingly +for mapping the sections or variables to either SRAM or FRAM. In our proposed mapping policy, +placing global variables is straightforward, i.e., mapping the respective variable to either SRAM or +FRAM based on the ILP decision. +We observed that from the linker script, we can map the whole stack section of each function +to either SRAM or FRAM. We analyzed the mappings of the stack section for each function by +modifying the linker script. We used the inbuilt attributes to differentiate mappings between SRAM +and FRAM; for instance, we used the inbuilt attribute (__𝑎𝑡𝑡𝑟𝑖𝑏𝑢𝑡𝑒__((𝑟𝑎𝑚𝑓𝑢𝑛𝑐)) that maps that +function to SRAM. If we want to place the stack section to SRAM, we modify the linker script by +replacing the default setting with " .stack: {} > RAM (HIGH) ". If we want to place the stack section +to FRAM, we modify the linker script by replacing the default setting with " .stack: {} > FRAM". +Similarly, for the text section, we observed that placing the text section in either SRAM or FRAM +shows an impact on EDP. This effect is because the majority of access in the text section are read +accesses, as we observed that the energy consumption for each read access to SRAM/FRAM differs. +Table 3 shows that approximately FRAM consumes 2x more read energy than SRAM. Thus, we +analyzed each application where to map the text section based on the free space available. If we +have enough space available in SRAM, we place the text section in SRAM itself; otherwise, we +place the text section in FRAM. We included the following four lines in our linker script to check +the above condition and map the text section. +(1) #𝑖𝑓 𝑛𝑑𝑒𝑓 __𝐿𝐴𝑅𝐺𝐸_𝐶𝑂𝐷𝐸_𝑀𝑂𝐷𝐸𝐿__ +(2) .text : {} > FRAM +(3) #else +(4) .text : {} » SRAM +We modified the linker script for mapping the data section by using the inbuilt compiler directives. +We followed the below three steps. +(1) Allocate a new memory block, for instance, 𝑁𝐸𝑊 _𝐷𝐴𝑇𝐴𝑆𝐸𝐶𝑇𝐼𝑂𝑁. We can declare the start +address and size of the data section in the linker script. +(2) Define a segment (.Localvars) which stores in this memory block (𝑁𝐸𝑊 _𝐷𝐴𝑇𝐴𝑆𝐸𝐶𝑇𝐼𝑂𝑁). + +Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing +11 +(3) Use #pragma 𝐷𝐴𝑇𝐴_𝑆𝐸𝐶𝑇𝐼𝑂𝑁 (𝑓𝑢𝑛𝑐𝑡_𝑛𝑎𝑚𝑒,𝑠𝑒𝑔_𝑛𝑎𝑚𝑒) in the program to define functions +in this segment. Where 𝑓𝑢𝑛𝑐𝑡_𝑛𝑎𝑚𝑒 is the function name and𝑠𝑒𝑔_𝑛𝑎𝑚𝑒 is the created segment +name. For instance, #pragma 𝐷𝐴𝑇𝐴_𝑆𝐸𝐶𝑇𝐼𝑂𝑁 (𝑓𝑢𝑛𝑐_1, .𝐿𝑜𝑐𝑎𝑙𝑣𝑎𝑟𝑠) +Once we are done with creating the different sections, we can allocate these sections to either +SRAM or FRAM based on ILP decisions. For instance, placing " 𝑁𝐸𝑊 _𝐷𝐴𝑇𝐴𝑆𝐸𝐶𝑇𝐼𝑂𝑁: {} > FRAM" +in the linker script, which maps the 𝑁𝐸𝑊 _𝐷𝐴𝑇𝐴𝑆𝐸𝐶𝑇𝐼𝑂𝑁 to FRAM. +5.3 +Support for Intermittent Computing +When the power is stable, everything works properly. Because of the static allocation scheme, we +map all functions/variables to SRAM/FRAM for the first time. During a power failure, SRAM and +registers lose all of their contents, including mapping information. When power is restored, we +don’t know what functions/variables were allocated to SRAM before the failure. As a result, we +must either restart the execution from the beginning or end up with incorrect results. Restarting +the application consumes extra energy and time, making our system inefficient in terms of energy +consumption and performance. +We propose a backup strategy during frequent power failures. FRAM was divided into 𝐹𝑅𝐴𝑀𝑛 +and 𝐹𝑅𝐴𝑀𝑏 as shown in the figure 3. 𝐹𝑅𝐴𝑀𝑛 has a size of 125 KB and is used for regular mappings. +𝐹𝑅𝐴𝑀𝑏 has a size of 3 KB that serves as a backup region (BR) during power failures. So, during a +power failure, we back up all register and SRAM contents to FRAM. Whenever power is restored, +we restore the register and SRAM contents from 𝐹𝑅𝐴𝑀𝑏 to SRAM and resume the application +execution. The proposed backup strategy reduces extra energy consumption and makes the system +more energy efficient. +6 +EXPERIMENTAL SETUP AND RESULTS +6.1 +Experimental Setup +We used TI’s MSP430FR6989 for all experiments. We experimented on mixed benchmarks, which +have both Mi-Bench [13] and TI-based benchmarks. We have shown the experimental setup in the +table 2. The development platform and experimental setup are shown in figure 5. We performed +experiments to determine the energy required for a single read/write to SRAM/FRAM, as shown in +the table 3. We collected the number of reads/writes for each global variable and functions as part +of a one-time characterization. We also used TI’s MSP430F5529 for comparing flash with FRAM. +We performed experiments to determine the energy required for a single read/write to flash, as +shown in the table 3. +Table 2. Experimental Setup +Component +Description +Target Board +TI MSP430FR6989 Launchpad +Core +MSP430 (1.8-3.6 V; 16 MHz) +Memory +2KB SRAM and 128KB FRAM +IDE +Code Composer Studio +Energy Profiling +Energy Trace++ +ILP Solver +LPSolve_IDE +Benchmarks +Mixed benchmarks (MiBench and TI-based) +MCU, which we experimented has MSP430 architecture, which is more suitable for IoT devices. +The majority of MSP430 software is written in C and compiled with one of TI’s recommended + +12 +S.J Badri, et al. +compilers ( IAR Embedded Code Bench, Code-Composer Studio (CCS), or msp430-gcc). The IAR +Embedded Code Bench and CCS compilers are part of integrated development environments (IDEs). +We used the widely used, freely available, and easily extended tool, i.e., CCS, for all experiments +in this article. EnergyTrace++ technology allows us to calculate energy and power consumption +directly. According to the datasheet for the MSP430FR6989, the number of cycles required to +read/write in FRAM is twice that of SRAM. +Table 3. Energy Values for each read/write to SRAM and FRAM +Memory +Per Read Energy (nJ) +Per Write Energy (nJ) +SRAM +5500 +5600 +FRAM +10325 +13125 +Flash +23876 +31198 +Fig. 5. (a) TI-based MSP430 Launchpad Development Boards (b) Working with EnergyTrace++ on CCS +6.2 +Evaluation Benchmarks +We chose benchmarks from both the MiBench suite and TI benchmarks. One of the primary +motivations for using the MiBench suite is that most of the TI-based benchmarks were small in size +and easily fit into either SRAM or FRAM. In these cases, we don’t require any hybrid memory design. +Most of the TI-based benchmarks have only one or two functions and 3-4 global variables, which is +not useful for the hybrid main-memory design. Thus we used mixed benchmarks consisting of 4 +TI-based benchmarks and 12 from the MiBench suite. +For the MiBench suite, we first make MCU-compatible benchmarks by adding MCU-related +header files and watchdog timers. All benchmarks may not be compatible with the MCU. Thus, we +need to choose the benchmarks from the MiBench suite, which are compatible with the MSP430 +boards. Once we have benchmarks, we execute them on board for the machine code. Using the +.asm file, we calculate the inputs that are required by the ILP solver, as shown in figure 4. +6.3 +Baseline Configurations +We chose five different memory configurations to compare with the proposed memory configuration. +We directly map all the functions/variables to FRAM in the baseline configuration 1, as shown in +figure 1. We use configuration-1 to compare our proposed memory configuration during stable and +unstable power scenarios. + +Code +Composer +Studio +MSP430F5529MSP430FR6989 +EnergyTrace++Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing +13 +We directly map all the functions/variables to SRAM in the baseline configuration 2, as shown in +figure 1. We use configuration-2 to compare our proposed memory configuration during stable and +unstable power scenarios. +In baseline configuration 3, we used the empirical method of Jayakumar et al. [20]. We compare +this configuration-3 with our proposed configuration during stable and unstable power scenarios +to observe the importance of the proposed than the existing work. +In baseline configuration 4, we used the proposed ILP technique for the flash-based msp430 +board [24]. We compare this configuration-4 with our proposed configuration during stable and +unstable power scenarios to observe the difference between FRAM and Flash technologies. +In baseline configuration 5, we only have a proposed memory mapping technique and no BR. +We compare this configuration-5 with our proposed configuration during frequent power failures +to observe the importance of BR. The overview of all baseline configurations is shown in table 4. +The experimental setup for all baseline configurations is the same as the one proposed. +Table 4. Overview of the Baseline Configurations +Configuration +FRAM +SRAM +Flash +Backup Region (BR) +ILP +Baseline-1 +✓ +✗ +✗ +✗ +✗ +Baseline-2 +✗ +✓ +✗ +✗ +✗ +Baseline-3 ( Jayakumar et al. [20]) +✓ +✓ +✗ +✗ +✗ +Baseline-4 +✗ +✓ +✓ +✗ +✓ +Baseline-5 +✓ +✓ +✗ +✗ +✓ +Proposed +✓ +✓ +✓ +✓ +✓ +✓- Supported , ✗- Not Supported +6.4 +Results +The proposed memory configuration is evaluated in this section under stable and unstable power. +The proposed memory configuration is compared with five baseline memory configurations as +discussed in the section 6.3. +6.4.1 +Under Stable Power: Our main objective of the proposed memory configuration is to +minimize the system’s EDP. All values shown in figure 6 are normalized with baseline-1. Compared +to baseline-1, the proposed gets 38.10% lesser EDP, as shown in figure 6. Because there are no +power interruptions in this scenario, this improvement is totally from the proposed ILP model. In +configuration-1, we place everything to FRAM, where FRAM consumes more energy and the number +of cycles than SRAM, as shown in the table 3. Our proposed memory configuration incorporates +the placement recommendation from the proposed ILP model and suggests utilizing both SRAM +and FRAM. +Under a stable power scenario, the proposed gets 9.30% less EDP than baseline-3, as shown in +figure 6. We discussed the author’s empirical model and assumptions in the previous section 3. The +authors assumed that the data section included all global variables, constants, and arrays. As a +result, our proposed ILP-based mapping differs from the author’s mapping in that our proposed +mapping outperforms the existing work. Under stable power, baseline-3 receives 24.57% less EDP +than baseline-1, as shown in figure 6. This advantage is primarily due to baseline-3’s hybrid memory. +In comparison to baseline-4, the proposed reduces EDP by 18.55%, as shown in figure 6. We +used flash+SRAM with our proposed ILP framework in baseline-4. As shown in table 3, the above +benefit is primarily due to FRAM because flash consumes more energy. Baseline-3 outperforms + +14 +S.J Badri, et al. +0 +0.2 +0.4 +0.6 +0.8 +1 +16bit_2dim +aes +basicmath_small +basicmath_large +bf +crc +dhrystone +dijkstra +fft +fir +matrix_mult +patricia +qsort_small +qsort_large +sha +susan +Normalized EDP (Normalized with +Baseline-1) +Benchmarks +Baseline-2 +Jayakumar et al. [20] +Baseline-4 +Proposed +Fig. 6. Comparison between Baseline configurations and the Proposed under Stable Power +baseline-4 during stable power. Because of FRAM in baseline-3, even our proposed ILP model is +ineffective in this case. We encountered that baseline-3 achieves 9.19% less EDP than baseline-4, and +this benefit is because of smaller applications. From figure 6, baseline-4 performs better for large +applications than baseline-4. Jayakumar et al. [20] empirical method suggests placing more content +on SRAM because SRAM is sufficient for placing the entire small-size application. As a result, the +performance of baseline 3 is dependent on the application size, as for large-size applications, even +FRAM does not outperform flash. +Baseline 2 outperforms the proposed and all other baselines under stable power conditions. +We noticed that this benefit is primarily due to SRAM, but it only applies to smaller applications. +Baseline 2 achieves 36.19% less EDP than the proposed for smaller applications, as shown in figure +6. We also looked at large applications where the proposed outperforms the baseline-2 by a small +margin. When the SRAM is full, the MCU must wait for the space to be released, which consumes +extra energy and cycles. For more extensive applications, baseline-2 achieves 2.94% more EDP than +proposed. +We also evaluated our proposed framework with another MSP430F5529 MCU with flash and +SRAM for completeness. This comparison assists the user in selecting the most appropriate NVM +technology, such as FRAM or flash, as needed. To be fair, we used the same sizes of SRAM (2 KB) +and Flash (128 KB) in this comparison. We compared FRAM-based and flash-based MCUs under +stable power conditions. We used the proposed frameworks and techniques in both MCUs. We +discovered that the proposed FRAM-based configuration outperforms the flash-based configuration. +Flash-based configurations consume 26.03% more EDP than FRAM-based configurations, as shown +in figure 7. Flash consumes more energy, as shown in table 3. +6.4.2 +Under Unstable power: We used the default TI-based compute through power loss (ctpl) +tool for migration. During a power failure, we need to migrate the SRAM contents to a FRAM-based +backup region (𝐹𝑅𝐴𝑀𝑏), i.e., the backup process. Whenever power comes back, we need to migrate +the (𝐹𝑅𝐴𝑀𝑏) contents to SRAM, i.e., the restoration process. So, all these migrations are done using +ctpl() functions. We introduce a power failure by changing the low power modes mentioned in +the MSP430FR6989 design document. We used ctpl() for creating power failures. We assume that +the number of power failures is spread equally within the execution period. For instance, if the + +Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing +15 +0 +0.2 +0.4 +0.6 +0.8 +1 +16bit_2dim +aes +basicmath_small +basicmath_large +bf +crc +dhrystone +dijkstra +fft +fir +matrix_mult +patricia +qsort_small +qsort_large +sha +susan +Normalized EDP (Normalized with +MSP430F5529) +Benchmarks +MSP430F5529 +MSP430FR6989 +Fig. 7. Comparison between MSP430FR6989 (FRAM-based MCU) and MSP430F5529 (Flash-based MCU) +under Stable Power +total execution period for an application is 20 milliseconds (ms), and let’s say the number of power +failures is four, then for every 5 ms, we experience a power failure. +We performed experiments under unstable power to compare the proposed memory configuration +with baseline configurations. All values shown in figure 8 are normalized with baseline-2. Compared +to baseline-2, the proposed gets 15.97% lesser EDP, as shown in figure 8. We observed that migration +overhead is less than the energy consumed to execute the application from FRAM, and this migration +overhead depends on the number of power failures. For instance, one backup migration consumes +approximately 16.88 mJ of energy, and one restore migration consumes approximately 11.606 mJ of +energy in a qsort application. The above benefit to our proposed configuration is using a hybrid +memory. +Under an unstable power scenario, the proposed gets 21.99% less EDP than baseline-3, as shown +in figure 8. We discussed the author’s empirical model and assumptions in the previous section 3. +As already stated, the Jayakumar et al. empirical method is more beneficial for small applications. +In contrast, the author’s empirical method suggests placing more content on SRAM because SRAM +is sufficient for placing the entire small-size application. Thus, for [20] work, backup/restore +operations take more energy during a power failure. Our proposed mapping outperforms the +existing work. During frequent power failures, baseline-3 receives 6.91% less EDP than baseline-1, +as shown in figure 8. This advantage is primarily due to baseline-3’s hybrid memory. +Compared to baseline-4, the proposed reduces EDP by 23.05%, as shown in figure 8. We used +flash+SRAM with our proposed ILP framework in baseline-4. As shown in table 3, the above benefit +is primarily due to FRAM because flash consumes more energy. Baseline-3 outperforms baseline-4 +during stable power. Because of FRAM in baseline-3, even our proposed ILP model is ineffective +for this comparison. We encountered that baseline-3 achieves 6.28% less EDP than baseline-4 for +smaller applications. The above benefit for baseline-3 is minimal because the size of backup/restores +increases, which even neutralizes the flash for some applications, as shown in figure 8. Baseline-4 +achieves 2.69% less EDP than baseline-3 for large applications, as shown in figure 8. As a result, the + +16 +S.J Badri, et al. +0 +0.2 +0.4 +0.6 +0.8 +1 +16bit_2dim +aes +basicmath_small +basicmath_large +bf +crc +dhrystone +dijkstra +fft +fir +matrix_mult +patricia +qsort_small +qsort_large +sha +susan +Normalized EDP (Normalized with +Baseline-2) +Benchmarks +Baseline-1 +Jayakumar et al. [20] +Baseline-4 +Baseline-5 +Proposed +Fig. 8. Comparison between Baseline configurations and the Proposed under Unstable Power +performance of baseline 3 is dependent on the application size, as for large-size applications, even +FRAM does not outperform flash. +The proposed outperforms all baselines under unstable power conditions. This benefit is primarily +due to a hybrid memory and the proposed mapping technique. Baseline 2 achieves 42.98% less EDP +than the proposed, as shown in figure 8. +When we remove BR, all the mapping information of SRAM is lost because our model is static. +We introduce a BR in the FRAM memory region to save this mapping information. During a power +failure, we migrate the SRAM contents to 𝐹𝑅𝐴𝑀𝑏, and whenever power comes back, we restore +the 𝐹𝑅𝐴𝑀𝑏 contents to the SRAM. +We experimented to know the importance of BR, where we compared the proposed memory +configuration with baseline-5. Compared to baseline-5, the proposed gets 23.94% lesser EDP, as +shown in figure 8. This benefit is because we need to re-execute the application four times from +the beginning, which consumes extra time and energy. The number of times re-executing the +application is equal to the number of power failures. +We also evaluated our proposed framework with another MSP430F5529 MCU, which consists of +flash and SRAM for completeness. This comparison assists the user in selecting the most appropriate +NVM technology, such as FRAM or flash, as needed. To be fair, we used the same sizes of SRAM (2 +KB) and Flash (128 KB) in this comparison. We also used BR for both baselines; the only difference +is that we replaced FRAM with the flash in the proposed configurations, and everything is the +same. We compared FRAM-based and flash-based MCUs under unstable power conditions. We +used the proposed frameworks and techniques in both MCUs. We discovered that the proposed +FRAM-based configuration outperforms the flash-based configuration. Flash-based configurations +consume 16.50% more EDP than FRAM-based configurations, as shown in figure 9. Flash consumes +more energy, as shown in table 3. +6.5 +Summary of the Proposed Mapping Technique +We outline the proposed ILP-based memory mapping technique in this section. Following all +of these analyses, we observed that the mappings shown below consume less EDP than other +design choices, as shown in the table. To keep things simple, we only showed the final mapping + +Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing +17 +0 +0.2 +0.4 +0.6 +0.8 +1 +16bit_2dim +aes +basicmath_small +basicmath_large +bf +crc +dhrystone +dijkstra +fft +fir +matrix_mult +patricia +qsort_small +qsort_large +sha +susan +Normalized EDP (Normalized with +MSP430F5529) +Benchmarks +MSP430F5529 +MSP430FR6989 +Fig. 9. Comparison between MSP430FR6989 (FRAM-based MCU) and MSP430F5529 (Flash-based MCU) +under Unstable Power +configurations for each application’s stack, data, and text sections, keeping out the final mappings +for global variables. +Table 5. Optimal Placement for different Applications in MSP430FR6989 +Benchmarks +Stack +Text +Data +16bit_2dim +SRAM +SRAM +SRAM +aes +SRAM +FRAM +FRAM +basicmath_small +SRAM +SRAM +FRAM +basicmath_large +SRAM +FRAM +FRAM +bf +SRAM +SRAM +FRAM +crc +SRAM +FRAM +SRAM +dhrystone +FRAM +SRAM +FRAM +dijkstra +SRAM +FRAM +SRAM +fft +SRAM +SRAM +FRAM +fir +SRAM +SRAM +FRAM +matrix_mult +SRAM +SRAM +SRAM +patricia +SRAM +FRAM +SRAM +qsort_small +SRAM +SRAM +FRAM +qsort_large +SRAM +FRAM +FRAM +sha +SRAM +FRAM +FRAM +susan +SRAM +FRAM +FRAM +Table 5 shows that, with the exception of the dhrystone application, the remaining three TI +benchmark applications (fir, matrix, and 16bit_2dim) are very small and can easily be placed in SRAM. +We don’t need FRAM for these types of smaller applications, but there is a disadvantage during +frequent power failures. Backup and restore sizes to FRAM are larger for these applications during +frequent power failures. As a result, our proposed backup/restore strategy should be intelligent + +18 +S.J Badri, et al. +enough to reduce EDP. The dhrystone application, on the other hand, has a larger stack section +that requires FRAM to accommodate the entire stack section. +As we can see from the table 5, many applications used both SRAM and FRAM for the Mi- +Bench applications. As a result, we can conclude that a hybrid main memory design is required +for many applications. Using a hybrid main memory design helps to reduce EDP during stable +power scenarios. Even so, determining how and where to backup the volatile contents can be +difficult during frequent power outages. However, our proposed memory mapping technique and +the framework suggest using a hybrid main memory design that supports intermittent computing. +7 +CONCLUSIONS +This paper proposed an ILP-based memory mapping technique that reduces the system’s energy- +delay product. For both global variables and functions, we formulated an ILP model. Functions +consist of data, stack, and code sections. Our ILP model suggests placing each section on either SRAM +or FRAM. Under both stable and unstable power scenarios, we compared the proposed memory +configuration to the baseline memory configurations. We evaluated our proposed frameworks and +techniques on actual boards. We added a backup region in FRAM to support intermittent computing. +We compared the proposed framework with the recent related work. +Under stable power, our proposed memory configuration consumes 38.10% less EDP than baseline- +1 and 9.30% less EDP than the existing work. Under unstable power, our proposed configuration +achieves 15.97% less EDP than baseline-1 and 21.99% less EDP than the existing work. Under stable +power, our proposed memory configuration consumes 18.55% less EDP than baseline-4. We also +compared FRAM-based MSP430FR6989 with flash-based MSP430F5529. 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IEEE Micro 39, 1 (2018), 24–32. + diff --git a/79FLT4oBgHgl3EQfAy4h/content/tmp_files/load_file.txt b/79FLT4oBgHgl3EQfAy4h/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..931fb6e802f6042d38094fb952e01214a88ff377 --- /dev/null +++ b/79FLT4oBgHgl3EQfAy4h/content/tmp_files/load_file.txt @@ -0,0 +1,869 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf,len=868 +page_content='Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing SATYAJASWANTH BADRI, MUKESH SAINI, and NEERAJ GOEL, Indian Institute of Technology, Ropar Battery-less technology evolved to replace battery usage in space, deep mines, and other environments to reduce cost and pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Non-volatile memory (NVM) based processors were explored for saving the system state during a power failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Such devices have a small SRAM and large non-volatile memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' To make the system energy efficient, we need to use SRAM efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' So we must select some portions of the application and map them to either SRAM or FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This paper proposes an ILP-based memory mapping technique for Intermittently powered IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Our proposed technique gives an optimal mapping choice that reduces the system’s Energy-Delay Product (EDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We validated our system using a TI-based MSP430FR6989 and MSP430F5529 development boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Our proposed memory configuration consumes 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='10% less EDP than the baseline configuration and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='30% less EDP than the existing work under stable power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Our proposed configuration achieves 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='97% less EDP than the baseline configuration and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='99% less EDP than the existing work under unstable power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This work supports intermittent computing and works efficiently during frequent power failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Additional Key Words and Phrases: NVM, MSP430FR6989, ILP, Intermittent power, Memory-Mapping 1 INTRODUCTION The Internet of Things (IoT) is a network of sensors and nodes that allows nearby objects to communicate and collaborate easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Batteries are the most common source of power for IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Because of the battery’s limited capacity and short lifespan [15], replacement is costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' IoT may consist of billions of sensors and systems by the end of 2050 [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Replacing and disposing billions of battery-operated devices is expensive and hazardous to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' As a result, we need battery-free IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Energy harvesters are a promising alternative to battery-powered devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The energy harvester collects energy from the environment and stores energy in capacitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Energy harvesting is un- reliable, power failures are unavoidable, and the application’s execution is irregular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This type of computing is known as intermittent computing [14, 27, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' For intermittently powered IoT devices, energy harvesting is the primary energy source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Energy harvesting sources like piezo-electric materials and radio-frequency devices extract a small amount of energy from their surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We must use energy efficiently in both stable and unstable power supply scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In order to utilize energy efficiently and to make the system energy efficient, we primarily have two choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The first choice is to reduce energy consumption by proposing new techniques that use energy efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The second choice is to increase the number of different energy harvesters, which will accumulate more energy while increasing maintenance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We need to maintain these many energy harvesters, which is not a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Thus, our main concern is to reduce energy consumption by proposing new techniques which help to design an energy-efficient system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' [10] mentioned energy as not an ideal metric for evaluating system efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' By simply reducing supply voltage or load capacitance, energy can be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Instead of using energy as a metric, they suggested using the Energy-Delay Product (EDP) as the energy-efficient design Authors’ address: SatyaJaswanth Badri, 2018CSZ0002@iitrpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Mukesh Saini, mukesh@iitrpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Neeraj Goel, neeraj@ iitrpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='in, Indian Institute of Technology, Ropar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='Ramanujan Block, IIT Ropar Main Campus, Ropar, Punjab, India, 140001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='11967v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='AR] 27 Jan 2023 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='J Badri, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The EDP considers both performance and energy simultaneously in a design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' If a design minimizes the EDP, we can call such a design energy-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We define EDP in the equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐸𝐷𝑃 = 𝐸𝑠𝑦𝑠𝑡𝑒𝑚 × 𝑁𝑢𝑚_𝑐𝑦𝑐𝑙𝑒𝑠 (1) Where 𝐸𝑠𝑦𝑠𝑡𝑒𝑚 is the system’s energy consumption, 𝑁𝑢𝑚_𝑐𝑦𝑐𝑙𝑒𝑠 is the number of CPU cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' During these frequent power failures, executing IoT applications becomes more difficult because all computed data may be lost, and the application’s execution must restart from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' During power failures, we need an additional procedure to backup/checkpoint the volatile memory contents to non-volatile memory (NVM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Flash memory was the prior NVM technology used by modern microcontrollers at the main memory level, such as MSP430F5529 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Flash is ineffective for frequent backups and checkpointing because its erase/write operations require a lot of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Emerging NVMs outperform flash, including spin-transfer-torque RAM (STT-RAM) [4, 28], phase-change memory (PCM) [25], resistive RAM (ReRAM), and ferroelectric RAM (FRAM) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Previous works have been demonstrated by incorporating these emerging NVMs into low-power-based microcontrollers (MCUs) [16, 18, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Recent non-volatile processors (NVPs), such as the flash-based MSP430F5529 and the FRAM-based MSP430FR6989, encourage the use of hybrid main memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The flash-based NVP, MSP430F5529, is made up of SRAM and flash, while the FRAM-based NVP, MSP430FR6989, is made up of SRAM and FRAM at the main memory level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The challenges associated with hybrid main memory-based NVPs, such as MSP430FR6989, are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' (1) FRAM consumes 2x times more energy and latency than SRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This design degrades system performance and consumes extra energy even during normal operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' (2) SRAM loses contents during a power failure and needs to execute the application from the beginning, which consumes extra energy and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' For large-size applications, this design will not be helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Anyway, using only SRAM performs better during regular operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' (3) We can design a hybrid main memory to get the benefits from both SRAM and FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The following questions need to be answered and analyzed to use the hybrid main memory design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' (a) How do we choose the appropriate sections of a program and map them to either SRAM or FRAM regions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' A significant challenge is mapping a program’s stack, code, and data sections to either SRAM or FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' (b) How and where should volatile contents be backed up to the NVM region during frequent power failures?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The main question is which section of an application should be placed in which memory region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' this is essentially a memory mapping problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Concerning all of the challenges mentioned earlier, this article makes the following contributions: To the best of our knowledge, this is the first work on the Integer-Linear Programming (ILP) based memory mapping technique for intermittently powered IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We formulated the memory mapping problem to cover all the possible design choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We also formulated our problem in such a way that it supports large-size applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We proposed a framework that efficiently consumes low energy during regular operation and frequent power failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Our proposed framework supports intermittent computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We evaluated the proposed techniques and frameworks in actual hardware boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Our proposed ILP model recommends placing each section in either SRAM or FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We com- pared the proposed memory configuration and techniques with the baseline memory configurations under both stable and unstable power scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Our proposed memory configuration consumes 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='10% less EDP than baseline-1 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='30% less EDP than the existing work under stable power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing 3 Our proposed configuration achieves 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='97% less EDP than baseline-1 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='99% less EDP than the existing work under unstable power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Paper organization: Section 2 discusses the background and related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Section 3 explains the motivation behind the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Section 4 explains the system model and gives an overview of the problem definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Section 5 explains about proposed ILP-based memory mapping technique and framework that supports during frequent power failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The experimental setup and results are described in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We conclude this work in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 2 BACKGROUND AND RELATED WORKS SRAM and DRAM are used to design registers, caches, and main memory in traditional processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' For an intermittently aware design, we replace a regular processor’s volatile memory model with an NVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' STT-RAM, PCM, flash, and FRAM are all relatively new NVM technologies [4– 6, 11, 17, 25, 28, 29, 31, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' FRAM consumes less energy than other NVM technologies, such as flash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' FRAM can be helpful for IoT devices that are operating at low power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' These NVM technologies motivated researchers because of their appealing characteristics, such as non-volatility, low cost, and high density [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Researchers started using real-time NVPs for intermittent computing [16, 24, 30, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Researchers observed that using only NVMs at the cache or main memory level degrades the system’s perfor- mance and consumes more energy, which gives an idea to explore hybrid memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Recent NVPs such as MSP430FR6989 [16] consists of both SRAM and FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We need to utilize the SRAM and FRAM efficiently and correctly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' otherwise, we may degrade system performance and consume extra energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' To make the system more efficient, we need to map the application contents to either SRAM or FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This is actually a memory mapping problem, similar to scratch-pad memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Researchers explored a similar mapping problem in scratch-pad memories (SPMs) [12, 26, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' [1] documented the existing and standard memory mapping techniques on SPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In earlier works, memory mapping was done mainly between SPMs and main memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Memory mapping can be done statically and dynamically [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In static memory mapping, either ILP or the compiler can assist in determining the best placement [12, 26, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' ILP-solver takes inputs obtained from profilers and memory sizes as constraints in ILP-based memory mapping works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The ILP-solver provides the best placement option based on the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In dynamic memory allocation [7, 8, 35, 36], either the user-defined program or the compiler will decide on an optimal placement choice at run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' However, our problem differs from the memory mapping techniques in SPMs because intermittent computing brings new constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' During intermittent computation, the challenges were the forward progress of an application, data consistency, environmental consistency, and concurrency between the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Due to these challenges, the execution model and development environment differ from the SPM-based memory mapping techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' As a result, we require a memory mapping technique that supports intermittent computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Researchers have explored memory mapping techniques and analysis for the MSP430FR6989 MCU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In FRAM-based MCUs, Jayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' [18] implement a checkpointing policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' They save the system state to FRAM during a power failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Jayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' [19, 20] propose an energy-efficient memory mapping technique for TI-based applications in FRAM-based MCUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' [23] present a detailed analysis of energy consumption for all memory sections in FRAM-based MCUs under different memory mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Earlier works investigated this problem by analyzing the possibilities to make the system efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The authors [19, 20, 23] have not covered all the design choices and possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In addition, there is significantly less contribution towards memory mappings in FRAM-based MCUs that supports 4 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='J Badri, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' intermittent computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Our work proposes an energy-efficient memory mapping technique for intermittently powered IoT devices that experience frequent power failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 3 MOTIVATION This section discusses the advantages of using hybrid SRAM and FRAM for these MSP430-based MCUs over unified SRAM or unified FRAM designs, as well as the importance of an efficient memory allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' SRAM is 2KB, and FRAM is 128KB in a FRAM-based MCU, MSP430FR6989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The first naive approach is to use the entire 128KB of FRAM in both stable and unstable power scenarios, resulting in longer execution cycles and higher energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Similarly, we have a second naive approach to use the entire 2KB SRAM for small applications (whichever fits within the SRAM size), which has advantages during regular operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Unfortunately, it loses all 2KB SRAM data during a power failure and takes more time to backup 2KB contents to FRAM during a power failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' These two approaches are treated as baselines 1 and 2 for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' As shown in figure 1, for the baseline-1 design, we map all three sections to FRAM and all three sections to SRAM for the baseline-2 design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' int glob1, glob2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=', globn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' func_1(){ local_variables } func_2(){ local_variables } func_n(){ local_variables } Text Data Stack For func_1 () Text Data Stack For func_2 () For func_n () Program Global_Variables Functions Consists of Local Variables For global_vars Data .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='bss Text Data Stack SRAM SRAM (2 KB) FRAM (128 KB) Memory Stack(func_1) Stack(func_n) Text(func_1) Data(func_1) Data(func_n) Text(func_n) Map to SRAM FRAM SRAM (2 KB) FRAM (128 KB) Memory Stack(func_1) Stack(func_n) Text(func_1) Data(func_1) Data(func_n) Text(func_n) Map to FRAM Baseline-1 Design Baseline-2 Design global_vars global_vars Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Overview of the Baseline-1 and Baseline-2 memory mappings in MSP430FR6989 Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing 5 We compared baseline-1 and baseline-2 in both stable and unstable power scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Baseline-1 performs better during frequent power failures, while baseline-2 performs better during regular operations (without any power failures), as shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' On average, baseline-1 consumes 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='9% more energy than baseline-2 during a stable power, as shown in figure 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' On average, baseline-2 consumes 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='7% more energy than baseline-1 during an unstable power, as shown in figure 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We also observed that MCU would pitch an error to either increase the SRAM space or use FRAM space for any computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' For large-size applications will not run using only SRAM, it requires FRAM as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Thus, large applications consume more energy in baseline-2 during a stable power scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' These two designs motivate us to propose a hybrid memory design that effectively uses both SRAM and FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We also encountered that baseline-2 is ineffective for larger applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' As a result, we had to use a hybrid memory and figure out how and where to place the sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' To the best of our knowledge, only one work explored the memory mapping issue for these MCUs [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We analyzed the mapping decisions using their empirical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Jayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' [20] calculated the energy consumption values for each configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The authors suggested that allocate the sections to either SRAM or FRAM based on the energy values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='8 1 16bit_2dim aes basicmath_small basicmath_large bf crc dhrystone dijkstra fft fir matrix_mult patricia qsort_small qsort_large sha susan Normalized Energy Consumption (Normalized with Baseline-1) Benchmarks Baseline-1 Baseline-2 (a) Under Stable Power 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='8 1 16bit_2dim aes basicmath_small basicmath_large bf crc dhrystone dijkstra fft fir matrix_mult patricia qsort_small qsort_large sha susan Normalized Energy Consumption (Normalized with Baseline-1) Benchmarks Baseline-1 Baseline-2 (b) Under Unstable Power Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Comparison between Baseline-1 and 2 configurations under Stable and Unstable Power Scenarios Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Analysis of the Empirical Methods Used by Jayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' [20] for qsort_small under stable and unstable power supply scenarios Configuration Text Data Stack 𝐸𝑛𝑒𝑟𝑔𝑦𝑠𝑡𝑎𝑏𝑙𝑒 (𝑚𝐽) 𝐸𝑛𝑒𝑟𝑔𝑦𝑢𝑛𝑠𝑡𝑎𝑏𝑙𝑒 (𝑚𝐽) 1 {SSS} SRAM SRAM SRAM 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='70 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='56 2 {SSF} SRAM SRAM FRAM 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='08 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='34 3 {SFS} SRAM FRAM SRAM 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='75 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='79 4 {SFF} SRAM FRAM FRAM 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='97 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='10 5 {FSS} FRAM SRAM SRAM 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='48 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='24 6 {FSF} FRAM SRAM FRAM 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='64 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='75 7 {FFS} FRAM FRAM SRAM 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='14 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='83 8 {FFF} FRAM FRAM FRAM 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='09 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='07 The empirical method used by the authors is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The authors considered functions as the basic unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' They explored all configurations and calculated the energy values, as shown in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The authors have eight configurations because they have two memory regions (SRAM or FRAM) and need to map three sections (stack, data, text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Using the author’s model, we calculated the 6 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='J Badri, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' energy values for the qsort_small application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' For instance, the SSS configuration performs better during a stable power supply, and during a power failure, SFS consumes less energy than all other configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' As a result, authors allocate text and stack sections to SRAM and data sections to FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We observed that this empirical method becomes ineffective as the number of configurations increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The authors considered all global variables, arrays, and constants as data sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Instead, why can’t we map each global variable or array to either SRAM or FRAM?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This increases the number of configurations, and calculating/tracking energy values is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Our design space grows enormously and makes our mapping problem challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This new set of challenges motivated us to propose an energy-efficient memory mapping tech- nique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Our proposed memory mapping framework supports large-size applications and covers all possible configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 4 SYSTEM MODEL AND PROBLEM DEFINITION This section discusses the system model for embedded MCUs and defines the mapping problem for these MCUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='1 System Model We consider a simple, customized RISC instruction set with a Von-Neumann architecture, where the instructions and data share the same address space that supports at least 16-bit addressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Base architecture doesn’t have a cache to avoid uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' To make things simple, we assume single cycle execution of the processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Base architecture has a small SRAM memory and a larger NVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The MSP430 is an example of such a processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Non-volatile memory sizes range from 1 kilobyte (KB) to 256 KB, while volatile RAM sizes range from 256 bytes to 2KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Both SRAM and NVM can be accessed by instructions using a compiler/linker script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We can modify the linker script to map memory according to the memory ranges specified by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' MSP430 doesn’t have any operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='2 Problem Definition Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='1: Optimal Memory Mapping Problem: Given a program that consists of various functions and global variables, sizes of SRAM and FRAM, the number of reads and writes for each function/variable, and the energy required per read/write to the SRAM/FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' What is the optimal memory mapping for these functions/variables in order to reduce the system’s EDP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The inputs are : Number of functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' number of global variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' energy per write to SRAM and FRAM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' energy per read to SRAM and FRAM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' SRAM and FRAM sizes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Number of CPU cycles per each function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' the number of reads;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' the number of writes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The output is: Mapping information for all functions and global variables, under which the system’s EDP is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='2: Support for Intermittent Computing: During power failures, we must safely backup the volatile contents to NVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' As previously stated, we must use SRAM efficiently for energy savings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' but again, how can we save the contents of SRAM?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' There are two significant issues with intermittent computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' First, during a power failure, all SRAM’s mapping information and register contents are lost, causing the system to become inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Second, how do we backup/restore the mapping information and register contents to ensure system consistency?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing 7 5 MAPI-PRO: AN ENERGY EFFICIENT MEMORY MAPPING FOR INTERMITTENT COMPUTING In this section, we discuss the details of the proposed mapping technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Our main objective is to pick the optimal mapping choice among all the design choices, which reduces the system’s EDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' To achieve this, we proposed an ILP-based mapping technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The overview of the proposed mapping technique is shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We also discuss how we support intermittent computing for these MCUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' int glob_1, glob_2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=', glob_n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' func_1(){ local_variables } func_2(){ local_variables } func_n(){ local_variables } Text Data Stack For func_1 () Text Data Stack For func_2 () For func_n () Program Global_Variables Functions Consists of Local Variables Proposed ILP based Mapping Technique For global_vars Data .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='bss Text Data Stack Placement Decision for SRAM Placement Decision for FRAM Stack(func_2) Stack(func_n) Text(func_1) Data(func_2) Text(func_n) Data(func_n) Data(func_1) Data(func_2) Stack(func_1) glob_3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='.,glob_n Text(func_1) Text(func_2) Data(func_n) SRAM (2 KB) FRAMn (125 KB) Memory Backup Region FRAMb (3 KB) glob_1, glob_2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Overview of the proposed memory mappings in MSP430FR6989 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='1 ILP Formulation for Data Mapping We present the ILP formulation for the memory mapping problem mentioned in definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We divide this ILP formulation into two parts, one is for global variables, and the second is for the functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We have shown the overview block diagram of the proposed ILP framework in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Application Profiling and one-time characterization Assembly Code Number of reads & writes to each variable Number of reads & writes to each function Energy per read/write to SRAM Energy per read/write to FRAM ILP Solver Number of CPU cycles required for eachfunctions and variable Number of Functions Number of Global variables SRAM and FRAM sizes Mapping Information for each Variable and Function MSP430FR6989 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Overview of the Proposed ILP Framework 8 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='J Badri, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' For Global Variables: Let the number of global variables in a program be ‘G’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Let the number of reads and writes to variable ‘i’ are 𝑟𝑖 and 𝑤𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We divided FRAM’s 128 KB into two regions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=', 𝐹𝑅𝐴𝑀𝑛 and 𝐹𝑅𝐴𝑀𝑏, 𝐹𝑅𝐴𝑀𝑛 memory region has 125 KB, and the 𝐹𝑅𝐴𝑀𝑏 memory region has 3 KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We have two memory regions represented as 𝑀𝑒𝑚𝑗 as shown in the equation 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' when j=1, we select the memory region as SRAM, and we use 𝐹𝑅𝐴𝑀𝑛 for j=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝑀𝑒𝑚𝑗 = � 𝑗 = 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' SRAM 𝑗 = 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐹𝑅𝐴𝑀𝑛 (2) Let the sizes of SRAM/FRAM as 𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗) as shown in equation 3, when j=1, we refer as SRAM memory size in bytes, and when j=2, we refer as 𝐹𝑅𝐴𝑀𝑛 memory size in bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗) = � 𝑗 = 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' SRAM 𝑗 = 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐹𝑅𝐴𝑀𝑛 (3) Let the energy required for each read/write to 𝑀𝑒𝑚𝑗 is 𝐸𝑟_𝑗 and 𝐸𝑤_𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Let the number of CPU cycles required to execute a global variable 𝑣𝑖 be 𝑁𝐶𝑣𝑖, where ∀𝑖 ∈ [1,𝐺]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Using one-time charac- terization and static profiling, we gathered data such as per read/write energy to SRAM/FRAM and the number of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We define a binary variable (BV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐼𝑗 (𝑣𝑖), which refers to a variable 𝑣𝑖 is allocated to memory region 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' If 𝐼𝑗 (𝑣𝑖)=1 then the variable 𝑣𝑖 is allocated and 𝐼𝑗 (𝑣𝑖)=0 indicates that the variable 𝑣𝑖 is not allocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐼𝑗 (𝑣𝑖), where (∀𝑗 ∈ [1, 𝑀𝑒𝑚𝑗], ∀𝑖 ∈ [1,𝐺]) is defined as shown in the equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐼𝑗 (𝑣𝑖) = � 1 𝑣𝑖 is allocated to memory region 𝑗 0 otherwise (4) Constraints: There are two constraints, one is for BV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐼𝑗 (𝑣𝑖) and one is a memory size constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In any case, a variable 𝑣𝑖 is allocated to only one memory region, which means 𝑣𝑖 is allocated to either SRAM or FRAM but not both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This constraint is defined in the equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝑀𝑒𝑚𝑗 ∑︁ 𝑗=1 𝐼𝑗 (𝑣𝑖) = 1 (∀𝑖 ∈ [1,𝐺]) (5) The other constraint is related to memory sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The allocated variables 𝑣𝑖 and its 𝑆𝑖𝑧𝑒(𝑣𝑖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' ∀𝑖 ∈ [1,𝐺]) should not be greater than the 𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This constraint is defined in the equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐺 ∑︁ 𝑖=1 𝐼𝑗 (𝑣𝑖) ∗ 𝑆𝑖𝑧𝑒(𝑣𝑖) ≤ 𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗) (∀𝑗 ∈ [1, 𝑀𝑒𝑚𝑗]) (6) Objective 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='1: The challenge of mapping global variables in a program to either SRAM or FRAM is to reduce EDP and improve system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐸𝑔𝑙𝑜𝑏𝑎𝑙 is defined in the equation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Where 𝐸𝑔𝑙𝑜𝑏𝑎𝑙 is the energy required to allocate global variables to either SRAM or FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐸𝑔𝑙𝑜𝑏𝑎𝑙 = 𝑀𝑒𝑚𝑗 ∑︁ 𝑗=1 𝐺 ∑︁ 𝑖=1 [𝐸𝑟_𝑗 × 𝑟𝑖 + 𝐸𝑤_𝑗 × 𝑤𝑖] (7) 𝐸𝐷𝑃𝑔𝑙𝑜𝑏𝑎𝑙 is defined in the equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Where 𝐸𝐷𝑃𝑔𝑙𝑜𝑏𝑎𝑙 is the energy-delay product required to allocate global variables to either SRAM or FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing 9 𝐸𝐷𝑃𝑔𝑙𝑜𝑏𝑎𝑙 = 𝑀𝑒𝑚𝑗 ∑︁ 𝑗=1 𝐺 ∑︁ 𝑖=1 𝐼𝑗 (𝑣𝑖) [𝐸𝑔𝑙𝑜𝑏𝑎𝑙 × 𝑁𝐶𝑣𝑖] (8) For Functions: Let the number of functions in a program be ‘𝑁 ′ 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Let the number of reads and writes to 𝑖𝑡ℎ function are 𝑟 (𝐹𝑖) and 𝑤(𝐹𝑖), where ∀𝑖 ∈ [1, 𝑁𝑓 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Functions consist of procedural parameters, local variables, and return variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Internally the code/data of functions are divided into the text, data, and stack sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We map at least one section among these three sections to either SRAM or FRAM regions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=', 𝑀𝑒𝑚𝑗 and 𝑆𝑒𝑐𝑘 (𝑖) defines section ‘k’ of 𝑖𝑡ℎ function as shown in the equation 9, when k=1, we refer to the text section of 𝑖𝑡ℎ function, when k=2, we refer to the data section of 𝑖𝑡ℎ function, and when k=3, we refer to the stack section of 𝑖𝑡ℎ function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝑆𝑒𝑐𝑘 (𝑖) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 𝑘 = 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Text 𝑘 = 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Data 𝑘 = 3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Stack ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='∀𝑖 ∈ [1, 𝑁𝑓 ] (9) We define a BV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)), which refers to a section 𝑆𝑒𝑐𝑘 of 𝑖𝑡ℎ function is allocated to only one memory region 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' If 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖))=1 then the section 𝑆𝑒𝑐𝑖 is allocated and 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖))=0 that indicates the section 𝑆𝑒𝑐𝑖 is not allocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)), where (∀𝑗 ∈ [1, 𝑀𝑒𝑚𝑗], ∀𝑖 ∈ [1, 𝑁𝑓 ]), ∀𝑘 ∈ [1,𝑆𝑒𝑐𝑘 (𝑖)]) is defined as shown in the equation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)) = � 1 𝑆𝑒𝑐𝑘 of 𝑖𝑡ℎ function is allocated to 𝑗 0 otherwise (10) Constraints: There are two constraints, one is for BV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)) and one is a memory size constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In any case, a 𝑆𝑒𝑐𝑘 of 𝑖𝑡ℎ function is allocated to only one memory region, which means 𝑆𝑒𝑐𝑘 of 𝑖𝑡ℎ function is either allocated to either SRAM or FRAM but not both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This constraint is defined in the equation 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 3 ∑︁ 𝑘=1 𝑀𝑒𝑚𝑗 ∑︁ 𝑗=1 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖))) = 1 (∀𝑖 ∈ [1, 𝑁𝑓 ]) (11) The other constraint is related to memory sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The allocated sections 𝑆𝑒𝑐𝑘 (𝑖) and its 𝑆𝑖𝑧𝑒(𝐹𝑖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' ∀𝑘 ∈ [1,𝑆𝑒𝑐𝑘 (𝑖)]), ∀𝑗 ∈ [1, 𝑀𝑒𝑚𝑗], ∀𝑖 ∈ [1, 𝑁𝑓 ] should not be greater than the 𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This constraint is defined in the equation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐺 ∑︁ 𝑖=1 𝐼𝑗 (𝑣𝑖) ∗ 𝑆𝑖𝑧𝑒(𝑣𝑖) + 3 ∑︁ 𝑘=1 𝑁𝑓 ∑︁ 𝑖=1 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)) ∗ 𝑆𝑖𝑧𝑒(𝐹𝑖) ≤ 𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗) (12) Objective 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='2: The challenge of mapping sections of these functions in a program to either SRAM or FRAM is to minimize EDP and improve system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐸𝑓 𝑢𝑛𝑐 is defined in the equation 13, where 𝑀𝑐𝑖 is the number of the times 𝑖𝑡ℎ functions called.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐸𝑓 𝑢𝑛𝑐 = 𝑀𝑒𝑚𝑗 ∑︁ 𝑗=1 𝑁𝑓 ∑︁ 𝑖=1 [𝐸𝑟_𝑗 × 𝑟 (𝐹𝑖) + 𝐸𝑤_𝑗 × 𝑤(𝐹𝑖)] × 𝑀𝑐𝑖 (13) 𝐸𝐷𝑃𝑓 𝑢𝑛𝑐 is defined in the equation 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Where 𝐸𝐷𝑃𝑓 𝑢𝑛𝑐 is the energy-delay product required to allocate all functions to either SRAM or FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Where 𝐸𝑓 𝑢𝑛𝑐 is the energy required to allocate 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='J Badri, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' functions to either SRAM or FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Where 𝑁𝐶𝐹𝑖 is the number of CPU cycles required to execute a function 𝐹𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐸𝐷𝑃𝑓 𝑢𝑛𝑐 = 3 ∑︁ 𝑘=1 𝑀𝑒𝑚𝑗 ∑︁ 𝑗=1 𝑁𝑓 ∑︁ 𝑖=1 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)) [𝐸𝑓 𝑢𝑛𝑐 × 𝑁𝐶𝐹𝑖] (14) The overall system EDP, 𝐸𝐷𝑃𝑠𝑦𝑠𝑡𝑒𝑚, is the sum of both 𝐸𝐷𝑃𝑔𝑙𝑜𝑏𝑎𝑙 and 𝐸𝐷𝑃𝑓 𝑢𝑛𝑐 as shown in the equation 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐸𝐷𝑃𝑠𝑦𝑠𝑡𝑒𝑚 = 𝐸𝐷𝑃𝑔𝑙𝑜𝑏𝑎𝑙 + 𝐸𝐷𝑃𝑓 𝑢𝑛𝑐 (15) Our objective function is shown in the equation 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Our main objective is to minimize the system’s EDP by choosing the optimal placement choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Objective Function: Minimize 𝐸𝐷𝑃𝑠𝑦𝑠𝑡𝑒𝑚 (16) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='2 Implementing Mapping Technique in MSP430FR6989 Once we obtain the placement information from the 𝐼𝐿𝑃_𝑠𝑜𝑙𝑣𝑒𝑟, we map the respective variables and the sections of a function to either SRAM or FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We modify the linker script accordingly for mapping the sections or variables to either SRAM or FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In our proposed mapping policy, placing global variables is straightforward, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=', mapping the respective variable to either SRAM or FRAM based on the ILP decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We observed that from the linker script, we can map the whole stack section of each function to either SRAM or FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We analyzed the mappings of the stack section for each function by modifying the linker script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We used the inbuilt attributes to differentiate mappings between SRAM and FRAM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' for instance, we used the inbuilt attribute (__𝑎𝑡𝑡𝑟𝑖𝑏𝑢𝑡𝑒__((𝑟𝑎𝑚𝑓𝑢𝑛𝑐)) that maps that function to SRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' If we want to place the stack section to SRAM, we modify the linker script by replacing the default setting with " .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='stack: {} > RAM (HIGH) ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' If we want to place the stack section to FRAM, we modify the linker script by replacing the default setting with " .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='stack: {} > FRAM".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Similarly, for the text section, we observed that placing the text section in either SRAM or FRAM shows an impact on EDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This effect is because the majority of access in the text section are read accesses, as we observed that the energy consumption for each read access to SRAM/FRAM differs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Table 3 shows that approximately FRAM consumes 2x more read energy than SRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Thus, we analyzed each application where to map the text section based on the free space available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' If we have enough space available in SRAM, we place the text section in SRAM itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' otherwise, we place the text section in FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We included the following four lines in our linker script to check the above condition and map the text section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' (1) #𝑖𝑓 𝑛𝑑𝑒𝑓 __𝐿𝐴𝑅𝐺𝐸_𝐶𝑂𝐷𝐸_𝑀𝑂𝐷𝐸𝐿__ (2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='text : {} > FRAM (3) #else (4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='text : {} » SRAM We modified the linker script for mapping the data section by using the inbuilt compiler directives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We followed the below three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' (1) Allocate a new memory block, for instance, 𝑁𝐸𝑊 _𝐷𝐴𝑇𝐴𝑆𝐸𝐶𝑇𝐼𝑂𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We can declare the start address and size of the data section in the linker script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' (2) Define a segment (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='Localvars) which stores in this memory block (𝑁𝐸𝑊 _𝐷𝐴𝑇𝐴𝑆𝐸𝐶𝑇𝐼𝑂𝑁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing 11 (3) Use #pragma 𝐷𝐴𝑇𝐴_𝑆𝐸𝐶𝑇𝐼𝑂𝑁 (𝑓𝑢𝑛𝑐𝑡_𝑛𝑎𝑚𝑒,𝑠𝑒𝑔_𝑛𝑎𝑚𝑒) in the program to define functions in this segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Where 𝑓𝑢𝑛𝑐𝑡_𝑛𝑎𝑚𝑒 is the function name and𝑠𝑒𝑔_𝑛𝑎𝑚𝑒 is the created segment name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' For instance, #pragma 𝐷𝐴𝑇𝐴_𝑆𝐸𝐶𝑇𝐼𝑂𝑁 (𝑓𝑢𝑛𝑐_1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='𝐿𝑜𝑐𝑎𝑙𝑣𝑎𝑟𝑠) Once we are done with creating the different sections, we can allocate these sections to either SRAM or FRAM based on ILP decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' For instance, placing " 𝑁𝐸𝑊 _𝐷𝐴𝑇𝐴𝑆𝐸𝐶𝑇𝐼𝑂𝑁: {} > FRAM" in the linker script, which maps the 𝑁𝐸𝑊 _𝐷𝐴𝑇𝐴𝑆𝐸𝐶𝑇𝐼𝑂𝑁 to FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='3 Support for Intermittent Computing When the power is stable, everything works properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Because of the static allocation scheme, we map all functions/variables to SRAM/FRAM for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' During a power failure, SRAM and registers lose all of their contents, including mapping information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' When power is restored, we don’t know what functions/variables were allocated to SRAM before the failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' As a result, we must either restart the execution from the beginning or end up with incorrect results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Restarting the application consumes extra energy and time, making our system inefficient in terms of energy consumption and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We propose a backup strategy during frequent power failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' FRAM was divided into 𝐹𝑅𝐴𝑀𝑛 and 𝐹𝑅𝐴𝑀𝑏 as shown in the figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐹𝑅𝐴𝑀𝑛 has a size of 125 KB and is used for regular mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 𝐹𝑅𝐴𝑀𝑏 has a size of 3 KB that serves as a backup region (BR) during power failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' So, during a power failure, we back up all register and SRAM contents to FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Whenever power is restored, we restore the register and SRAM contents from 𝐹𝑅𝐴𝑀𝑏 to SRAM and resume the application execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The proposed backup strategy reduces extra energy consumption and makes the system more energy efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 6 EXPERIMENTAL SETUP AND RESULTS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='1 Experimental Setup We used TI’s MSP430FR6989 for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We experimented on mixed benchmarks, which have both Mi-Bench [13] and TI-based benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We have shown the experimental setup in the table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The development platform and experimental setup are shown in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We performed experiments to determine the energy required for a single read/write to SRAM/FRAM, as shown in the table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We collected the number of reads/writes for each global variable and functions as part of a one-time characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We also used TI’s MSP430F5529 for comparing flash with FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We performed experiments to determine the energy required for a single read/write to flash, as shown in the table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Experimental Setup Component Description Target Board TI MSP430FR6989 Launchpad Core MSP430 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='8-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='6 V;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 16 MHz) Memory 2KB SRAM and 128KB FRAM IDE Code Composer Studio Energy Profiling Energy Trace++ ILP Solver LPSolve_IDE Benchmarks Mixed benchmarks (MiBench and TI-based) MCU, which we experimented has MSP430 architecture, which is more suitable for IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The majority of MSP430 software is written in C and compiled with one of TI’s recommended 12 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='J Badri, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' compilers ( IAR Embedded Code Bench, Code-Composer Studio (CCS), or msp430-gcc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The IAR Embedded Code Bench and CCS compilers are part of integrated development environments (IDEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We used the widely used, freely available, and easily extended tool, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=', CCS, for all experiments in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' EnergyTrace++ technology allows us to calculate energy and power consumption directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' According to the datasheet for the MSP430FR6989, the number of cycles required to read/write in FRAM is twice that of SRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Energy Values for each read/write to SRAM and FRAM Memory Per Read Energy (nJ) Per Write Energy (nJ) SRAM 5500 5600 FRAM 10325 13125 Flash 23876 31198 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' (a) TI-based MSP430 Launchpad Development Boards (b) Working with EnergyTrace++ on CCS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='2 Evaluation Benchmarks We chose benchmarks from both the MiBench suite and TI benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' One of the primary motivations for using the MiBench suite is that most of the TI-based benchmarks were small in size and easily fit into either SRAM or FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In these cases, we don’t require any hybrid memory design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Most of the TI-based benchmarks have only one or two functions and 3-4 global variables, which is not useful for the hybrid main-memory design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Thus we used mixed benchmarks consisting of 4 TI-based benchmarks and 12 from the MiBench suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' For the MiBench suite, we first make MCU-compatible benchmarks by adding MCU-related header files and watchdog timers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' All benchmarks may not be compatible with the MCU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Thus, we need to choose the benchmarks from the MiBench suite, which are compatible with the MSP430 boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Once we have benchmarks, we execute them on board for the machine code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Using the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='asm file, we calculate the inputs that are required by the ILP solver, as shown in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='3 Baseline Configurations We chose five different memory configurations to compare with the proposed memory configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We directly map all the functions/variables to FRAM in the baseline configuration 1, as shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We use configuration-1 to compare our proposed memory configuration during stable and unstable power scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Code Composer Studio MSP430F5529MSP430FR6989 EnergyTrace++Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing 13 We directly map all the functions/variables to SRAM in the baseline configuration 2, as shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We use configuration-2 to compare our proposed memory configuration during stable and unstable power scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In baseline configuration 3, we used the empirical method of Jayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We compare this configuration-3 with our proposed configuration during stable and unstable power scenarios to observe the importance of the proposed than the existing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In baseline configuration 4, we used the proposed ILP technique for the flash-based msp430 board [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We compare this configuration-4 with our proposed configuration during stable and unstable power scenarios to observe the difference between FRAM and Flash technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In baseline configuration 5, we only have a proposed memory mapping technique and no BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We compare this configuration-5 with our proposed configuration during frequent power failures to observe the importance of BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The overview of all baseline configurations is shown in table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The experimental setup for all baseline configurations is the same as the one proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Overview of the Baseline Configurations Configuration FRAM SRAM Flash Backup Region (BR) ILP Baseline-1 ✓ ✗ ✗ ✗ ✗ Baseline-2 ✗ ✓ ✗ ✗ ✗ Baseline-3 ( Jayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' [20]) ✓ ✓ ✗ ✗ ✗ Baseline-4 ✗ ✓ ✓ ✗ ✓ Baseline-5 ✓ ✓ ✗ ✗ ✓ Proposed ✓ ✓ ✓ ✓ ✓ ✓- Supported , ✗- Not Supported 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='4 Results The proposed memory configuration is evaluated in this section under stable and unstable power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The proposed memory configuration is compared with five baseline memory configurations as discussed in the section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='1 Under Stable Power: Our main objective of the proposed memory configuration is to minimize the system’s EDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' All values shown in figure 6 are normalized with baseline-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Compared to baseline-1, the proposed gets 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='10% lesser EDP, as shown in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Because there are no power interruptions in this scenario, this improvement is totally from the proposed ILP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In configuration-1, we place everything to FRAM, where FRAM consumes more energy and the number of cycles than SRAM, as shown in the table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Our proposed memory configuration incorporates the placement recommendation from the proposed ILP model and suggests utilizing both SRAM and FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Under a stable power scenario, the proposed gets 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='30% less EDP than baseline-3, as shown in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We discussed the author’s empirical model and assumptions in the previous section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The authors assumed that the data section included all global variables, constants, and arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' As a result, our proposed ILP-based mapping differs from the author’s mapping in that our proposed mapping outperforms the existing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Under stable power, baseline-3 receives 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='57% less EDP than baseline-1, as shown in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This advantage is primarily due to baseline-3’s hybrid memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In comparison to baseline-4, the proposed reduces EDP by 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='55%, as shown in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We used flash+SRAM with our proposed ILP framework in baseline-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' As shown in table 3, the above benefit is primarily due to FRAM because flash consumes more energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Baseline-3 outperforms 14 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='J Badri, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='8 1 16bit_2dim aes basicmath_small basicmath_large bf crc dhrystone dijkstra fft fir matrix_mult patricia qsort_small qsort_large sha susan Normalized EDP (Normalized with Baseline-1) Benchmarks Baseline-2 Jayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' [20] Baseline-4 Proposed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Comparison between Baseline configurations and the Proposed under Stable Power baseline-4 during stable power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Because of FRAM in baseline-3, even our proposed ILP model is ineffective in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We encountered that baseline-3 achieves 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='19% less EDP than baseline-4, and this benefit is because of smaller applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' From figure 6, baseline-4 performs better for large applications than baseline-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Jayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' [20] empirical method suggests placing more content on SRAM because SRAM is sufficient for placing the entire small-size application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' As a result, the performance of baseline 3 is dependent on the application size, as for large-size applications, even FRAM does not outperform flash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Baseline 2 outperforms the proposed and all other baselines under stable power conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We noticed that this benefit is primarily due to SRAM, but it only applies to smaller applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Baseline 2 achieves 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='19% less EDP than the proposed for smaller applications, as shown in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We also looked at large applications where the proposed outperforms the baseline-2 by a small margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' When the SRAM is full, the MCU must wait for the space to be released, which consumes extra energy and cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' For more extensive applications, baseline-2 achieves 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='94% more EDP than proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We also evaluated our proposed framework with another MSP430F5529 MCU with flash and SRAM for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This comparison assists the user in selecting the most appropriate NVM technology, such as FRAM or flash, as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' To be fair, we used the same sizes of SRAM (2 KB) and Flash (128 KB) in this comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We compared FRAM-based and flash-based MCUs under stable power conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We used the proposed frameworks and techniques in both MCUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We discovered that the proposed FRAM-based configuration outperforms the flash-based configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Flash-based configurations consume 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='03% more EDP than FRAM-based configurations, as shown in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Flash consumes more energy, as shown in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='2 Under Unstable power: We used the default TI-based compute through power loss (ctpl) tool for migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' During a power failure, we need to migrate the SRAM contents to a FRAM-based backup region (𝐹𝑅𝐴𝑀𝑏), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=', the backup process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Whenever power comes back, we need to migrate the (𝐹𝑅𝐴𝑀𝑏) contents to SRAM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=', the restoration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' So, all these migrations are done using ctpl() functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We introduce a power failure by changing the low power modes mentioned in the MSP430FR6989 design document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We used ctpl() for creating power failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We assume that the number of power failures is spread equally within the execution period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' For instance, if the Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='8 1 16bit_2dim aes basicmath_small basicmath_large bf crc dhrystone dijkstra fft fir matrix_mult patricia qsort_small qsort_large sha susan Normalized EDP (Normalized with MSP430F5529) Benchmarks MSP430F5529 MSP430FR6989 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Comparison between MSP430FR6989 (FRAM-based MCU) and MSP430F5529 (Flash-based MCU) under Stable Power total execution period for an application is 20 milliseconds (ms), and let’s say the number of power failures is four, then for every 5 ms, we experience a power failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We performed experiments under unstable power to compare the proposed memory configuration with baseline configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' All values shown in figure 8 are normalized with baseline-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Compared to baseline-2, the proposed gets 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='97% lesser EDP, as shown in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We observed that migration overhead is less than the energy consumed to execute the application from FRAM, and this migration overhead depends on the number of power failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' For instance, one backup migration consumes approximately 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='88 mJ of energy, and one restore migration consumes approximately 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='606 mJ of energy in a qsort application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The above benefit to our proposed configuration is using a hybrid memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Under an unstable power scenario, the proposed gets 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='99% less EDP than baseline-3, as shown in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We discussed the author’s empirical model and assumptions in the previous section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' As already stated, the Jayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' empirical method is more beneficial for small applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In contrast, the author’s empirical method suggests placing more content on SRAM because SRAM is sufficient for placing the entire small-size application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Thus, for [20] work, backup/restore operations take more energy during a power failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Our proposed mapping outperforms the existing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' During frequent power failures, baseline-3 receives 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='91% less EDP than baseline-1, as shown in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This advantage is primarily due to baseline-3’s hybrid memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Compared to baseline-4, the proposed reduces EDP by 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='05%, as shown in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We used flash+SRAM with our proposed ILP framework in baseline-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' As shown in table 3, the above benefit is primarily due to FRAM because flash consumes more energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Baseline-3 outperforms baseline-4 during stable power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Because of FRAM in baseline-3, even our proposed ILP model is ineffective for this comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We encountered that baseline-3 achieves 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='28% less EDP than baseline-4 for smaller applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The above benefit for baseline-3 is minimal because the size of backup/restores increases, which even neutralizes the flash for some applications, as shown in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Baseline-4 achieves 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='69% less EDP than baseline-3 for large applications, as shown in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' As a result, the 16 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='J Badri, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='8 1 16bit_2dim aes basicmath_small basicmath_large bf crc dhrystone dijkstra fft fir matrix_mult patricia qsort_small qsort_large sha susan Normalized EDP (Normalized with Baseline-2) Benchmarks Baseline-1 Jayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' [20] Baseline-4 Baseline-5 Proposed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Comparison between Baseline configurations and the Proposed under Unstable Power performance of baseline 3 is dependent on the application size, as for large-size applications, even FRAM does not outperform flash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The proposed outperforms all baselines under unstable power conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This benefit is primarily due to a hybrid memory and the proposed mapping technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Baseline 2 achieves 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='98% less EDP than the proposed, as shown in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' When we remove BR, all the mapping information of SRAM is lost because our model is static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We introduce a BR in the FRAM memory region to save this mapping information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' During a power failure, we migrate the SRAM contents to 𝐹𝑅𝐴𝑀𝑏, and whenever power comes back, we restore the 𝐹𝑅𝐴𝑀𝑏 contents to the SRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We experimented to know the importance of BR, where we compared the proposed memory configuration with baseline-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Compared to baseline-5, the proposed gets 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='94% lesser EDP, as shown in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This benefit is because we need to re-execute the application four times from the beginning, which consumes extra time and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The number of times re-executing the application is equal to the number of power failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We also evaluated our proposed framework with another MSP430F5529 MCU, which consists of flash and SRAM for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' This comparison assists the user in selecting the most appropriate NVM technology, such as FRAM or flash, as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' To be fair, we used the same sizes of SRAM (2 KB) and Flash (128 KB) in this comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We also used BR for both baselines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' the only difference is that we replaced FRAM with the flash in the proposed configurations, and everything is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We compared FRAM-based and flash-based MCUs under unstable power conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We used the proposed frameworks and techniques in both MCUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We discovered that the proposed FRAM-based configuration outperforms the flash-based configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Flash-based configurations consume 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='50% more EDP than FRAM-based configurations, as shown in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Flash consumes more energy, as shown in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='5 Summary of the Proposed Mapping Technique We outline the proposed ILP-based memory mapping technique in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Following all of these analyses, we observed that the mappings shown below consume less EDP than other design choices, as shown in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' To keep things simple, we only showed the final mapping Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing 17 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='8 1 16bit_2dim aes basicmath_small basicmath_large bf crc dhrystone dijkstra fft fir matrix_mult patricia qsort_small qsort_large sha susan Normalized EDP (Normalized with MSP430F5529) Benchmarks MSP430F5529 MSP430FR6989 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Comparison between MSP430FR6989 (FRAM-based MCU) and MSP430F5529 (Flash-based MCU) under Unstable Power configurations for each application’s stack, data, and text sections, keeping out the final mappings for global variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Optimal Placement for different Applications in MSP430FR6989 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='Benchmarks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='Stack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='16bit_2dim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='patricia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='SRAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='FRAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='SRAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='qsort_small ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='SRAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='SRAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='FRAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='qsort_large ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='SRAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='FRAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='FRAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='sha ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='SRAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='FRAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='FRAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='susan ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='SRAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='FRAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='FRAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='Table 5 shows that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' with the exception of the dhrystone application,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' the remaining three TI benchmark applications (fir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' and 16bit_2dim) are very small and can easily be placed in SRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We don’t need FRAM for these types of smaller applications, but there is a disadvantage during frequent power failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Backup and restore sizes to FRAM are larger for these applications during frequent power failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' As a result, our proposed backup/restore strategy should be intelligent 18 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='J Badri, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' enough to reduce EDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The dhrystone application, on the other hand, has a larger stack section that requires FRAM to accommodate the entire stack section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' As we can see from the table 5, many applications used both SRAM and FRAM for the Mi- Bench applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' As a result, we can conclude that a hybrid main memory design is required for many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Using a hybrid main memory design helps to reduce EDP during stable power scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Even so, determining how and where to backup the volatile contents can be difficult during frequent power outages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' However, our proposed memory mapping technique and the framework suggest using a hybrid main memory design that supports intermittent computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 7 CONCLUSIONS This paper proposed an ILP-based memory mapping technique that reduces the system’s energy- delay product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' For both global variables and functions, we formulated an ILP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Functions consist of data, stack, and code sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Our ILP model suggests placing each section on either SRAM or FRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Under both stable and unstable power scenarios, we compared the proposed memory configuration to the baseline memory configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We evaluated our proposed frameworks and techniques on actual boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We added a backup region in FRAM to support intermittent computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We compared the proposed framework with the recent related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Under stable power, our proposed memory configuration consumes 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='10% less EDP than baseline- 1 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='30% less EDP than the existing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Under unstable power, our proposed configuration achieves 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='97% less EDP than baseline-1 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='99% less EDP than the existing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Under stable power, our proposed memory configuration consumes 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='55% less EDP than baseline-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' We also compared FRAM-based MSP430FR6989 with flash-based MSP430F5529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Compared to the flash, the FRAM-based hybrid main memory design consumes less EDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' FRAM-based design consumes 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='03% less EDP than flash-based design during stable power and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content='50% less EDP than flash based during frequent power failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' REFERENCES [1] Prasenjit Chakraborty, Preeti Ranjan Panda, and Sandeep Sen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 2016.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In Proceedings of the fourth annual IEEE international workshop on workload characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' IEEE, IEEE, 3–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' [14] Josiah Hester and Jacob Sorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' The future of sensing is batteryless, intermittent, and awesome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' [30] Keni Qiu, Mengying Zhao, Zhenge Jia, Jingtong Hu, Chun Jason Xue, Kaisheng Ma, Xueqing Li, Yongpan Liu, and Vijaykrishnan Narayanan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Design insights of non-volatile processors and accelerators in energy harvesting systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' In Proceedings of the 2020 on Great Lakes Symposium on VLSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 369–374.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Towards a formal foundation of intermittent computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Proceedings of the ACM on Programming Languages 4, OOPSLA (2020), 1–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' [35] Sumesh Udayakumaran, Angel Dominguez, and Rajeev Barua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} +page_content=' Dynamic 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79FLT4oBgHgl3EQfAy4h/content/2301.11967v1.pdf'} diff --git a/9NE4T4oBgHgl3EQfdgy1/content/tmp_files/2301.05092v1.pdf.txt b/9NE4T4oBgHgl3EQfdgy1/content/tmp_files/2301.05092v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..edc62884519cf5c37c81108a86512ccb87cf5787 --- /dev/null +++ b/9NE4T4oBgHgl3EQfdgy1/content/tmp_files/2301.05092v1.pdf.txt @@ -0,0 +1,3087 @@ +arXiv:2301.05092v1 [gr-qc] 12 Jan 2023 +, +Slowly rotating Kerr metric derived from the Einstein equations in affine-null +coordinates +Thomas M¨adler +∗ +Escuela de Obras Civiles and Instituto de Estudios Astrof´ısicos, +Facultad de Ingenier´ıa y Ciencias, Universidad Diego Portales, +Avenida Ej´ercito Libertador 441, Casilla 298-V, Santiago, Chile. +Emanuel Gallo +† +FaMAF, UNC; Instituto de F´ısica Enrique Gaviola (IFEG), CONICET, +Ciudad Universitaria, (5000) C´ordoba, Argentina. +Using a quasi-spherical approximation of an affine-null metric adapted to an asymptotic Bondi +inertial frame, we present high order approximations of the metric functions in terms of the specific +angular momentum for a slowly rotating stationary and axi-symmetric vacuum spacetime. +The +metric is obtained by following the procedure of integrating the hierarchy of Einstein equations in a +characteristic formulation utilizing master functions for the perturbations. It is further verified its +equivalence with the Kerr metric in the slowly rotation approximation by carrying out an explicit +transformation between the Boyer-Lindquist coordinates to the employed affine-null coordinates. +PACS numbers: +I. +INTRODUCTION +At the dawn of the ’Golden Era of General Relativity’ +in the 60ties of the last century, two important space- +time metrics were found, the Bondi-Sachs metric [1–3] +and the Kerr metric [4, 5]. The first settled the question +that an isolated system looses mass via gravitational radi- +ation and that this effect is a non-linear effect of General +Relativity; while the second describes a stationary and +rotating isolated black hole that is expected to be the +end product of a gravitational collapse of a massive star +or a merger of two compact objects. +One of the defining features of the Bondi-Sachs metric +is that one coordinate is constant along a family of null +hypersurfaces while a radial coordinate along these null +hypersurfaces is an areal distance that can be related to a +luminosity distance [6]. As such, the first long term sta- +ble evolution of black hole space times were made using a +Bondi-Sachs metric in a null cone-world tube formalism +[7], also see [8, 9] for review. Apart from usage in numer- +ical relativity simulations, the Bondi-Sachs metric is now +frequently used in high energy physics addressing ques- +tions of the AdS/CFT correspondence [10] (and citations +thereof). It also became popular to discuss gravitational +wave memory effects [11–15]. A pleasant property of the +Bondi-Sachs formalism is that the Einstein equation can +be solved in a hierarchical manner when initial data on +a null hypersurface and boundary conditions at a world +tube or vertex are given. However, the radial coordinate +of the Bondi-Sachs metric has the unpleasant property +∗Electronic address: thomas.maedler˙.at.˙mail.udp.cl +†Electronic address: egallo˙.at.˙unc.edu.ar +that it breaks down when an apparent horizon forms due +to the focusing of the surface-forming null rays and their +vanishing expansion. +This can be overcome in choos- +ing an affine parameter as radial coordinate, because an +affine parameter only becomes singular at a caustic. But, +the Einstein equations resulting from an affine-null met- +ric do not provide the hierarchical structure as the Bondi +Sachs metric [9] and the hierarchical structure needs to +be reestablished by various new definitions of variables +[16–18]. Moreover, it turns out that also the hierarchy +of equations in the affine-null metric formulation breaks +down in the events of apparent horizon formation, but +fortunately the equations can be regularized so that it is +possible to follow up the formation of black holes up to +singularity [19, 20]. +Despite the success and popularity of the Bondi-Sachs +metric in the various areas, an explicit closed analytical +representation of the Kerr metric in Bondi-Sachs form +without bad behaviour in the exterior region or related +metrics with one or two null coordinates is missing. Var- +ious attempts have been made to derive a null metric +representation, numerically [21, 22] as well as analyti- +cally [23–25]. In all of the approaches, the authors start +out with the Kerr metric and then calculate the respec- +tive null metric via a coordinate transformation. After +these transformations the resulting metric can still posses +a conical singularity at the axis of symmetry (see [22] +for a complete discussion). In addition, the final met- +ric is determined by integrals of non-elementary func- +tions. Arga˜naraz and Moreschi’s approach [22] differs to +the aforementioned ones that the authors aim to find a +double–null representation of the Kerr metric by geomet- +rically adopting the coordinates to in- and outgoing null +geodesics adapted to the center of mass [27]. In this way, +the authors were successful in finding null coordinates + +2 +that are not only regular at every point of the external +communication region (unlike the previous formulations) +but also that they are regular at the event horizon, thus +allowing a way to study the evolution of different matter +fields (as scalar fields) in such background even when they +cross the event horizon[26]. Unfortunately, even in their +construction arises a differential equation that needs to +be solved numerically and an explicit closed form repre- +sentation of the double null version of the Kerr metric is +not possible. The work of Bai and collaborators [28, 29] +also starts with the Kerr metric (in Boyer-Lindquist co- +ordinates) and then makes coordinate transformation to +a Bondi-Sachs metric valid near future null infinity (in a +compactified version of the metric). The authors are able +to calculate the Newman-Penrose quantities and multi- +poles at large distances and show the peeling property of +the Weyl tensor at large radii and the vanishing of the +so-called Newman-Penrose constants. +In this article, in contrast to all the previous works +which start with the Kerr metric expressed in Boyer- +Lindquist coordinates and attempt to find a null coordi- +nate version of it, we will directly solve the Einstein equa- +tions in a characteristic formulation based on an affine- +null metric formulation of the Einstein equations. In ad- +dition, inspired by the Hartle-Thorne methods for obtain- +ing solutions for slowly rotating compact stars [30], we +will employ a quasi-spherical approximation of the field +equations to find a high order approximation of the Kerr +metric in out-going polar null coordinates. +To obtain +our solution, we assume stationarity and axial symme- +try. We further require an asymptotic inertial observer +as well as that that Weyl scalar Ψ0 is regular everywhere +where the background solution is regular. A study of vac- +uum stationary metrics with a smooth future null infinity +in affine-null coordinates has recently be carried out by +Tafel in [31] by considering power series of the metric +components in terms of the inverse affine distance. +Throughout the article, we will use signature +2, units +G = c = 1 and the Einstein sum convention for indices +as well as products of associated Legendre polynomials. +The article is organised as follows: +Sec. II recalls +the affine-null metric formulation, makes the necessary +symmetry assumptions for archiving our goal and de- +fines the perturbative variables; in Sec. III, we determine +the background model (Sec. III A), define useful recur- +sively re-appearing functions in the perturbation analysis +(Sec. III B), solve the perturbation equations (Sec. III C- +III F) and in Sec. III G the affine-null metric functions +for the null are expressed in terms of the mass and spe- +cific angular momentum, in Sec. IV, to verify our re- +sults, we calculate the affine-null version of Kerr metric +in a Bondi frame via a coordinate transformation with a +method adopted from [28], in Sec. V position of the outer +ergosphere and event (past) horizon of the black hole we +discuss the and Sec. VI contains the final discussion of our +work. The article finishes with two appendices: App. A +lists relations between associated Legendre polynomials +and App. B presents a derivation of the expression of the +Komar charges relevant for this work. +II. +AFFINE-NULL METRIC FORMULATION +FOR STATIONARY AND AXIAL SYMMETRIC +SPACETIMES +Here we review the necessary properties of character- +istic initial value formulation of the Einstein equations +in affine-null coordinates, discuss the implications of the +imposed symmetry assumptions and present the notation +used in our analysis. +Taking coordinates xa = (u, λ, xA), where u is an out– +going null coordinate, λ an affine parameter, and xA are +angular coordinates, a generic line element for an affine- +null metric defined with respect to a family of outgoing +null hypersurfaces u = const is [16–18, 32] +gabdxadxb = −Wdu2 − 2dudλ ++R2hAB(dxA − W Adu)(dxB − W Bdu). (2.1) +The determinant det(hAB) = det(qAB) = sin2 θ is the +determinant of a round unit sphere metric qAB. +Con- +sequently hAB is transverse-traceless and has only two +degrees of freedom. Thus, the function R relates to the +area of cuts du = dλ = 0. The inverse metric is given by +guλ = −1 , gλλ = W , gλA = −W A , gAB = hAB +R2 , +(2.2) +where W A = (W θ, W φ) and hABhBC = δC +A and in par- +ticular [33] +hABdxAdxB = +� +e2γdθ2 + sin2 θ +e2γ dφ2� +cosh(2δ) ++2 sinθ sinh(2δ)dθdφ . +(2.3) +A complex null dyad to represent the 2-metric hAB like +hAB = m(A ¯mB) with mAmBhAB = mA ¯mBhAB − 1 = 0 +is +mA∂A = +1 +√ +2eγ +� +cosh δ − i sinh δ +� +∂y ++ +ieγ +√ +2 sin θ +� +cosh δ + i sinh δ +� +∂φ, +(2.4) +Like in any Bondi-Sachs type metric [9], the vacuum field +equations Rab = 0 with Rab being the Ricci tensor can +be grouped into supplementary equations Si = 0 with +Si = (Ruu, Ruθ, Ruφ), +(2.5) +one trivial equation Ruλ = 0 and the six main equations +H(γ) +K += 0, K ∈ (1, 2, 3, 4)) and H(δ) +k += 0, k ∈ (1, 2)) with +H(γ) +K += +� +Rλλ, Rλθ, hABRAB, ℜe(mAmBRAB) +� +, +H(δ) +k += +� +Rλφ, ℑm(mAmBRAB) +� +, +(2.6) +with ℜe(x) and ℑm(x) the real an imaginary part of x re- +spectively. We assume that the spacetime is axisymmet- +ric and stationary with associated Killing vectors fields + +3 +∂u and ∂φ. Therefore the metric functions do not depend +on u and φ. The Killing symmetries imply two conserved +quantities, the Komar mass, Km, and the Komar angu- +lar momentum, KL, which can be calculated from their +respective integrals (also see App. B) +Km := K(∂u) = 1 +8π lim +λ→∞ +� � +W,λ−R2hABW AW B +,λ +� +R2d2q +(2.7) +while for the axial Killing vector we have +KL := K(∂φ) = − 1 +16π lim +λ→∞ +� � +R4hφBW B +,λ +� +d2q +(2.8) +where dq = sin θdθdφ is the surface area element of the +unit sphere. +Let us assume there is a smooth one parameter family +of stationary and axially symmetric metrics gab(ε), where +ε is a small dimensionless parameter such that ε = 0 is a +corresponds to a (static) spherically symmetric spacetime +solution of the vacuum Einstein equations. Then there is +an expansion of the metric fields like +R(λ, θ) = r(λ) + R[1](λ, θ)ε + R[2](λ, θ)ε2 + R[3](λ, θ)ε3 + O(ε4), +(2.9a) +W(λ, θ) = V (λ) + W[1](λ, θ)ε + W[2](λ, θ)ε2 + W[3](λ, θ)ε3 + O(ε4), +(2.9b) +W A(λ, θ) = W A +[1](λ, θ)ε + W A +[2](λ, θ)ε2 + W A +[3](λ, θ)ε3 + O(ε4), +(2.9c) +γ(λ, θ) = γ[1](λ, θ)ε + γ[2](λ, θ)ε2 + γ[3](λ, θ)ε3 + O(ε4), +(2.9d) +δ(λ, θ) = δ[1](λ, θ)ε + δ[2](λ, θ)ε2 + δ[3](λ, θ)ε3 + O(ε4). +(2.9e) +Inserting (2.9) in (2.7) and (2.8) implies Km = O(ε0) and +KL = O(ε). We make the requirements +Km(ε) = Km(−ε) , KL(ε) = −KL(−ε). +(2.10) +These +conditions +imply +that +under +the +change +ε → −ε the sense of rotation is reversed (recall that +K(∂φ) = −K(∂(−φ))). +From the metric (2.1), we see +that the 2-surfaces with u = u0 and λ = λ0, defined +such that R(u0, λ0, θ) =const have the induced metric +R2hABdxAdxB with area 4πR2(u0, λ0). +We assume +that the area of these 2-surfaces is invariant under +the change ε → −ε, which implies that R2 is an even +function of ε. Therefore R is either an even or an odd +function of ε. +However, if R were an odd function, +we had R(ε = 0) = 0, which is a non admissible solu- +tion. +In addition, ds2(∂φ, ∂φ) and ds2(∂θ, ∂θ) must be +independent of the sense of rotation implying that hφφ +and hθθ are even. However, due to the frame dragging +effect ds2(∂θ, ∂φ) must depend on the sense of rotation. +Therefore hθφ is an odd function of ε. +Using similar +arguments, because the Komar angular momentum KL +is an odd function of ε and taking into account (2.8) +and the parity behaviour of hAB and R2, we have that +W θ is even and W φ odd. Similarly, since Km must be a +even function of ε, W must be even in ε. Therefore, +R[2n+1] = W[2n+1] = 0, +(2.11a) +W θ +[2n+1] = 0, +(2.11b) +W φ +[2n] = 0, +(2.11c) +γ[2n+1] = δ[2n] = 0. +(2.11d) +To arrive at the last conditions (2.11d) we have taken +into account the odd parity of hθφ, which gives us +sinh(δ(ε)) = − sinh(δ(−ε)). +Hence, δ must be odd in +ε. Similarly, for hθθ and hφφ be even, γ(ε) must satisfies +e2γ(ε) = e2γ(−ε), which implies that γ is a even function +of ε. +We conclude +R = r + R[2]ε2 + +R[4]ε4 + O(ε6), +(2.12a) +W = V + W[2]ε2 + W[4]ε4 + O(ε6), +(2.12b) +W θ = W θ +[2]ε2 + W θ +[4]ε4 + O(ε4), +(2.12c) +W φ = W φ +[1]ε + W φ +[3]ε3 + O(ε5), +(2.12d) +γ = γ[2]ε2 + γ[4]ε4 + O(ε6), +(2.12e) +δ = δ[1]ε + δ[3]ε3 + O(ε5). +(2.12f) +A similar expansion was made by Hartle [30] in the +derivation of a metric for slowly rotating stars using a +a 3+1 decomposition of the metric. From (2.9) follows +that the Ricci tensor has the expansions +Rab = R[0]ab + R[1]abε + R[2]abε2 + R[3]abε3 + ... (2.13) +In fact, with the notation f[i] ∈ {γ[i], δ[i], R[i], W A +[i], W[i]}, +it turns out for a perturbation at order n > 1 that +S[n]i = ˆSi(f[n]) + s[i](f[m 1 coefficients +while using (3.37) gives us +0 =ℓ(ℓ + 1)Bδ +[1.ℓ] +(3.39) +which implies +Bδ +[1.ℓ] = 0 +: +∀ℓ > 1 . +(3.40) +Furthermore, requiring an asymptotic Bondi frame (a +non-rotating inertial observer at large distances), i.e. +gabdxadxb → −du2 − dλdu + λ2qABdxAdxB +(3.41) +annuls the integration constants, +W φ +[0.1] = Bδ +[0.ℓ] = 0. +(3.42) +From the above requirements, the final solution of the +linear perturbations are +δ[1](y, λ) = 0 , +W φ +[1](y, λ) = − B +3λ3 +P 1 +ℓ (y) +s += − B +3λ3 +y +s, +(3.43) +where we redefined B := Bφ +[3,1] for notational convenience +because it is the only remaining integration constant. +D. +Quadratic perturbations +Using the notation of Sec. III B, the relevant main +equations (i.e. only those containing γ[2], R[2], W y +[2] and +W[2]) for the quadratic perturbations are found to be +0 = ˆS1(W[2], W y +[2]) + B2s2 +2λ6 +� +1 − A +2λ +� +(3.44a) +0 = ˆS2(W[2], W y +[2]) +(3.44b) +0 = ˆH(γ) +1 +(R[2]) +(3.44c) +0 = ˆH(γ) +2 +(R[2], γ[2], W y +[2]) +(3.44d) +0 = ˆH(γ) +3 +(W[2], R[2], γ[2], W y +[2]) − B2s2 +4λ4 +(3.44e) +0 = ˆH(γ) +4 +(γ[2], W y +[2]) + B2s2 +4λ4 +(3.44f) +The first hypersurface equation (3.44c) is readily inte- +grated +R[2] = CR20(y) + CR11(y)r. +(3.45) +Similarily to (3.9b), we can deduce a master equation for +γ[2] +0 =M(γ[2]) − s2R2,λλyy − 5B2 +2λ5 s2. +(3.46) +For finding a solution of the remaining fields γ[2], W y +[2] +and W[2], we need to solve the master equation (3.46). +Defining +ψ[2] = (λγ[2]),λλ +(3.47) +with Legendre decomposition +ψ[2] = ψ[2.ℓ](λ)P 2 +ℓ (y) +(3.48) +while using (3.44c) gives us after insertion of (3.45), +(3.47) and (3.48) into (3.46) +0 = +� +−1 +2r(A − 2λ)d2ψ[2.ℓ] +dλ2 ++ +� +4r − A +2 +� dψ[2.ℓ] +dy ++ +� +2 − ℓ(ℓ + 1) + A +2λ +� +ψ[2.ℓ] +� +P 2 +ℓ − 5B2 +2λ5 s2 +(3.49) +To fully factor out the Legendre polynomials P 2 +ℓ , we re- +call that P 2 +2 (y) = 3s2. This allows us to write +0 = +�� +−1 +2r(A − 2λ)d2ψ[2.ℓ] +dλ2 ++ +� +4r − A +2 +� dψ[2.ℓ] +dy ++ +� +2 − ℓ(ℓ + 1) + A +2λ +� +ψ[2.ℓ] +� +δℓ′ +ℓ − 5B2 +6λ5 δℓ′ +2 +� +P 2 +ℓ′(y) +(3.50) +We can see that (3.50) resembles (3.21) if B = 0. It +is in fact a inhomogeneous version of (3.21). We seek +solutions of (3.50) as a superposition of a homogeneous +solution, ψ(hom) +[2.ℓ] +for B = 0, and a particular solution +ψ(part) +[2.ℓ] +for B ̸= 0, i.e. +ψ[2.ℓ] = ψ(hom) +[2.ℓ] ++ ψ(part) +[2.ℓ] +. +(3.51) + +8 +The homogeneous solution ψ(hom) +[2.ℓ] +will be like (3.25). +Also note that a particular solution needs to be found +for the ℓ = 2 mode, only. We find ψ(part) +[2.2] += −B2/(9Aλ4). +Hence, +ψ[2.ℓ](λ) =A +� +C[1.ℓ]P 2 +ℓ +� 4λ +A − 1 +� ++ C[2.ℓ]Q2 +ℓ +� 4λ +A − 1 +� +2A − 4λ +� ++ +� +− B2 +9Aλ4 +� +δ2 +ℓ. +(3.52) +It follows by the same regularity arguments like in the +discussion for (3.25) that in order the Weyl curvature +scalar Ψ0 does not blow up at the horizon of the un- +perturbed solution and towards null infinity we must set +C[1.ℓ] = C[2.ℓ] = 0. +Consequently a solution for the ψ[2.ℓ]–modes is +ψ[2.ℓ](λ) = +� +− B2 +9Aλ4 +� +δ2 +ℓ . +(3.53) +Setting +γ[2](λ, y) = γ[2.ℓ](λ)P 2 +ℓ (y), +(3.54) +we find after integration of (3.47) +γ[2.ℓ](λ, y) = Cγ +[0.ℓ] + +Cγ +[1.ℓ] +λ +− +B2 +54Aλ3 δ2 +ℓ +(3.55) +Insertion of (3.55) and (3.45) into (3.44d) gives us +0 = +(λ4W y +[2],r) +2sλ2 ++ sCR20,y +λ2 ++ +�dγ[2.ℓ] +dλ +� 1 +s +d +dy +� +s2P 2 +ℓ +� +. +(3.56) +using (A5) we find +0 = +� +λ4 W y +[2],r +s +� +,λ ++ 2sCR20,y − 2λ2Kℓ +�dγ[2.ℓ] +dλ +� +P 1 +ℓ (y) +(3.57) +which indicates that the angular behaviour of W y +[2]/s and +sCR20,y are dictated by the associated Legendre polyno- +mials P 1 +ℓ (y). As of (A7), we set (note Pℓ(y) = P 0 +ℓ (y)) +R[2](λ, y) =R[2.ℓ](λ)Pℓ(y) = +� +CR +[20.ℓ] + CR +[21.ℓ]λ +� +Pℓ(y), +(3.58) +W y +[2](λ, y) =W y +[2.ℓ](λ)s(y)P 1 +ℓ (y) +(3.59) +This gives us +0 = d +dλ +� +λ4 d +dλW y +[2.ℓ] +� +− 2CR +[20.ℓ] − 2λ2Kℓ +�dγ[2.ℓ] +dλ +� +, +(3.60) +Integrating (3.60) yields +W y +[2.ℓ] =Cy +[0.ℓ] + +KℓCγ +[1.ℓ] − CR +[20.ℓ] +λ2 +− +Cy +[3.ℓ] +3λ3 − +B2 +9Aλ4 δ2 +ℓ +(3.61) +where we set the integration constants Cy +[0.0] = Cy +[3.0] = 0, +because P 1 +0 (y) = 0. +Considering (3.7d) with (3.54), +(3.59), (A8) and s2 = P 2 +ℓ (y)/3 gives us +� +λ2 +� +1 − A +2λ +� +γ[2.ℓ],r +� +,λ += 1 +2 +� +λ2W y +[2.ℓ] +� +,λ + B2 +12λ4 δ2 +ℓ +(3.62) +so that after insertion of (3.55) and (3.61), we obtain +λCy +[0.ℓ] = +A +2λ2 +� +Cγ +[1.ℓ] − +Cy +[3.ℓ] +3A +� ++ +B2 +9Aλ3 +� +1 + Kℓ +4 +� +δ2 +ℓ +(3.63) +implying for any ℓ ≥ 2 +Cy +[0.ℓ] = 0 , Cy +[3.ℓ] = 3ACγ +[1.ℓ] +(3.64) +Next, proceed with the hypersurface equation (3.44e) for +W[2]. Insertion of (3.54), (3.58) and (3.59) into (3.44e) +gives us +(λW[2]),λ = +� +− +�� +1 − A +2λ +� +(λR[2.ℓ]),λ +� +,λ ++ ℓ(ℓ + 1) +� +R[2.ℓ] +λ ++ +(λ4W y +[2.ℓ]),λ +2λ2 +− Kℓγ[2,ℓ] +�� +P 0 +ℓ (y) +− B2s2 +4λ4 +(3.65) +Using +s2 = 1 − y2 = 2 +3[P 0 +ℓ (y) − P 0 +2 (y)] +(3.66) +as well as setting +W[2](λ, y) = W[2.ℓ](λ)P 0 +ℓ (y) +(3.67) +yields +(λW[2.ℓ]),λ = − +�� +1 − A +2λ +� +(λR[2.ℓ]),λ +� +,λ ++ ℓ(ℓ + 1) +� +R[2.ℓ] +λ ++ +(λ4W y +[2.ℓ]),λ +2λ2 +− Kℓγ[2,ℓ] +� +− B2 +6λ4 (δ0 +ℓ − δ2 +ℓ) +(3.68) +Since R[2.ℓ], W y +[2.ℓ] and γ[2.ℓ] are known, we find after +integration +W[2.ℓ] = −KℓCR +[21.ℓ] − ℓ(ℓ + 1)KℓCγ +[0.ℓ] + +CW +[1.ℓ] +λ ++ +ACR +[20.ℓ] +2λ2 ++ +ℓ(ℓ + 1)Cy +[3.ℓ] +6λ2 ++ +� 2B2 +9Aλ3 − B2 +18λ4 +� +δ2 +ℓ + B2 +18λ4 δ0 +ℓ +(3.69) + +9 +where CW +[1.ℓ] are integration constants. +Calculation of (3.44a) and (3.44b) while using (3.58), +(3.59), (3.66), (A1) and (A10) gives us +0 = +� +1 − A +2λ +� � +λ2W[2.ℓ],r + AR[2.ℓ] +λ +� +,λ +− ℓ(ℓ + 1) +� +W[2.ℓ] + A +2 W y +[2,ℓ] +� +(3.70) +0 = +1 +2λ2 +� +1 − A +2λ +� +(λ4W y +[2.ℓ]),λ − 1 +2W[2.ℓ],r + W y +[2.ℓ] +(3.71) +and insertion of the respective coefficient solutions +(3.58),(3.61) and (3.69) yields +0 = ℓ(ℓ + 1) +� +− +CW +[1.ℓ] +λ ++ Kℓ +� +Cγ +[1.ℓ] − CR +[21.ℓ] +� ++ +Cy +[3.ℓ] − 3ACγ +[1.ℓ] +6λ2 +� +(3.72) +0 = +CW +[1.ℓ] +2λ2 + +(Cy +[3.ℓ] − 3ACγ +[1.ℓ])Kℓ +6λ3 +, +∀ℓ ≥ 1 +(3.73) +Therefore, +CW +[1.ℓ] =0 , ∀ℓ ≥ 1 +(3.74a) +Cy +[3.ℓ] =3ACγ +[1.ℓ] , ∀ℓ ≥ 2 +(3.74b) +CR +[21.ℓ] =Cγ +[1.ℓ] , ∀ℓ ≥ 2 +(3.74c) +Note, (3.74b) is consistent with (3.64). The requirement +of an asymptotic inertial observer leads to +Cγ +[0.ℓ] = CR +[21.ℓ] = 0 +(3.75) +which gives with (3.74) that Cγ +[1.ℓ] = Cy +[3.ℓ] = 0. Thus, +redefining C := CW +[1.1], the quadratic perturbations are +γ[2](λ, y) = +� +− +B2 +54Aλ3 δ2 +ℓ +� +P 2 +ℓ (y) +(3.76) +R[2](λ, y) = 0 +(3.77) +W y +2 (λ, y) = +� +− B2 +9Aλ4 δ2 +ℓ +� +s(y)P 1 +ℓ (y) +(3.78) +W[2](λ, y) = C +λ + B2 +18λ4 + +� 2B2 +9Aλ3 − B2 +18λ4 +� +P 0 +2 (y) . +(3.79) +E. +Third order perturbations +Similarly, expressions for the higher order perturba- +tions quantities f[i] can be obtained using the same pro- +cedere as in the previous sections. In this and in the next +subsection we show the fundamental results without re- +peating intermediate steps. +The relevant equations for the third perturbations are +0 = ˆS3(δ[3], W φ +[3]) − B3s4 +6Aλ6 +(3.80) +0 = ˆH(δ) +1 (δ[3], W φ +[3]) − B3s4 +6Aλ6 +(3.81) +0 = ˆH(δ) +2 (δ[3], W φ +[3]) + 2B3ys2 +3Aλ5 +(3.82) +Similarily to (3.9b), we can deduce a master equation +for δ[3] +0 =M(δ[3]) − 40B3 +3Aλ6 s2y +(3.83a) +Using P 2 +3 (y) = 15ys2 and following the steps of Sec. III C, +we find +δ[3](λ, y) = +� +− +B3 +162A2λ4 +� +P 2 +3 (y) +(3.84) +W[3](λ, y) = +� +− D +3λ3 − +2B3 +135Aλ6 +� P 1 +1 (y) +s(y) ++ +� +B3 +405Aλ6 − +4B3 +81A2λ5 +� P 1 +3 (y) +s(y) +(3.85) +where D is the only free new remaining integration con- +stant that appears at this order. +F. +Fourth order perturbations +Here the relevant main equations are those containing +γ[4], R[4], W y +[4] and W[4] which are +0 = ˆS1(W[4], W y +4]) + +� 14 +9Aλ − +1 +12λ2 − 35 +6A2 +� B4s4 +3λ8 ++ +�� +1 − A +2λ +� +D − CB +2A + +� 16 +Aλ2 − +7 +3λ3 +� B3 +9A +� Bs2 +λ6 +− +8B4 +27λ8A2 + CB2 +3Aλ6 +(3.86a) +0 = ˆS2(W[4], W y +[4]) + +�(7A + 120λ)s2 +12λ +− 8 +� B4ys +9A2λ7 ++ 2ysCB2 +3Aλ5 +(3.86b) +0 = ˆH(γ) +1 +(R[4]) − +B4s4 +18A2λ8 +(3.86c) +0 = ˆH(γ) +2 +(R[2], γ[2], W y +[2]) + B4ys3 +27A2λ7 +(3.86d) +0 = ˆH(γ) +3 +(W[2], R[2], γ[2], W y +[2]) + (A − 14r)B4s4 +36A2λ7 ++ 2B4s2 +9A2λ6 + DBs4 +2λ4 +(3.86e) +0 = ˆH(γ) +4 +(γ[2], W y +[2]) + +� +14 + A +2r +� B4s4 +9A2λ6 ++ +� B2C +2Aλ4 − BD +2λ4 − +38B2 +27A2λ6 +� +s2 +(3.86f) + +10 +The first hypersurface equation (3.86c) is readily inte- +grated +R[4](λ, y) =ER0(y) + ER1(y)λ − +s4B4 +1080A2λ5 +(3.87) +or expressing in terms of the Legendre polynomials P 0 +ℓ (y) +R[4](λ, y) = +� +ER +[0.ℓ] + ER +[1.ℓ]λ +� +P 0 +ℓ (y) +− +B4 +135A2λ5 +�P 0 +0 (y) +15 +− 2P 0 +2 (y) +21 ++ P 0 +4 (y) +35 +� +(3.88) +Similarily to (3.9a) we can deduce a master equation +for γ[4] +0 =M(γ[4]) − s2R[4],λλyy − 5B2Cs2 +Aλ5 +− 5BDs2 +λ5 ++ +� +358 − s2 +� +397 + A +λ +�� B4s2 +9A2λ7 +(3.89a) +Using the methods of Sec. III D together with the in- +verted Legendre relations +1 =P 0 +0 (y) = −P 1 +1 (y) +s +(3.90a) +y =P 0 +1 (y) = −P 1 +2 (y) +3s +(3.90b) +y2 =1 +3 − 2 +3P 0 +2 (y) = 1 − 1 +3P 2 +2 (y) +(3.90c) +y3 = − 2P 1 +4 (y) +35s ++ P 1 +2 (y) +7s +(3.90d) +y4 =1 +5 − 4P 0 +2 (y) +7 ++ 8P 0 +4 (y) +35 += 1 − 8 +21P 2 +2 (y) − +2 +105P 2 +4 (y) +(3.90e) +we deduce the following solution for the fourth order per- +turbation +R[4] = − +B4 +135A2λ5 +�P 0 +0 +15 − 2P 0 +2 +21 + P 0 +4 +35 +� +(3.91a) +γ[4] = +� BD +27Aλ3 − +B2C +27A2λ3 − +B4 +1134A2λ6 +� +P 2 +2 ++ +� +B4 +405A3λ5 + +B4 +17010A2λ6 +� +P 2 +4 +(3.91b) +W y +[4] = +� 2BD +9Aλ4 − 2B2C +9A2λ4 − +2B4 +2835A2λ7 +� +P 1 +2 ++ +� +2B4 +81A3λ6 + +B4 +4725A2λ7 +� +P 1 +4 +(3.91c) +W[4] =E +λ − BD +9λ4 + +4B4 +405A2λ6 − +B4 +675Aλ7 ++ +� +2B4 +945Aλ7 − +2B4 +81A2λ6 + 4B2C +9A2λ3 − 4BD +9Aλ3 + BD +9λ4 +� +P 0 +2 ++ +� +− +8B4 +81A3λ5 + +2B4 +135A2λ6 − +B4 +1575Aλ7 +� +P 0 +4 +(3.91d) +Note that E is the only remaining new integration con- +stant, all other vanish because of the reasons mentioned +in Sec. III D. +G. +Perturbations in terms of Komar quantities +The solution of the perturbation involve the free inte- +gration constants A, B, C, D and E. These free constants +determine the Komar mass, Km, and the Komar angular +momentum, KL, which can be found by calculation of +(2.7) and (2.8) +m := Km = A +4 − C +2 ε2 − E +2 ε4 + O(ε5) +(3.92) +L := KL = −B +6 ε + D +6 ε3 + O(ε5) +(3.93) +If ε = 0, Km = A/4 corresponds to the mass m0 of the +unperturbed system. Furthermore, we can see that L = +O(ε). This allows us relate ε with the angular momentum +L of the system. To do that we have to solve the cubic +equation +0 = D +6 ε3 − B +6 ε + L +(3.94) +for ε. This equation also shows that in order to make the +substitution of ε by L, we seek the solution ε(L) = O(L). +The root of (3.94) which fulfils this requirement is +ε = − 6 +B L − 216D +B4 L3 + O(L5) +(3.95) +Subsequent insertion of this expansion into (3.92) and +solving for A gives us +A = 4m + 72C +B2 L2 + 2592EB − 2CD +B5 +L4 + O(L6) (3.96) +The relations (3.95) and (3.96) allow us to substitute +A and ε, by the physical quantities m and L. Insertion +of (3.95) and (3.96) into the solution of the perturba- +tions and subsequent expansion up to O(L4) allows us to +eliminate the integration constants C, D and E from the +perturbations. This gives us + +11 +R(λ, y) = λ − +�P 0 +0 +5 − 2P 0 +2 +7 ++ 3P 0 +4 +35 +� +L4 +5m2λ5 + O(L6) +(3.97a) +W(λ, y) = 1 − 2m +λ + +� 2 +λ4 + +� +2 +mλ3 − 2 +λ4 +� +P 0 +2 +� +L2 + +� +4 +5m2λ6 − +12 +25λ7 + +� +24 +35mλ7 − 2L4 +m2λ6 +� +P 0 +2 ++ +� +− +2 +m3λ5 + +6 +5m2λ6 − +36 +175mλ7 +� +P 0 +4 +� +L4 + O(L6) +(3.97b) +W y(λ, y) = +� +− +1 +mλ4 (sP 1 +2 ) +� +L2 + +� +− +2 +35m2λ7 (sP 1 +2 ) + +� +1 +2m3λ6 + +3 +175m2λ7 +� +(sP 1 +4 ) +� +L4 + O(L6) +(3.97c) +W φ(λ, y) = +� +− 2 +λ3 +P 1 +1 +s +� +L + +� +4 +5mλ6 +P 1 +1 +s + +� +2 +3m2λ5 − +2 +15mλ6 +� P 1 +3 +s +� +L3 + O(L5) +(3.97d) +γ(λ, y) = +� +− +1 +6mλ3 P 2 +2 +� +L2 + +� +− +1 +14m2λ6 P 2 +2 + +� +1 +20m3λ5 + +1 +210m2λ6 +� +P 2 +4 +� +L4 + O(L6) +(3.97e) +δ(λ, y) = +� +1 +12m2λ4 P 2 +3 +� +L3 + O(L5) +(3.97f) +We see in (3.97) that the perturbations are determined by the mass and angular momentum, i.e. the solution has two +hairs. To show that this solutions represents the Kerr solution in affine-null coordinates, we introduce the specific +angular momentum, a := L/m. In terms of a, (3.97) read after changing to the angular coordinate θ +R(λ, θ) = λ − 3m2 sin4 θ +40λ5 +a4 + O(a6) +(3.98a) +W(λ, θ) = 1 − 2m +λ + +�2m +λ3 + +� +−3m +λ3 + 3m2 +λ4 +� +sin2 θ +� +a2 ++ +� +−2m +λ5 + +�10m +λ5 +− 3m2 +λ6 +� +sin2 θ + +� +−35m +4λ5 + 21m2 +4λ6 − 9m3 +10λ7 +� +sin4 θ +� +a4 + O(a6) +(3.98b) +W θ(λ, θ) = +� +−3m +λ4 a2 + +�5m +λ6 − +�35m +4λ6 + 3m2 +10λ7 +� +sin2 θ +� +a4 +� +sin θ cos θ + O(a6) +(3.98c) +W φ(λ, θ) = 2m +λ3 a + +� +−4m +λ5 + +�5m +λ5 − m2 +λ6 +� +sin2 θ +� +a3 + O(a5) +(3.98d) +γ(λ, θ) = +� +−m sin2 θ +2λ3 +� +a2 + +�9m sin2 θ +4λ5 ++ +� +−21m +8λ5 − m2 +4λ6 +� +sin4 θ +� +a4 + O(a6) +(3.98e) +δ(λ, θ) = −5m cosθ sin2 θ +4λ4 +a3 + O(a5) +(3.98f) +Comparing with [28], we find agreement for R which corresponds to their areal coordinate r. Calculation of the metric + +12 +components gab using (3.98) gives us +guu(λ, θ) = −1 + 2m +λ + +��3m +λ3 + m2 +λ4 +� +sin2 θ − 2m +λ3 +� +a2 ++ +�2m +λ5 − +�10m +λ5 ++ 4m2 +λ6 +� +sin2 θ + +�35m +4λ5 + 23m2 +4λ6 + 9m3 +10λ7 +� +sin4 θ +� +a4 + O(a6) +(3.99a) +guλ(λ, θ) = −1 +(3.99b) +guθ(λ, θ) = +��3m +λ2 +� +a2 + +� +−5m +λ4 + +�35m +λ4 ++ 23m2 +10λ5 +� +sin2 θ +� +a4 +� +sin θ cos θ + O(a6) +(3.99c) +guφ(λ, θ) = +�� +−2m +λ +� +a + +�4m +λ3 − +�5m +λ3 + m2 +λ4 +� +sin2 θ +� +a3 +� +sin2 θ + O(a5) +(3.99d) +gθθ(λ, θ) = λ2 + +� +−m sin2 θ +λ +� +a2 + +� 9m +2λ3 sin2 θ − +�21m +4λ3 + 3m2 +20λ4 +� +sin4 θ +� +a4 + O(a6) +(3.99e) +gθφ(λ, θ) = +� +−5m sin3 θ cos θ +2λ2 +� +a3 + O(a5) +(3.99f) +gφφ(λ, θ) = +� +λ2 + +�m sin2 θ +λ +� +a2 + +� +− 9m +2λ3 sin2 θ + +�21m +4λ3 + 17m2 +20λ4 +� +sin4 θ +� +a4 +� +sin2 θ + O(a6) +(3.99g) +Eqs.(3.99) constitute our final expression for the slowly +rotating stationary and axially symmetric (Kerr) metric +adapted to null coordinates which asymptotically match +an inertial Bondi frame. At difference of all previous ap- +proaches, it was obtained as an explicit solution of the +Einstein equations. After comparison with [28], we find +agreement up to a typo in their equation for gθφ. We also +note care should be taken when comparing [28]’s expres- +sions with ours. First, [28] present a Bondi-Sachs form of +the metric, while we have an affine-null metric approach- +ing a Bondi frame, the difference is in the choice of radial +coordinate, and the two agree only up to O(λ−4) with one +another. Second, [28] make a large λ expansion while we +make a small a expansion, this results in powers of λ−k +absorbed by order symbols in [28]. A slowly rotating ver- +sion of the Kerr metric in null affine coordinates at second +order in a was also obtained by Dozmorov who made a +null tetrad rotations starting with the Kerr metric as ex- +pressed in Boyer-Lindquist coordinates [35]. In the next +section we show an alternative procedure to recover the +slowly rotating Kerr metric components as expressed in +(3.99) by doing appropriate coordinates transformations. +IV. +APPROXIMATED AFFINE-NULL METRIC +DERIVED FROM THE KERR-METRIC +Here, starting with the Kerr metric expressed in Boyer- +Lindquist coordinates (BL) {ˆt, ˆr, ˆθ, ˆφ}, we present an +explicit transformation to affine-null coordinates up to +fourth order in a. The Kerr metric in BL coordinates +reads: +ds2 = gˆtˆtdˆt2+gˆtˆφdˆtdˆφ+gˆrˆrdˆr2+gˆrˆrdˆr2+gˆθˆθdˆθ2+g ˆφˆφdˆφ2; +(4.1) +with +gˆtˆt = − +� +1 − 2mˆr +Σ +� +, +(4.2) +gˆt ˆφ = −2maˆr sin2 ˆθ +Σ +, +(4.3) +gˆrˆr = Σ +∆, +(4.4) +gˆθˆθ = Σ, +(4.5) +g ˆφˆφ = +� +ˆr2 + a2 + 2ma2ˆr sin2 ˆθ +Σ +� +sin2 ˆθ, +(4.6) +with ∆ = ˆr2 − 2mˆr + a2 and Σ = ˆr2 + a2 cos2 ˆθ. The u +null coordinate must satisfy the eikonal equation, +gab∇au∇bu = 0, +(4.7) +Inspired by [28], we propose the following expansion for +u, +u = ˆt − ˆr − 2m ln +� ˆr − 2m +2m +� ++ +∞ +� +i=1 +fi(ˆr, ˆθ)ai. +(4.8) +Note that for a = 0 this expression reduces to the stan- +dard outgoing Schwarzschild null coordinate. By replac- +ing (4.8) into (4.7), we obtain a set of differential equa- +tions for fi(ˆr, ˆθ) that can be solved iteratively. Conserv- +ing terms up to fourth order in a we find that only the +even coefficients f2n(ˆr, ˆθ) are non–vanishing with: + +13 +f2(ˆr, ˆθ) = +5ˆr − 2m +4ˆr(2m − ˆr) + cos 2ˆθ +4ˆr +− ln(1 − 2m +ˆr ) +2m +,(4.9) +f4(ˆr, ˆθ) = (2ˆr + m) +16ˆr4 +sin4(2ˆθ) − 3 ln(1 − 2m +ˆr ) +8m3 +−4m2 − 9mˆr + 3ˆr2 +4m2ˆr(ˆr − 2m)2 , +(4.10) +Similarly, affine-null coordinates {λ, θ, φ} can be ob- +tained from the requirements +gab∇au∇bλ = −1, +(4.11a) +gab∇au∇bθ = gab∇au∇bφ = 0, +(4.11b) +by assuming relations of the form: +λ = ˆr + +∞ +� +i=1 +ˆΛi(ˆθ, ˆr)ai, +(4.12) +θ = ˆθ + +∞ +� +i=1 +ˆΘi(ˆθ, ˆr)ai, +(4.13) +φ = ˆφ + +∞ +� +i=1 +ˆΦi(ˆθ, ˆr)ai, +(4.14) +and replacing into the set (4.11), the coefficients functions +ˆΛi, ˆΘi, ˆΦi can be obtained in the same way as u. After +that, the resulting relations can be inverted in order to +express the BL coordinates in terms of the affine-null +coordinates. Following these steps up to fourth order, +the final transformation coordinates reads: +ˆt = u + λ + 2m ln( λ +2m − 1) + +�ln(1 − 2m +λ ) +2m ++ 3 m cos(2 θ) + 4 λ − 3 m +(4 λ − 8 m) λ +� +a2 ++ +� +−m +� +175 λ2 − 224 mλ − 72 m2� +(cos (2 θ))2 +320 (λ − 2 m)2 λ4 +− m +� +25 λ2 + 64 mλ + 72 m2� +cos (2 θ) +160 (λ − 2 m)2 λ4 ++3 ln +� +1 − 2 m +λ +� +8 m3 ++ 240 λ5 − 720 λ4m + 320 λ3m2 + 225 λ2m3 − 96 λ m4 + 72 m5 +320 m2λ4 (λ − 2 m)2 +� +a4 + O(a6) +(4.15) +ˆr = λ − (λ + m) sin2 θ +2λ2 +a2 + +�sin2 θ(5 cos 2θ + 3)) +16λ3 ++ m sin2 θ(7 cos 2θ + 1) +16λ4 +− m2 sin4 θ +5λ5 +� +a4 + O(a6) +(4.16) +ˆθ = θ − sin (2 θ) +4 λ2 +a2 + sin (2 θ) (3 λ cos (2 θ) + m cos(2 θ) − m) +16 λ5 +a4 + O(a6) +(4.17) +ˆφ = φ + +� 1 +λ + ln(1 − 2m +λ ) +2m +� +a + +�ln(1 − 2m +λ ) +4m ++ m(2m + 5λ) cos 2θ +8(λ − 2m)λ4 +−6m4 − m3λ + 8m2λ2 + 12mλ3 − 12λ4 +24m2(λ − 2m)λ4 +� +a3 + O(a5) +(4.18) +Finally, with these transformations in hand, we obtain +the same metric components in affine-null coordinates +up to fourth order in a as given in (3.99) in the previous +Section. +V. +LOCALIZING THE EVENT HORIZON AND +ERGOSPHERE IN AFFINE-NULL CORDINATES +In this Section we show that the affine-null coordinates +for the slowly rotating Kerr metric cover the ergosphere +and the (past) event horizon r+. In order to find them in +a consistent way, they must be localized at O(a4). Recall +that in BL coordinates the Kerr metric has the external + +14 +ergosphere placed at +ˆrerg =m + +� +m2 − a2 cos2 ˆθ +=2m − a2 cos2 ˆθ +m +− a4 cos4 ˆθ +8m3 ++ O(a6), +(5.1) +and the event horizon at +r+ = m + +� +m2 − a2 = 2m − a2 +2m − a4 +8m3 + O(a6). (5.2) +The boundary of the external ergosphere is obtained by +looking for the timelike surface Γ where the stationary +Killing vector field ∂u becomes a null vector field that is +where +guu|Γ = 0. +(5.3) +Taking into account the expression for guu as found in the +first equation of (3.99), the ergosphere will be located at +a given λ = λerg(θ), with +λerg(θ) = +2 +� +i=0 +λerg[2i](θ)a2i + O(a6). +(5.4) +where the even expansion is a consequence of the symme- +try assumption of Sec. II. Inserting (5.4) into (5.3), and +after re-expanding in powers of a we find +λerg(θ) =2 m − +� +7 cos2 θ − 3 +� +8m +a2 +− +� +51 cos4 θ − 2 cos2 θ + 31 +� +640 m3 +a4 + O(a6), +(5.5) +which gives the location of the (external) ergosphere in +affine-null coordinates. +By replacing (5.5) into (4.16), +and after a re-expansion in powers of a it can be checked +that the standard fourth order expression for the BL ex- +pression of the ergosphere as given by (5.1) is recovered. +Similarly, for the (Killing) event horizon we search +a null surface Σ described in affine-null coordinates by +Σ(λ, θ) = λ − λH(θ) = 0. +Hence, its normal vector +Na = ∇aΣ must satisfy N aNa = 0 which implies the +following differential equation for λH(θ) = 0, +gabNaNb = W + 2W θ ∂λH(θ) +∂θ ++ hφφ +R2 +�∂λH(θ) +∂θ +�2 += 0. +(5.6) +Let us assume an expansion for λH(θ) similar to (5.4), +i.e. +λH(θ) = +2 +� +i=0 +λH[2i](θ)a2i + O(a6); +(5.7) +with λH[0] = 2m (the Schwarzschild value for the loca- +tion of the horizon). +Introducing (5.7) into (5.6); re- +expanding again in powers of a, we find (omitting the +O(a6) term) +0 = +�λH[2] +2m + 3 cos2 θ + 1 +16m2 +� +a2 ++ +� +(λH[2],θ)2 +4m2 +− 3 sin θ cos θ +8m3 +λH[2],θ − +λ2 +H[2] +4m2 − 3(cos2 θ + 1) +16m3 +λH[2] − 127 cos4 θ − 320m3λH[4] − 84 cos2 θ − 3 +640m4 +� +a4. +(5.8) +So that solving for the coefficient λH[2] and λH[4] gives +us +λH(θ) =2m − (1 + 3 cos2 θ) +8m +a2 ++ (29 cos4 θ − 78 cos2 θ − 31) +640m3 +a4 + O(a6), +(5.9) +which gives the location of the (past) event horizon in +affine-null coordinates. By replacing into (4.16) and af- +ter a reexpansion in a up to fourth order, the well known +value (5.2) for the BL radial coordinate of the event hori- +zon is recovered. At this location, the resulting compo- +nents of the metric are regular. +VI. +SUMMARY +We have derived high-order slow rotation approxima- +tion of the Kerr metric in affine-null coordinates. +To +achieve this aim a metric in affine-null coordinates was +expanded off a spherically symmetric background metric +that corresponds to a Schwarzschild metric in outgoing +Eddington Finkelstein coordinates. This quasi-spherical +expansion was done with respect to a general smallness +parameter ε. Subject to stationarity and axial symmetry + +15 +the perturbations did not depend on the u and φ coor- +dinate. +Moreover, requiring even parity of the Komar +integral of stationary (giving the mass of the system) +and odd parity of the Komar integral of axial symme- +try (giving the angular momentum of the system), we +argued that on the one hand the metric functions γ, R, +W θ and W have only even perturbations in ε while on +the other hand the metric fields δ and W φ have only odd +perturbations in ε. This fact significantly simplifies the +integration of the perturbation equations resulting form +the quasi-spherical expansion of the Ricci tensor. In ad- +dition, we find that the integration of the perturbation +equations follows an alternative hierarchical structure. +Meaning with the spherically symmetric background so- +lution at hand, the linear perturbations only involve the +functions δ and W φ and its integration provides (after +application of the boundary condition of an asymptotic +inertial observer) one free integration constant B. +At +next order, the quadratic perturbations turn out to be +a linear combination of the derivatives of functions γ, +R, W θ and W together with nonlinear terms containing +the integration constants A of the background model and +the free integration constant B of the linear perturbation. +Their integration also yields a free integration constant, +C. Following up the next order, there only differential +equations involving the cubic perturbations of δ and W φ +as well as the integration constants A, B and C charac- +terizing the lower order perturbations. This alternative +scheme between the perturbations of (δ, W φ) and those of +(γ, R, W θ, W) continues up to any order and is in fact a +result of the symmetry assumptions. A common feature +in solving for the even and odd-parity modes of ε, is that +at any order there is a fourth order master equation for ei- +ther the perturbation in γ or the perturbation in δ. With +the solution of this master equation, the remaining per- +turbations can be solve by mere integration. After having +obtained the perturbed solution and calculation of the +Komar mass and Komar angular momentum, the arising +free integration constants A, B, C, ... can expressed by the +Komar mass and Komar angular momentum or by mass +and specific angular momentum. Hence, the solution de- +pends only on two free physical parameters. Since the +Komar angular momentum is O(ε), it turns out that the +formal expansion parameter ε relates to the specific angu- +lar momentum and the previously made quasi-spherical +approximation is in fact a slow rotation approximation, +like those of Hartle and Thorne [30, 36]. By successively +solving Einstein equations, we thus derived a slow rota- +tion approximation of the Kerr-metric up to fourth order +in the specific angular momentum. This solution is fur- +ther verified for correctness using a ’standard’ approach +by obtaining a different representation of a given metric +in another coordinate chart via a coordinate transforma- +tion. The slowly rotating Kerr metric presented here also +obeys the peeling property, which can be seen considering +the Weyl scalars in (6.1) +Ψ0 = +�3ma2 +λ5 ++ i15ma3 +λ6 +cos θ +� +sin2 θ + O(λ−7) +(6.1a) +Ψ1 = i3 +√ +2ma +2λ4 +sin θ + O(λ−5) +(6.1b) +Ψ2 = m +λ3 + i3ma cosθ +λ4 ++ O(λ−5) +(6.1c) +Ψ3 = i3 +√ +2ma +4λ4 +sin θ + O(λ−5) +(6.1d) +Ψ4 = 3 +√ +2ma2 +4λ5 +sin2 θ + O(λ−6) +(6.1e) +Moreover it is easily checked that the (only) conserved +Newman Penrose constant [21] vanishes [28]. +What is is interesting to remark is that up to the con- +sidered order of approximation of our work and those of +[28], the small a expansion and the large λ expansion +coincide. It would be interesting to see up until which +order this is the case. Such analysis might give insight +on the validity and universality of general small param- +eter expansions of the Kerr spacetime in relation to null +coordinates. +It may also give insight if a closed form +solution of the Kerr metric with a surface forming null +coordinate can be obtained at all. The method presented +here offers the possibility to calculate any type of approx- +imate rotating null-metric solution that is stationary, ax- +ially symmetric and has a known spherically symmetric +background, like e.g. +those to describe compact mat- +ter systems or with a cosmological constant. Indeed, the +study presented here (solving the characteristic equations +in this affine-null, metric formulation for vacuum space- +times) is the natural starting point for further studying +matter system under the given symmetry assumptions. +Some of such questions we are currently investigating. +Acknowledgements +The authors thanks J. Winicour, L. Lehner, N. Ster- +gioulas, E. M¨uller and G. Dotti for discussions at (early) +stages of the project. T.M acknowledges financial sup- +port from the FONDECYT de iniciaci´on 2019 (Project +No. 11190854) of the ”Agencia Nacional de Investigaci´on +y Desarrollo” in Chile. E.G gratefully acknowledges the +hospitality extended to him during his stay at the Fac- +ultad de Ingenier´ıa, Universidad Diego Portales and the +financial support from CONICET and SeCyT-UNC. +Appendix A: Useful Relations between Legendre +Polynomials +For completeness, we list some properties of the Legen- +dre differential equations and relations between the Leg- +endre polynomials. + +16 +The Legendre differential equation for the Legendre +polynomials Pℓ(y) is +d +dy +� +(1 − y2)dPℓ +dy +� ++ ℓ(ℓ + 1)Pℓ = 0 +(A1) +where the Legendre Polynomials Pℓ(y) are defined +Pℓ(y) = +1 +2ℓℓ! +dℓ +dyℓ (y2 − 1)ℓ +(A2) +The associated Legendre differential equation is +d +dy +� +(1 − y2)dP m +ℓ +dy +� ++ +� +ℓ(ℓ + 1) − +m2 +1 − y2 +� +P m +ℓ += 0 (A3) +where P m +ℓ (y) are the associated Legendre polynomials, +defined via +P m +ℓ (y) =(−)m(1 − y2)m/2 dmPℓ(y) +dxm +=(−)m +2ℓℓ! (1 − y2)m/2 dℓ+m +dyℓ+m (y2 − 1)ℓ +(A4) +which also shows P 0 +ℓ (y) = Pℓ(y). From these definitions, +some useful identities can be derived +d +dy +� +(1 − y2)P 2 +ℓ (y) +� += [ℓ(ℓ+1)−2](1−y2)1/2P 1 +ℓ (y) (A5) +d +dy +� +(1 − y2)2 dP 2 +ℓ +dy +� +1 − y2 +− 2P 2 +ℓ += ℓ(ℓ + 1)(ℓ + 2)(ℓ − 1)Pℓ(y) +(A6) +P 1 +ℓ += −(1 − y2)1/2 dP 0 +ℓ +dy +(A7) +P 2 +ℓ += (1 − y2)d2P 0 +ℓ +dy2 +(A8) +d +dy +� +(1 − y2)2 d +dy +P 1 +ℓ +(1 − y2)1/2 +� += [2−ℓ(ℓ+1)](1−y2)1/2P 1 +ℓ +(A9) +d +dy(1 − y2)1/2P 1 +ℓ = ℓ(ℓ + 1)P 0 +ℓ +(A10) +Appendix B: Komar charges +Depending on the Killing vector Xa ∈ {∂u, ∂φ}, we +take the Komar charges to be +KX = −kX +8π +� +∇[aXb]dΣab +(B1) +with kX = 1, −1/2 for a timelike ( e.g. +∂u) or rota- +tional Killing vector (e.g. ∂φ), respectively. Consider the +general null metric with the nonzero contravariant com- +ponents g01, g11, g1A and gAB. The corresponding line +element is +gabdxadxb = (g11 + gABg1Ag1B) +�dx0 +g01 +�2 ++ 2 +�dx0 +g01 +� +dx1 +− 2gABg1AdxB +�dx0 +g01 +� ++ gABdxAdxB +(B2) +where gACgCB = δB +A. Defining the null vectors +l = −dx0 = −g01∂1 +(B3) +and +n = −1 +2 +g11 +(g01)2 dx0 + dx1 +g01 = ∂0 + 1 +2 +g11 +g01 ∂1 + g1A +g01 ∂A (B4) +which obey lana + 1 = lala = nana = 0, the surface +element follows as +dΣab = 2k[anb] +� +det(gAB)dx2dx3 +(B5) +with xA = (x2, x3) being any angular coordinates for the +units sphere. Setting gAB = R2hAB with hAB having +the determinant of the unit sphere metric qAB, q(xC) := +det(hAB). The corresponding volume element is defined +as d2q := √qdx2dx3 and we have � d2q = 4π. Hence, +dΣab = 2l[anb]R2d2q . +This allows us to write the Komar integal as +K(X) = −kX +8π +� +(2lanb∂[aXb])R2d2q , +(B6) +Since +2lanb∂[aXb] = 2l[anb]Xb,a +(B7) += (lanb − lbna)Xb,a +(B8) += l1(nbXb,1) − l1(nbX1,b) +(B9) += −g01[(nbXb,1) − (nbX1,b)] , (B10) +we have +K(X) = kX +8π +� � +(nbXb,1) − (nbX1,b) +� +g01R2d2q, (B11) +Taking the Killing vector to be X = Xa∂a and specifica- +tion to an affine null metric +g01 = ǫ , g1A = ǫW A , g11 = W , +g0A = −R2hABW B , gAB = R2hAB +(B12) +and ǫ2 = 1 gives us +2lanb∂[aXb] = − +� +W,1 − R2hABW AW B +,1 +� +X0 ++ R2� +hABW B +,1 XA − 2hABW BXA +,1 +� ++ ǫ(X1 +,1 − X0 +,0) − WX0 +,1 − W AX0 +,A +(B13) + +17 +Assuming the timelike Killing vector X = ∂0 gives us +2lanb∂[aXb] = − +� +W,1 − R2hABW AW B +,1 . +� +Thus for the above form of the Killing vector we have the +related Komar charge using kX = 1 +K(∂0) = 1 +8π +� � +− ǫ +� +W,1 − R2hABW AW B +,1 +�� +R2d2q. +(B14) +With the rotational Killing X = ∂3, we have +2lanb∂[aXb] = R2h3BW B +,1 +so that the Komar charge is with kX = − 1 +2 +K(∂3) = − ǫ +16π +� � +R4h3BW B +,1 +� +d2q. +(B15) +[1] H. 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J. 153, 807 +(1968). + diff --git a/9NE4T4oBgHgl3EQfdgy1/content/tmp_files/load_file.txt b/9NE4T4oBgHgl3EQfdgy1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..62d88b4e98d7472afaa84c26f349be768676cc39 --- /dev/null +++ b/9NE4T4oBgHgl3EQfdgy1/content/tmp_files/load_file.txt @@ -0,0 +1,1015 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf,len=1014 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='05092v1 [gr-qc] 12 Jan 2023 , Slowly rotating Kerr metric derived from the Einstein equations in affine-null coordinates Thomas M¨adler ∗ Escuela de Obras Civiles and Instituto de Estudios Astrof´ısicos, Facultad de Ingenier´ıa y Ciencias, Universidad Diego Portales, Avenida Ej´ercito Libertador 441, Casilla 298-V, Santiago, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Emanuel Gallo † FaMAF, UNC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Instituto de F´ısica Enrique Gaviola (IFEG), CONICET, Ciudad Universitaria, (5000) C´ordoba, Argentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Using a quasi-spherical approximation of an affine-null metric adapted to an asymptotic Bondi inertial frame, we present high order approximations of the metric functions in terms of the specific angular momentum for a slowly rotating stationary and axi-symmetric vacuum spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' The metric is obtained by following the procedure of integrating the hierarchy of Einstein equations in a characteristic formulation utilizing master functions for the perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' It is further verified its equivalence with the Kerr metric in the slowly rotation approximation by carrying out an explicit transformation between the Boyer-Lindquist coordinates to the employed affine-null coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' PACS numbers: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' INTRODUCTION At the dawn of the ’Golden Era of General Relativity’ in the 60ties of the last century, two important space- time metrics were found, the Bondi-Sachs metric [1–3] and the Kerr metric [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' The first settled the question that an isolated system looses mass via gravitational radi- ation and that this effect is a non-linear effect of General Relativity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' while the second describes a stationary and rotating isolated black hole that is expected to be the end product of a gravitational collapse of a massive star or a merger of two compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' One of the defining features of the Bondi-Sachs metric is that one coordinate is constant along a family of null hypersurfaces while a radial coordinate along these null hypersurfaces is an areal distance that can be related to a luminosity distance [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' As such, the first long term sta- ble evolution of black hole space times were made using a Bondi-Sachs metric in a null cone-world tube formalism [7], also see [8, 9] for review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Apart from usage in numer- ical relativity simulations, the Bondi-Sachs metric is now frequently used in high energy physics addressing ques- tions of the AdS/CFT correspondence [10] (and citations thereof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' It also became popular to discuss gravitational wave memory effects [11–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' A pleasant property of the Bondi-Sachs formalism is that the Einstein equation can be solved in a hierarchical manner when initial data on a null hypersurface and boundary conditions at a world tube or vertex are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' However, the radial coordinate of the Bondi-Sachs metric has the unpleasant property ∗Electronic address: thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='maedler˙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='˙mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='udp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='cl †Electronic address: egallo˙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='˙unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='ar that it breaks down when an apparent horizon forms due to the focusing of the surface-forming null rays and their vanishing expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' This can be overcome in choos- ing an affine parameter as radial coordinate, because an affine parameter only becomes singular at a caustic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' But, the Einstein equations resulting from an affine-null met- ric do not provide the hierarchical structure as the Bondi Sachs metric [9] and the hierarchical structure needs to be reestablished by various new definitions of variables [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Moreover, it turns out that also the hierarchy of equations in the affine-null metric formulation breaks down in the events of apparent horizon formation, but fortunately the equations can be regularized so that it is possible to follow up the formation of black holes up to singularity [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Despite the success and popularity of the Bondi-Sachs metric in the various areas, an explicit closed analytical representation of the Kerr metric in Bondi-Sachs form without bad behaviour in the exterior region or related metrics with one or two null coordinates is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Var- ious attempts have been made to derive a null metric representation, numerically [21, 22] as well as analyti- cally [23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' In all of the approaches, the authors start out with the Kerr metric and then calculate the respec- tive null metric via a coordinate transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' After these transformations the resulting metric can still posses a conical singularity at the axis of symmetry (see [22] for a complete discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' In addition, the final met- ric is determined by integrals of non-elementary func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Arga˜naraz and Moreschi’s approach [22] differs to the aforementioned ones that the authors aim to find a double–null representation of the Kerr metric by geomet- rically adopting the coordinates to in- and outgoing null geodesics adapted to the center of mass [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' In this way, the authors were successful in finding null coordinates 2 that are not only regular at every point of the external communication region (unlike the previous formulations) but also that they are regular at the event horizon, thus allowing a way to study the evolution of different matter fields (as scalar fields) in such background even when they cross the event horizon[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Unfortunately, even in their construction arises a differential equation that needs to be solved numerically and an explicit closed form repre- sentation of the double null version of the Kerr metric is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' The work of Bai and collaborators [28, 29] also starts with the Kerr metric (in Boyer-Lindquist co- ordinates) and then makes coordinate transformation to a Bondi-Sachs metric valid near future null infinity (in a compactified version of the metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' The authors are able to calculate the Newman-Penrose quantities and multi- poles at large distances and show the peeling property of the Weyl tensor at large radii and the vanishing of the so-called Newman-Penrose constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' In this article, in contrast to all the previous works which start with the Kerr metric expressed in Boyer- Lindquist coordinates and attempt to find a null coordi- nate version of it, we will directly solve the Einstein equa- tions in a characteristic formulation based on an affine- null metric formulation of the Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' In ad- dition, inspired by the Hartle-Thorne methods for obtain- ing solutions for slowly rotating compact stars [30], we will employ a quasi-spherical approximation of the field equations to find a high order approximation of the Kerr metric in out-going polar null coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' To obtain our solution, we assume stationarity and axial symme- try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' We further require an asymptotic inertial observer as well as that that Weyl scalar Ψ0 is regular everywhere where the background solution is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' A study of vac- uum stationary metrics with a smooth future null infinity in affine-null coordinates has recently be carried out by Tafel in [31] by considering power series of the metric components in terms of the inverse affine distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Throughout the article, we will use signature +2, units G = c = 1 and the Einstein sum convention for indices as well as products of associated Legendre polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' The article is organised as follows: Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' II recalls the affine-null metric formulation, makes the necessary symmetry assumptions for archiving our goal and de- fines the perturbative variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' III, we determine the background model (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' III A), define useful recur- sively re-appearing functions in the perturbation analysis (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' III B), solve the perturbation equations (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' III C- III F) and in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' III G the affine-null metric functions for the null are expressed in terms of the mass and spe- cific angular momentum, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' IV, to verify our re- sults, we calculate the affine-null version of Kerr metric in a Bondi frame via a coordinate transformation with a method adopted from [28], in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' V position of the outer ergosphere and event (past) horizon of the black hole we discuss the and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' VI contains the final discussion of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' The article finishes with two appendices: App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' A lists relations between associated Legendre polynomials and App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' B presents a derivation of the expression of the Komar charges relevant for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' AFFINE-NULL METRIC FORMULATION FOR STATIONARY AND AXIAL SYMMETRIC SPACETIMES Here we review the necessary properties of character- istic initial value formulation of the Einstein equations in affine-null coordinates, discuss the implications of the imposed symmetry assumptions and present the notation used in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Taking coordinates xa = (u, λ, xA), where u is an out– going null coordinate, λ an affine parameter, and xA are angular coordinates, a generic line element for an affine- null metric defined with respect to a family of outgoing null hypersurfaces u = const is [16–18, 32] gabdxadxb = −Wdu2 − 2dudλ +R2hAB(dxA − W Adu)(dxB − W Bdu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='1) The determinant det(hAB) = det(qAB) = sin2 θ is the determinant of a round unit sphere metric qAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Con- sequently hAB is transverse-traceless and has only two degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Thus, the function R relates to the area of cuts du = dλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' The inverse metric is given by guλ = −1 , gλλ = W , gλA = −W A , gAB = hAB R2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='2) where W A = (W θ, W φ) and hABhBC = δC A and in par- ticular [33] hABdxAdxB = � e2γdθ2 + sin2 θ e2γ dφ2� cosh(2δ) +2 sinθ sinh(2δ)dθdφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='3) A complex null dyad to represent the 2-metric hAB like hAB = m(A ¯mB) with mAmBhAB = mA ¯mBhAB − 1 = 0 is mA∂A = 1 √ 2eγ � cosh δ − i sinh δ � ∂y + ieγ √ 2 sin θ � cosh δ + i sinh δ � ∂φ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='4) Like in any Bondi-Sachs type metric [9], the vacuum field equations Rab = 0 with Rab being the Ricci tensor can be grouped into supplementary equations Si = 0 with Si = (Ruu, Ruθ, Ruφ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='5) one trivial equation Ruλ = 0 and the six main equations H(γ) K = 0, K ∈ (1, 2, 3, 4)) and H(δ) k = 0, k ∈ (1, 2)) with H(γ) K = � Rλλ, Rλθ, hABRAB, ℜe(mAmBRAB) � , H(δ) k = � Rλφ, ℑm(mAmBRAB) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='6) with ℜe(x) and ℑm(x) the real an imaginary part of x re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' We assume that the spacetime is axisymmet- ric and stationary with associated Killing vectors fields 3 ∂u and ∂φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Therefore the metric functions do not depend on u and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' The Killing symmetries imply two conserved quantities, the Komar mass, Km, and the Komar angu- lar momentum, KL, which can be calculated from their respective integrals (also see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' B) Km := K(∂u) = 1 8π lim λ→∞ � � W,λ−R2hABW AW B ,λ � R2d2q (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='7) while for the axial Killing vector we have KL := K(∂φ) = − 1 16π lim λ→∞ � � R4hφBW B ,λ � d2q (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='8) where dq = sin θdθdφ is the surface area element of the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Let us assume there is a smooth one parameter family of stationary and axially symmetric metrics gab(ε), where ε is a small dimensionless parameter such that ε = 0 is a corresponds to a (static) spherically symmetric spacetime solution of the vacuum Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Then there is an expansion of the metric fields like R(λ, θ) = r(λ) + R[1](λ, θ)ε + R[2](λ, θ)ε2 + R[3](λ, θ)ε3 + O(ε4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='9a) W(λ, θ) = V (λ) + W[1](λ, θ)ε + W[2](λ, θ)ε2 + W[3](λ, θ)ε3 + O(ε4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='9b) W A(λ, θ) = W A [1](λ, θ)ε + W A [2](λ, θ)ε2 + W A [3](λ, θ)ε3 + O(ε4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='9c) γ(λ, θ) = γ[1](λ, θ)ε + γ[2](λ, θ)ε2 + γ[3](λ, θ)ε3 + O(ε4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='9d) δ(λ, θ) = δ[1](λ, θ)ε + δ[2](λ, θ)ε2 + δ[3](λ, θ)ε3 + O(ε4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='9e) Inserting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='9) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='8) implies Km = O(ε0) and KL = O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' We make the requirements Km(ε) = Km(−ε) , KL(ε) = −KL(−ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='10) These conditions imply that under the change ε → −ε the sense of rotation is reversed (recall that K(∂φ) = −K(∂(−φ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' From the metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='1), we see that the 2-surfaces with u = u0 and λ = λ0, defined such that R(u0, λ0, θ) =const have the induced metric R2hABdxAdxB with area 4πR2(u0, λ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' We assume that the area of these 2-surfaces is invariant under the change ε → −ε, which implies that R2 is an even function of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Therefore R is either an even or an odd function of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' However, if R were an odd function, we had R(ε = 0) = 0, which is a non admissible solu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' In addition, ds2(∂φ, ∂φ) and ds2(∂θ, ∂θ) must be independent of the sense of rotation implying that hφφ and hθθ are even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' However, due to the frame dragging effect ds2(∂θ, ∂φ) must depend on the sense of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Therefore hθφ is an odd function of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Using similar arguments, because the Komar angular momentum KL is an odd function of ε and taking into account (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='8) and the parity behaviour of hAB and R2, we have that W θ is even and W φ odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Similarly, since Km must be a even function of ε, W must be even in ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Therefore, R[2n+1] = W[2n+1] = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='11a) W θ [2n+1] = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='11b) W φ [2n] = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='11c) γ[2n+1] = δ[2n] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='11d) To arrive at the last conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='11d) we have taken into account the odd parity of hθφ, which gives us sinh(δ(ε)) = − sinh(δ(−ε)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Hence, δ must be odd in ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' Similarly, for hθθ and hφφ be even, γ(ε) must satisfies e2γ(ε) = e2γ(−ε), which implies that γ is a even function of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' We conclude R = r + R[2]ε2 + +R[4]ε4 + O(ε6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='12a) W = V + W[2]ε2 + W[4]ε4 + O(ε6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='12b) W θ = W θ [2]ε2 + W θ [4]ε4 + O(ε4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='12c) W φ = W φ [1]ε + W φ [3]ε3 + O(ε5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='12d) γ = γ[2]ε2 + γ[4]ε4 + O(ε6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='12e) δ = δ[1]ε + δ[3]ε3 + O(ε5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='12f) A similar expansion was made by Hartle [30] in the derivation of a metric for slowly rotating stars using a a 3+1 decomposition of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='9) follows that the Ricci tensor has the expansions Rab = R[0]ab + R[1]abε + R[2]abε2 + R[3]abε3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfdgy1/content/2301.05092v1.pdf'} +page_content='13) In fact, with the notation f[i] ∈ {γ[i], δ[i], R[i], W A [i], W[i]}, it turns out for a perturbation at order n > 1 that S[n]i = ˆSi(f[n]) + s[i](f[m 2 s duration limit. GRB 210731A triggered the +Fermi Gamma-ray Burst Monitor (GBM; Meegan et al. 2009; +Lesage et al. 2021) one second earlier than Swift/BAT, with the +10-1000 keV light curve showing a single pulse with duration1 +T90 = 25.9±5.3 s, in agreement with the BAT duration. The time- +averaged spectra for both BAT and GBM were best-fitted with a +power law function and exponential high-energy cutoff with pho- +ton indices of −0.25±0.59 and −0.1±0.1, and cutoff energies of +(107±27) keV and (175±11) keV, respectively (Stamatikos et al. +2021; Lesage et al. 2021). The 10-1000 keV GBM fluence2 inte- +grated over the burst duration was (3.05±0.06)×10−6 erg cm−2. +Using a measured afterglow redshift of z = 1.2525 obtained by +X-Shooter on the Very Large Telescope at 1.19 days post-trigger +(Kann et al. 2021), this corresponds to an isotropic-equivalent +γ-ray energy of Eγ,iso = (1.29 ± 0.03) × 1052 erg. +We take the Swift/BAT trigger time as T0 for this burst and +reference all future times with respect to this T0. +2.2. X-ray observations +The Swift X-Ray Telescope (XRT; Burrows et al. 2005) started +observing the field of GRB 210731A 201 seconds post-trigger, +finding a bright new X-ray source consistent with the BAT po- +sition (Troja et al. 2021). The initial 62 seconds of data were +obtained in windowed timing (WT) mode after which Swift had +to slew away. Data capture recommenced in photon counting +(PC) mode at 3.3 hours post-trigger. We obtained the X-ray light +curve and spectra from the online Swift-XRT GRB Catalogue3 +(Evans et al. 2007, 2009). The Burst Analyser (Evans et al. 2010) +count-rate light curve showed that the X-ray flux was decreasing +rapidly during the WT-mode observations with a spectrum that +hardened from a photon index of ΓX = 3.2 to 2.2 over 60 sec- +onds. Once data capture resumed in PC mode at 10 ks, the X-ray +light curve was in a shallow decay phase before declining more +steeply at ∼20 ks post-trigger. +We fitted the PC-mode spectrum with a photoelectrically ab- +sorbed power-law model (tbabs*ztbabs*pow) in Xspec ver- +sion 12.12.0, fixing the source redshift at 1.2525 and Galactic +hydrogen column density at NGal +H += 1.15 × 1021 cm−2 for consis- +tency with the online fit. The fitted spectrum was characterised +by a photon index of Γ = 2.00+0.11 +−0.11 with a host galaxy column +density of Nhost +H += 2.46+1.99 +−1.56 × 1021 cm−2 and C-stat 182.1 for +213 degrees of freedom. There were insufficient photons in the +PC-mode light curve for time-resolved analysis and to test for +spectral evolution. We converted the PC-mode count-rate light +curve to a 1 keV flux density light curve using the spectral in- +dex from our PC-mode spectral fit of βX ≡ 1 − ΓX ≈ −1.00 and +the online unabsorbed count-to-flux conversion factor of 4.36 × +10−11 erg cm−2 ct−1. We performed a similar procedure for the +WT-mode data using the spectral parameters in the automated fit +on the Swift website, with a photon index of 3.07 and unabsorbed +count-to-flux conversion factor of 4.69 × 10−11 erg cm−2 ct−1. +1 Obtained from the online Fermi GBM Burst Catalog (von Kienlin +et al. 2020). +2 See footnote 1. +3 The Burst Analyser for GRB 210731A is available on the UK Swift +Data Science Centre website. +Article number, page 2 of 20 + +S. de Wet et al.: The triple-peaked afterglow of GRB 210731A from X-ray to radio frequencies +2.3. Optical/near-infrared observations +The fully robotic, 60 cm MeerLICHT optical telescope (Bloe- +men et al. 2016) was automatically triggered by Swift/BAT and +began observing the field of GRB 210731A 286 seconds after +the BAT trigger, taking 60 second exposures in the u, g, r, i, z, +and q optical bands (where the q band is roughly equivalent to +g + r), following the sequence quqgqrqiqz in order to obtain +high cadence coverage in the wider and more sensitive q band +with quasi-simultaneous multi-colour coverage of the evolving +afterglow. Comparison of the first q-band image with an exist- +ing MeerLICHT reference image revealed a new transient can- +didate at α = 20h01m13.19s, δ = −28d03m40.10s (J2000). This +position was 0.3′′ away from the refined XRT position (D’Ai +et al. 2021), confirming the new source as the optical afterglow +of GRB 210731A (de Wet et al. 2021). These observations con- +tinued until the target set, approximately 4.29 hours post-trigger. +Four cycles of the same filter sequence were obtained the fol- +lowing night in two time intervals separated by ∼2 hours. Since +the afterglow was by this point in a declining phase and below +the 60 second single-exposure detection limit, repeated q-band +exposures were taken on the nights of 2021 August 2 and 3 in +order to track the optical light curve. The MeerLICHT pipeline +(Vreeswijk et al., in prep) was used to perform standard charge- +coupled device (CCD) reduction tasks as well as astrometry and +point-spread function (PSF) photometry, producing a catalogue +file containing all 5σ source detections for each image. For im- +ages where the afterglow was fainter than 5σ above the back- +ground we used forced photometry to obtain magnitudes that +were at least at the 3σ level. Images from the night of 2021 +August 1 onwards were co-added to produce more significant +detections or deeper upper limits. +The Swift UltraViolet and Optical Telescope (UVOT; +Roming et al. 2005) took a single 61.7 second exposure in +the white filter beginning 210.4 seconds after the BAT trig- +ger but did not continue observing the field of the GRB until +3.27 hours after the trigger, whereafter it was observed with mul- +tiple filters intermittently over the next five days (Troja et al. +2021; Kuin et al. 2021). We performed aperture photometry +on the Swift/UVOT data using standard analysis tools from the +HEASoft (Nasa High Energy Astrophysics Science Archive Re- +search Center (Heasarc) 2014) Swift FTOOLS software pack- +age (version 6.29c4). We extracted magnitudes using the tool +uvotsource with a 3.5′′ radius aperture centred on the after- +glow position, and a nearby background aperture with a 10′′ ra- +dius. A total of 64 individual exposures in all seven UVOT filters +were taken over the course of the follow-up campaign. We co- +added exposures in the same filter with clear detections but taken +close to each other temporally using uvotimsum in order to pro- +duce more significant detections, and once the afterglow became +too faint to detect in individual exposures we co-added images +within wider time baselines in order to provide the deepest lim- +iting magnitudes. +The afterglow of GRB 210731A was observed simultane- +ously in the g′r′i′z′JHK bands with the Gamma-Ray Burst +Optical Near-Infrared Detector (GROND; Greiner et al. 2008) +mounted at the 2.2 m MPG telescope at the European Southern +Observatory (ESO) La Silla Observatory in Chile. The afterglow +was clearly detected in all bands in the first epoch of observations +taken 4.2 hours after the GRB trigger (Nicuesa Guelbenzu et al. +2021a). A further three epochs were obtained at 1.225, 2.214, +and 5.253 days post-trigger (Nicuesa Guelbenzu et al. 2021b). +We also obtained deep host-galaxy observations at 285 days that +4 Available at https://heasarc.gsfc.nasa.gov/docs/software/lheasoft/ +yielded detections in the g′ and r′ bands, and which we regard as +the host-galaxy flux. The multi-colour GROND data were anal- +ysed through standard PSF photometry using DAOPHOT (Stet- +son 1987) and ALLSTAR tasks of IRAF (Tody 1993), in a simi- +lar way to the procedure described in Krühler et al. (2008). The +optical data were calibrated against the Pan-STARRS catalogue5 +(Chambers et al. 2016), while for the near-infrared (NIR) bands, +photometric calibration was performed against the 2MASS cat- +alogue (Skrutskie et al. 2006), resulting in a typical absolute ac- +curacy of 0.04 mag in g′r′i′z′, 0.06 mag in JH and 0.08 mag in +K. +The 76 cm Katzman Automatic Imaging Telescope located +at the Lick Observatory (KAIT; Filippenko et al. 2001) obtained +20 × 60 second exposures in the clear band (similar to R; see Li +et al. 2003), starting ∼9.04 hours after the BAT trigger (Zheng +et al. 2021). All images were reduced and co-added using a +custom pipeline (Ganeshalingam et al. 2010; Stahl et al. 2019), +whereafter PSF photometry was performed on the co-added im- +age using DAOPHOT. Several nearby stars were chosen from the +Pan-STARRS1 catalogue for flux calibration, with their magni- +tudes transformed into Landolt magnitudes following Eq. 6 of +Tonry et al. (2012). The optical afterglow of GRB 210731A was +clearly detected in the co-added image. +Images in the SDSS g, r, and z filters were obtained at a +single epoch 1.18 days post-trigger with the acquisition camera +of the X-shooter spectrograph, mounted on the ESO Very Large +Telescope (VLT) UT3 (Melipal). Reduction was carried out us- +ing standard procedures. For the z-band image, a fringe correc- +tion was applied, using a template fringe pattern provided by +the observatory. We also observed the afterglow at two epochs +in the r and z bands with the Nordic Optical Telescope (NOT) +equipped with the ALFOSC imager. The images were reduced +following standard procedures including subtraction of a master +bias and correction with sky flats. Magnitudes were measured +using aperture photometry, and photometric calibration was car- +ried out against the Pan-STARRS catalogue. +We show all UV/optical/NIR photometry separated by in- +strument and filter in Fig. 1. +2.4. Radio observations +We obtained three epochs of radio continuum observations with +the MeerKAT radio telescope (Jonas & MeerKAT Team 2016) +in the L band (1.4 GHz) through director’s discretionary time +(DDT) proposal DDT-20120810-SD-01 (PI de Wet). Each ob- +servation had a total integration time on source of 0.78 hours, us- +ing J1939–6342 as the flux and bandpass calibrator and J1924– +2914 as the gain calibrator. All data were reduced using the oxkat +pipeline6 (Heywood 2020). No radio afterglow was detected at +1.4 GHz across the three epochs at 10.8, 34.1 and 59.7 days post- +trigger. The RMS noise was ≈ 14 µJy at the 1σ-level in each +image. We take upper limits on the afterglow flux as three times +the RMS noise. +Radio continuum observations were also obtained with the +Karl G. Jansky Very Large Array (JVLA; Perley et al. 2011) in +the C and X bands (centred on 6 and 10 GHz) through DDT +proposals 21B-333 and 21B-342 (PI de Wet) at four epochs +spanning 18.2 to 118 days post-trigger. The total integration +time per observation was 0.44 hours in each band, with 3C286 +used as the flux and bandpass calibrator and J1924–2914 as +5 See http://archive.stsci.edu/panstarrs/search.php. +6 See https://github.com/IanHeywood/oxkat/blob/master/README.md +and references therein. +Article number, page 3 of 20 + +A&A proofs: manuscript no. aas +102 +103 +104 +105 +Time since trigger (s) +10−3 +10−2 +10−1 +100 +101 +Time since trigger (days) +10 +12 +14 +16 +18 +20 +22 +24 +26 +28 +30 +AB magnitude +MeerLICHT/u + 2 +MeerLICHT/g + 1 +MeerLICHT/q +MeerLICHT/r − 1 +MeerLICHT/i − 2 +MeerLICHT/z − 3 +UVOT/uvw2 + 5 +UVOT/uvm2 + 4 +UVOT/uvw1 + 3 +UVOT/u + 2 +UVOT/b + 1 +UVOT/v + 0.5 +UVOT/white + 2 +GROND/g′ + 1 +GROND/r′ − 1 +GROND/i′ − 2 +GROND/z′ − 3 +GROND/J − 4 +GROND/H − 5 +GROND/K − 6 +VLT/g + 1 +VLT/r − 1 +VLT/z − 3 +NOT/r − 1 +NOT/z − 3 +KAIT/clear − 1 +Fig. 1. Combined UV/optical/NIR photometry of GRB 210731A. We only show detections where all magnitudes are in the AB system and have +not been corrected for Galactic extinction. Times are relative to the Swift/BAT trigger time. +the complex gain calibrator. We performed preliminary imag- +ing on the pipeline-calibrated measurements sets using standard +Common Astronomy Software Applications (CASA; McMullin +et al. 2007) procedures and detected the radio afterglow to GRB +210731A in all four epochs at 10 GHz and in all but the first +epoch at 6 GHz. +The first epoch National Radio Astronomy Observatory +(NRAO) -calibrated measurement set failed to pass internal qual- +ity thresholds for science usability so we chose to calibrate +manually from the raw data to obtain more accurate flux mea- +surements. We used CASA version 5.8.0 and performed imag- +ing with the task tclean and flux measurements with the task +imfit. We obtained satisfactory results with our X-band cal- +ibration, but the C-band calibration contained persistent phase +errors as a result of de-correlation problems during observing, +and the measured flux of all sources in the field was substan- +tially lower than in subsequent epochs. Despite these problems, +imaging on the re-calibrated data showed a radio source at the af- +terglow position, in contrast with preliminary imaging. We there- +fore adopted the following approach to calculate the flux of the +afterglow. We identified six point-like sources present at all four +epochs in the C band and measured their fluxes. For each source +a flux correction factor could then be calculated between the first +epoch flux and the flux in each subsequent epoch. The correct- +ing factors ranged from 1.2 to 2.2, with a mean value of 1.77 +and standard deviation of 0.26. No obvious trends in the cor- +recting factor were found as a function of source brightness or +offset from the image centre across all epochs, so we took the +mean correcting factor as the flux correcting factor to apply to +the measured afterglow flux in the first epoch image. We incor- +porate the standard deviation of the correction factor as an addi- +tional source of systematic uncertainty in the flux measurement +from the first epoch in the C band. +The third epoch X-band measurement set also had major +phase issues, so we performed the same procedure as for the C- +band first epoch data, calibrating the raw data and determining +a mean flux correcting factor to apply to our afterglow measure- +ment. Only two sources were used in determining the mean cor- +recting factor as the X-band images had far fewer sources than +the C-band images. The correcting factors ranged from 1.3 to +1.6, with a mean value of 1.47 and standard deviation of 0.1. +All X-ray, optical, and radio flux measurements associated +with GRB 210731A are presented in Table B.1. +3. Afterglow temporal and spectral analysis +We interpret our combined multi-wavelength data in the frame- +work of synchrotron radiation emitted by electrons accelerated +to a power-law distribution in energy behind the forward shock, +with N(γ) ∝ γ−p for γ > γmin, with γmin being the minimum +Lorentz factor of electrons in the distribution and p being the +electron energy index, which we assume to be bounded between +2 and 3, though values of less than 2 have been suggested in +the literature (Dai & Cheng 2001; Panaitescu & Kumar 2002). +The synchrotron spectra are characterised by power-law seg- +ments that join at a number of break frequencies, namely the +synchrotron self-absorption frequency νsa, the characteristic syn- +chrotron frequency νm corresponding to emission from γmin elec- +Article number, page 4 of 20 + +S. de Wet et al.: The triple-peaked afterglow of GRB 210731A from X-ray to radio frequencies +trons, and the cooling frequency νc. The orderings of the spectral +breaks depend on the hydrodynamic evolution of the forward +shock, which is described by the Blandford & McKee (1976) +spherical self-similar solution of an adiabatic relativistic blast +wave expanding into a cold medium with a circumburst density +profile varying as a power-law with radius: n(r) = n0r−k. We con- +sider two density profiles: the constant k = 0 case corresponding +to an interstellar-medium-like density profile; and the k = 2 case +corresponding to a stellar wind from a massive star progenitor. +The synchrotron forward shock model is described in Sari et al. +(1998), Chevalier & Li (2000), and Granot & Sari (2002), and +we follow the convention Fν(t) ∝ tανβ throughout. +3.1. Optical/X-ray temporal evolution +The most striking feature of our GRB 210731A dataset is the +three peaks in our early-time optical data (see Fig. 1). To char- +acterise this light curve further, we created a composite R-band +light curve by combining our q-, r-, and R-band data since they +are the most well-sampled optical bands and also have similar +central wavelengths. We also included the KAIT clear flux mea- +surement since it is calibrated to the R band. We used an op- +tical spectral index of βopt = −0.81 ± 0.05 derived from the +first GROND epoch (see Sect. 3.2) to transform the data to an +R-band central wavelength of 700 nm7. The composite R-band +light curve (see Fig. 2) exhibits three distinct peaks occurring +within the first 0.3 days of the GRB trigger, each with rising and +decaying segments of varying steepness and smoothness. We in- +vestigate the nature of the optical peaks in Sect. 5. After the last +peak at ∼0.2 days, the light curve entered a final declining phase +until the last optical observation at 6.2 days post-trigger. +We follow two approaches to fit the data: first we fitted a +single power-law to each rising and decaying segment8 directly +in order to get an indication of the steepness of each segment (we +use these in Sect. 5.1.5); then we performed an empirical fit as +in Li et al. (2012) by decomposing the light curve into separate +components, each of which may arise from different emission +sites or physical mechanisms. Since we have three clear peaks +in our light curve, we employed a model that comprises the sum +of three broken power-law (BPL) components (Beuermann et al. +1999; Price et al. 2001; Zeh et al. 2004), each characterised by +a normalising flux level, F0, rise and decay indices, α1 and α2, +break time, tb , and break smoothness, ω, according to +F(t) = F0 +�� t +tb +�−α1ω ++ +� t +tb +�−α2ω�−1/ω +. +(1) +If α1 is positive and α2 is negative, the light curve peaks at a +time, tp, between rising and decaying segments; tb = tp in Eq. 1. +We also include a constant term to account for the host-galaxy +r′-band brightness of 24.7 ± 0.2 mag measured at 285 days by +GROND. Considering values of 1, 3, 5 and 9 for ω, we find that +a smoother break with a value of 1 produces a fit with a χ2 +r value +closer to 1. Our fit allows us to compare the temporal evolution +in each optical band, which we do in Sect. 3.2. +Examining the X-ray light curve, the WT-mode data within +the first 300 seconds shows a steep decline with αX = −3.52 ± +0.36 and a photon index that hardened from 3.2 to 2.2, as taken +from the online Burst Analyser. The most likely explanation is +7 For direct comparison with the sample in Li et al. (2012) and Liang +et al. (2013). +8 We determine the boundary between each segment by eye. These are +shown as vertical dotted lines in Fig. 2. +high latitude prompt emission, as the temporal and spectral in- +dices agree broadly with the α = −2 + β curvature effect re- +lation (Kumar & Panaitescu 2000; Willingale et al. 2010). It is +unfortunate that we have no X-ray data during the time of the +first two optical peaks, rendering a direct comparison between +the two bands unfeasible. There is, however, X-ray data from +∼0.13 days onwards starting when the optical light curve was +rising to its final peak. The X-ray light curve started in a shallow +decay or plateau phase before steepening, which coincided with +the final decaying phase in the optical. We fit a BPL according to +Eq. 1 to determine the break time and temporal slopes, fixing the +break smoothness at 1, 3, 5, or 9. Each fit had a similar reduced +χ2 +r value (∼0.4); therefore, we employed a break smoothness of +ω = 1 to match the value used in the optical fit, though this re- +sults in a pre-break index that is poorly constrained. The R-band +and 1 keV light curves are presented in Fig. 2 along with their +fits, and we present the results of the fits in Table 1. +3.2. Achromatic optical/X-ray spectral evolution +We now investigate if there is evidence for spectral evolution in +the optical data. In Fig. 3 we show the optical spectral energy dis- +tributions (SEDs) formed using data from the first three of five +GROND epochs corrected for a Galactic extinction of AV = 0.24 +mag in the direction of the GRB (Schlafly & Finkbeiner 2011). +We fitted the data with power-laws in frequency, with the first +epoch yielding a spectral index of βopt = −0.81 ± 0.05. There +does not appear to be substantial spectral evolution between the +first two epochs at 0.184 and 1.225 days, particularly in the opti- +cal g′, r′, i′, and z′ bands. It is unclear why there is excess emis- +sion in the near infrared J, H, and K bands during the second +and third epochs. A possible explanation is that there is contam- +inating emission from the host galaxy, in which case we would +expect the light curves to flatten towards a constant value. The +observed decline to below detection levels at 5.25 and 285 days +appears to rule out this possibility. It could also be the case that +there is an additional unaccounted-for source of systematic pho- +tometric error. +For our early-time data, we took our composite R-band light +curve fit (Fig. 2) and fitted the flux in each of the optical bands +with this model, which amounts to shifting the R-band fit light +curve vertically until it fits the data in a given band. The spectral +slope in the blue and UV bands (u through uvw2) was steeper +than in the optical owing to Galactic extinction and damping +by Lyα absorption at the redshift of the burst. We therefore +shifted the MeerLICHT u-band data to the UVOT u band us- +ing an approximate spectral index of βUV ≈ −4 measured from +the UVOT/u, uvw1, uvm1, and uvw2 data. Figure 4 shows that the +data in each of the UV and optical bands is reasonably well fitted +by the R-band light curve. Deviations from the fit at earlier times +(< 0.2 days) are visible in the u, g, i, and z bands but they do not +appear statistically significant - only 7.5% of the UV/optical/NIR +data points deviate by 2σ or more from each model fit. Overall +the optical evolution appears achromatic and consistent with a +constant optical spectrum, which points towards a hydrodynam- +ical rather than spectral origin to the complex early-time optical +light curve. +As mentioned in Sect. 2.2, insufficient X-ray photons were +collected during the PC-mode observations to create time-sliced +spectra. The photon index from the online Burst Analyser was +fairly constant during PC mode, however, with a mean value of +ΓX = 1.84 ± 0.27. This is indicative of insubstantial spectral +evolution in the X-ray band during PC mode. The X-ray spectral +index of βX = −1.00 ± 0.11 from our spectral fit in Sect. 2.2 +Article number, page 5 of 20 + +A&A proofs: manuscript no. aas +103 +104 +105 +Time since trigger (s) +100 +101 +102 +103 +Fν (µJy) +t1.16 +t−0.77 +t0.64 +t−0.11 +t0.48 +t−1.65 +t−3.52 +Optical R-band +BPL component +Combined fit +XRT 1 keV ×101.5 +BPL fit +10−2 +10−1 +100 +Time since trigger (days) +0.5 +1.0 +1.5 +Ratio +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +AB magnitude +Fig. 2. Composite R-band light curve and X-ray 1 keV light curve for GRB 210731A. For each rising and decaying segment of the optical light +curve, we show the power-law slope (tα) as an indicator of the steepness of the light curve between each pair of adjacent vertical dotted lines. We +also show the fit comprising the sum of three BPL components and a constant term equal to the r′-band host galaxy flux measured by GROND +at 285 days (solid red line), along with each individual BPL component (dashed blue lines). For the X-ray light curve, we indicate the steepness +of the WT-mode segment and we show the BPL fit to the PC-mode data as a dashed-dotted red line. The results of the X-ray and optical fits are +presented in Table 1. The ratio of observed flux to fitted flux is shown in the lower panel. +Table 1. Parameters derived from fits to the composite R-band light curve and X-ray 1 keV light curve, as shown in Fig. 2. The R-band light curve +was fitted with the sum of three BPLs, while the X-ray light curve was fitted with a single BPL. +α1 +α2 +tp (days) +χ2/dof +Optical BPL 1 +1.39 ± 0.36 +−2.58 ± 0.75 +0.0088 ± 0.0013 +1.32 +Optical BPL 2 +1.19 ± 0.62 +−5.16 ± 2.58 +0.066 ± 0.007 +- +Optical BPL 3 +0.99 ± 0.16 +−1.84 ± 0.04 +0.27 ± 0.01 +- +X-ray 1 keV BPL +0.44 ± 0.62 +−1.69 ± 0.19 +0.26 ± 0.09 +0.39 +(with a mean photon arrival time of 0.42 days) is steeper than +the first GROND epoch (at 0.18 days) spectral index of βopt = +−0.81 ± 0.05, by ∆β = 0.19 ± 0.12. This difference is suggestive +of a spectral break lying between the X-ray and optical bands. +If the break was the cooling break, we would expect ∆β = 0.5, +however. This discrepancy could be explained by the fact that the +cooling break is always smooth (Uhm & Zhang 2014) or that νc +may lie near to either the optical or X-ray bands. We investigate +this further in Sect. 3.3. +3.3. Closure relation analysis +Within the synchrotron forward shock model the ‘closure rela- +tions’ relate the spectral index β with the temporal index α in +Article number, page 6 of 20 + +S. de Wet et al.: The triple-peaked afterglow of GRB 210731A from X-ray to radio frequencies +400 +800 +1200 +1600 +2000 +2400 +Wavelength (nm) +100 +101 +102 +103 +Fν (µJy) +0.18 days, β = −0.81 ± 0.05 +1.23 days, β = −0.79 ± 0.06 +2.21 days, β = −0.60 ± 0.14 +5.25 days +285 days, host galaxy +17 +18 +19 +20 +21 +22 +23 +24 +25 +AB magnitude +Fig. 3. Power-law fits to the GROND optical/NIR data at three epochs, +corrected for a Galactic extinction of AV = 0.24 mag. The uncorrected +magnitudes are shown in a lighter shade below each corrected data +point. We also show the fourth epoch, which only had detections in the +g′ and r′ bands, as well as the detections (shown as diamonds) and lim- +its from deep observations of the host galaxy at 285 days. Upper limits +are shown as upside-down triangles. +the convention Fν ∝ tανβ and depend on the physical regime, +spectral regime, and external medium density profile (Zhang & +Mészáros 2004; Zhang et al. 2006). The closure relations can +be adapted to describe a variety of alternative scenarios to the +standard self-similar deceleration phase, including the post-jet +break scenario, whether there is energy injection involved, the +reverse shock crossing phase, or the Newtonian/non-relativistic +phase (see the comprehensive review in Gao et al. 2013). It is +important to note that the spectral breaks are inherently smooth, +so that a transitioning spectral break or spectral break near to an +observing band may define a ‘grey zone’ where the α − β rela- +tions are not strictly satisfied (Zhang et al. 2006; Uhm & Zhang +2014). +If we assume that the final declining phase of our optical and +X-ray light curves arises from standard forward shock emission +in the slow cooling regime (νm < νc), which is usually the case +at later times, we can determine which spectral regime and den- +sity profile best fits our data by performing a closure relation +analysis. Our X-ray and optical data both have negative spec- +tral slopes, with fluxes that decline with increasing frequency. +There are two spectral regimes that give rise to negative spec- +tral slopes: νc < ν (Regime I) with β = −p/2, or νm < ν < νc +(Regime II) with β = (1 − p)/2. For the optical spectral index +of βopt = −0.81 ± 0.05 derived from the first GROND epoch we +would have p = 1.62 ± 0.10 or p = 2.62 ± 0.10 in Regime I and +II, respectively. The decay index for both a wind or interstellar +medium (ISM) environment in Regime I is α = (2 − 3p)/4 re- +sulting in α = −0.72±0.08 with p = 1.62±0.10, too shallow for +the observed decay rate of αopt = −1.84 ± 0.04 from our three +BPL-component fit. In Regime II, we have α = 3(1 − p)/4 or +α = (1 − 3p)/4 for an ISM or wind density profile, respectively. +With p = 2.62±0.10 we get α = −1.22±0.08 or α = −1.72±0.08 +for the ISM or wind profiles. Clearly, the observed optical tem- +poral index of αopt = −1.84 ± 0.04 is most consistent with the +optical spectral index in Regime II for a wind profile. +103 +104 +105 +Time since trigger (s) +10−2 +10−1 +100 +101 +102 +103 +104 +Fν (µJy) +XRT 1 keV +uvw2 × 10−0.8 +uvm2 × 10−0.7 +uvw1 × 10−0.6 +u × 10−0.5 +b × 10−0.3 +g × 10−0.1 +v × 100.1 +q × 100.3 +r × 100.5 +i × 100.7 +z × 100.9 +J × 101.0 +H × 101.1 +K × 101.2 +10−2 +10−1 +100 +Time since trigger (days) +0 +1 +2 +Ratio +Fig. 4. Fits to the data in each optical band using the best-fit model to +the composite R-band light curve in Fig. 2. We also fitted this model to +the X-ray PC-mode data. The lower panel presents the ratio of measured +flux to model flux. +In a wind medium the cooling break moves to higher fre- +quencies as t1/2, with light curves that are shallower by ∆α = +0.25 above νc. With the cooling break between the optical and X- +rays one could therefore expect the X-ray light curve to decline +more slowly than the optical light curve. Our fits to the X-ray +and optical light curves in Sect. 3.1 result in ∆α = 0.15 ± 0.19, +which agrees within uncertainties with the predicted difference +of ∆α = 0.25. We also see, however, that this ∆α value is con- +sistent with a zero difference within 1σ and is supported by the +fact that the composite R-band light curve provides a good fit +to the X-ray light curve, as shown in Fig. 4. It is therefore not +possible to conclusively say whether νc lies between the optical +and X-ray bands from our data, though we note that the slightly +different temporal and spectral (see Sect. 3.2) indices between +the two bands does hint at this possibility. In summary, our op- +tical and X-ray data can be accommodated within the standard +closure relations in a wind medium. +3.4. Broadband SED evolution +The optical and X-ray data alone can only place weak con- +straints on the location of νm and the peak flux of the evolving +synchrotron spectrum. Our late-time radio observations, which +probe the low-frequency end of the synchrotron spectrum, can +provide valuable constraints on the location of νm and the peak +flux, and can therefore lead to an estimation of the intrinsic blast +wave parameters (see Sect. 4). +Our three epochs of L-band (at 1.4 GHz) observations all +yielded non-detections, whereas our four epochs of C-band and +X-band data yielded detections spanning 18 to 118 days post- +trigger. The flux across the first two epochs at 18.2 and 34.2 days +was fairly constant in both the C and X bands, which is consis- +Article number, page 7 of 20 + +A&A proofs: manuscript no. aas +tent with the predicted evolution of t0 for the spectral ordering +νsa < ν < νm in a wind medium undergoing slow cooling. The +spectral slope between the first epoch C- and X-band detections +of β6−10 GHz = 0.57 ± 0.27 is also close to the predicted value +of ν1/3 for the same spectral segment. The subsequent decline +in flux across the last two epochs can be interpreted as the pas- +sage of νm through 6 and 10 GHz, whereafter both bands lie in +Regime II of the synchrotron spectrum in which the flux declines +with time and the spectrum declines with increasing frequency. +This is seen in our last two epochs where the C-band flux is +in fact brighter than the X-band flux. At 34.2 days we have +quasi-simultaneous flux measurements at 1.4, 6 and 10 GHz +(see Fig. 6). The spectral index between the C and X bands is +β6−10 GHz = 0.69±0.23, which is closer to the optically thin spec- +tral slope of ν1/3 than the synchrotron self-absorbed slope of ν2. +Based on the C-band detection and L-band upper limit, we place +a lower limit on the spectral slope of β1.4−6 GHz > 0.84. It there- +fore may be the case that synchrotron self-absorption is respon- +sible for the non-detections at 1.4 GHz, since our L-band limit +is consistent with a ν2 spectrum. In that case, the self-absorption +frequency could lie between 1.4 and 10 GHz at 34.2 days. +In a wind medium, νm moves to lower frequencies as t−3/2 +with the corresponding peak flux of the synchrotron spectrum +declining as t−1/2 for the spectral break ordering νsa < νm < +νc. From our GROND SED at t = 0.184 days (Fig. 3), we know +that νm lies below the K band with a peak flux greater than the +measured K-band flux of 555 µJy. If we assume that νm passes +through the radio X band (10 GHz) at 34.2 days with a peak flux +of ∼250 µJy, we would have expected νm to be at 1.48 × 1012 Hz +with a peak flux of 1320 µJy at 1.22 days, the time of our second +GROND epoch. This frequency lies below the K-band frequency +of 1.38 × 1014 Hz, as expected. The spectral index between this +expected peak flux value and the measured K-band flux value +at 1.22 days results in β ≈ −0.7, which is in agreement with +the measured optical spectral index of βopt = −0.79 ± 0.06 at +this time. We also note that the X-ray to optical R-band spectral +index at ∼0.38 days (βopt,X ≈ −0.95) is between the X-ray-only ( +βX = −1.00±0.11) and optical-only index (βopt = −0.81±0.05), +demonstrating that both observing bands can be accommodated +via a forward shock model. +With these basic considerations, we attempted to find a first- +guess set of blast wave parameters that can explain our data. We +have the following assumed constraints: (i) νm passes through +10 GHz at 34.2 days; (ii) the corresponding peak flux at this +frequency and time is ∼250 µJy; (iii) νc lies between the optical +and X-ray bands at early times (i.e. νc ≈ 1017 Hz at 0.3 days); +and (iv) νsa lies between 1.4 and 10 GHz at 34.2 days. +With these four constraints we can attempt to solve the sys- +tem of four equations describing the locations of the spectral +breaks and their corresponding flux densities in a wind medium, +given in Table 2 of Granot & Sari (2002). Our solution given the +above constraints results in an unphysical value of ϵe > 1, which +is driven primarily by the requirement that the self-absorption +frequency lies between 1.4 and 10 GHz at 34.2 days. Lowering +νsa to a frequency of ∼107 Hz results in a physical solution for all +of the blast wave parameters. Our L-band limits therefore pose a +challenge to the interpretation of our multi-wavelength afterglow +data. We return to this point in Sect. 5.2. +3.5. Early jet-break scenario +Our optical light curve during the final declining phase had a +temporal index of αopt = −1.84 ± 0.04 from the combined fit or +αopt = −1.65 ± 0.04 from the direct fit to the late-time data only, +which is steep for normal pre-jet break evolution (see Fig. 4 in +Wang et al. 2018). An alternative scenario to explain the steep +final declining phase in the optical and X-ray light curves is post- +jet break decay. If the jet break is due to a purely geometric edge +effect (Panaitescu et al. 1998), the light curves within all spectral +regimes should steepen by t−3/4 for the ISM case and t−1/2 for +the wind case once the ejecta has slowed down such that the +relativistic beaming angle 1/Γ is greater than the jet half-opening +angle θj, assuming a top-hat jet. Sideways expansion of a conical +jet would result in a steeper jet break decay of approximately t−p +in Regimes I and II for an ISM (Rhoads 1999; Sari et al. 1999). +Considering the edge effect only, a jet break will not change +the temporal evolution of the synchrotron spectral break frequen- +cies. If an early jet break occurred we would expect our radio +data to show declining light curves that decay as t−1/2 in a wind +medium under the assumption that νsa < ν6,10 GHz < νm. Taking +into account sideways expansion, the evolution of the break fre- +quencies is altered, though we would still expect declining light +curves at radio frequencies (Sari et al. 1999). The rising radio +light curves in the C and X bands until ∼34 days are therefore +inconsistent with an early jet break. This implies that the steep +optical and X-ray decline is normal pre-jet break decay in a wind +medium, supporting our analysis in Sect. 3.3. +4. Theoretical modelling +We have shown in the previous sections that our X-ray, opti- +cal, and radio data after the last optical peak can be reconciled +within the synchrotron forward shock model in a wind medium +with p ≈ 2.6 if we exclude our L-band limits. We now proceed to +find a set of blast wave parameters that can describe our data by +employing the smoothly connected power-law spectra outlined +in Granot & Sari (2002) and fitting for the forward shock param- +eters p, ϵe, ϵB, A⋆, and EK,iso, where EK,iso is the total isotropic- +equivalent kinetic energy in the blast wave; ϵe and ϵB are the +fractions of shock internal energy given to the electrons and the +magnetic fields, respectively; and A⋆ = A/(5 × 1011 g cm−1) is +the wind density parameter as defined in Chevalier & Li (2000). +We correct the data for Galactic extinction with AV = 0.24 mag. +We perform a Markov chain Monte Carlo (MCMC) analy- +sis with emcee (Foreman-Mackey et al. 2013) using 512 walk- +ers and 2000 steps, discarding the initial 250 steps as burn-in. +The details of our implementation are described in Laskar et al. +(2013, 2014). The host galaxy extinction, AV,host, is a free pa- +rameter in our model. We include the effects of Klein-Nishina +(KN) corrections (G. McCarthy & T. Laskar in prep) using pre- +scriptions from Nakar et al. (2009) and Jacovich et al. (2021). +We used uniform, uninformative priors flat in log space, and re- +stricted ϵe + ϵB < 1, although this limit is not reached. We did +not include data before the inferred time of the last optical peak +at ∼0.3 days in the modelling, and discuss these data further +in Sect. 5. We also did not include the MeerKAT 1.4 GHz ob- +servations, as we do not expect these to be fit with this model +(Sect. 3.4). We present theoretical modelling including the L- +band limits in Appendix A. For completeness, we also include +the possibility of a jet break following Rhoads (1999). We set a +lower limit on the jet break time of tjet ≳ 34 days since there is +no evidence for an earlier jet break in the data, as discussed in +Sect. 3.5. +The physical parameters for the highest-likelihood model +and those derived from the MCMC analysis are presented in Ta- +ble 2, while the corresponding light curves are presented in Fig. +7. For these parameters, both inverse Compton and KN effects +are important at early times (≲ 1 day), with Compton Y ≈ 4 at +Article number, page 8 of 20 + +S. de Wet et al.: The triple-peaked afterglow of GRB 210731A from X-ray to radio frequencies +1014 +1015 +1016 +1017 +1018 +Frequency (Hz) +10−1 +100 +101 +102 +103 +Fν (µJy) +Unabsorbed model SED +Absorbed model SED +XRT PC-mode spectrum +UV/Optical/NIR photometry +Fig. 5. Optical to X-ray SED at 0.3 days along with the highest- +likelihood theoretical model SED, both absorbed (dashed line) and un- +absorbed (solid line). The optical photometric data points were derived +from the light curve fits to each observing band (see Fig. 4) through +interpolating each fit to 0.3 days. The X-ray PC-mode spectrum had a +mean photon arrival time of 0.43 days, so we used the X-ray BPL light +curve fit in Fig. 2 to determine a correcting factor to shift the spectrum +to the expected flux level at 0.3 days. +∼0.3 days. At this time, the relevant spectral break frequencies +are located at νm ≈ 4 × 1013 Hz, νc ≈ ˆνc ≈ 1016 Hz, and ˆνm ≈ +1021 Hz, resulting in the spectral ordering νopt < νc ≈ ˆνc ≲ νX, +where ˆνc and ˆνm are KN spectral breaks as outlined in Nakar +et al. (2009). The cooling frequency passes through the X-ray +band between ≈ 0.6 to 12 days, consistent with the discussion +in Sects. 3.2 and 3.3. In this regime, the spectral index in the X- +rays is expected to be intermediate between (1 − p)/2 ≈ −0.88 +and −p/2 ≈ −1.34, which is consistent with the observed X- +ray spectral index of βX = −1.00 ± 0.11. Taking into account +the 1σ confidence intervals from the MCMC analysis, the de- +rived value of p = 2.75 ± 0.03 is consistent with the value +of p = 2.62 ± 0.10 inferred from our closure relation analysis +in Sect. 3.3. This model also requires an intrinsic extinction of +AV,host ≈ 0.2 mag, consistent with the observed UV suppression +(Fig. 5). The corner plot in Fig. 8 shows that there are strong cor- +relations between some pairs of parameters, especially those in- +volving ϵe, ϵB, A⋆, and tjet. The model over-predicts the 1.4 GHz +flux by a factor of ≈ 3 with respect to our MeerKAT upper limits +(Fig. 6); even when taking scintillation into account, the upper +limits are all more than 4σ below the model flux. We return to +this point in Sect. 5.2. +Our shock microphysics parameters are fairly typical. Our +value of p = 2.75 ± 0.03 is within the 1σ range of the sample in +Wang et al. (2015), for which they find p = 2.33 ± 0.48. Santana +et al. (2014) collect ϵe and ϵB values in the literature and find that +ϵe is narrowly distributed across one order of magnitude between +∼0.02 to 0.6 with a median of value 0.22. Our value of ≈ 0.1 is a +normal value within their sample. For the magnetic field equipar- +tition factor, they find a much wider distribution varying across +almost 5 orders of magnitude from ∼3.5×10−5 to 0.33. Our value +Table 2. Parameters derived from our multi-wavelength theoretical +modelling of the afterglow data after 0.3 days. We show both the +highest-likelihood model parameters from a maximum-likelihood (ML) +estimation and the median values along with their corresponding 1σ +confidence intervals from the MCMC marginalised posterior distribu- +tions presented in Fig. 8. The beaming-corrected prompt γ-ray and ki- +netic energies are given in the lower panel of the table, where we have +placed a lower limit on the opening angle of the jet based on a limit of +tjet ≳ 118 days. +Parameter +ML estimate +MCMC results +p +2.75 +2.75 ± 0.03 +ϵe +9.7 × 10−2 +9.6+1.3 +−1.0 × 10−2 +ϵB +1.7 × 10−3 +2.2+1.9 +−0.9 × 10−3 +A⋆ +7.1 × 10−2 +6.4+1.6 +−1.5 × 10−2 +EK,iso (1052 erg) +63 +69+32 +−19 +AV,host (mag) +0.22 +0.23 ± 0.01 +tjet (days) +64 +55+23 +−14 +θjet (deg) +≳ 6 +- +Eγ (erg) +≳ 7.07 × 1049 +- +EK (erg) +≳ 3.45 × 1051 +- +of ϵB ≈ 0.2 × 10−2 is close to their median value of 1.0 × 10−2. +Additionally, Beniamini & van der Horst (2017) use radio light +curve peaks to determine the distribution of ϵe and find a value +of log10ϵe = −0.88±0.26 for a wind medium. Again, our derived +valued is consistent with their sample value. +The model requires a jet break at tjet ≈ 64 days, which is +driven by the declining radio light curves after this time. For +the highest-likelihood parameters, this corresponds to a jet open- +ing angle of θjet ≈ 5◦. In the absence of a jet break, the model +over-predicts the final X-band detection by ≈ 3.5σ. However, +we note that the evidence in support of a jet break is fairly +weak and this inferred opening angle should be interpreted with +caution. For a limit of tjet ≳ 118 days (the last radio detec- +tion), the highest-likelihood model yields a lower limit on the +opening angle of θjet ≳ 6◦. The corresponding beaming correc- +tion of fb = (1 − cos θjet) ≳ 5.48 × 10−3 implies constraints on +the true γ-ray and kinetic energy of Eγ ≳ 7.07 × 1049 erg and +EK ≳ 3.45 × 1051 erg, respectively, where we have used the +maximum-likelihood (ML) estimate from Table 2 for EK,iso. +5. Discussion +The optical light curve of GRB 210731A is unusual for show- +ing three distinct peaks of similar brightness within the first +five hours of the GRB trigger. Multiple peaks in optical light +curves have been observed before (e.g. GRBs 060904B, 080928, +100621A, 100814A, 100901A; Klotz et al. 2008; Rossi et al. +2011; Greiner et al. 2013; Nardini et al. 2014; Laskar et al. +2015). We investigate a number of explanations proposed in the +literature for peaks and re-brightenings in afterglow light curves, +including the passage of a spectral break, flaring behaviour, sec- +ondary jets, an off-axis viewing angle, energy injection into the +forward shock, and the onset of afterglow. We also consider the +implications of our L-band upper limits. +5.1. The nature of the optical re-brightenings +5.1.1. Passage of a spectral break +The passage of the spectral break associated with the peak of +the synchrotron spectrum (νm) through the optical bands could +Article number, page 9 of 20 + +A&A proofs: manuscript no. aas +108 +109 +1010 +1011 +101 +102 +103 +Fν (µJy) +t = 10.8 days +108 +109 +1010 +1011 +Frequency (Hz) +t = 34.2 days +108 +109 +1010 +1011 +t = 59.7 days +Fig. 6. Radio SEDs at the times of the three MeerKAT epochs. We show the highest-likelihood unabsorbed model SED along with the effects of +Galactic scintillation at the 1σ level as derived from the NE2001 model (Cordes & Lazio 2002) for the GRB line of sight through the Milky Way. +For the second epoch we show the C- and X-band detections obtained quasi-simultaneously at 34.2 days, while for the third epoch we show the +C- and X-band detections obtained 7.4 days after the third MeerKAT epoch. +in principle give rise to a peak in the optical light curves. Since +νm moves to lower frequencies with increasing time, we would +expect the spectral index to transition from a positive to negative +value over the rise and fall of the light curve. We have shown +in Sect. 3.2 and in Fig. 4, however, that our optical evolution is +achromatic, ruling out a spectral break origin to any of the peaks. +5.1.2. Flaring behaviour +The peaks in the early light curve are too smooth and long-lived +to be due to flares. Hence, the observed structure in the optical +light curve cannot be due to any of the mechanisms that are typ- +ically invoked to explain flares, such as late-time central engine +activity, density fluctuations, or reverse shock emission. +5.1.3. Off-axis viewing angle +It is possible to obtain a rising light curve if the GRB jet is +viewed from an angle outside the cone of the jet (i.e. θobs ≳ θjet; +Granot et al. 2002). The peak of the light curve corresponds to +the time when the Lorentz factor of the jet is ∼ 1/θobs, where- +after the light curve evolves in a post-jet break manner. Our +radio data and theoretical modelling does not support an early +jet break however, as discussed in Sects. 3.5 and 4. Although +it would be possible to explain a single peak in the light curve +through viewing angle effects, it is difficult to interpret all three +light curve peaks within such a scenario. Due to the relativis- +tic beaming effect, one would also expect to observe negligible +prompt γ-ray emission when the viewing angle is outside the +jet cone, resulting in an orphan afterglow (Rhoads 1999; Gra- +not et al. 2002; Zou et al. 2007). The prompt γ-ray observations +of GRB 210731A therefore do not support an off-axis viewing +angle interpretation. +5.1.4. Two-component jet model +The two-component jet model (Peng et al. 2005) has been +invoked to explain chromatic behaviour and late-time re- +brightenings observed in a number of GRB afterglows (Berger +et al. 2003; Racusin et al. 2008; Filgas et al. 2011; Nicuesa Guel- +benzu et al. 2011; Kann et al. 2018). In this model a fast, narrow +inner jet powers the prompt emission and early afterglow emis- +sion, while a slower, wider jet powers the late-time afterglow +evolution, with both jets viewed on-axis. This model was pre- +ferred by Liang et al. (2013) to explain the re-brightenings ob- +served in their sample of optical afterglows, in which the deceler- +ation of the slow jet explains the re-brightening peaks, analogous +to the onset peaks. They claim that the similar rising and decay- +ing indices of the onset and re-brightening peaks supports this in- +terpretation. Furthermore, they find that the properties of the re- +brightenings are not correlated with the prompt emission prop- +erties (contrary to the onset peaks), and so they are likely inde- +pendent emission components. The fact that the GRB 210731A +optical light curve shows three distinct peaks appears to rule out +the two-component jet model. A three-component jet might be +able to explain the three light curve peaks, but we consider it +beyond the scope of this work. +5.1.5. Energy injection +The most straightforward explanation for the optical light curve +evolution is energy injection into the forward shock. Energy in- +jection, or a refreshed shock, has been invoked to explain the +plateaus and shallow decay segments seen in X-ray (Campana +et al. 2005; Vaughan et al. 2006; Nousek et al. 2006; Zhang +et al. 2006; Liang et al. 2007; Evans et al. 2009) and optical light +curves (Mangano et al. 2007; Greiner et al. 2009; Swenson et al. +2013). In this framework, the blast wave energy increases with +time during the energy injection period, rather than remaining +constant. Two physical mechanisms have been proposed: the first +is a long-lasting central engine that continuously injects a Poynt- +ing flux into the blast wave as in the case of a spin-down millisec- +ond magnetar (Dai & Lu 1998; Zhang & Mészáros 2001), where +the central engine luminosity is described as a power law in time +with L(t) = L0(t/t0)−q. The injected energy Einj is essentially +constant when q ≥ 1, while the total energy in the blast wave +can only increase significantly with time when q < 1. Once the +injected energy begins to exceed the original energy in the blast +wave, the total energy will scale as Etot ∝ t1−q. The second case +could arise from an impulsive central engine injection episode, +producing a stratified ejecta distribution where the ejecta mass +above a certain Lorentz factor Γ is described as a power law, +for example M(> Γ) ∝ Γ−s. The energy in a shell with Lorentz +factor Γ is added to the blast wave when the blast wave bulk +Article number, page 10 of 20 + +S. de Wet et al.: The triple-peaked afterglow of GRB 210731A from X-ray to radio frequencies +103 +104 +105 +106 +107 +Time since trigger (s) +10−2 +10−1 +100 +101 +102 +Time since trigger (days) +10−2 +102 +106 +1010 +1014 +1018 +1022 +1026 +Fν (µJy) +1 keV ×1023 +uvw2 × 1020 +uvm2 × 1019 +uvw1 × 1018 +white × 1017 +u × 1016 +b × 1015 +g × 1014 +v × 1013 +q × 1012 +r × 1011 +i × 1010 +z × 109 +J × 108 +H × 107 +K × 106 +X ×102 +C ×101 +L ×100 +Fig. 7. Light curves from our highest-likelihood theoretical model +shown for each observing band, spanning X-ray, optical, and radio fre- +quencies. Only data points after 0.3 days were used in the modelling, +and we show these as filled-in data points, in contrast to the earlier time +data points shown as empty circles. Upper limits are shown as upside- +down triangles. The shaded regions surrounding the three radio bands +(X, C, and L) represent the effects of Galactic scintillation at the 1σ +level. The optical g- and r-band model fits plateau towards the mea- +sured host-galaxy flux levels at 285 days, shown as diamonds. +Lorentz factor has slowed down to Γ, so that the energy in the +blast wave scales as E(> Γ) ∝ Γ1−s (Rees & Mészáros 1998; +Sari & Mészáros 2000). For both cases the blast wave scaling +laws can be derived and applied to the synchrotron spectra to +obtain closure relations that depend on either q or s. Both forms +of energy injection can be cast in an equivalent form, and simple +relations between q and s can be derived (Zhang et al. 2006). +From a closure relation analysis with energy injection alone, it is +therefore impossible to distinguish energy injection from a long- +lasting central engine or from injection of a stratified ejecta. +The R-band light curve segments between the first and final +peaks have temporal indices of α = [−0.77, 0.64, −0.11, 0.48] +(from the direct fit to each segment in Fig. 2). Adopting the q- +formalism, we can determine the energy injection index q for +each segment by making use of the closure relations for a wind +medium in the slow cooling regime (Zhang et al. 2006). We +make the assumption that the R band remained in the spectral +regime satisfying νm < νR < νc during all four segments of en- +ergy injection, which is valid since our data supports achromatic +optical evolution. In this regime, α = (2−2p)−(p+1)q +4 +, so employ- +ing p = 2.75 from our theoretical modelling we derive values of +q = [−0.11, −1.62, −0.82, −1.45] for each light curve segment +after the first peak. The energy increase during a time period +from t0 to t1 is calculated as +EK,iso,1 = EK,iso,0 +�t1 +t0 +�1−q +. +(2) +Assuming that the blast wave kinetic energy evolves according +to Eq. 2 during each of the four power law segments, we can +determine the energy at the time of the first peak, EK,iso,0, using +EK,iso,f = EK,iso,0 +�t1 +t0 +�1−q1 �t2 +t1 +�1−q2 �t3 +t2 +�1−q3 �t4 +t3 +�1−q4 +(3) +along with the start and end times of each segment9, our values +calculated for q above, and a final blast wave energy of EK,iso,f = +6.3 × 1053 erg from our theoretical modelling. We find that the +blast wave energy at the time of the first optical peak is equal to +7.3×1050 erg, smaller by a factor of ∼1000 compared to the final +kinetic energy, indicative of substantial energy injection. +Laskar et al. (2015) argue that the significant X-ray and op- +tical re-brightenings seen in a sample of GRB afterglows are +best explained by the stratified ejecta model, since energy in- +jection from a spinning-down millisecond magnetar should not +lead to a significant increase in the blast wave energy (i.e. +q ≥ 1). They also exclude fall-back accretion onto a black +hole as the theoretically-predicted accretion rate is insufficient +to power plateaus or re-brightenings. In the stratified ejecta for- +malism, there is a gap between the initial blast wave shell and +the fast outer shell of the stratified ejecta that is moving with +some maximum Lorentz factor, Γmax. As the initial shell slows +down, the stratified ejecta deposits energy into the blast wave +until the slowest shell moving with Lorentz factor Γmin has de- +posited its energy, whereafter the afterglow evolves following +the standard framework. From their study of a sample of after- +glows exhibiting later-time re-brightenings, Laskar et al. (2015) +showed that a large amount of the kinetic energy deposited into +the blast wave comes from the slowest-moving ejecta. They also +find that the GRBs with significant energy injection have low +radiative efficiencies, consistent with the prompt γ-ray emission +being produced by the fastest-moving ejecta and a large amount +of kinetic energy being carried by the slower ejecta. With our +value of Eγ,iso = 1.29 × 1052 erg and EK,iso = 63 × 1052 erg, we +calculate a radiative efficiency using η ≡ Eγ,iso/(Eγ,iso + EK,iso) +of η ≈ 0.02, a low radiative efficiency consistent with significant +energy injection. We also note that the energy released in γ rays +is ≈ 18 times greater than the kinetic energy in the blast wave +derived at the time of the first optical peak. This is unphysical if +the kinetic energy at that time is the true energy reservoir from +which the prompt emission is drawn, and adds support to the +final kinetic energy being the true energy reservoir. +5.1.6. Afterglow onset and initial Lorentz factor, Γ0 +The synchrotron forward shock model predicts a smooth onset +peak in optical light curves as the expanding blast wave is slowed +down by the circumburst medium (Sari & Piran 1999; Kobayashi +& Zhang 2007). Such onset peaks are not uncommon in early +optical afterglows and have been studied by a number of authors +(Zhang et al. 2003; Molinari et al. 2007; Xue et al. 2009; Liang +et al. 2010, 2013). For an ISM density profile, the deceleration +9 We use the vertical dashed lines in Fig. 2 as the start and end times of +each segment. These have values of t0 = 0.0077, t1 = 0.021, t2 = 0.057, +t3 = 0.09, and t4 = 0.22 days. +Article number, page 11 of 20 + +A&A proofs: manuscript no. aas +p = 2.75+0.03 +−0.03 +−1.50 +−1.25 +−1.00 +−0.75 +log(ϵe) +log(ϵe) = −1.02+0.06 +−0.05 +−4 +−3 +−2 +−1 +log(ϵB) +log(ϵB) = −2.66+0.26 +−0.23 +−2.5 +−2.0 +−1.5 +−1.0 +−0.5 +log(A⋆) +log(A⋆) = −1.19+0.09 +−0.11 +1.2 +1.5 +1.8 +2.1 +2.4 +log(EK,iso,52) +log(EK,iso,52) = 1.74+0.15 +−0.13 +0.20 +0.22 +0.24 +0.26 +0.28 +AV,host +AV,host = 0.23+0.01 +−0.01 +2.58 +2.64 +2.70 +2.76 +2.82 +p +1.6 +2.4 +3.2 +4.0 +log(tjet) +−1.50 +−1.25 +−1.00 +−0.75 +log(ϵe) +−4 +−3 +−2 +−1 +log(ϵB) +−2.5 +−2.0 +−1.5 +−1.0 +−0.5 +log(A⋆) +1.2 +1.5 +1.8 +2.1 +2.4 +log(EK,iso,52) +0.20 +0.22 +0.24 +0.26 +0.28 +AV,host +1.6 +2.4 +3.2 +4.0 +log(tjet) +log(tjet) = 1.84+0.16 +−0.14 +Fig. 8. Corner plot showing the marginalised posterior distributions for each model parameter along with the 2D marginalised posterior distribu- +tions for each pair of model parameters, from our MCMC analysis. Contours are at the 1σ, 2σ, and 3σ levels, and the red lines denote the median +values derived from the marginalised posterior distributions for each model parameter. +time (or onset peak time) is most sensitive to the bulk Lorentz +factor of the ejecta and depends weakly on other parameters. +Onset peaks can therefore provide a valuable way of constraining +the initial Lorentz factor Γ0 of the GRB blast wave. For a non- +ISM density profile, however, the dependence of the deceleration +time on other parameters becomes stronger. +The first optical detection associated with GRB 210731A +was made in the UVOT white filter starting 210 seconds post- +trigger, with subsequent detections showing a steady rise to a +smooth peak around 700 seconds. Thereafter, the light curve en- +tered a declining phase before starting to rise steadily again at +∼1700 seconds. Li et al. (2012) and Liang et al. (2013) define +an onset peak as a smooth hump peaking within one hour post- +trigger that is followed by a normal power-law decay component. +By comparing our onset peak with the sample of 38 onset peaks +in Liang et al. (2013), we can assess how likely it is that our +first peak is indeed the onset of afterglow. From our combined +R-band fit (Fig. 2 and Table 1) we derive a number of additional +properties from each BPL component. These include the peak +flux (Fp), peak time (tp), full width at half maximum (w), peak +R-band luminosity (LR,p), and the isotropic energy release in the +interval [tp/5, 5tp], as shown in Table 3. +Liang et al. (2013) find that the onset peak times of their +sample span a range of 30-3000 seconds, with typical rising +and decaying indices of 1.5 and −1.15 in ranges of [0.3,4] and +[−1.8,−0.6], respectively. Our peak time of 760 ± 111 seconds +and rising index of 1.39±0.36 are typical values within this sam- +ple, but our decaying index of −2.58 ± 0.75 from BPL 1 in Table +1 is steeper than average, though it does fall within the sample +range within uncertainty. It should be noted that the decaying +slope of our first peak is not well determined owing to the un- +certain contribution of the second BPL component at this time. +Furthermore, the authors derive a number of empirical relations +between pairs of properties of onset peaks (see their Figs. 7 and +9): the width of a peak is strongly correlated with the peak time; +the R-band peak luminosity is anti-correlated with the rest-frame +Article number, page 12 of 20 + +S. de Wet et al.: The triple-peaked afterglow of GRB 210731A from X-ray to radio frequencies +Table 3. Parameters derived from each BPL component of our R-band light curve fit. +Fp (10−12 erg cm−2 s−1) +tp (s) +w (s) +LR,p (1045 erg s−1) +ER,iso (1048 erg) +1.92 ± 0.27 +760 ± 111 +956 +15.69 ± 2.22 +8.28 +0.75 ± 0.27 +5704 ± 645 +5385 +6.11 ± 2.21 +16.96 +2.62 ± 0.10 +23161 ± 1179 +39824 +21.41 ± 0.78 +464.53 +time; and the peak luminosity and energy are correlated with the +isotropic γ-ray energy. We find that our measured values in Table +3 agree closely with their empirical relations, lending support to +the onset peak claim. +Under the assumption that GRB 210731A occurred in a stel- +lar wind medium (k = 2), we calculate the initial Lorentz factor +following Zhang (2018) as +Γ0 ≃ 1.31/4 +�3EK,iso(1 + z) +8πAc3tdec +�1/4 +≃ 120t−1/4 +dec +�1 + z +2 +�1/4 +E1/4 +52 A−1/4 +⋆ +, +(4) +where after the last equality, tdec, is measured in seconds, E52 in +units of 1052 erg, and A⋆ is the wind density parameter as de- +scribed previously. Equation 4 depends on the assumption of an +impulsive fireball, that is, the thin shell regime. The isotropic- +equivalent kinetic energy of the blast wave EK,iso can be inferred +from theoretical modelling of the afterglow, which we do in Sect. +4. However, the value calculated in Sect. 4 was only applicable +to the late-time light curve, so we employ the value for EK,iso of +7.3 × 1050 erg at the time of the first optical peak calculated in +Sect. 5.1.5. Using the A⋆ value from our theoretical modelling +and the peak time of our first peak, we calculate an initial Lorentz +factor of Γ0 ≈ 24. As mentioned previously, the dependence of +the deceleration time on other parameters is stronger for a non- +ISM density profile. This is also the case if there is energy injec- +tion during deceleration, which may be the case during our early +optical observations (Sect. 5.1.5). We therefore caution that our +calculation of Γ0 is an estimate. +5.2. Suppressed L-band flux +Our highest-likelihood theoretical model can provide an ade- +quate fit to all of our late-time data except for the L band, where +the model over-predicts the flux by a factor of ≈ 3 compared to +our MeerKAT upper limits. We did not expect our model to fit +the L-band data since we found that requiring the synchrotron +self-absorption frequency to lie above the L band at 34.2 days +resulted in an unphysical value of ϵe > 1. We therefore need to +consider an additional source of opacity at these observing fre- +quencies to explain our L-band limits. +One possible source of additional opacity is a thermal elec- +tron population within the GRB shock front. Ressler & Laskar +(2017) modelled afterglow spectra and light curves while con- +sidering the effect of such a population and find that it has +two effects on the spectra: an excess of flux near the peak syn- +chrotron frequency of the thermal electrons that fades with time +as the electrons cool; and additional opacity in the optically thick +portion of the spectrum compared to the case with only non- +thermal electrons. The latter effect is consistent with a higher +self-absorption frequency by a factor of 10-100. It could there- +fore be the case that our suppressed L-band flux points towards a +population of thermal electrons. We leave the detailed modelling +including a thermal population of electrons to a future work. +6. Conclusion +GRB 210731A was a long-duration burst discovered by +Swift/BAT. Observations with the optical telescope MeerLICHT +starting 286 seconds post-trigger found an unusual optical light +curve evolution with three peaks of similar brightness within the +first 4.3 hours; afterwards, the burst entered a declining phase. +We find that the early optical evolution is consistent with a con- +stant optical spectrum, pointing towards a hydrodynamical ori- +gin. A closure relation analysis based on the optical SED and +temporal decay after the last peak showed a preference for a +stellar wind environment, consistent with the long GRB duration +and therefore a massive star origin. We find that the first optical +peak can be explained as the onset of afterglow, while energy in- +jection into the forward shock from a stratified ejecta is a natural +explanation for the two subsequent re-brightenings. We estimate +that the blast wave kinetic energy increased by a factor of ∼1000 +from the first optical peak to the last peak. Detailed theoretical +modelling of the optical, X-ray, and radio data after the last opti- +cal peak at ∼0.3 days resulted in typical blast wave and shock mi- +crophysics parameters. Our MeerKAT L-band upper limits could +not be reconciled with our model, however, possibly implying a +thermal electron population within the shocked region that pro- +vided an additional source of opacity. Future multi-wavelength +modelling of GRB afterglows, especially at millimetre and radio +frequencies, will shed light on the electron distribution in GRB +shocks. +Acknowledgements. The MeerLICHT consortium is a partnership between Rad- +boud University, the University of Cape Town, the Netherlands Organisation +for Scientific Research (NWO), the South African Astronomical Observatory +(SAAO), the University of Oxford, the University of Manchester and the Uni- +versity of Amsterdam, in association with and, partly supported by, the South +African Radio Astronomy Observatory (SARAO), the European Research Coun- +cil and the Netherlands Research School for Astronomy (NOVA). We acknowl- +edge the use of the Inter-University Institute for Data Intensive Astronomy +(IDIA) data intensive research cloud for data processing. IDIA is a South African +university partnership involving the University of Cape Town, the University of +Pretoria and the University of the Western Cape. SdW and PJG are supported +by NRF SARChI Grant 111692. Part of the funding for GROND (both hardware +and personnel) was generously granted from the Leibniz-Prize to G. Hasinger +(DFG grant HA 1850/28-1) and by the Thüringer Landessternwarte Tautenburg. +AVF is grateful for financial assistance from the Christopher R. Redlich Fund +and numerous individual donors. KAIT and its ongoing operation were made +possible by donations from Sun Microsystems, Inc., the Hewlett-Packard Com- +pany, AutoScope Corporation, Lick Observatory, the U.S. NSF, the University of +California, the Sylvia & Jim Katzman Foundation, and the TABASGO Founda- +tion. Research at Lick Observatory is partially supported by a generous gift from +Google. This work made use of data supplied by the UK Swift Science Data +Centre at the University of Leicester. This publication made use of the python +package corner.py (Foreman-Mackey 2016). +References +Barthelmy, S. D., Barbier, L. M., Cummings, J. R., et al. 2005, Space Sci. Rev., +120, 143 +Beniamini, P. & van der Horst, A. J. 2017, MNRAS, 472, 3161 +Berger, E., Kulkarni, S. 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Same as Table 2, but for the fit including the L-band upper limits. +Parameter +ML estimate +MCMC results +p +2.76 +2.76+0.03 +−0.04 +ϵe +1.2 × 10−2 +1.2+0.1 +−0.2 × 10−1 +ϵB +3.1 × 10−3 +5.7+17.7 +−2.5 × 10−3 +A⋆ +4.1 × 10−2 +3.1+0.8 +−1.3 × 10−2 +EK,iso (1052 erg) +39 +30+11 +−10 +AV,host (mag) +0.22 +0.23 ± 0.01 +tjet (days) +125 +186+37 +−87 +106 +107 +Time since trigger (s) +101 +102 +Time since trigger (days) +101 +102 +103 +104 +105 +Fν (µJy) +X ×102 +C ×101 +L ×100 +Fig. A.1. Highest-likelihood model for the fit that includes the L-band data (dashed lines) and the fit that excludes the L-band data (solid lines). +The fit to the optical and X-ray light curves is similar in both models. +Appendix A: Theoretical modelling with MeerKAT L-band limits +We repeated the theoretical modelling following the procedure outlined in Sect. 4, but including the MeerKAT L-band limits. The +radio light curves from the highest-likelihood model are shown in Fig. A.1, and the MCMC parameter distributions are presented in +Fig. A.2 and Table A.1. +Including the MeerKAT L-band limits leads to a poorer fit. This is demonstrated by the highest-likelihood light curve under- +predicting the C- and X-band fluxes while still over-predicting the L-band flux, and the actual likelihood value of this fit being lower +than for the fit excluding the MeerKAT limits. We measured a maximum log likelihood of 236.9 for the fit without the limits vs +225.1 for the fit including the MeerKAT limits. As we expected, including the MeerKAT data does not lead to a model with a steep +enough spectral index to accommodate the non-detections. +Appendix B: Flux measurements +Table B.1 contains all the X-ray, optical, and radio observations used in this work. +Table B.1. X-ray/optical/radio flux measurements of GRB 210731A. +∆t +(days) +Telescope +Band/Filter +Frequency +(Hz) +Flux +(µJy) +Uncertainty +(µJy) +Detection? +(1 = yes) +0.00241 +Swift/XRT +1 keV +2.42e+17 +177.589 +27.077 +1 +0.00243 +Swift/XRT +1 keV +2.42e+17 +144.408 +21.527 +1 +0.00245 +Swift/XRT +1 keV +2.42e+17 +173.501 +26.453 +1 +0.00247 +Swift/XRT +1 keV +2.42e+17 +173.423 +27.678 +1 +0.00249 +Swift/XRT +1 keV +2.42e+17 +135.870 +20.254 +1 +Article number, page 15 of 20 + +A&A proofs: manuscript no. aas +Table B.1. Continued. +∆t +(days) +Telescope +Band/Filter +Frequency +(Hz) +Flux +(µJy) +Uncertainty +(µJy) +Detection? +(1 = yes) +0.00251 +Swift/XRT +1 keV +2.42e+17 +166.523 +25.390 +1 +0.00254 +Swift/XRT +1 keV +2.42e+17 +104.861 +16.356 +1 +0.00257 +Swift/XRT +1 keV +2.42e+17 +122.493 +18.260 +1 +0.00259 +Swift/XRT +1 keV +2.42e+17 +118.424 +17.654 +1 +0.00262 +Swift/XRT +1 keV +2.42e+17 +132.423 +20.190 +1 +0.00264 +Swift/XRT +1 keV +2.42e+17 +115.389 +17.201 +1 +0.00267 +Swift/XRT +1 keV +2.42e+17 +119.856 +19.577 +1 +0.00270 +Swift/XRT +1 keV +2.42e+17 +102.343 +17.113 +1 +0.00273 +Swift/XRT +1 keV +2.42e+17 +87.334 +13.938 +1 +0.00276 +Swift/XRT +1 keV +2.42e+17 +106.299 +16.965 +1 +0.00279 +Swift/XRT +1 keV +2.42e+17 +94.213 +15.036 +1 +0.00283 +Swift/XRT +1 keV +2.42e+17 +93.477 +13.935 +1 +0.00287 +Swift/XRT +1 keV +2.42e+17 +72.925 +13.765 +1 +0.00290 +Swift/XRT +1 keV +2.42e+17 +89.665 +14.310 +1 +0.00294 +Swift/XRT +1 keV +2.42e+17 +85.993 +14.046 +1 +0.00298 +Swift/XRT +1 keV +2.42e+17 +64.979 +11.965 +1 +0.00302 +Swift/XRT +1 keV +2.42e+17 +96.119 +15.340 +1 +0.00305 +Swift/XRT +1 keV +2.42e+17 +70.237 +13.596 +1 +0.00309 +Swift/XRT +1 keV +2.42e+17 +64.675 +13.529 +1 +0.13866 +Swift/XRT +1 keV +2.42e+17 +0.507 +0.116 +1 +0.20459 +Swift/XRT +1 keV +2.42e+17 +0.438 +0.114 +1 +0.20839 +Swift/XRT +1 keV +2.42e+17 +0.538 +0.093 +1 +0.27115 +Swift/XRT +1 keV +2.42e+17 +0.415 +0.109 +1 +0.27459 +Swift/XRT +1 keV +2.42e+17 +0.455 +0.103 +1 +0.33652 +Swift/XRT +1 keV +2.42e+17 +0.441 +0.115 +1 +0.33905 +Swift/XRT +1 keV +2.42e+17 +0.378 +0.100 +1 +0.34243 +Swift/XRT +1 keV +2.42e+17 +0.285 +0.075 +1 +0.34705 +Swift/XRT +1 keV +2.42e+17 +0.287 +0.076 +1 +0.35107 +Swift/XRT +1 keV +2.42e+17 +0.427 +0.087 +1 +0.40418 +Swift/XRT +1 keV +2.42e+17 +0.280 +0.073 +1 +0.40918 +Swift/XRT +1 keV +2.42e+17 +0.283 +0.074 +1 +0.41349 +Swift/XRT +1 keV +2.42e+17 +0.233 +0.061 +1 +0.42004 +Swift/XRT +1 keV +2.42e+17 +0.282 +0.060 +1 +0.56853 +Swift/XRT +1 keV +2.42e+17 +0.234 +0.061 +1 +0.60579 +Swift/XRT +1 keV +2.42e+17 +0.165 +0.043 +1 +0.61160 +Swift/XRT +1 keV +2.42e+17 +0.207 +0.046 +1 +0.68902 +Swift/XRT +1 keV +2.42e+17 +0.125 +0.027 +1 +0.74695 +Swift/XRT +1 keV +2.42e+17 +0.133 +0.035 +1 +0.77010 +Swift/XRT +1 keV +2.42e+17 +0.150 +0.032 +1 +2.50026 +Swift/XRT +1 keV +2.42e+17 +0.018 +0.005 +1 +0.00368 +MeerLICHT +q +5.169e+14 +59.71 +4.40 +1 +0.00485 +MeerLICHT +u +7.889e+14 +74.48 +13.03 +1 +0.00607 +MeerLICHT +q +5.169e+14 +106.67 +3.93 +1 +0.00719 +MeerLICHT +g +6.246e+14 +97.28 +5.38 +1 +0.00844 +MeerLICHT +q +5.169e+14 +106.67 +3.93 +1 +0.00962 +MeerLICHT +r +4.789e+14 +128.24 +7.09 +1 +0.01086 +MeerLICHT +q +5.169e+14 +97.28 +3.58 +1 +0.01210 +MeerLICHT +i +3.919e+14 +167.50 +13.88 +1 +0.01320 +MeerLICHT +q +5.169e+14 +80.91 +3.73 +1 +0.01447 +MeerLICHT +z +3.276e+14 +124.75 +28.72 +1 +0.01575 +MeerLICHT +q +5.169e+14 +74.48 +4.12 +1 +0.01823 +MeerLICHT +q +5.169e+14 +57.55 +3.71 +1 +0.01950 +MeerLICHT +g +6.246e+14 +50.59 +5.59 +1 +0.02076 +MeerLICHT +q +5.169e+14 +56.50 +3.64 +1 +0.02193 +MeerLICHT +r +4.789e+14 +68.55 +7.58 +1 +0.02311 +MeerLICHT +q +5.169e+14 +58.62 +3.78 +1 +0.02438 +MeerLICHT +i +3.919e+14 +77.27 +14.23 +1 +0.02564 +MeerLICHT +q +5.169e+14 +61.38 +3.96 +1 +0.02691 +MeerLICHT +z +3.276e+14 +100.01 +32.24 +1 +0.02818 +MeerLICHT +q +5.169e+14 +69.19 +5.10 +1 +Article number, page 16 of 20 + +S. de Wet et al.: The triple-peaked afterglow of GRB 210731A from X-ray to radio frequencies +Table B.1. Continued. +∆t +(days) +Telescope +Band/Filter +Frequency +(Hz) +Flux +(µJy) +Uncertainty +(µJy) +Detection? +(1 = yes) +0.02936 +MeerLICHT +u +7.889e+14 +47.87 +15.87 +1 +0.03061 +MeerLICHT +q +5.169e+14 +67.30 +4.34 +1 +0.03189 +MeerLICHT +g +6.246e+14 +67.30 +7.44 +1 +0.03318 +MeerLICHT +q +5.169e+14 +64.87 +3.58 +1 +0.03438 +MeerLICHT +r +4.789e+14 +78.71 +7.97 +1 +0.03559 +MeerLICHT +q +5.169e+14 +84.73 +4.68 +1 +0.03684 +MeerLICHT +i +3.919e+14 +86.30 +12.72 +1 +0.03811 +MeerLICHT +q +5.169e+14 +82.42 +4.55 +1 +0.03942 +MeerLICHT +z +3.276e+14 +128.24 +35.43 +1 +0.04071 +MeerLICHT +q +5.169e+14 +87.91 +4.05 +1 +0.04199 +MeerLICHT +u +7.889e+14 +55.47 +17.37 +1 +0.04321 +MeerLICHT +q +5.169e+14 +86.30 +3.97 +1 +0.04450 +MeerLICHT +g +6.246e+14 +77.99 +6.46 +1 +0.04577 +MeerLICHT +q +5.169e+14 +90.37 +4.16 +1 +0.04696 +MeerLICHT +r +4.789e+14 +88.72 +8.17 +1 +0.04815 +MeerLICHT +q +5.169e+14 +93.76 +5.18 +1 +0.04939 +MeerLICHT +i +3.919e+14 +154.18 +12.78 +1 +0.05064 +MeerLICHT +q +5.169e+14 +92.90 +4.28 +1 +0.05189 +MeerLICHT +z +3.276e+14 +197.71 +38.24 +1 +0.05312 +MeerLICHT +q +5.169e+14 +97.28 +3.58 +1 +0.05430 +MeerLICHT +u +7.889e+14 +76.56 +18.33 +1 +0.05548 +MeerLICHT +q +5.169e+14 +106.67 +4.91 +1 +0.05675 +MeerLICHT +g +6.246e+14 +89.54 +6.60 +1 +0.05801 +MeerLICHT +q +5.169e+14 +103.76 +4.78 +1 +0.05919 +MeerLICHT +r +4.789e+14 +125.90 +9.28 +1 +0.06037 +MeerLICHT +q +5.169e+14 +101.87 +4.69 +1 +0.06149 +MeerLICHT +i +3.919e+14 +124.75 +12.64 +1 +0.06275 +MeerLICHT +q +5.169e+14 +101.87 +5.63 +1 +0.06398 +MeerLICHT +z +3.276e+14 +164.45 +46.95 +1 +0.06522 +MeerLICHT +q +5.169e+14 +100.01 +5.53 +1 +0.06766 +MeerLICHT +q +5.169e+14 +100.93 +6.51 +1 +0.06893 +MeerLICHT +g +6.246e+14 +75.86 +7.69 +1 +0.07020 +MeerLICHT +q +5.169e+14 +98.18 +5.43 +1 +0.07143 +MeerLICHT +r +4.789e+14 +115.88 +11.74 +1 +0.07275 +MeerLICHT +q +5.169e+14 +99.09 +5.48 +1 +0.07403 +MeerLICHT +i +3.919e+14 +151.37 +16.73 +1 +0.07527 +MeerLICHT +q +5.169e+14 +100.93 +5.58 +1 +0.07651 +MeerLICHT +z +3.276e+14 +210.88 +40.79 +1 +0.08293 +MeerLICHT +q +5.169e+14 +97.28 +6.27 +1 +0.08413 +MeerLICHT +r +4.789e+14 +114.82 +14.81 +1 +0.08532 +MeerLICHT +q +5.169e+14 +103.76 +6.69 +1 +0.08659 +MeerLICHT +i +3.919e+14 +129.43 +23.84 +1 +0.08914 +MeerLICHT +z +3.276e+14 +307.63 +62.33 +1 +0.09540 +MeerLICHT +q +5.169e+14 +120.23 +7.75 +1 +0.09662 +MeerLICHT +r +4.789e+14 +121.35 +14.53 +1 +0.10654 +MeerLICHT +g +6.246e+14 +60.82 +9.52 +1 +0.10781 +MeerLICHT +q +5.169e+14 +92.90 +6.85 +1 +0.10899 +MeerLICHT +r +4.789e+14 +114.82 +10.58 +1 +0.11016 +MeerLICHT +q +5.169e+14 +93.76 +6.05 +1 +0.11138 +MeerLICHT +i +3.919e+14 +110.67 +17.33 +1 +0.11252 +MeerLICHT +q +5.169e+14 +98.18 +6.33 +1 +0.11377 +MeerLICHT +z +3.276e+14 +157.05 +40.50 +1 +0.11503 +MeerLICHT +q +5.169e+14 +99.09 +6.39 +1 +0.11754 +MeerLICHT +q +5.169e+14 +104.72 +6.75 +1 +0.11877 +MeerLICHT +g +6.246e+14 +69.83 +10.29 +1 +0.12003 +MeerLICHT +q +5.169e+14 +125.90 +6.96 +1 +0.12121 +MeerLICHT +r +4.789e+14 +140.61 +14.25 +1 +0.12248 +MeerLICHT +q +5.169e+14 +108.65 +10.01 +1 +0.13362 +MeerLICHT +r +4.789e+14 +131.83 +14.57 +1 +0.13482 +MeerLICHT +q +5.169e+14 +114.82 +9.52 +1 +Article number, page 17 of 20 + +A&A proofs: manuscript no. aas +Table B.1. Continued. +∆t +(days) +Telescope +Band/Filter +Frequency +(Hz) +Flux +(µJy) +Uncertainty +(µJy) +Detection? +(1 = yes) +0.13604 +MeerLICHT +i +3.919e+14 +134.28 +25.97 +1 +0.13725 +MeerLICHT +q +5.169e+14 +113.77 +11.53 +1 +0.13958 +MeerLICHT +q +5.169e+14 +100.01 +11.05 +1 +0.14195 +MeerLICHT +q +5.169e+14 +127.07 +8.19 +1 +0.14305 +MeerLICHT +g +6.246e+14 +97.28 +11.65 +1 +0.14430 +MeerLICHT +q +5.169e+14 +125.90 +6.96 +1 +0.14550 +MeerLICHT +r +4.789e+14 +159.97 +14.73 +1 +0.14670 +MeerLICHT +q +5.169e+14 +136.78 +7.56 +1 +0.14797 +MeerLICHT +i +3.919e+14 +169.05 +26.47 +1 +0.15050 +MeerLICHT +z +3.276e+14 +275.44 +48.20 +1 +0.16412 +MeerLICHT +q +5.169e+14 +157.05 +21.70 +1 +0.16650 +MeerLICHT +q +5.169e+14 +141.91 +27.45 +1 +0.16775 +MeerLICHT +g +6.246e+14 +119.13 +24.14 +1 +0.913 +MeerLICHT +r +4.789e+14 +25.12 +4.40 +1 +0.915 +MeerLICHT +q +5.169e+14 +24.21 +1.56 +1 +0.915 +MeerLICHT +g +6.246e+14 +16.75 +3.70 +1 +0.918 +MeerLICHT +i +3.919e+14 +41.31 +8.37 +1 +1.018 +MeerLICHT +r +4.789e+14 +21.09 +4.66 +1 +1.018 +MeerLICHT +q +5.169e+14 +23.12 +1.28 +1 +1.020 +MeerLICHT +g +6.246e+14 +17.70 +2.77 +1 +1.023 +MeerLICHT +i +3.919e+14 +32.21 +8.31 +1 +2.03 +MeerLICHT +q +5.169e+14 +7.52 +0.83 +1 +3.00 +MeerLICHT +q +5.169e+14 +3.53 +0.68 +1 +0.00277 +Swift/UVOT +white +7.488e+14 +16.29 +4.80 +1 +0.13848 +Swift/UVOT +v +5.511e+14 +120.23 +19.93 +1 +0.20686 +Swift/UVOT +b +6.848e+14 +92.90 +7.70 +1 +0.27287 +Swift/UVOT +uvm2 +1.319e+15 +13.43 +2.60 +1 +0.340 +Swift/UVOT +uvw1 +1.115e+15 +16.00 +2.21 +1 +0.349 +Swift/UVOT +u +8.583e+14 +47.43 +3.93 +1 +0.408 +Swift/UVOT +uvw2 +1.401e+15 +5.15 +1.23 +1 +0.418 +Swift/UVOT +v +5.511e+14 +87.10 +12.03 +1 +0.560 +Swift/UVOT +b +6.848e+14 +36.31 +7.69 +1 +0.606 +Swift/UVOT +uvm2 +1.319e+15 +3.53 +1.40 +1 +0.650 +Swift/UVOT +uvw1 +1.115e+15 +5.30 +1.37 +1 +0.694 +Swift/UVOT +u +8.583e+14 +13.80 +3.94 +1 +0.747 +Swift/UVOT +uvw2 +1.401e+15 +3.40 +1.13 +0 +0.757 +Swift/UVOT +v +5.511e+14 +27.04 +11.21 +1 +0.802 +Swift/UVOT +b +6.848e+14 +21.09 +7.19 +1 +2.79 +Swift/UVOT +uvm2 +1.319e+15 +1.91 +0.64 +0 +2.90 +Swift/UVOT +uvw1 +1.115e+15 +2.25 +0.75 +0 +3.20 +Swift/UVOT +u +8.583e+14 +7.24 +2.41 +0 +3.21 +Swift/UVOT +b +6.848e+14 +15.42 +5.14 +0 +3.21 +Swift/UVOT +white +7.488e+14 +3.56 +1.19 +0 +3.21 +Swift/UVOT +v +5.511e+14 +35.32 +11.77 +0 +3.71 +Swift/UVOT +uvw2 +1.401e+15 +1.79 +0.60 +0 +0.18419 +GROND +g′ +6.536e+14 +116.96 +1.08 +1 +0.18419 +GROND +r′ +4.820e+14 +148.60 +1.37 +1 +0.18419 +GROND +i′ +3.924e+14 +188.81 +1.74 +1 +0.18419 +GROND +z′ +3.335e+14 +227.00 +2.09 +1 +0.18437 +GROND +J +2.418e+14 +307.63 +5.67 +1 +0.18437 +GROND +H +1.820e+14 +428.57 +7.89 +1 +0.18437 +GROND +K +1.381e+14 +554.66 +15.33 +1 +1.225 +GROND +g′ +6.536e+14 +13.68 +0.13 +1 +1.225 +GROND +r′ +4.820e+14 +18.20 +0.17 +1 +1.225 +GROND +i′ +3.924e+14 +22.91 +0.42 +1 +1.225 +GROND +z′ +3.335e+14 +27.29 +0.50 +1 +1.225 +GROND +J +2.418e+14 +46.99 +2.60 +1 +1.225 +GROND +H +1.820e+14 +57.02 +5.78 +1 +1.225 +GROND +K +1.381e+14 +58.62 +17.82 +1 +2.21 +GROND +g′ +6.536e+14 +5.11 +0.09 +1 +Article number, page 18 of 20 + +S. de Wet et al.: The triple-peaked afterglow of GRB 210731A from X-ray to radio frequencies +Table B.1. Continued. +∆t +(days) +Telescope +Band/Filter +Frequency +(Hz) +Flux +(µJy) +Uncertainty +(µJy) +Detection? +(1 = yes) +2.21 +GROND +r′ +4.820e+14 +6.25 +0.17 +1 +2.21 +GROND +i′ +3.924e+14 +7.24 +0.40 +1 +2.21 +GROND +z′ +3.335e+14 +9.64 +0.53 +1 +2.21 +GROND +J +2.418e+14 +20.14 +2.78 +1 +2.21 +GROND +H +1.820e+14 +22.29 +5.75 +1 +2.21 +GROND +K +1.381e+14 +59.71 +19.90 +0 +5.25 +GROND +g′ +6.536e+14 +1.51 +0.14 +1 +5.25 +GROND +r′ +4.820e+14 +1.45 +0.17 +1 +5.25 +GROND +i′ +3.924e+14 +1.14 +0.38 +0 +5.25 +GROND +z′ +3.335e+14 +1.57 +0.52 +0 +5.25 +GROND +J +2.418e+14 +8.32 +2.77 +0 +5.25 +GROND +H +1.820e+14 +15.56 +5.19 +0 +5.25 +GROND +K +1.381e+14 +49.21 +16.40 +0 +285.0 +GROND +g′ +6.536e+14 +0.40 +0.07 +1 +285.0 +GROND +r′ +4.820e+14 +0.48 +0.09 +1 +285.0 +GROND +i′ +3.924e+14 +0.76 +0.25 +0 +285.0 +GROND +J +2.418e+14 +6.31 +2.10 +0 +285.0 +GROND +H +1.820e+14 +10.97 +3.66 +0 +285.0 +GROND +K +1.381e+14 +27.54 +9.18 +0 +0.385 +KAIT +clear +4.722e+14 +104.72 +4.82 +1 +1.183 +VLT +r +4.830e+14 +18.54 +0.34 +1 +1.186 +VLT +g +6.394e+14 +13.18 +0.24 +1 +1.189 +VLT +z +3.124e+14 +28.84 +0.80 +1 +2.09 +NOT +r +4.830e+14 +6.79 +0.25 +1 +2.10 +NOT +z +3.124e+14 +12.94 +1.07 +1 +6.08 +NOT +r +4.830e+14 +1.05 +0.18 +1 +10.8 +MeerKAT +L +1.4e+09 +42.0 +14.0 +0 +18.2 +VLA +C +6.0e+09 +136.0 +32.0 +1 +18.2 +VLA +X +1.0e+10 +182.0 +8.0 +1 +34.1 +MeerKAT +L +1.4e+09 +41.1 +13.7 +0 +34.2 +VLA +C +6.0e+09 +140.0 +10.0 +1 +34.2 +VLA +X +1.0e+10 +199.0 +9.0 +1 +59.7 +MeerKAT +L +1.4e+09 +40.2 +13.4 +0 +67.1 +VLA +C +6.0e+09 +102.0 +9.0 +1 +67.1 +VLA +X +1.0e+10 +50.0 +15.0 +1 +118.0 +VLA +C +6.0e+09 +66.0 +11.0 +1 +118.0 +VLA +X +1.0e+10 +37.0 +7.0 +1 +Notes. All times are relative to the Swift/BAT trigger time. X-ray, optical and radio data are separated by horizontal lines. Detections are all at least +at the 3σ level except for the first UVOT/white detection, which was at the 2.3σ level. Optical and radio upper limits are at the 3σ level. +Article number, page 19 of 20 + +A&A proofs: manuscript no. aas +p = 2.76+0.03 +−0.04 +−1.35 +−1.20 +−1.05 +−0.90 +−0.75 +log(ϵe) +log(ϵe) = −0.93+0.04 +−0.08 +−4 +−3 +−2 +−1 +log(ϵB) +log(ϵB) = −2.25+0.61 +−0.25 +−2.4 +−2.0 +−1.6 +−1.2 +−0.8 +log(A⋆) +log(A⋆) = −1.50+0.10 +−0.23 +1.2 +1.6 +2.0 +2.4 +log(EK,iso,52) +log(EK,iso,52) = 1.48+0.13 +−0.18 +0.200 +0.225 +0.250 +0.275 +AV,host +AV,host = 0.23+0.01 +−0.01 +2.64 +2.70 +2.76 +2.82 +2.88 +p +1.6 +2.4 +3.2 +4.0 +4.8 +log(tjet) +−1.35 +−1.20 +−1.05 +−0.90 +−0.75 +log(ϵe) +−4 +−3 +−2 +−1 +log(ϵB) +−2.4 +−2.0 +−1.6 +−1.2 +−0.8 +log(A⋆) +1.2 +1.6 +2.0 +2.4 +log(EK,iso,52) +0.200 +0.225 +0.250 +0.275 +AV,host +1.6 +2.4 +3.2 +4.0 +4.8 +log(tjet) +log(tjet) = 2.26+0.46 +−0.27 +Fig. A.2. Same as Fig. 8, but for the fit including the L-band upper limits. +Article number, page 20 of 20 + diff --git a/AdFLT4oBgHgl3EQfEy_H/content/tmp_files/load_file.txt b/AdFLT4oBgHgl3EQfEy_H/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..579b5860317d34dbd32f93a459656fab18bef473 --- /dev/null +++ b/AdFLT4oBgHgl3EQfEy_H/content/tmp_files/load_file.txt @@ -0,0 +1,2992 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf,len=2991 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' aas ©ESO 2023 January 31, 2023 The triple-peaked afterglow of GRB 210731A from X-ray to radio frequencies⋆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' de Wet1,⋆⋆, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Laskar2, 3, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Groot1, 3, 4, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Cavallaro1, 5, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Nicuesa Guelbenzu6, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Chastain7, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Izzo8, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Levan3, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Malesani3, 9, 10, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Monageng1, 4, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' van der Horst7, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Zheng11, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Bloemen3, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Filippenko11, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Kann12, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Klose6, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Pieterse3, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Rau13, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Vreeswijk3, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Woudt1, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Zhu14 1 Inter-University Institute for Data Intensive Astronomy & Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch, 7701, South Africa 2 Department of Physics & Astronomy, University of Utah, Salt Lake City, UT 84112, USA 3 Department of Astrophysics/IMAPP, Radboud University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='O.' metadata={'source': 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+page_content=' Denmark 9 Cosmic Dawn Center (DAWN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Denmark 10 Niels Bohr Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' University of Copenhagen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Rådmandsgade 62-64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Copenhagen N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Denmark 11 Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' CA 94720-3411,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' USA 12 Hessian Research Cluster ELEMENTS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Giersch Science Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Max-von-Laue-Strasse 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Goethe University Frankfurt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Campus Riedberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 60438 Frankfurt am Main,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Germany 13 Max-Planck-Institut für Extraterrestrische Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Giessenbachstraße,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Germany 14 Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Huazhong University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Wuhan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 430074,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' China Received January 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' accepted January 31, 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' GRB 210731A was a long-duration (T90 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 s) gamma-ray burst discovered by the Burst Alert Telescope (BAT) aboard the Neil Gehrels Swift observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Swift triggered the wide-field, robotic MeerLICHT optical telescope in Sutherland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' it began observing the BAT error circle 286 seconds after the Swift trigger and discovered the optical afterglow of GRB 210731A in its first 60-second q-band exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Multi-colour observations of the afterglow with MeerLICHT revealed a light curve that showed three peaks of similar brightness within the first four hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The unusual optical evolution prompted multi-wavelength follow-up observations that spanned from X-ray to radio frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We present the results of our follow-up campaign and interpret our observations in the framework of the synchrotron forward shock model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We performed temporal and spectral fits to determine the spectral regime and external medium density profile, and per- formed detailed multi-wavelength theoretical modelling of the afterglow following the last optical peak at ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 days to determine the intrinsic blast wave parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We find a preference for a stellar wind density profile consistent with a massive star origin, while our theoretical modelling results in fairly typical shock microphysics parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Based on the energy released in γ rays and the kinetic energy in the blast wave, we determine a low radiative efficiency of η ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The first peak in the optical light curve is likely the onset of the afterglow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We find that energy injection into the forward shock offers the simplest explanation for the subsequent light curve evolution, and that the blast wave kinetic energy increasing by a factor of ∼1000 from the first peak to the last peak is indicative of substantial energy injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Our highest-likelihood theoretical model over-predicts the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 GHz flux by a factor of approximately three with respect to our upper limits, possibly implying a population of thermal electrons within the shocked region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Gamma-ray burst: individual: GRB 210731A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Introduction Gamma-ray bursts (GRBs) are the most energetic explosions in the Universe, with isotropic γ-ray energies of up to ∼1055 erg (Kumar & Zhang 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Long-duration bursts are typically as- sociated with the core collapse of massive stars (Colgate 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' ⋆ Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1 is available in electronic form at the CDS via anonymous ftp to cdsarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='fr (130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5) or via https://cdsarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='fr/cgi-bin/qcat?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='J/A+A ⋆⋆ Corresponding author e-mail: DWTSIM002@myuct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='za Woosley 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' van Paradijs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Galama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 1998a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Woosley & Bloom 2006), where the compact object remnant acts as a central engine powering a collimated relativistic jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' In the fireball shock model (Rees & Meszaros 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Meszaros & Rees 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Mészáros & Rees 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Sari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 1998), internal shocks within the expanding ejecta are the source of the prompt γ-ray emission and the interaction of the relativistic outflow with the circumburst medium (the external forward shock) is the source of the afterglow emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Article number, page 1 of 20 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='11985v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='HE] 27 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' aas Prior to the launch of the Neil Gehrels Swift Observatory (Gehrels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2004) in 2004, observations of afterglows showed broad agreement with the basic external shock model (Galama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 1998b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Panaitescu & Kumar 2001, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Yost et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Rapid X-ray and UV/optical follow-up of GRB triggers with Swift have revealed rich features in early-time X-ray and opti- cal light curves that challenge the standard theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Similar to the ‘canonical’ X-ray light curve (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Nousek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2006), optical light curves have been decomposed into a number of distinct components that arise from different emis- sion sites and physical mechanisms (see the synthetic light curve in Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' In addition to the normal and jet break de- cay segments explained by the standard forward shock model, onset bumps, steep decay segments, flares, and late-time re- brightenings have been observed in optical afterglows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Early on- set bumps with a smooth transition to the normal decay seg- ment are regarded as the onset of afterglow and were predicted within the standard theory (Sari & Piran 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Kobayashi & Zhang 2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' steep decay segments early on (some with an addi- tional steep rise) have been attributed to reverse shock emission (Mészáros & Rees 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Sari & Piran 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Yi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' similar to X-ray light curves, shallow decay and plateau segments and flares have been observed in a number of bursts (Mangano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Greiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Swenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' late-time re-brightenings have also been observed in some optical afterglows (Nardini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2013), with a few having rarer simultaneous X-ray re-brightenings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' GRB 120326A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Melandri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Urata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Laskar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Furthermore, small-scale bumps and wiggles have been seen superposed over the larger-scale light curve features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' GRBs 021004 and 030329;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Holland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Lipkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Proposed explanations for these additional features include inhomogeneities in the circumburst medium, multiple-component jets, structured jets, varying mi- crophysical parameters, and energy injection into the forward shock (Lazzati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Dai & Wu 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Mundell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Racusin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Filgas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Mészáros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Zhang & Mészáros 2002a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Rossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Ioka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Fan & Piran 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Granot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Sari & Mészáros 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Zhang & Mészáros 2002b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Björnsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Laskar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Here we report on multi-wavelength observations of the long-duration gamma-ray burst GRB 210731A discovered by Swift (Gehrels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Our early-time MeerLICHT optical observations show a complex light curve evolution with an ini- tial smooth bump followed by two further re-brightenings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We combine our observations with X-ray and radio data that span 200 seconds to 118 days post-trigger and interpret our observa- tions in the traditional synchrotron forward shock framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We investigate the nature of the early optical light curve evolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We adopt a Λ cold dark matter cosmology with Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='31, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='69, and H0 = 68 km s−1 Mpc−1 (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' All reported magnitudes are in the AB magnitude system unless stated otherwise, and errors are reported at the 1σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Observations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Prompt gamma-ray emission The Swift Burst Alert Telescope (BAT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Barthelmy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2005) was triggered by GRB 210731A at 22:21:08 UT on 2021 July 31, with the mask-weighted 15–350 keV light curve showing a single-pulse structure of duration T90 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='8 s (Stamatikos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2021), making GRB 210731A a long-duration GRB under the traditional > 2 s duration limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' GRB 210731A triggered the Fermi Gamma-ray Burst Monitor (GBM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Meegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Lesage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2021) one second earlier than Swift/BAT, with the 10-1000 keV light curve showing a single pulse with duration1 T90 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='9±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 s, in agreement with the BAT duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The time- averaged spectra for both BAT and GBM were best-fitted with a power law function and exponential high-energy cutoff with pho- ton indices of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='59 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1, and cutoff energies of (107±27) keV and (175±11) keV, respectively (Stamatikos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Lesage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The 10-1000 keV GBM fluence2 inte- grated over the burst duration was (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='06)×10−6 erg cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Using a measured afterglow redshift of z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2525 obtained by X-Shooter on the Very Large Telescope at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='19 days post-trigger (Kann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2021), this corresponds to an isotropic-equivalent γ-ray energy of Eγ,iso = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='03) × 1052 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We take the Swift/BAT trigger time as T0 for this burst and reference all future times with respect to this T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' X-ray observations The Swift X-Ray Telescope (XRT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Burrows et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2005) started observing the field of GRB 210731A 201 seconds post-trigger, finding a bright new X-ray source consistent with the BAT po- sition (Troja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The initial 62 seconds of data were obtained in windowed timing (WT) mode after which Swift had to slew away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Data capture recommenced in photon counting (PC) mode at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 hours post-trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We obtained the X-ray light curve and spectra from the online Swift-XRT GRB Catalogue3 (Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2007, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The Burst Analyser (Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2010) count-rate light curve showed that the X-ray flux was decreasing rapidly during the WT-mode observations with a spectrum that hardened from a photon index of ΓX = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 over 60 sec- onds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Once data capture resumed in PC mode at 10 ks, the X-ray light curve was in a shallow decay phase before declining more steeply at ∼20 ks post-trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We fitted the PC-mode spectrum with a photoelectrically ab- sorbed power-law model (tbabs*ztbabs*pow) in Xspec ver- sion 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0, fixing the source redshift at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2525 and Galactic hydrogen column density at NGal H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='15 × 1021 cm−2 for consis- tency with the online fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The fitted spectrum was characterised by a photon index of Γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='11 with a host galaxy column density of Nhost H = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='46+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='99 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='56 × 1021 cm−2 and C-stat 182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1 for 213 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' There were insufficient photons in the PC-mode light curve for time-resolved analysis and to test for spectral evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We converted the PC-mode count-rate light curve to a 1 keV flux density light curve using the spectral in- dex from our PC-mode spectral fit of βX ≡ 1 − ΓX ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00 and the online unabsorbed count-to-flux conversion factor of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='36 × 10−11 erg cm−2 ct−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We performed a similar procedure for the WT-mode data using the spectral parameters in the automated fit on the Swift website, with a photon index of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='07 and unabsorbed count-to-flux conversion factor of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='69 × 10−11 erg cm−2 ct−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 1 Obtained from the online Fermi GBM Burst Catalog (von Kienlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2 See footnote 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3 The Burst Analyser for GRB 210731A is available on the UK Swift Data Science Centre website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Article number, page 2 of 20 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' de Wet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' : The triple-peaked afterglow of GRB 210731A from X-ray to radio frequencies 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Optical/near-infrared observations The fully robotic, 60 cm MeerLICHT optical telescope (Bloe- men et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2016) was automatically triggered by Swift/BAT and began observing the field of GRB 210731A 286 seconds after the BAT trigger, taking 60 second exposures in the u, g, r, i, z, and q optical bands (where the q band is roughly equivalent to g + r), following the sequence quqgqrqiqz in order to obtain high cadence coverage in the wider and more sensitive q band with quasi-simultaneous multi-colour coverage of the evolving afterglow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Comparison of the first q-band image with an exist- ing MeerLICHT reference image revealed a new transient can- didate at α = 20h01m13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='19s, δ = −28d03m40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='10s (J2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' This position was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3′′ away from the refined XRT position (D’Ai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2021), confirming the new source as the optical afterglow of GRB 210731A (de Wet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' These observations con- tinued until the target set, approximately 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='29 hours post-trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Four cycles of the same filter sequence were obtained the fol- lowing night in two time intervals separated by ∼2 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Since the afterglow was by this point in a declining phase and below the 60 second single-exposure detection limit, repeated q-band exposures were taken on the nights of 2021 August 2 and 3 in order to track the optical light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The MeerLICHT pipeline (Vreeswijk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=', in prep) was used to perform standard charge- coupled device (CCD) reduction tasks as well as astrometry and point-spread function (PSF) photometry, producing a catalogue file containing all 5σ source detections for each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' For im- ages where the afterglow was fainter than 5σ above the back- ground we used forced photometry to obtain magnitudes that were at least at the 3σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Images from the night of 2021 August 1 onwards were co-added to produce more significant detections or deeper upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The Swift UltraViolet and Optical Telescope (UVOT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Roming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2005) took a single 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='7 second exposure in the white filter beginning 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 seconds after the BAT trig- ger but did not continue observing the field of the GRB until 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='27 hours after the trigger, whereafter it was observed with mul- tiple filters intermittently over the next five days (Troja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Kuin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We performed aperture photometry on the Swift/UVOT data using standard analysis tools from the HEASoft (Nasa High Energy Astrophysics Science Archive Re- search Center (Heasarc) 2014) Swift FTOOLS software pack- age (version 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='29c4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We extracted magnitudes using the tool uvotsource with a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5′′ radius aperture centred on the after- glow position, and a nearby background aperture with a 10′′ ra- dius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' A total of 64 individual exposures in all seven UVOT filters were taken over the course of the follow-up campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We co- added exposures in the same filter with clear detections but taken close to each other temporally using uvotimsum in order to pro- duce more significant detections, and once the afterglow became too faint to detect in individual exposures we co-added images within wider time baselines in order to provide the deepest lim- iting magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The afterglow of GRB 210731A was observed simultane- ously in the g′r′i′z′JHK bands with the Gamma-Ray Burst Optical Near-Infrared Detector (GROND;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Greiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2008) mounted at the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 m MPG telescope at the European Southern Observatory (ESO) La Silla Observatory in Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The afterglow was clearly detected in all bands in the first epoch of observations taken 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 hours after the GRB trigger (Nicuesa Guelbenzu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' A further three epochs were obtained at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='225, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='214, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='253 days post-trigger (Nicuesa Guelbenzu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We also obtained deep host-galaxy observations at 285 days that 4 Available at https://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='gov/docs/software/lheasoft/ yielded detections in the g′ and r′ bands, and which we regard as the host-galaxy flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The multi-colour GROND data were anal- ysed through standard PSF photometry using DAOPHOT (Stet- son 1987) and ALLSTAR tasks of IRAF (Tody 1993), in a simi- lar way to the procedure described in Krühler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The optical data were calibrated against the Pan-STARRS catalogue5 (Chambers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2016), while for the near-infrared (NIR) bands, photometric calibration was performed against the 2MASS cat- alogue (Skrutskie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2006), resulting in a typical absolute ac- curacy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='04 mag in g′r′i′z′, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='06 mag in JH and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='08 mag in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The 76 cm Katzman Automatic Imaging Telescope located at the Lick Observatory (KAIT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Filippenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2001) obtained 20 × 60 second exposures in the clear band (similar to R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' see Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2003), starting ∼9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='04 hours after the BAT trigger (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' All images were reduced and co-added using a custom pipeline (Ganeshalingam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Stahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2019), whereafter PSF photometry was performed on the co-added im- age using DAOPHOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Several nearby stars were chosen from the Pan-STARRS1 catalogue for flux calibration, with their magni- tudes transformed into Landolt magnitudes following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 6 of Tonry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The optical afterglow of GRB 210731A was clearly detected in the co-added image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Images in the SDSS g, r, and z filters were obtained at a single epoch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='18 days post-trigger with the acquisition camera of the X-shooter spectrograph, mounted on the ESO Very Large Telescope (VLT) UT3 (Melipal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Reduction was carried out us- ing standard procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' For the z-band image, a fringe correc- tion was applied, using a template fringe pattern provided by the observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We also observed the afterglow at two epochs in the r and z bands with the Nordic Optical Telescope (NOT) equipped with the ALFOSC imager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The images were reduced following standard procedures including subtraction of a master bias and correction with sky flats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Magnitudes were measured using aperture photometry, and photometric calibration was car- ried out against the Pan-STARRS catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We show all UV/optical/NIR photometry separated by in- strument and filter in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Radio observations We obtained three epochs of radio continuum observations with the MeerKAT radio telescope (Jonas & MeerKAT Team 2016) in the L band (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 GHz) through director’s discretionary time (DDT) proposal DDT-20120810-SD-01 (PI de Wet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Each ob- servation had a total integration time on source of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='78 hours, us- ing J1939–6342 as the flux and bandpass calibrator and J1924– 2914 as the gain calibrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' All data were reduced using the oxkat pipeline6 (Heywood 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' No radio afterglow was detected at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 GHz across the three epochs at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='8, 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1 and 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='7 days post- trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The RMS noise was ≈ 14 µJy at the 1σ-level in each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We take upper limits on the afterglow flux as three times the RMS noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Radio continuum observations were also obtained with the Karl G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Jansky Very Large Array (JVLA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Perley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2011) in the C and X bands (centred on 6 and 10 GHz) through DDT proposals 21B-333 and 21B-342 (PI de Wet) at four epochs spanning 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 to 118 days post-trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The total integration time per observation was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='44 hours in each band, with 3C286 used as the flux and bandpass calibrator and J1924–2914 as 5 See http://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='edu/panstarrs/search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='php.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 6 See https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='com/IanHeywood/oxkat/blob/master/README.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='md and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Article number, page 3 of 20 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' aas 102 103 104 105 Time since trigger (s) 10−3 10−2 10−1 100 101 Time since trigger (days) 10 12 14 16 18 20 22 24 26 28 30 AB magnitude MeerLICHT/u + 2 MeerLICHT/g + 1 MeerLICHT/q MeerLICHT/r − 1 MeerLICHT/i − 2 MeerLICHT/z − 3 UVOT/uvw2 + 5 UVOT/uvm2 + 4 UVOT/uvw1 + 3 UVOT/u + 2 UVOT/b + 1 UVOT/v + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 UVOT/white + 2 GROND/g′ + 1 GROND/r′ − 1 GROND/i′ − 2 GROND/z′ − 3 GROND/J − 4 GROND/H − 5 GROND/K − 6 VLT/g + 1 VLT/r − 1 VLT/z − 3 NOT/r − 1 NOT/z − 3 KAIT/clear − 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Combined UV/optical/NIR photometry of GRB 210731A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We only show detections where all magnitudes are in the AB system and have not been corrected for Galactic extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Times are relative to the Swift/BAT trigger time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' the complex gain calibrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We performed preliminary imag- ing on the pipeline-calibrated measurements sets using standard Common Astronomy Software Applications (CASA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' McMullin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2007) procedures and detected the radio afterglow to GRB 210731A in all four epochs at 10 GHz and in all but the first epoch at 6 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The first epoch National Radio Astronomy Observatory (NRAO) -calibrated measurement set failed to pass internal qual- ity thresholds for science usability so we chose to calibrate manually from the raw data to obtain more accurate flux mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We used CASA version 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 and performed imag- ing with the task tclean and flux measurements with the task imfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We obtained satisfactory results with our X-band cal- ibration, but the C-band calibration contained persistent phase errors as a result of de-correlation problems during observing, and the measured flux of all sources in the field was substan- tially lower than in subsequent epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Despite these problems, imaging on the re-calibrated data showed a radio source at the af- terglow position, in contrast with preliminary imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We there- fore adopted the following approach to calculate the flux of the afterglow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We identified six point-like sources present at all four epochs in the C band and measured their fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' For each source a flux correction factor could then be calculated between the first epoch flux and the flux in each subsequent epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The correct- ing factors ranged from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2, with a mean value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='77 and standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' No obvious trends in the cor- recting factor were found as a function of source brightness or offset from the image centre across all epochs, so we took the mean correcting factor as the flux correcting factor to apply to the measured afterglow flux in the first epoch image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We incor- porate the standard deviation of the correction factor as an addi- tional source of systematic uncertainty in the flux measurement from the first epoch in the C band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The third epoch X-band measurement set also had major phase issues, so we performed the same procedure as for the C- band first epoch data, calibrating the raw data and determining a mean flux correcting factor to apply to our afterglow measure- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Only two sources were used in determining the mean cor- recting factor as the X-band images had far fewer sources than the C-band images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The correcting factors ranged from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6, with a mean value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='47 and standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' All X-ray, optical, and radio flux measurements associated with GRB 210731A are presented in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Afterglow temporal and spectral analysis We interpret our combined multi-wavelength data in the frame- work of synchrotron radiation emitted by electrons accelerated to a power-law distribution in energy behind the forward shock,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' with N(γ) ∝ γ−p for γ > γmin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' with γmin being the minimum Lorentz factor of electrons in the distribution and p being the electron energy index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' which we assume to be bounded between 2 and 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' though values of less than 2 have been suggested in the literature (Dai & Cheng 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Panaitescu & Kumar 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The synchrotron spectra are characterised by power-law seg- ments that join at a number of break frequencies, namely the synchrotron self-absorption frequency νsa, the characteristic syn- chrotron frequency νm corresponding to emission from γmin elec- Article number, page 4 of 20 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' de Wet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' : The triple-peaked afterglow of GRB 210731A from X-ray to radio frequencies trons, and the cooling frequency νc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The orderings of the spectral breaks depend on the hydrodynamic evolution of the forward shock, which is described by the Blandford & McKee (1976) spherical self-similar solution of an adiabatic relativistic blast wave expanding into a cold medium with a circumburst density profile varying as a power-law with radius: n(r) = n0r−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We con- sider two density profiles: the constant k = 0 case corresponding to an interstellar-medium-like density profile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' and the k = 2 case corresponding to a stellar wind from a massive star progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The synchrotron forward shock model is described in Sari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (1998), Chevalier & Li (2000), and Granot & Sari (2002), and we follow the convention Fν(t) ∝ tανβ throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Optical/X-ray temporal evolution The most striking feature of our GRB 210731A dataset is the three peaks in our early-time optical data (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' To char- acterise this light curve further, we created a composite R-band light curve by combining our q-, r-, and R-band data since they are the most well-sampled optical bands and also have similar central wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We also included the KAIT clear flux mea- surement since it is calibrated to the R band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We used an op- tical spectral index of βopt = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='05 derived from the first GROND epoch (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2) to transform the data to an R-band central wavelength of 700 nm7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The composite R-band light curve (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2) exhibits three distinct peaks occurring within the first 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 days of the GRB trigger, each with rising and decaying segments of varying steepness and smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We in- vestigate the nature of the optical peaks in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' After the last peak at ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 days, the light curve entered a final declining phase until the last optical observation at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 days post-trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We follow two approaches to fit the data: first we fitted a single power-law to each rising and decaying segment8 directly in order to get an indication of the steepness of each segment (we use these in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' then we performed an empirical fit as in Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2012) by decomposing the light curve into separate components, each of which may arise from different emission sites or physical mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Since we have three clear peaks in our light curve, we employed a model that comprises the sum of three broken power-law (BPL) components (Beuermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Zeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2004), each characterised by a normalising flux level, F0, rise and decay indices, α1 and α2, break time, tb , and break smoothness, ω, according to F(t) = F0 �� t tb �−α1ω + � t tb �−α2ω�−1/ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (1) If α1 is positive and α2 is negative, the light curve peaks at a time, tp, between rising and decaying segments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' tb = tp in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We also include a constant term to account for the host-galaxy r′-band brightness of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 mag measured at 285 days by GROND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Considering values of 1, 3, 5 and 9 for ω, we find that a smoother break with a value of 1 produces a fit with a χ2 r value closer to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Our fit allows us to compare the temporal evolution in each optical band, which we do in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Examining the X-ray light curve, the WT-mode data within the first 300 seconds shows a steep decline with αX = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='36 and a photon index that hardened from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2, as taken from the online Burst Analyser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The most likely explanation is 7 For direct comparison with the sample in Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2012) and Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 8 We determine the boundary between each segment by eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' These are shown as vertical dotted lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' high latitude prompt emission, as the temporal and spectral in- dices agree broadly with the α = −2 + β curvature effect re- lation (Kumar & Panaitescu 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Willingale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' It is unfortunate that we have no X-ray data during the time of the first two optical peaks, rendering a direct comparison between the two bands unfeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' There is, however, X-ray data from ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='13 days onwards starting when the optical light curve was rising to its final peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The X-ray light curve started in a shallow decay or plateau phase before steepening, which coincided with the final decaying phase in the optical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We fit a BPL according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 1 to determine the break time and temporal slopes, fixing the break smoothness at 1, 3, 5, or 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Each fit had a similar reduced χ2 r value (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' therefore, we employed a break smoothness of ω = 1 to match the value used in the optical fit, though this re- sults in a pre-break index that is poorly constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The R-band and 1 keV light curves are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2 along with their fits, and we present the results of the fits in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Achromatic optical/X-ray spectral evolution We now investigate if there is evidence for spectral evolution in the optical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3 we show the optical spectral energy dis- tributions (SEDs) formed using data from the first three of five GROND epochs corrected for a Galactic extinction of AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='24 mag in the direction of the GRB (Schlafly & Finkbeiner 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We fitted the data with power-laws in frequency, with the first epoch yielding a spectral index of βopt = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' There does not appear to be substantial spectral evolution between the first two epochs at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='184 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='225 days, particularly in the opti- cal g′, r′, i′, and z′ bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' It is unclear why there is excess emis- sion in the near infrared J, H, and K bands during the second and third epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' A possible explanation is that there is contam- inating emission from the host galaxy, in which case we would expect the light curves to flatten towards a constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The observed decline to below detection levels at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='25 and 285 days appears to rule out this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' It could also be the case that there is an additional unaccounted-for source of systematic pho- tometric error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' For our early-time data, we took our composite R-band light curve fit (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2) and fitted the flux in each of the optical bands with this model, which amounts to shifting the R-band fit light curve vertically until it fits the data in a given band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The spectral slope in the blue and UV bands (u through uvw2) was steeper than in the optical owing to Galactic extinction and damping by Lyα absorption at the redshift of the burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We therefore shifted the MeerLICHT u-band data to the UVOT u band us- ing an approximate spectral index of βUV ≈ −4 measured from the UVOT/u, uvw1, uvm1, and uvw2 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Figure 4 shows that the data in each of the UV and optical bands is reasonably well fitted by the R-band light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Deviations from the fit at earlier times (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 days) are visible in the u, g, i, and z bands but they do not appear statistically significant - only 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5% of the UV/optical/NIR data points deviate by 2σ or more from each model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Overall the optical evolution appears achromatic and consistent with a constant optical spectrum, which points towards a hydrodynam- ical rather than spectral origin to the complex early-time optical light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' As mentioned in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2, insufficient X-ray photons were collected during the PC-mode observations to create time-sliced spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The photon index from the online Burst Analyser was fairly constant during PC mode, however, with a mean value of ΓX = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' This is indicative of insubstantial spectral evolution in the X-ray band during PC mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The X-ray spectral index of βX = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='11 from our spectral fit in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 Article number, page 5 of 20 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' aas 103 104 105 Time since trigger (s) 100 101 102 103 Fν (µJy) t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='16 t−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='77 t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='64 t−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='11 t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='48 t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='65 t−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='52 Optical R-band BPL component Combined fit XRT 1 keV ×101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 BPL fit 10−2 10−1 100 Time since trigger (days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 Ratio 14 15 16 17 18 19 20 21 22 23 24 25 AB magnitude Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Composite R-band light curve and X-ray 1 keV light curve for GRB 210731A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' For each rising and decaying segment of the optical light curve, we show the power-law slope (tα) as an indicator of the steepness of the light curve between each pair of adjacent vertical dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We also show the fit comprising the sum of three BPL components and a constant term equal to the r′-band host galaxy flux measured by GROND at 285 days (solid red line), along with each individual BPL component (dashed blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' For the X-ray light curve, we indicate the steepness of the WT-mode segment and we show the BPL fit to the PC-mode data as a dashed-dotted red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The results of the X-ray and optical fits are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The ratio of observed flux to fitted flux is shown in the lower panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Parameters derived from fits to the composite R-band light curve and X-ray 1 keV light curve, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The R-band light curve was fitted with the sum of three BPLs, while the X-ray light curve was fitted with a single BPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' α1 α2 tp (days) χ2/dof Optical BPL 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='36 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0088 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0013 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='32 Optical BPL 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='62 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='16 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='066 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='007 Optical BPL 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='16 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='01 X-ray 1 keV BPL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='62 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='39 (with a mean photon arrival time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42 days) is steeper than the first GROND epoch (at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='18 days) spectral index of βopt = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='05, by ∆β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' This difference is suggestive of a spectral break lying between the X-ray and optical bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' If the break was the cooling break, we would expect ∆β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' This discrepancy could be explained by the fact that the cooling break is always smooth (Uhm & Zhang 2014) or that νc may lie near to either the optical or X-ray bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We investigate this further in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Closure relation analysis Within the synchrotron forward shock model the ‘closure rela- tions’ relate the spectral index β with the temporal index α in Article number, page 6 of 20 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' de Wet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' : The triple-peaked afterglow of GRB 210731A from X-ray to radio frequencies 400 800 1200 1600 2000 2400 Wavelength (nm) 100 101 102 103 Fν (µJy) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='18 days, β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='23 days, β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='21 days, β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='25 days 285 days, host galaxy 17 18 19 20 21 22 23 24 25 AB magnitude Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Power-law fits to the GROND optical/NIR data at three epochs, corrected for a Galactic extinction of AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='24 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The uncorrected magnitudes are shown in a lighter shade below each corrected data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We also show the fourth epoch, which only had detections in the g′ and r′ bands, as well as the detections (shown as diamonds) and lim- its from deep observations of the host galaxy at 285 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Upper limits are shown as upside-down triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' the convention Fν ∝ tανβ and depend on the physical regime, spectral regime, and external medium density profile (Zhang & Mészáros 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The closure relations can be adapted to describe a variety of alternative scenarios to the standard self-similar deceleration phase, including the post-jet break scenario, whether there is energy injection involved, the reverse shock crossing phase, or the Newtonian/non-relativistic phase (see the comprehensive review in Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' It is important to note that the spectral breaks are inherently smooth, so that a transitioning spectral break or spectral break near to an observing band may define a ‘grey zone’ where the α − β rela- tions are not strictly satisfied (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Uhm & Zhang 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' If we assume that the final declining phase of our optical and X-ray light curves arises from standard forward shock emission in the slow cooling regime (νm < νc), which is usually the case at later times, we can determine which spectral regime and den- sity profile best fits our data by performing a closure relation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Our X-ray and optical data both have negative spec- tral slopes, with fluxes that decline with increasing frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' There are two spectral regimes that give rise to negative spec- tral slopes: νc < ν (Regime I) with β = −p/2, or νm < ν < νc (Regime II) with β = (1 − p)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' For the optical spectral index of βopt = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='05 derived from the first GROND epoch we would have p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='10 or p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='10 in Regime I and II, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The decay index for both a wind or interstellar medium (ISM) environment in Regime I is α = (2 − 3p)/4 re- sulting in α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='08 with p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='10, too shallow for the observed decay rate of αopt = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='04 from our three BPL-component fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' In Regime II, we have α = 3(1 − p)/4 or α = (1 − 3p)/4 for an ISM or wind density profile, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' With p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='10 we get α = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='22±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='08 or α = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='08 for the ISM or wind profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Clearly, the observed optical tem- poral index of αopt = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='04 is most consistent with the optical spectral index in Regime II for a wind profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 103 104 105 Time since trigger (s) 10−2 10−1 100 101 102 103 104 Fν (µJy) XRT 1 keV uvw2 × 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='8 uvm2 × 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='7 uvw1 × 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6 u × 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 b × 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 g × 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1 v × 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1 q × 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 r × 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 i × 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='7 z × 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='9 J × 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 H × 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1 K × 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 10−2 10−1 100 Time since trigger (days) 0 1 2 Ratio Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Fits to the data in each optical band using the best-fit model to the composite R-band light curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We also fitted this model to the X-ray PC-mode data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The lower panel presents the ratio of measured flux to model flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' In a wind medium the cooling break moves to higher fre- quencies as t1/2, with light curves that are shallower by ∆α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='25 above νc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' With the cooling break between the optical and X- rays one could therefore expect the X-ray light curve to decline more slowly than the optical light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Our fits to the X-ray and optical light curves in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1 result in ∆α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='19, which agrees within uncertainties with the predicted difference of ∆α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We also see, however, that this ∆α value is con- sistent with a zero difference within 1σ and is supported by the fact that the composite R-band light curve provides a good fit to the X-ray light curve, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' It is therefore not possible to conclusively say whether νc lies between the optical and X-ray bands from our data, though we note that the slightly different temporal and spectral (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2) indices between the two bands does hint at this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' In summary, our op- tical and X-ray data can be accommodated within the standard closure relations in a wind medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Broadband SED evolution The optical and X-ray data alone can only place weak con- straints on the location of νm and the peak flux of the evolving synchrotron spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Our late-time radio observations, which probe the low-frequency end of the synchrotron spectrum, can provide valuable constraints on the location of νm and the peak flux, and can therefore lead to an estimation of the intrinsic blast wave parameters (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Our three epochs of L-band (at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 GHz) observations all yielded non-detections, whereas our four epochs of C-band and X-band data yielded detections spanning 18 to 118 days post- trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The flux across the first two epochs at 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 days was fairly constant in both the C and X bands, which is consis- Article number, page 7 of 20 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' aas tent with the predicted evolution of t0 for the spectral ordering νsa < ν < νm in a wind medium undergoing slow cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The spectral slope between the first epoch C- and X-band detections of β6−10 GHz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='27 is also close to the predicted value of ν1/3 for the same spectral segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The subsequent decline in flux across the last two epochs can be interpreted as the pas- sage of νm through 6 and 10 GHz, whereafter both bands lie in Regime II of the synchrotron spectrum in which the flux declines with time and the spectrum declines with increasing frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' This is seen in our last two epochs where the C-band flux is in fact brighter than the X-band flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' At 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 days we have quasi-simultaneous flux measurements at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4, 6 and 10 GHz (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The spectral index between the C and X bands is β6−10 GHz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='69±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='23, which is closer to the optically thin spec- tral slope of ν1/3 than the synchrotron self-absorbed slope of ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Based on the C-band detection and L-band upper limit, we place a lower limit on the spectral slope of β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4−6 GHz > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' It there- fore may be the case that synchrotron self-absorption is respon- sible for the non-detections at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 GHz, since our L-band limit is consistent with a ν2 spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' In that case, the self-absorption frequency could lie between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 and 10 GHz at 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' In a wind medium, νm moves to lower frequencies as t−3/2 with the corresponding peak flux of the synchrotron spectrum declining as t−1/2 for the spectral break ordering νsa < νm < νc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' From our GROND SED at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='184 days (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3), we know that νm lies below the K band with a peak flux greater than the measured K-band flux of 555 µJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' If we assume that νm passes through the radio X band (10 GHz) at 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 days with a peak flux of ∼250 µJy, we would have expected νm to be at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='48 × 1012 Hz with a peak flux of 1320 µJy at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='22 days, the time of our second GROND epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' This frequency lies below the K-band frequency of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='38 × 1014 Hz, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The spectral index between this expected peak flux value and the measured K-band flux value at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='22 days results in β ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='7, which is in agreement with the measured optical spectral index of βopt = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='06 at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We also note that the X-ray to optical R-band spectral index at ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='38 days (βopt,X ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='95) is between the X-ray-only ( βX = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='11) and optical-only index (βopt = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='05), demonstrating that both observing bands can be accommodated via a forward shock model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' With these basic considerations, we attempted to find a first- guess set of blast wave parameters that can explain our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We have the following assumed constraints: (i) νm passes through 10 GHz at 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 days;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (ii) the corresponding peak flux at this frequency and time is ∼250 µJy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (iii) νc lies between the optical and X-ray bands at early times (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' νc ≈ 1017 Hz at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 days);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' and (iv) νsa lies between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 and 10 GHz at 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' With these four constraints we can attempt to solve the sys- tem of four equations describing the locations of the spectral breaks and their corresponding flux densities in a wind medium, given in Table 2 of Granot & Sari (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Our solution given the above constraints results in an unphysical value of ϵe > 1, which is driven primarily by the requirement that the self-absorption frequency lies between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 and 10 GHz at 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Lowering νsa to a frequency of ∼107 Hz results in a physical solution for all of the blast wave parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Our L-band limits therefore pose a challenge to the interpretation of our multi-wavelength afterglow data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We return to this point in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Early jet-break scenario Our optical light curve during the final declining phase had a temporal index of αopt = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='04 from the combined fit or αopt = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='04 from the direct fit to the late-time data only, which is steep for normal pre-jet break evolution (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 4 in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' An alternative scenario to explain the steep final declining phase in the optical and X-ray light curves is post- jet break decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' If the jet break is due to a purely geometric edge effect (Panaitescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 1998), the light curves within all spectral regimes should steepen by t−3/4 for the ISM case and t−1/2 for the wind case once the ejecta has slowed down such that the relativistic beaming angle 1/Γ is greater than the jet half-opening angle θj, assuming a top-hat jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Sideways expansion of a conical jet would result in a steeper jet break decay of approximately t−p in Regimes I and II for an ISM (Rhoads 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Sari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Considering the edge effect only, a jet break will not change the temporal evolution of the synchrotron spectral break frequen- cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' If an early jet break occurred we would expect our radio data to show declining light curves that decay as t−1/2 in a wind medium under the assumption that νsa < ν6,10 GHz < νm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Taking into account sideways expansion, the evolution of the break fre- quencies is altered, though we would still expect declining light curves at radio frequencies (Sari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The rising radio light curves in the C and X bands until ∼34 days are therefore inconsistent with an early jet break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' This implies that the steep optical and X-ray decline is normal pre-jet break decay in a wind medium, supporting our analysis in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Theoretical modelling We have shown in the previous sections that our X-ray, opti- cal, and radio data after the last optical peak can be reconciled within the synchrotron forward shock model in a wind medium with p ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6 if we exclude our L-band limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We now proceed to find a set of blast wave parameters that can describe our data by employing the smoothly connected power-law spectra outlined in Granot & Sari (2002) and fitting for the forward shock param- eters p, ϵe, ϵB, A⋆, and EK,iso, where EK,iso is the total isotropic- equivalent kinetic energy in the blast wave;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' ϵe and ϵB are the fractions of shock internal energy given to the electrons and the magnetic fields, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' and A⋆ = A/(5 × 1011 g cm−1) is the wind density parameter as defined in Chevalier & Li (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We correct the data for Galactic extinction with AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='24 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We perform a Markov chain Monte Carlo (MCMC) analy- sis with emcee (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2013) using 512 walk- ers and 2000 steps, discarding the initial 250 steps as burn-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The details of our implementation are described in Laskar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2013, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The host galaxy extinction, AV,host, is a free pa- rameter in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We include the effects of Klein-Nishina (KN) corrections (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' McCarthy & T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Laskar in prep) using pre- scriptions from Nakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2009) and Jacovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We used uniform, uninformative priors flat in log space, and re- stricted ϵe + ϵB < 1, although this limit is not reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We did not include data before the inferred time of the last optical peak at ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 days in the modelling, and discuss these data further in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We also did not include the MeerKAT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 GHz ob- servations, as we do not expect these to be fit with this model (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We present theoretical modelling including the L- band limits in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' For completeness, we also include the possibility of a jet break following Rhoads (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We set a lower limit on the jet break time of tjet ≳ 34 days since there is no evidence for an earlier jet break in the data, as discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The physical parameters for the highest-likelihood model and those derived from the MCMC analysis are presented in Ta- ble 2, while the corresponding light curves are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' For these parameters, both inverse Compton and KN effects are important at early times (≲ 1 day), with Compton Y ≈ 4 at Article number, page 8 of 20 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' de Wet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' : The triple-peaked afterglow of GRB 210731A from X-ray to radio frequencies 1014 1015 1016 1017 1018 Frequency (Hz) 10−1 100 101 102 103 Fν (µJy) Unabsorbed model SED Absorbed model SED XRT PC-mode spectrum UV/Optical/NIR photometry Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Optical to X-ray SED at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 days along with the highest- likelihood theoretical model SED, both absorbed (dashed line) and un- absorbed (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The optical photometric data points were derived from the light curve fits to each observing band (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 4) through interpolating each fit to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The X-ray PC-mode spectrum had a mean photon arrival time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='43 days, so we used the X-ray BPL light curve fit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2 to determine a correcting factor to shift the spectrum to the expected flux level at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' At this time, the relevant spectral break frequencies are located at νm ≈ 4 × 1013 Hz, νc ≈ ˆνc ≈ 1016 Hz, and ˆνm ≈ 1021 Hz, resulting in the spectral ordering νopt < νc ≈ ˆνc ≲ νX, where ˆνc and ˆνm are KN spectral breaks as outlined in Nakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The cooling frequency passes through the X-ray band between ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6 to 12 days, consistent with the discussion in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' In this regime, the spectral index in the X- rays is expected to be intermediate between (1 − p)/2 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='88 and −p/2 ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='34, which is consistent with the observed X- ray spectral index of βX = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Taking into account the 1σ confidence intervals from the MCMC analysis, the de- rived value of p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='03 is consistent with the value of p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='10 inferred from our closure relation analysis in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' This model also requires an intrinsic extinction of AV,host ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 mag, consistent with the observed UV suppression (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The corner plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 8 shows that there are strong cor- relations between some pairs of parameters, especially those in- volving ϵe, ϵB, A⋆, and tjet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The model over-predicts the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 GHz flux by a factor of ≈ 3 with respect to our MeerKAT upper limits (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' even when taking scintillation into account, the upper limits are all more than 4σ below the model flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We return to this point in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Our shock microphysics parameters are fairly typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Our value of p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='03 is within the 1σ range of the sample in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2015), for which they find p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Santana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2014) collect ϵe and ϵB values in the literature and find that ϵe is narrowly distributed across one order of magnitude between ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='02 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6 with a median of value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Our value of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1 is a normal value within their sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' For the magnetic field equipar- tition factor, they find a much wider distribution varying across almost 5 orders of magnitude from ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5×10−5 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Our value Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Parameters derived from our multi-wavelength theoretical modelling of the afterglow data after 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We show both the highest-likelihood model parameters from a maximum-likelihood (ML) estimation and the median values along with their corresponding 1σ confidence intervals from the MCMC marginalised posterior distribu- tions presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The beaming-corrected prompt γ-ray and ki- netic energies are given in the lower panel of the table, where we have placed a lower limit on the opening angle of the jet based on a limit of tjet ≳ 118 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Parameter ML estimate MCMC results p 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='03 ϵe 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='7 × 10−2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 × 10−2 ϵB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='7 × 10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='9 × 10−3 A⋆ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 × 10−2 EK,iso (1052 erg) 63 69+32 −19 AV,host (mag) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='01 tjet (days) 64 55+23 −14 θjet (deg) ≳ 6 Eγ (erg) ≳ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='07 × 1049 EK (erg) ≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='45 × 1051 of ϵB ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 × 10−2 is close to their median value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 × 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Additionally, Beniamini & van der Horst (2017) use radio light curve peaks to determine the distribution of ϵe and find a value of log10ϵe = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='88±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='26 for a wind medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Again, our derived valued is consistent with their sample value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The model requires a jet break at tjet ≈ 64 days, which is driven by the declining radio light curves after this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' For the highest-likelihood parameters, this corresponds to a jet open- ing angle of θjet ≈ 5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' In the absence of a jet break, the model over-predicts the final X-band detection by ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' However, we note that the evidence in support of a jet break is fairly weak and this inferred opening angle should be interpreted with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' For a limit of tjet ≳ 118 days (the last radio detec- tion), the highest-likelihood model yields a lower limit on the opening angle of θjet ≳ 6◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The corresponding beaming correc- tion of fb = (1 − cos θjet) ≳ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='48 × 10−3 implies constraints on the true γ-ray and kinetic energy of Eγ ≳ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='07 × 1049 erg and EK ≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='45 × 1051 erg, respectively, where we have used the maximum-likelihood (ML) estimate from Table 2 for EK,iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Discussion The optical light curve of GRB 210731A is unusual for show- ing three distinct peaks of similar brightness within the first five hours of the GRB trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Multiple peaks in optical light curves have been observed before (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' GRBs 060904B, 080928, 100621A, 100814A, 100901A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Klotz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Rossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Greiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Nardini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Laskar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We investigate a number of explanations proposed in the literature for peaks and re-brightenings in afterglow light curves, including the passage of a spectral break, flaring behaviour, sec- ondary jets, an off-axis viewing angle, energy injection into the forward shock, and the onset of afterglow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We also consider the implications of our L-band upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The nature of the optical re-brightenings 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Passage of a spectral break The passage of the spectral break associated with the peak of the synchrotron spectrum (νm) through the optical bands could Article number, page 9 of 20 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' aas 108 109 1010 1011 101 102 103 Fν (µJy) t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='8 days 108 109 1010 1011 Frequency (Hz) t = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 days 108 109 1010 1011 t = 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='7 days Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Radio SEDs at the times of the three MeerKAT epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We show the highest-likelihood unabsorbed model SED along with the effects of Galactic scintillation at the 1σ level as derived from the NE2001 model (Cordes & Lazio 2002) for the GRB line of sight through the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' For the second epoch we show the C- and X-band detections obtained quasi-simultaneously at 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 days, while for the third epoch we show the C- and X-band detections obtained 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 days after the third MeerKAT epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' in principle give rise to a peak in the optical light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Since νm moves to lower frequencies with increasing time, we would expect the spectral index to transition from a positive to negative value over the rise and fall of the light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We have shown in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 4, however, that our optical evolution is achromatic, ruling out a spectral break origin to any of the peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Flaring behaviour The peaks in the early light curve are too smooth and long-lived to be due to flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Hence, the observed structure in the optical light curve cannot be due to any of the mechanisms that are typ- ically invoked to explain flares, such as late-time central engine activity, density fluctuations, or reverse shock emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Off-axis viewing angle It is possible to obtain a rising light curve if the GRB jet is viewed from an angle outside the cone of the jet (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' θobs ≳ θjet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Granot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The peak of the light curve corresponds to the time when the Lorentz factor of the jet is ∼ 1/θobs, where- after the light curve evolves in a post-jet break manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Our radio data and theoretical modelling does not support an early jet break however, as discussed in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Although it would be possible to explain a single peak in the light curve through viewing angle effects, it is difficult to interpret all three light curve peaks within such a scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Due to the relativis- tic beaming effect, one would also expect to observe negligible prompt γ-ray emission when the viewing angle is outside the jet cone, resulting in an orphan afterglow (Rhoads 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Gra- not et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The prompt γ-ray observations of GRB 210731A therefore do not support an off-axis viewing angle interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Two-component jet model The two-component jet model (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2005) has been invoked to explain chromatic behaviour and late-time re- brightenings observed in a number of GRB afterglows (Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Racusin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Filgas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Nicuesa Guel- benzu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Kann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' In this model a fast, narrow inner jet powers the prompt emission and early afterglow emis- sion, while a slower, wider jet powers the late-time afterglow evolution, with both jets viewed on-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' This model was pre- ferred by Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2013) to explain the re-brightenings ob- served in their sample of optical afterglows, in which the deceler- ation of the slow jet explains the re-brightening peaks, analogous to the onset peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' They claim that the similar rising and decay- ing indices of the onset and re-brightening peaks supports this in- terpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Furthermore, they find that the properties of the re- brightenings are not correlated with the prompt emission prop- erties (contrary to the onset peaks), and so they are likely inde- pendent emission components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The fact that the GRB 210731A optical light curve shows three distinct peaks appears to rule out the two-component jet model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' A three-component jet might be able to explain the three light curve peaks, but we consider it beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Energy injection The most straightforward explanation for the optical light curve evolution is energy injection into the forward shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Energy in- jection, or a refreshed shock, has been invoked to explain the plateaus and shallow decay segments seen in X-ray (Campana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Vaughan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Nousek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2009) and optical light curves (Mangano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Greiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Swenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' In this framework, the blast wave energy increases with time during the energy injection period, rather than remaining constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Two physical mechanisms have been proposed: the first is a long-lasting central engine that continuously injects a Poynt- ing flux into the blast wave as in the case of a spin-down millisec- ond magnetar (Dai & Lu 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Zhang & Mészáros 2001), where the central engine luminosity is described as a power law in time with L(t) = L0(t/t0)−q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The injected energy Einj is essentially constant when q ≥ 1, while the total energy in the blast wave can only increase significantly with time when q < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Once the injected energy begins to exceed the original energy in the blast wave, the total energy will scale as Etot ∝ t1−q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The second case could arise from an impulsive central engine injection episode, producing a stratified ejecta distribution where the ejecta mass above a certain Lorentz factor Γ is described as a power law, for example M(> Γ) ∝ Γ−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The energy in a shell with Lorentz factor Γ is added to the blast wave when the blast wave bulk Article number, page 10 of 20 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' de Wet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' : The triple-peaked afterglow of GRB 210731A from X-ray to radio frequencies 103 104 105 106 107 Time since trigger (s) 10−2 10−1 100 101 102 Time since trigger (days) 10−2 102 106 1010 1014 1018 1022 1026 Fν (µJy) 1 keV ×1023 uvw2 × 1020 uvm2 × 1019 uvw1 × 1018 white × 1017 u × 1016 b × 1015 g × 1014 v × 1013 q × 1012 r × 1011 i × 1010 z × 109 J × 108 H × 107 K × 106 X ×102 C ×101 L ×100 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Light curves from our highest-likelihood theoretical model shown for each observing band, spanning X-ray, optical, and radio fre- quencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Only data points after 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 days were used in the modelling, and we show these as filled-in data points, in contrast to the earlier time data points shown as empty circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Upper limits are shown as upside- down triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The shaded regions surrounding the three radio bands (X, C, and L) represent the effects of Galactic scintillation at the 1σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The optical g- and r-band model fits plateau towards the mea- sured host-galaxy flux levels at 285 days, shown as diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Lorentz factor has slowed down to Γ, so that the energy in the blast wave scales as E(> Γ) ∝ Γ1−s (Rees & Mészáros 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Sari & Mészáros 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' For both cases the blast wave scaling laws can be derived and applied to the synchrotron spectra to obtain closure relations that depend on either q or s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Both forms of energy injection can be cast in an equivalent form, and simple relations between q and s can be derived (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' From a closure relation analysis with energy injection alone, it is therefore impossible to distinguish energy injection from a long- lasting central engine or from injection of a stratified ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The R-band light curve segments between the first and final peaks have temporal indices of α = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='77, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='64, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='11, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='48] (from the direct fit to each segment in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Adopting the q- formalism, we can determine the energy injection index q for each segment by making use of the closure relations for a wind medium in the slow cooling regime (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We make the assumption that the R band remained in the spectral regime satisfying νm < νR < νc during all four segments of en- ergy injection, which is valid since our data supports achromatic optical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' In this regime, α = (2−2p)−(p+1)q 4 , so employ- ing p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='75 from our theoretical modelling we derive values of q = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='11, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='62, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='82, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='45] for each light curve segment after the first peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The energy increase during a time period from t0 to t1 is calculated as EK,iso,1 = EK,iso,0 �t1 t0 �1−q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2) Assuming that the blast wave kinetic energy evolves according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2 during each of the four power law segments, we can determine the energy at the time of the first peak, EK,iso,0, using EK,iso,f = EK,iso,0 �t1 t0 �1−q1 �t2 t1 �1−q2 �t3 t2 �1−q3 �t4 t3 �1−q4 (3) along with the start and end times of each segment9, our values calculated for q above, and a final blast wave energy of EK,iso,f = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 × 1053 erg from our theoretical modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We find that the blast wave energy at the time of the first optical peak is equal to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3×1050 erg, smaller by a factor of ∼1000 compared to the final kinetic energy, indicative of substantial energy injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Laskar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2015) argue that the significant X-ray and op- tical re-brightenings seen in a sample of GRB afterglows are best explained by the stratified ejecta model, since energy in- jection from a spinning-down millisecond magnetar should not lead to a significant increase in the blast wave energy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' q ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' They also exclude fall-back accretion onto a black hole as the theoretically-predicted accretion rate is insufficient to power plateaus or re-brightenings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' In the stratified ejecta for- malism, there is a gap between the initial blast wave shell and the fast outer shell of the stratified ejecta that is moving with some maximum Lorentz factor, Γmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' As the initial shell slows down, the stratified ejecta deposits energy into the blast wave until the slowest shell moving with Lorentz factor Γmin has de- posited its energy, whereafter the afterglow evolves following the standard framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' From their study of a sample of after- glows exhibiting later-time re-brightenings, Laskar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2015) showed that a large amount of the kinetic energy deposited into the blast wave comes from the slowest-moving ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' They also find that the GRBs with significant energy injection have low radiative efficiencies, consistent with the prompt γ-ray emission being produced by the fastest-moving ejecta and a large amount of kinetic energy being carried by the slower ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' With our value of Eγ,iso = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='29 × 1052 erg and EK,iso = 63 × 1052 erg, we calculate a radiative efficiency using η ≡ Eγ,iso/(Eγ,iso + EK,iso) of η ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='02, a low radiative efficiency consistent with significant energy injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We also note that the energy released in γ rays is ≈ 18 times greater than the kinetic energy in the blast wave derived at the time of the first optical peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' This is unphysical if the kinetic energy at that time is the true energy reservoir from which the prompt emission is drawn, and adds support to the final kinetic energy being the true energy reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Afterglow onset and initial Lorentz factor, Γ0 The synchrotron forward shock model predicts a smooth onset peak in optical light curves as the expanding blast wave is slowed down by the circumburst medium (Sari & Piran 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Kobayashi & Zhang 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Such onset peaks are not uncommon in early optical afterglows and have been studied by a number of authors (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Molinari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2010, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' For an ISM density profile, the deceleration 9 We use the vertical dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2 as the start and end times of each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' These have values of t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0077, t1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='021, t2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='057, t3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='09, and t4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='22 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Article number, page 11 of 20 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' aas p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='75+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='03 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='50 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='25 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='75 log(ϵe) log(ϵe) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='05 −4 −3 −2 −1 log(ϵB) log(ϵB) = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='66+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='23 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 log(A⋆) log(A⋆) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='19+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 log(EK,iso,52) log(EK,iso,52) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='74+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='28 AV,host AV,host = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='23+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='58 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='76 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='82 p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 log(tjet) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='50 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='25 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='75 log(ϵe) −4 −3 −2 −1 log(ϵB) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 log(A⋆) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 log(EK,iso,52) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='28 AV,host 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 log(tjet) log(tjet) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='84+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Corner plot showing the marginalised posterior distributions for each model parameter along with the 2D marginalised posterior distribu- tions for each pair of model parameters, from our MCMC analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Contours are at the 1σ, 2σ, and 3σ levels, and the red lines denote the median values derived from the marginalised posterior distributions for each model parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' time (or onset peak time) is most sensitive to the bulk Lorentz factor of the ejecta and depends weakly on other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Onset peaks can therefore provide a valuable way of constraining the initial Lorentz factor Γ0 of the GRB blast wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' For a non- ISM density profile, however, the dependence of the deceleration time on other parameters becomes stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The first optical detection associated with GRB 210731A was made in the UVOT white filter starting 210 seconds post- trigger, with subsequent detections showing a steady rise to a smooth peak around 700 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Thereafter, the light curve en- tered a declining phase before starting to rise steadily again at ∼1700 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2012) and Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2013) define an onset peak as a smooth hump peaking within one hour post- trigger that is followed by a normal power-law decay component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' By comparing our onset peak with the sample of 38 onset peaks in Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2013), we can assess how likely it is that our first peak is indeed the onset of afterglow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' From our combined R-band fit (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2 and Table 1) we derive a number of additional properties from each BPL component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' These include the peak flux (Fp), peak time (tp), full width at half maximum (w), peak R-band luminosity (LR,p), and the isotropic energy release in the interval [tp/5, 5tp], as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (2013) find that the onset peak times of their sample span a range of 30-3000 seconds, with typical rising and decaying indices of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 and −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='15 in ranges of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3,4] and [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='8,−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Our peak time of 760 ± 111 seconds and rising index of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='36 are typical values within this sam- ple, but our decaying index of −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='75 from BPL 1 in Table 1 is steeper than average, though it does fall within the sample range within uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' It should be noted that the decaying slope of our first peak is not well determined owing to the un- certain contribution of the second BPL component at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Furthermore, the authors derive a number of empirical relations between pairs of properties of onset peaks (see their Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 7 and 9): the width of a peak is strongly correlated with the peak time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' the R-band peak luminosity is anti-correlated with the rest-frame Article number, page 12 of 20 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' de Wet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' : The triple-peaked afterglow of GRB 210731A from X-ray to radio frequencies Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Parameters derived from each BPL component of our R-band light curve fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Fp (10−12 erg cm−2 s−1) tp (s) w (s) LR,p (1045 erg s−1) ER,iso (1048 erg) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='27 760 ± 111 956 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='69 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='22 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='27 5704 ± 645 5385 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='11 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='21 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='10 23161 ± 1179 39824 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='78 464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='53 time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' and the peak luminosity and energy are correlated with the isotropic γ-ray energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We find that our measured values in Table 3 agree closely with their empirical relations, lending support to the onset peak claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Under the assumption that GRB 210731A occurred in a stel- lar wind medium (k = 2), we calculate the initial Lorentz factor following Zhang (2018) as Γ0 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='31/4 �3EK,iso(1 + z) 8πAc3tdec �1/4 ≃ 120t−1/4 dec �1 + z 2 �1/4 E1/4 52 A−1/4 ⋆ , (4) where after the last equality, tdec, is measured in seconds, E52 in units of 1052 erg, and A⋆ is the wind density parameter as de- scribed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Equation 4 depends on the assumption of an impulsive fireball, that is, the thin shell regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The isotropic- equivalent kinetic energy of the blast wave EK,iso can be inferred from theoretical modelling of the afterglow, which we do in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' However, the value calculated in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 4 was only applicable to the late-time light curve, so we employ the value for EK,iso of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 × 1050 erg at the time of the first optical peak calculated in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Using the A⋆ value from our theoretical modelling and the peak time of our first peak, we calculate an initial Lorentz factor of Γ0 ≈ 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' As mentioned previously, the dependence of the deceleration time on other parameters is stronger for a non- ISM density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' This is also the case if there is energy injec- tion during deceleration, which may be the case during our early optical observations (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We therefore caution that our calculation of Γ0 is an estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Suppressed L-band flux Our highest-likelihood theoretical model can provide an ade- quate fit to all of our late-time data except for the L band, where the model over-predicts the flux by a factor of ≈ 3 compared to our MeerKAT upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We did not expect our model to fit the L-band data since we found that requiring the synchrotron self-absorption frequency to lie above the L band at 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 days resulted in an unphysical value of ϵe > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We therefore need to consider an additional source of opacity at these observing fre- quencies to explain our L-band limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' One possible source of additional opacity is a thermal elec- tron population within the GRB shock front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Ressler & Laskar (2017) modelled afterglow spectra and light curves while con- sidering the effect of such a population and find that it has two effects on the spectra: an excess of flux near the peak syn- chrotron frequency of the thermal electrons that fades with time as the electrons cool;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' and additional opacity in the optically thick portion of the spectrum compared to the case with only non- thermal electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The latter effect is consistent with a higher self-absorption frequency by a factor of 10-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' It could there- fore be the case that our suppressed L-band flux points towards a population of thermal electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We leave the detailed modelling including a thermal population of electrons to a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Conclusion GRB 210731A was a long-duration burst discovered by Swift/BAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Observations with the optical telescope MeerLICHT starting 286 seconds post-trigger found an unusual optical light curve evolution with three peaks of similar brightness within the first 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 hours;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' afterwards, the burst entered a declining phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We find that the early optical evolution is consistent with a con- stant optical spectrum, pointing towards a hydrodynamical ori- gin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' A closure relation analysis based on the optical SED and temporal decay after the last peak showed a preference for a stellar wind environment, consistent with the long GRB duration and therefore a massive star origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We find that the first optical peak can be explained as the onset of afterglow, while energy in- jection into the forward shock from a stratified ejecta is a natural explanation for the two subsequent re-brightenings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We estimate that the blast wave kinetic energy increased by a factor of ∼1000 from the first optical peak to the last peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Detailed theoretical modelling of the optical, X-ray, and radio data after the last opti- cal peak at ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 days resulted in typical blast wave and shock mi- crophysics parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Our MeerKAT L-band upper limits could not be reconciled with our model, however, possibly implying a thermal electron population within the shocked region that pro- vided an additional source of opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Future multi-wavelength modelling of GRB afterglows, especially at millimetre and radio frequencies, will shed light on the electron distribution in GRB shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The MeerLICHT consortium is a partnership between Rad- boud University, the University of Cape Town, the Netherlands Organisation for Scientific Research (NWO), the South African Astronomical Observatory (SAAO), the University of Oxford, the University of Manchester and the Uni- versity of Amsterdam, in association with and, partly supported by, the South African Radio Astronomy Observatory (SARAO), the European Research Coun- cil and the Netherlands Research School for Astronomy (NOVA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We acknowl- edge the use of the Inter-University Institute for Data Intensive Astronomy (IDIA) data intensive research cloud for data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' IDIA is a South African university partnership involving the University of Cape Town, the University of Pretoria and the University of the Western Cape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' SdW and PJG are supported by NRF SARChI Grant 111692.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Part of the funding for GROND (both hardware and personnel) was generously granted from the Leibniz-Prize to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Hasinger (DFG grant HA 1850/28-1) and by the Thüringer Landessternwarte Tautenburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' AVF is grateful for financial assistance from the Christopher R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Redlich Fund and numerous individual donors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' KAIT and its ongoing operation were made possible by donations from Sun Microsystems, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=', the Hewlett-Packard Com- pany, AutoScope Corporation, Lick Observatory, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' NSF, the University of California, the Sylvia & Jim Katzman Foundation, and the TABASGO Founda- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Research at Lick Observatory is partially supported by a generous gift from Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' This publication made use of the python package corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='py 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=', & KAIT GRB Team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2021, GRB Coordinates Network, 30582, 1 Zou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=', Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=', & Dai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 2007, A&A, 461, 115 Article number, page 14 of 20 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' de Wet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' : The triple-peaked afterglow of GRB 210731A from X-ray to radio frequencies Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Same as Table 2, but for the fit including the L-band upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Parameter ML estimate MCMC results p 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='76 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='76+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='04 ϵe 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 × 10−1 ϵB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1 × 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='7+17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='7 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='5 × 10−3 A⋆ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1 × 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3 × 10−2 EK,iso (1052 erg) 39 30+11 −10 AV,host (mag) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='01 tjet (days) 125 186+37 −87 106 107 Time since trigger (s) 101 102 Time since trigger (days) 101 102 103 104 105 Fν (µJy) X ×102 C ×101 L ×100 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Highest-likelihood model for the fit that includes the L-band data (dashed lines) and the fit that excludes the L-band data (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The fit to the optical and X-ray light curves is similar in both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Appendix A: Theoretical modelling with MeerKAT L-band limits We repeated the theoretical modelling following the procedure outlined in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 4, but including the MeerKAT L-band limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' The radio light curves from the highest-likelihood model are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1, and the MCMC parameter distributions are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 and Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Including the MeerKAT L-band limits leads to a poorer fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' This is demonstrated by the highest-likelihood light curve under- predicting the C- and X-band fluxes while still over-predicting the L-band flux, and the actual likelihood value of this fit being lower than for the fit excluding the MeerKAT limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' We measured a maximum log likelihood of 236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='9 for the fit without the limits vs 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1 for the fit including the MeerKAT limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' As we expected, including the MeerKAT data does not lead to a model with a steep enough spectral index to accommodate the non-detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Appendix B: Flux measurements Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1 contains all the X-ray, optical, and radio observations used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' X-ray/optical/radio flux measurements of GRB 210731A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' ∆t (days) Telescope Band/Filter Frequency (Hz) Flux (µJy) Uncertainty (µJy) Detection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (1 = yes) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00241 Swift/XRT 1 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42e+17 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='589 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='077 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00243 Swift/XRT 1 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42e+17 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='408 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='527 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00245 Swift/XRT 1 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42e+17 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='501 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='453 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00247 Swift/XRT 1 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42e+17 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='423 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='678 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00249 Swift/XRT 1 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42e+17 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='870 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='254 1 Article number, page 15 of 20 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' aas Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' ∆t (days) Telescope Band/Filter Frequency (Hz) Flux (µJy) Uncertainty (µJy) Detection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (1 = yes) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00251 Swift/XRT 1 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42e+17 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='523 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='390 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00254 Swift/XRT 1 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42e+17 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='861 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='356 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00257 Swift/XRT 1 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42e+17 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='493 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='260 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00259 Swift/XRT 1 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42e+17 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='424 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='654 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00262 Swift/XRT 1 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42e+17 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='423 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='190 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00264 Swift/XRT 1 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42e+17 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='389 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='201 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00267 Swift/XRT 1 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42e+17 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='856 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='577 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00270 Swift/XRT 1 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42e+17 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='343 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='113 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00273 Swift/XRT 1 keV 2.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00279 Swift/XRT 1 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42e+17 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='213 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='036 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='00283 Swift/XRT 1 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42e+17 93.' metadata={'source': 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page 17 of 20 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' aas Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' ∆t (days) Telescope Band/Filter Frequency (Hz) Flux (µJy) Uncertainty (µJy) Detection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (1 = yes) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='13604 MeerLICHT i 3.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='225 GROND i′ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='924e+14 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='42 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='225 GROND z′ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='335e+14 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} 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+page_content='820e+14 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='78 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='225 GROND K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='381e+14 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='62 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='82 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' ∆t (days) Telescope Band/Filter Frequency (Hz) Flux (µJy) Uncertainty (µJy) Detection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' (1 = yes) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='21 GROND r′ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='820e+14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='25 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='53 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='21 GROND J 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='418e+14 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='78 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='21 GROND H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='820e+14 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='75 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='21 GROND K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='381e+14 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='71 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='90 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='25 GROND g′ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='536e+14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='14 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='25 GROND r′ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='820e+14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='17 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='25 GROND i′ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='924e+14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='38 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='25 GROND z′ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='09 1 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 GROND i′ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='924e+14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='25 0 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 GROND J 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='418e+14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='10 0 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 GROND H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='820e+14 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='97 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='66 0 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 GROND K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='381e+14 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='54 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='18 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='385 KAIT clear 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='722e+14 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' All times are relative to the Swift/BAT trigger time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' X-ray, optical and radio data are separated by horizontal lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Detections are all at least at the 3σ level except for the first UVOT/white detection, which was at the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='3σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Optical and radio upper limits are at the 3σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Article number, page 19 of 20 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' aas p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='76+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='04 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='35 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='20 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='75 log(ϵe) log(ϵe) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='93+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='08 −4 −3 −2 −1 log(ϵB) log(ϵB) = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='25+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='61 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='25 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='8 log(A⋆) log(A⋆) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='50+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 log(EK,iso,52) log(EK,iso,52) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='48+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='275 AV,host AV,host = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='23+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='76 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='88 p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='8 log(tjet) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='35 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='20 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='75 log(ϵe) −4 −3 −2 −1 log(ϵB) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='8 log(A⋆) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 log(EK,iso,52) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='275 AV,host 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='8 log(tjet) log(tjet) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='27 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' 8, but for the fit including the L-band upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} +page_content=' Article number, page 20 of 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFLT4oBgHgl3EQfEy_H/content/2301.11985v1.pdf'} diff --git a/BNE5T4oBgHgl3EQfTA98/content/tmp_files/2301.05533v1.pdf.txt b/BNE5T4oBgHgl3EQfTA98/content/tmp_files/2301.05533v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..caede3e77ce1eed2641e9a392854eb62983076ca --- /dev/null +++ b/BNE5T4oBgHgl3EQfTA98/content/tmp_files/2301.05533v1.pdf.txt @@ -0,0 +1,421 @@ + + + + + + +Investigation of radiation hardness of silicon semiconductor +detectors under irradiation with fission products of 252Cf +nuclide. +N V Bazlov1,2, A V Derbin1, I S Drachnev1, I M Kotina1, O I Konkov1,3, I S +Lomskaya1, M S Mikulich1, V N Muratova1, D A Semenov1, M V Trushin1 and +E V Unzhakov1 +1 NRC "Kurchatov Institute" - PNPI, Gatchina, Russia +2 Saint-Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, Russia +3 Ioffe Physical-Technical Institute of the Russian Academy of Sciences, St. +Petersburg, Russia + +e-mail: trushin_mv@pnpi.nrcki.ru +Abstract. Influence of the prolonged irradiation by fission products of 252Cf radionuclide on +the operational parameters of silicon-lithium Si(Li) p-i-n detectors, Si surface barrier detectors +and Si planar p+n detector was investigated. The obtained results revealed a linear shift of the +fission fragment peaks positions towards the lower energies with increase of the irradiation +dose for all investigated detectors. The rate of the peaks shift was found to depend strongly on +the detector type and the strength of the electric field in the detector’s active region, but not on +the temperature of irradiation (room or liquid nitrogen temperature). Based on the obtained +results, the possibility of integration of the investigated types of Si semiconductor detectors in +a radionuclide neutron calibration source is considered. +1. Introduction +Heavy nuclides subjected to spontaneous fission decay accompanied by emission of several fast +neutrons can be utilized as a compact neutron calibration source. The most common spontaneous +fission source is 252Cf which undergoes α-decay and spontaneous fission with a branching ratio of +97:3, whereas each spontaneous fission event liberates 3.8 neutrons and 9.7 gamma-ray photons on +average [1]. The timing of the moment of neutron production can be fixed by detecting the fission +fragments signal with a semiconductor detector. +Semiconductor detectors possess sufficiently high energy resolution for detection of the high- +energy heavy ions. The main obstacle for the integration of such detectors in the neutron calibration +source could be their limited lifetime under the influence of the nuclide radiation [2]. Degradation of +the detector’s operational parameters effectively proceeds just in case of irradiation by alpha particles +and fission fragments (FF), which are capable of transferring a significant fraction of their energy to +the atoms of the detector lattice. Therefore, the degradation of the semiconductor detector will limit +the maximum neutron source activity and/or the source expiration period. +This article is devoted to the investigations of degradation of the operational parameters of several +types of silicon semiconductor detectors under prolonged irradiation with fission products of 252Cf (- + + + + + + + +particles and fission fragments). The main issue was to study the rate of degradation of different +detector types under irradiation by 252Cf fission products at various irradiation conditions. Irradiation +was performed at room and liquid nitrogen temperatures as well as with different detector’s +operational biases, i.e. with different electric field strength in the detectors active regions. Results of +the preceding investigations were presented in previous articles [3-5]. +2. Detectors and experimental setup +Three types of silicon semiconductor detectors were under investigations. Detectors of the first type +are SiLi p-i-n detectors produced from p-type silicon ingot with resistivity of 2.5 kΩ×cm and carrier +lifetime of 1000 µs. Two similar detectors with a sensitive region of 20 mm in diameter and 4 mm +thick were produced using standard Li drift technology [6]. The thickness of the undrifted p-type layer +in these detectors (i.e. the entrance window thickness) usually amounts to 300-500 nm [7], which is +kept to suppress the excessive growth of the leakage current at high operation reverse voltage [8]. +Detectors of the second type were two surface-barrier (SB) detectors fabricated from p-type boron- +doped silicon wafer of (111) orientation and 10 mm in diameter. The resistivity and the carrier lifetime +were 1 kΩ×cm and 1000 µs, respectively. The front side of the wafers was covered by a thin layer of +amorphous silicon which served as a passivation coating [9]. The ohmic contact was made by +sputtering of Pd layer on the whole rear side of the wafer, whereas the rectifying one – by evaporation +of Al dot with diameter of 7 mm in the center of the wafer’s front side. Detector of the third type was +p+n planar detector with the thickness of 300 m produced in Ioffe Physical-Technical Institute +(entrance window thickness was about 50 nm and the voltage of full depletion – nearly 150 V). +Irradiation by a 252Cf source was performed in vacuum cryostat typically during 10-20 days. The +source representing a stainless steel substrate covered by an active layer under the thin protective +coating was mounted 1 cm above the detector front surface that was collimated in order to exclude +side surface effects of incomplete charge collection. The spectra of the fission products of 252Cf were +recorded continuously during the whole irradiation period in short 1-hour series, what allowed us to +observe the spectra evolution directly. Detector reverse current was also monitored during the whole +irradiation period on 5-second basis with the following averaging on 1-hour measurement series. +Details of the measurement setup were presented in [3-5]. + + + + +Figure 1. (a) The first and the last spectra measured by SB2 detector in the beginning and at the end of +the prolonged irradiation period. The following peaks are marked: constant amplitude generator peak +(g), peak of -particles at 6.118 MeV (), peak at doubled energy of -particles (2) and the peaks +due to FFs of light (LF) and heavy (HF) groups. (b) Dependence of the light and heavy FF peaks +visible energies on exposure by FFs. + +a) +106 +α +first spectrum +last spectrum +g +105 +Counts per hour +104 +103 +2α +HF +102 +101 +100 +0 +20 +40 +60 +80 +100 +Energy, MeVb) +Heavy Fragments +80 +Light Fragments +Peak Position, MeV +75 +70 +65 +60 +55 +0 +1 +2 +3 +4 +Exposure, FF*107 + + + + + +3. Experimental results +In order to study the influence of temperature of irradiation on the degradation of the detector’s +parameters, the irradiation of identical SiLi detectors was performed at room (SiLi1 detector) and +liquid nitrogen (SiLi2 detector) temperature, respectively. To study the influence of external electric +field strength on the detector’s parameters degradation, two identical SB detectors were subjected to +the irradiation with different applied reverse biases, i.e. with different electric field strengths in their +active regions. The operating biases applied to the respective detector during the irradiation period, the +corresponding surface electric field strengths and the total exposures are collected in Table 1. +For all investigated detectors the similar signs of operational parameters degradation as a result of +the prolonged irradiation by 252Cf fission products were revealed. As an example, Figure 1a represents +the spectra recorded by SB2 detector at the beginning and at the end of the prolonged irradiation +period. The peak at 6.1 MeV corresponding to α-particles and another peak at doubled energy of the α- +particles caused by their accidental coincidences were used as reference points for the calibration of +the energy scale. Two broad unresolved peaks appearing at higher energies correspond to fission +fragments of light and heavy groups, respectively. +The main effect of the detector degradation is a gradual shift of fission fragments visible energy +towards the lower values, see Figure 1a. The positions of the peaks corresponding to heavy (HF) and +light (LF) fission fragments were approximated using the Gaussian function for each 1-hour series. +The dependences of the peaks positions with exposure by fission fragments can be well described by +linear functions (Figure 1b) for any masses of fission fragments and for all investigated detectors. The +obtained slope coefficients are summarized in Table 1. It is interesting to note, that the obtained +coefficients for the peaks of light and heavy fission fragments groups differ approximately by the +factor of 2 – this holds for all types of investigated detectors and for all irradiation conditions. In more +details this fact will be discussed separately in the next paper. A similar approximation of the positions +of α-peaks didn’t reveal any measurable shift with the irradiation dose for all studied detectors [4-5]. +Another sign of the detector’s operational parameters degradation under irradiation is the rapid +increase of the leakage current which proceeds linear with the number of absorbed fission products +[3]. The obtained slope coefficients of the leakage current growth are also collected in Table 1. +4. Discussion +It could be noted in Figure 1 that the peak energies of light and heavy groups of fission fragments are +below the predicted values of 104 MeV and 79 MeV [10], respectively, even on the spectrum +measured by non-irradiated detectors. The same is true for all other investigated detectors. This effect +is known as pulse-height defect (PHD) in heavy charged particles spectroscopy by semiconductor +detectors implying that the measured pulse height amplitude for heavy charged particles is somewhat +lower than that for -particles of the same energy [1]. It is generally considered that PHD is caused by +a combination of energy losses (i) in the detector dead layer/entrance window, (ii) due to the atomic +collisions and (iii) due to recombination of the electron-hole pairs created by the incident heavy +particle. Whereas energy losses by (i) and (ii) mechanisms are well understood, the full understanding +of the charge losses due to recombination is still missing. Two models were suggested supposing that +enhanced carrier recombination proceeds either in the bulk region on the radiation-induced defects +created by incident FFs [11], or at the surface states of the semiconductor [12]. The later model is +consistent with the TRIM [13] simulation results (Figure 2) showing that the density of electron-hole +pairs generated by fission fragments reaches the maximum in the near-surface region of the detector +and then gradually drops down towards the bulk, suggesting therefore that decisive influence on PHD +would have the carrier recombination at the surface states. +Previously, the PHD of about 7-10 MeV was reported for 252Cf fission fragments detection by +semiconductor detectors not subjected to the prolonged irradiation [10]. These PHD values are close to +those ones obtained for the investigated planar and SB1 detectors operated at high reverse bias – see +Table 1. We believe, that higher PHD values in non-irradiated SiLi are related with rather thick +entrance window in these detectors. Whereas the increase of PHD for SB2 detector operated at lower + + + + + + + +Table 1. Irradiation conditions and the degradation of the operational parameters of the investigated +detectors: Ub – applied bias during irradiation; Fs – surface electric field strength; PHDLF/ PHDHF – +pulse-height defects for light and heavy fragments peaks registered by non-irradiated detectors; NFF +and N – exposure by fission fragments and -particles, respectively; ∆EHF/∆NFF – slope coefficient +describing the linear shift of heavy fission fragment maximum; ∆ELF/∆NFF – slope coefficient +describing the linear shift of light fission fragment maximum; ∆I/∆N – rate of the reverse current +increase relative to the total number of the registered fission products (wasn’t measured for SiLi2 +detector); NFFmax – maximal permissible exposure by fission fragments; t – expected active operation +period of the detector in a neutron source. + + +p+n planar +SB1 +SB2 +SiLi1 +SiLi2 +Ub, V +150 +200 +30 +400 +400 +Fs, kV/cm +8.5 +40 +17 +1.5 +1.5 +PHDLF/PHDHF, MeV +8/10 +9/11 +18/19 +28/29 +35/37 +NFF *108 +1.1 +0.45 +0.43 +3.4 +1 +N* +0.5 +0.20 +0.19 +1.5 +0.44 +∆EHF/∆NFF*10-5, keV/FF +-0.9 +-1.8 +-8.9 +-3.6 +-5.7 +∆ELF/∆NFF*10-5, keV/FF +-1.9 +-3.9 +-20 +-6.2 +-12 +∆I/∆N*10-16, A/ion +8.9 +14 +8.0 +4.4 +- +NFFmax *108 +22 +12 +2.2 +6.9 +4.7 +t, years +11.6 +6.3 +1.2 +3.6 +2.5 + + +electric field (Table 1) reflects the influence of the electric field strength on the charge carrier +collection efficiency, i.e. on the recombination of the generated electron-hole pairs (note that the +active layer thickness in SB2 detector exceeds the projection range of incident FFs even at 30V). +As a result of the prolonged irradiation by 252Cf fission products, the linear shift of FF peaks +positions, i.e. the linear increase of PHD for fission fragments peaks, was revealed. Since the task of +semiconductor detector operating as a part of neutron calibration source is the reliable detection of +fission fragments signal, the irradiated detector could be considered to be "degraded" when the +spectrum of the heavy fission fragment overlaps with much more intense signal at double energy of α- +peak, what prevents us from discrimination between them [3]. The values of maximal “permissible” +exposure by fission fragments NFFmax corresponding to the beginning of the peaks overlap at three +standard deviations from their maxima were estimated for each detector using the corresponding slope +coefficients derived for HF peak and the results are presented in Table 1. + + + +Figure 2. TRIM simulated vacancies +distribution profiles (solid lines) and +linear densities of electron-hole pairs +(dashed lines) generated by light and +heavy FFs with mean energies and +masses of 104 MeV and 79 MeV, 106 +amu and 142 amu, respectively. + +Heavy Fragments +300 +2 +Light Fragments +250 +200 +150 +100 +50 +0 +0 +0 +5 +10 +15 +20 +Depth, μm + + + + + +Permissible exposure values NFFmax for the investigated detectors appeared to vary approximately +by one order of magnitude. The highest NFFmax values were found for planar and SB1 detector operated +at 200V. Reduction of the operating bias and thus the electric field strength in case of SB2 detector has +led to considerable decrease of the expected permissible exposure value. Therefore, the electric field +strength affects not only the PHD on non-irradiated detector, but also the value of the expected +maximal exposure. However the NFFmax exposure values for SiLi detectors – which operated with +lowest electric field as compared with other detectors – are significantly higher than that for SB2 +detector. Thus the expected maximal exposure appeared to be more sensitive to the electric field +strength in the surface barrier detectors and less sensitive in SiLi and planar detectors. It follows then +that not only the electric field strength, but also a detector’s internal structure defines the PHD growth +under irradiation and the maximal permissible exposure. +According to TRIM simulations, irradiation of Si detectors by fission fragments will lead to the +creation of vacancy-interstitial pairs and therefore to the formation of high density of radiation- +induced defects in the region from detector surface till the depth of 17 μm with the maxima at 14-16 +μm (Figure 2). Additionally TRIM indicates, that the energy of FFs is high enough to damage the +detector surface by sputtering. Therefore, prolonged irradiation with fission fragments will lead to an +increase of the carrier recombination rate both in Si bulk and on the surface of the semiconductor, thus +contributing to the PHD growth. +The transition region in the detectors produced by planar and by SiLi technology (p+n and p-i +transition regions, respectively) is located inside the crystalline matrix at the typical depths of 50-500 +nm from the surface. Apparently, the contribution of the surface recombination to the charge carrier +losses will be more significant for surface-barrier detectors than for SiLi and planar ones, whereas the +contribution of bulk defects – approximately similar in all detectors, what may be the reason for +different sensitivity of NFFmax exposure to the electric field strength in these detectors. Additional +investigations are needed to determine the dominant charge loss channel. +Suggested neutron calibration source should operate also at cryogenic temperatures (liquid nitrogen +or slightly above). Performed irradiation of SiLi2 detector at liquid nitrogen temperature has shown, +that in contrast to the electric field, temperature of irradiation seems to have no or only minor +influence on the expected value of maximal exposure as it could be concluded from the comparable +NFFmax values obtained for SiLi detectors irradiated at different temperatures. Somewhat smaller NFFmax +exposure obtained for SiLi2 detector is probably related with thicker entrance window in this detector. +Knowing the maximal expected exposure values NFFmax, it is possible to estimate the duration of +active “lifetime” of neutron calibration source. For the operation of neutron calibration source the +reasonable neutron activity would be the around 20 neutrons/s and taking into account that each +spontaneous fission releases in average 3.7 fast neutrons, the activity of 20 neutrons/s would +correspond to ~6 spontaneous fissions per second. Therefore, considering the maximal exposure value +from Table 1, the duration of active “lifetime” of such neutron calibration source will be 1.2-11.6 +years (without taking into account the decay of the radiation source). +During this operation period, a significant increase of leakage current up to ~100 μA can be +expected at room temperature, as can be calculated from the obtained coefficients of leakage current +growth (Table 1). Such high reverse current is unacceptable and therefore the detector cooling in order +to reduce the reverse current during the neutron source operation will be required. The coefficients of +current growth upon irradiation by fission products of 252Cf appeared to be an order of magnitude +higher than the corresponding coefficients of 7-17×10–17 A/α determined by us earlier for the identical +detectors subjected to long-term irradiation by -particles [4]. This fact confirms that prolonged +irradiation by FFs leads to the creation of the effective recombination-generation defect centers +participating in charge carrier recombination and the reverse current growth. +5. Conclusions +Prolonged irradiation of three different types of Si semiconductor detectors by fission products of 252Cf +nuclide has led to a gradual increase of pulse-height defect for the fission fragments peaks in all + + + + + + + +investigated detectors. This will eventually lead to the overlap with more intense -peak and therefore +to the impossibility of further reliable detection of fission fragments by the semiconductor detector and +thus to the limitation of the operation period of neutron calibration source. Obtained experimental +results suggest, that in order to assure the longest operation period of the neutron calibration source it +is worth to use the semiconductor detectors with lowest surface recombination rate and with highest +possible electric field strength in their active region. Among the investigated detectors, the planar one +most fully meets these requirements, whereas in relatively thick SiLi detectors it is difficult to achieve +the high electric field strength and surface-barrier detectors may suffer from high surface +recombination. With properly chosen semiconductor detector the expected active operation period of +252Cf-based neutron calibration source may reach up to 12 years. + +Acknowledgements +The reported study was funded by RFBR, project number 20-02-00571 +References +[1] +Knoll G F 2000 Radiation Detection and Measurement, 3rd ed. (New York: John Wiley and +Sons) ISBN 978-0-471-07338-3, 978-0-471-07338-3 +[2] +Moll M 2018 IEEE Transactions on Nuclear Science 65 1561–1582 +[3] +Bakhlanov S V, Derbin A V, Drachnev I S, Konkov O I, Kotina I M, Kuzmichev A M, +Lomskaya I S, Mikulich M S, Muratova N V, Niyazova N V, Semenov D A, Trushin M V +and Unzhakov E V 2021 Journal of Physics: Conference Series 2103 012138 +[4] +Bakhlanov S V, Bazlov N V, Chernobrovkin I D, Derbin A V, Drachnev I S, Kotina I M, +Konkov O I, Kuzmichev A M, Mikulich M S, Muratova N V, Trushin M V and Unzhakov E +V 2021 Journal of Physics: Conference Series 2103 012139 +[5] +Bazlov N V, Bakhlanov S V, Derbin A V, Drachnev I S, Eremin V K, Kotina I M, Muratova V +N, Pilipenko N V, Semenov D A, Unzhakov E V and Chmel E A 2018 Instruments and +Experimental Techniques 61 323 +[6] +Bazlov N V, Bakhlanov S V, Derbin A V, Drachnev I S, Izegov G A, Kotina I M, Muratova V +N, Niyazova N V, Semenov D A, Trushin M V, Unzhakov E V, Chmel E A 2020 Instrum. +Exp. Tech. 63(1) 25 +[7] +Alekseev I E, Bakhlanov S V, Derbin A V, Drachnev I S, Kotina I M, Lomskaya I S, Muratova +V N, Niyazova N V, Semenov D A, Trushin M V, Unzhakov E V 2020 Physical Review C +102 064329 +[8] +Kozai M, Fuke H, Yamada M, Perez K, Erjavec T, Hailey C J, Madden N, Rogers F, Saffold N, +Seyler D, Shimizu Y, Tokuda K, Xiao M 2019 Nuclear Inst. and Methods in Physics +Research A 947 162695 +[9] +Kotina I M, Danishevskii A M, Konkov O I, Terukov E I, Tuhkonen L M 2014 Semiconductors +48(9) 1167 +[10] Paasch K, Krause H, Scobel W 1984 Nuclear Inst. and Methods in Physics Research 221 558 +[11] Eremin V K, Il’yashenko I N, Strokan N B, Shmidt B 1995 Fiz. Tekh. Poluprovodn. 29(1) 79 +[in Russian] +[12] Tsyganov Y S 2013 Physics of Particles and Nuclei 44(1) 92 +[13] Ziegler J F, Biersack J P, Ziegler M D SRIM – Stopping and Range of Ions in Matter +www.srim.org (accessed: May 2022) + + diff --git a/CdFJT4oBgHgl3EQftC3b/content/tmp_files/2301.11616v1.pdf.txt b/CdFJT4oBgHgl3EQftC3b/content/tmp_files/2301.11616v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b4fe7ac428b4dc826c5f1e8afa0955e2c9e40d72 --- /dev/null +++ b/CdFJT4oBgHgl3EQftC3b/content/tmp_files/2301.11616v1.pdf.txt @@ -0,0 +1,1537 @@ +A Systematic Mapping Study on Responsible AI +Risk Assessment +Boming Xia∗†, Qinghua Lu∗†, Harsha Perera∗, Liming Zhu∗†, Zhenchang Xing∗, Yue Liu∗†, Jon Whittle∗ +∗CSIRO’s Data61, Sydney, Australia +†University of New South Wales, Sydney, Australia +Abstract—The rapid development of artificial intelligence (AI) +has led to increasing concerns about the capability of AI systems +to make decisions and behave responsibly. Responsible AI (RAI) +refers to the development and use of AI systems that benefit +humans, society, and the environment while minimising the risk +of negative consequences. To ensure responsible AI, the risks as- +sociated with AI systems’ development and use must be identified, +assessed and mitigated. Various AI risk assessment frameworks +have been released recently by governments, organisations, and +companies. However, it can be challenging for AI stakeholders to +have a clear picture of the available frameworks and determine +the most suitable ones for a specific context. Additionally, there +is a need to identify areas that require further research or +development of new frameworks. To fill the gap, we present a +mapping study of 16 existing RAI risk assessment frameworks +from the industry, governments, and non-government organiza- +tions (NGOs). We identify key characteristics of each framework +and analyse them in terms of RAI principles, stakeholders, +system lifecycle stages, geographical locations, targeted domains, +and assessment methods. Our study provides a comprehensive +analysis of the current state of the frameworks and highlights +areas of convergence and divergence among them. We also +identify the deficiencies in existing frameworks and outlines the +essential characteristics a concrete framework should possess. +Our findings and insights can help relevant stakeholders choose +suitable RAI risk assessment frameworks and guide the design +of future frameworks towards concreteness. +Index Terms—artificial intelligence, machine learning, risk +assessment, impact assessment, responsible AI, risk mitigation, +pattern +I. INTRODUCTION +The adoption of artificial intelligence (AI) in various appli- +cation domains has led to numerous advantages, such as im- +proved efficiency and reduced cost in manufacturing. However, +the risks associated with AI systems have also attracted signif- +icant attention from both industry and academia [1]–[3]. For +example, an AI system may make biased decisions that lead +to unintended discrimination [4]–[6]. Also, the AI system’s +dataset may contain sensitive information, risking violation of +laws such as EU General Data Protection Regulation (GDPR)1 +and EU AI Act (proposed)2. The AI incident database3 has +collected over 2200 (as of January 2023) reported real-world +incidents caused by AI systems. +Responsible AI (RAI) is developing and applying AI sys- +tems that benefit humans, society, and the environment (HSE) +1https://gdprinfo.eu/ +2https://artificialintelligenceact.eu/ +3https://incidentdatabase.ai/ +while minimizing the associated risks. A number of RAI +principle frameworks that AI systems and stakeholders should +adhere to have been released recently, such as Australia’s +AI Ethics Principles4 and European Commission’s Ethics +guidelines for trustworthy AI5. Many organizations have de- +veloped principle-driven RAI risk assessment frameworks to +implement RAI based on the RAI principles, m (e.g., US NIST +AI risk management framework [7], EU Assessment List for +Trust AI framework [8]). These frameworks are designed to +help organizations and individuals systematically assess and +mitigate potential risks associated with AI systems. Despite +the availability of these RAI risk assessment frameworks, AI +system stakeholders need to gain a holistic view of the existing +frameworks to choose the most appropriate one for their +context. Also, it is unclear how effective these frameworks +are at assessing and mitigating RAI risks. +To bridge the gaps, we have performed a systematic map- +ping study on the existing RAI risk assessment frameworks. +The main objectives of this study are: 1) to provide a summary +of the current available higher-quality AI risk frameworks to +which researchers and practitioners can refer; 2) to investigate +the capabilities and limitations of the RAI risk assessment +frameworks; and 3) to provide insights for future research and +development on concrete AI risk assessment frameworks. +The main contributions of this study are: +• We present a comprehensive qualitative and quantitative +analysis and synthesis of 16 state-of-practice RAI risk +assessment frameworks selected from the grey literature. +• We provide empirical findings and insights on the capa- +bilities and limitations of the existing frameworks and +highlight the essentials for developing concrete RAI risk +assessment frameworks. +The remainder of this paper is organized as follows: Sec- +tion II presents the methodology and research questions (RQ) +followed by the results and findings for each RQ in section +III. Section IV discusses RAI risk assessment framework +“concreteness” and threats to validity. Then, section V lists +related work while section VI concludes the paper with a +summary and the future work of this study. +4https://www.industry.gov.au/publications/australias-artificial-intelligence- +ethics-framework/australias-ai-ethics-principles +5https://ec.europa.eu/digital-single-market/en/news/ethics-guidelines- +trustworthy-ai +arXiv:2301.11616v1 [cs.SE] 27 Jan 2023 + +Fig. 1: Methodology overview. +II. METHODOLOGY +We perform the systematic mapping study following +Kitchenham’s guideline [9] on conducting literature reviews +in software engineering. The overall methodology is presented +in Figure 1. To investigate the capabilities of the existing AI +risk assessment frameworks, we derived the following RQs: +• RQ1: Who have published RAI risk assessment frame- +works? +• RQ2: What are the characteristics of the existing RAI +risk assessment frameworks? +– RQ2.1 What RAI principles are addressed? +– RQ2.2 Who are the stakeholders? +∗ RQ2.2.1 Who conducts the assessment? +∗ RQ2.2.2 Whose activities are assessed? +– RQ2.3 What is the scope of the frameworks? +∗ RQ2.3.1 Which development stages are covered +by the frameworks? +∗ RQ2.3.2 Where can the frameworks be applied? +∗ RQ2.3.3 Which domains/sectors are the frame- +works designed for? +• RQ3: How are the RAI risks assessed? +– RQ3.1: What are the inputs? +– RQ3.2: What is the assessment process? +– RQ3.3: What are the outputs? +The data sources include ACM, IEEE, Science Direct, +Springer, Google scholar for academic papers and Google +Search for industrial frameworks. The search was conducted +in November 2022 and the search term is (“artificial in- +telligence” OR “machine learning” OR AI OR ML) AND +(impact OR risk) AND (assess OR assessment OR assessing +OR evaluate OR evaluation OR evaluating OR measure OR +measurement OR measuring OR mitigate OR mitigation OR +mitigating OR manage OR managing OR management). The +study only includes frameworks with relatively concrete AI +risk assessment solutions and excludes frameworks discussing +high-level AI risks. +Although the literature search included academic databases, +academic papers and frameworks are excluded considering the +following reasons: a) one of our inclusion criteria is to include +frameworks that are currently being used in practice (e.g., +governmental/industrial/international organizations consulting +extensively with practitioners and extracting proven uses); b) +based on the inclusion and exclusion criteria, the academic pa- +pers we collected are either discussing the identified industrial +frameworks, or lack of details on concrete AI risk assessment +solutions. In the end, we selected 16 industrial frameworks to +be investigated. The complete research protocol is available +online6. We adopt Australia’s AI ethics principles in this +study: Human, societal and environmental (HSE) wellbeing, +Human-centered values, Fairness, Privacy protection and se- +curity, Reliability and safety, Transparency and explainability, +Contestability, and Accountability. +III. RESEARCH RESULTS +This section presents the results and findings of each RQ. +A. RQ1: Who have published RAI risk assessment frame- +works? +As illustrated in Table I and Fig. 2, we identified 16 frame- +works to be included in this study. These frameworks have +been published by organizations based in the United States +(US), the United Kingdom (UK), the European Union (EU), +Canada (CA), Australia (AU), Singapore (SA), the Netherlands +(NL), and Germany (DE). Additionally, one framework has +been released by the World Economic Forum, an international +(INT) organization. Although we did not set a specific time +limit for the literature search, as shown in Fig. 2b, the majority +of the frameworks (10 out of 16, 62.5%) were published +or last updated (some frameworks tend to be updated over +time) in 2022, followed by 1 framework (6.25%) published in +2021, 3 frameworks (18.75%) published in 2020, 0 framework +published in 2019, and 2 (12.5%) frameworks published in +2018. Also, a significant proportion of the frameworks (9 +out of 16, 56.25%) were published by government agencies +worldwide, with 6 of them last updated in 2022. This suggests +that the issue of RAI risks has been gaining significant +attention worldwide, particularly among government agencies. +In terms of the number of published frameworks (Fig. 2a +and Fig. 2c), the US leads in research and development on RAI +risk assessment and published 6 related frameworks, including +3 frameworks from US government agencies (i.e., I1 by +National Institute of Standards and Technology (NIST), and I8 +6https://docs.google.com/document/d/1F sAmRI7zvJBYyiF96cn5oNtG3O8 +psyB/edit?usp=sharing&ouid=111846093034327217492&rtpof=true&sd=true + +Retrieved +Academic: 10988 +Grey: 3160 +Tentative +Academic: 104 +Grey: 198 +Tentative +Academic: 104+39=143 +Grey: 198+24=222 +Selected +Research protocol +Academic: 0 +Grey: 16 +ReportingTABLE I: Industrial AI risk assessment frameworks (collected in November 2022). +Demographics (RQ1) +Characteristics (RQ2) +Processes (RQ3) +No. +Frameworks +Region +Affiliation +Affiliation type First release Last update +RAI Princples +Stakeholders +Stages +Region +Sector +Type +Mitigation +*Risk factors +I1 +AI risk management framework +(AI RMF) [7] +US +National Institute of Standards +and technology (NIST) +Government +2022.05 +2022.08 +All principles +Specified +All stages +Region- +agnostic +Sector- +agnostic +Descriptive +*Yes +Hazard, +exposure, +vulnerability +I2 +Assessment list for trustworthy AI +(ALTAI) [8] +EU +European Commission High- +Level Expert Group on AI +Government +2019.06 +2020.07 +All principles +Specified +All stages +Region- +agnostic +Sector- +agnostic +Procedural +Yes +Hazard, +exposure, +vulnerability +I3 +Algorithm Impact Assessment +tool (AIA) [10] +CA +Government of Canada +Government +2019 +2022.11 +Not specified +Not specified +Planning & +requirements +analysis, +design, testing +Region- +agnostic +Sector- +agnostic +Procedural +Yes +Hazard, +exposure, +vulnerability +I4 +Fundamental rights and algorithm +impact assessment (FRAIA) [11] +NL +Ministry of the Interior and +Kingdom Relations (BZK) +Government +2022.03 +N/A +Not specified +Specified +All stages +Region- +agnostic +Public +sectors +Procedural +*Yes +Hazard, +exposure, +vulnerability +I5 +AI and data protection risk toolkit +[12] +UK +Information Commissioner’s +Office (ICO) +Government +2021 +2022.05 +HSE wellbeing, human- +centered values, fairness, +privacy protection & +security, reliability & +safety, transparency & +explainability, +accountability +Specified +All stages +Reusable +anywhere +with +adjustments +Sector- +agnostic +Procedural +No +Hazard, +exposure, +vulnerability +I6 +Model AI governance framework +[13] +SA +Personal Data Protection +Commission (PDPC) +Government +2019.01 +2020.01 +HSE wellbeing, human- +centered values, fairness, +transparency & +explainability, reliability +& safety +Not specified +Not specified +Region- +agnostic +Sector- +agnostic +Descriptive +Yes +Hazard, +exposure, +vulnerability +I7 +NSW artificial intelligence +assurance framework [14] +AU +NSW Government +Government +2022.03 +N/A +HSE wellbeing, human- +centered values, fairness, +privacy protection & +security, reliability & +safety, transparency & +explainability, +accountability +Specified +All stages +Reusable +anywhere +with +adjustments +Sector- +agnostic +Procedural +*Yes +Hazard, +exposure, +vulnerability, +mitigation risk +I8 +Ethics & algorithms toolkit [15] +US +GovEX, the City and County +of San Francisco, Harvard +DataSmart, and Data +Community DC +Government +involved +2018 +N/A +HSE wellbeing, human- +centered values, fairness, +privacy protection +& security, reliability & +safety, transparency & +explainability, +accountability +Specified +Not specified +Region- +agnostic +Sector- +agnostic +Procedural +No +Hazard, +exposure, +vulnerability +I9 +RFD-BUS012A artificial +intelligence assessment tool [16] +US +Pennsylvania Office of +Administration +Government +2018.09 +2022.08 +Not specified +Not specified +Planning & +requirements +analysis, design +Region- +agnostic +Sector- +agnostic +Procedural +No +Vulnerability +I10 +Model rules on impact assessment +of algorithmic decision-making +systems used by public +administration [17] +EU +European Law Institute +(ELI) +NGO +2022.01 +N/A +Not specified +Not specified +Not specified +Region- +agnostic +Public +sectors +Procedural +Yes +Hazard, +exposure, +vulnerability +I11 Artificial intelligence for children +toolkit [18] +INT +World Economic Forum +(WEF) +NGO +2022.03 +N/A +HSE wellbeing, human- +centered values, fairness, +privacy protection & +security, reliability & +safety, transparency & +explainability, +contestability +Specified +Not specified +Region- +agnostic +Children +& youth +Procedural +Yes +Hazard, +exposure, +vulnerability +I12 +Recommended practices for +assessing the impact of +autonomous and intelligent +systems on human well-being [19] +US +IEEE +NGO +2020.05 +N/A +HSE wellbeing, human- +centered values +Specified +All stages +Region- +agnostic +Sector- +agnostic +Descriptive +*Yes +Hazard, +exposure, +vulnerability +I13 Responsible AI impact assessment +template [20] +US +Microsoft +Industry +2022.06 +N/A +All principles +Not specified +Not specified +Region- +agnostic +Sector- +agnostic +Procedural +No +Hazard, +exposure, +vulnerability +I14 Algorithmic impact assessments +(AIAs) in healthcare [21] +UK +Ada Lovelace Institute +NGO +2022.01 +N/A +HSE wellbeing, human +centered values, fairness, +privacy protection & +security, reliability & +safety, transparency & +explainability +Specified +Planning & +requirements +analysis, +design, tetsing, +deployment, +monitoring +UK only +UK +healthcare Procedural +*Yes +Hazard, +exposure, +vulnerability +I15 Artificial intelligence impact +assessment [22] +NL +ECP, Platform for the +Information Society +NGO +2018 +N/A +Not specified +Specified +Not specified +Region- +agnostic +Sector- +agnostic +Procedural +*Yes +Hazard, +exposure, +vulnerability, +mitigation risk +I16 +Automated decision-making +systems in the public sector: an +impact assessment +tool for public authorities [23] +DE +Algorithm Watch +NGO +2021.06 +N/A +All principles +Not specified +Not specified +Region- +agnostic +Public +sectors +Procedural +*Yes +Hazard, +exposure, +vulnerability + +(a) Percentage by region. +(b) Number by year. +(c) Number by year and region. +Fig. 2: Demographics of collected frameworks. +and I9 by two state/city government agencies) and 2 from US- +based organizations (i.e., I12 by IEEE and I13 by Microsoft). +The UK, EU, and Netherlands rank second with 2 frameworks +developed. In the UK, one framework was developed by a +government agency (Information Commissioner’s Office, ICO) +and Ada Lovelace Institute published the other specifically +for the proposed National Medical Imaging Platform of the +National Health Service (NHS). The EU published 2 frame- +works on RAI risk assessment in 2020 (last updated) and +2022, respectively. Notably, the EU has drafted its AI Act, +which marks a significant step towards operationalizing RAI +by legislation. The Netherlands published 2 frameworks, one +in 2018 by a non-government organization (ECP) and the other +in 2022 by its Ministry of the Interior and Kingdom Relations +(BZK). Australia’s New South Wales government published +the nation’s first AI Assurance Framework in 2022. Singapore +had its AI governance framework published in 2019, while +it launched AI Verify in May 2022 to objectively assess AI +systems in a verifiable way. The Canadian government released +its Algorithm impact assessment tool in 2021. The World +Economic Forum (WEF) published a toolkit for managing +RAI risks to children. Lastly, one framework published by +a German-based organization, Algorithm Watch, is identified +in this study. +Finding to RQ1: The growing number of RAI risk +assessment frameworks worldwide indicates increasing +global concern about the risks associated with the +development and use of AI systems and a growing +recognition of RAI approaches to assess and mitigate +RAI risks. +B. RQ2: What are the characteristics of the existing AI risk +assessment frameworks? +This subsection discusses the characteristics of the collected +frameworks based on the following aspects: RAI principles, +stakeholders, software development lifecycle stages, geograph- +ical locations, and targeted sectors. +To improve presentation, we first classify the frameworks +based on whether they have clear specifications on differ- +ent characteristics (e.g., whether RAI principles/stakeholder- +s/stages are specified). Then, we further categorize them to see +whether they have formulated the assessment and mitigation +based on different sub-categories of those characteristics (e.g., +different RAI principles). +1) RQ2.1 What RAI principles are addressed?: This sub- +RQ aims to investigate the RAI principles (i.e., the correspond- +ing risk category) addressed by the identified frameworks. We +have mapped the various principles from different frameworks +to Australia’s AI ethics principles (see Table I). +As illustrated in Fig. 3a, among the 16 identified frame- +works, 11 frameworks (I1, I2, I5-I8, I11-I14, I16) have speci- +fied their guiding principles. 5 frameworks (I3, I4, I9, I10, I15) +do not explicitly state their corresponding principles, although +they may implicitly encompass these principles through their +framework description and introduction (e.g., I3) or references +to other existing frameworks, standards, and guidelines (e.g., +I4). Among the 11 frameworks with specified guiding princi- +ples, only 5 frameworks (i.e., I1, I2, I11, I13, I16) organize +their sets of RAI risk assessment questions or checklists based +on different RAI principles. +All the 11 frameworks that explicitly specify the guid- +ing principles or targeted risks consider HSE wellbeing and +human-centred values. Out of these 11 frameworks, 10 frame- +works cover fairness, reliability & safety, transparency & +explainability. The only exception is framework I12, which +focuses mainly on HSE wellbeing. Privacy protection & se- +curity is covered by 9 (I1, I2, I5, I7, I8, I11, I13, I14, I16) +and accountability is covered by 8 frameworks (I1, I2, I5, I7, +I8, I11, I13, I16). Only 5 frameworks (I1, I2, I11, I13, I16) +include contestability (Fig. 3b). +2) RQ2.2: Who are the stakeholders?: This subsection +examines the stakeholders involved in the frameworks from +two perspectives: the framework user(s) who are responsible +for conducting the risk assessment (i.e., assessor), and those +whose activities are being assessed (i.e., assessee). The stake- +holders classification is based on [24], where the stakeholders +are categorized into three levels: industry-level, organization- +level, and team-level (see Fig. 4). + +INT, 1 +DE,1 +SA, 1 +US, 5 +CA, 1 +AU,1 +NL, 2 +EU,2 +UK, 21 +3 +6 +1 +1 +2 +11 +1 +1 +1 +2 +1 +1 +1 +1 +3 +1 +1 +1(a) Principle overview. +(b) Principle coverage. +Fig. 3: Ethical principles covered by the identified frameworks. +Fig. 4: Stakeholders classification [24]. +Fig. 5 shows that 10 of the collected frameworks (I1, I2, +I4, I5, I7, I8, I11, I12, I14, I15) have mentioned their targeted +stakeholders. For example, NIST’s AI RMF (I1) specifies +the framework is intended for “AI actors” defined by the +Organisation for Economic Co-operation and Development +(OECD), while EU’s ALTAI (I2) has listed the example +stakeholders in its guide on “How to complete ALTAI7”. +However, only the Netherlands BZK’s FRAIA (I4) has clearly +specified the different stakeholders associated with different +assessment stages to answer stage-specific questions. +The data synthesis results show that all 16 frameworks +involve the participation of RAI governors as the assessors and +development teams (e.g., data scientists, system developers) as +the assessee. RAI governors are those who set and enforce RAI +7https://altai.insight-centre.org/Home/HowToComplete +Fig. 5: Stakeholders of collected frameworks. +policies within an organization or community, and they can be +internal or external. +One issue identified through the data synthesis process is +the lack of consideration of more diverse and inclusive (i.e., +comprehensive) roles of stakeholders from different levels. For +example, industry-level procurers are largely neglected, with +only I1, I2 and I7 considering this aspect. Team-level speaking, +all 10 frameworks with identified stakeholders require input +from AI system development teams (i.e., assessees) on infor- +mation such as intended use, data source, data privacy, and +algorithm transparency. The assessees typically include prod- +uct managers, project managers, team leaders, data scientists, +and system developers. However, 9 out of 10 frameworks fail +to consider more diverse roles of assessees (e.g., architects, +UI/UX designers [7], [24]). The US NIST’s AI RMF (I1) is +distinguished by its inclusion of a broader range of stake- +holders involved in various stages of AI system development +and post-development, such as procurement, deployment, and +operations. However, I1 does not explicitly present categorized +assessments and mitigations based on different stakeholders. +Finding to RQ2.1 & RQ2.2: The current RAI risk as- +sessment frameworks are developed with ad-hoc scope +and focus, making them difficult for organizations to +use effectively in practice. This can be seen in the lack +of consideration for certain key stakeholders, lifecycle +stages, or ethical principles in their assessment and +mitigation, failing to identify and mitigate important +risks. +3) RQ2.3 What is the scope of the frameworks?: With the +RQ, we aim to explore the scope of the existing AI risk +assessment frameworks. +a) RQ2.3.1: Which development stages are covered by the +frameworks? +This RQ aims to investigate the stages covered by the +collected frameworks in the AI system development lifecycle +(AI-SDLC). By referencing several existing sources with AI- +SDLC [1], [7], [12], [25], we first summarized the typical +stages included in (AI) SDLC (i.e., planning & requirement +analysis, design, implementation, testing, deployment, opera- +tion & monitoring). Then, we adapted the tasks in each stage + +uestions +tlassified, +Principles +Principles +5 +not +specified, 11 +Questions +specified, +not +5 +classified +611 +11 +10 +9 +10 +10 +5 +8Industry-level stakeholders + AI technology producers/procurers +Al impacted subjects +AI solution producers/procurers +RAI governors +AI users/consumers + RAI tool producers/procurers +Organization-level stakeholders +Employees +Board members ·Executives +Managers· +Team-level stakeholders + Product managers · Project managers · Team leaders +Business analysts · Architects +·UX/UI designers +Data scientists · Developers · Testers · OperatorsQuestions +not +Stakeholder +classified, 9 +Stakeholder +specified, 10 +not +specified, 6with additional AI-specific context and derived an AI-SDLC +(Fig. 6). The detailed results of the AI-SDLC stages covered +by the collected frameworks are presented in Table I. +Fig. 7 shows that 7 of the collected frameworks do not +specify AI system lifecycle stages. Although the other 9 +frameworks (I1-I5, I7, I9, I12, I14) have clarified when they +can be applied during the AI system lifecycle, The UK ICO’s +AI and Data Protection Risk toolkit (I5) is the only one that has +categorized AI risk assessment and evaluation processes based +on different stages of the AI system lifecycle. Netherlands +BZK’s FRAIA (I4) is similarly structured in a more coarse- +grained way in that the assessment is conducted based on three +stages: input, throughput, and output. +6 (I1, I2, I4, I5, I7, I12) out of 9 frameworks with specified +AI system lifecycle stages can be used to evaluate potential +risks throughout the entire AI system lifecycle. The other 3 +frameworks (I3, I9, I14) focus on the initial stages of ideation +(i.e., planning & requirements analysis, design). In addition, +I3 covers the testing stage, while I14 covers the testing, +deployment and follow-up monitoring stages. +b) RQ2.3.2: Where can the frameworks be applied? +With the RQ, we aim to explore whether there are geograph- +ical constraints to applying the existing AI risk assessment +frameworks. +The government-developed frameworks (i.e., I1-I9) can be +applied anywhere, although some of them may require adjust- +ments considering region-specific elements in the frameworks. +For example, The UK ICO’s AI and Data Protection Risk +toolkit (I5) is aligned with the UK’s General Data Protection +Regulation (GDPR), and the AU NSW’s AI Assurance Frame- +work (I7) references relevant policies in the Australian state +of New South Wales. 6 out of 7 frameworks developed by +NGOs and industrial companies (I10-I13, I15-I16) are region- +agnostic. At the same time, I14 is specially designed for UK +NHS’s planned National Medical Imaging platform. +c) RQ2.3.2 Which domains/sectors are the frameworks +designed for? +This RQ intends to investigate the domains and sectors +where the frameworks can be applied. +Most of the collected frameworks are generally designed +across various domains. However, 5 frameworks have been +designed for specific purposes. FRAIA (I4), Model Rules (I10) +and Impact Assessment Tool for Public Authorities (I16) are +intended for evaluating the development and deployment of +AI systems in the public sector. The AI for Children toolkit +(I11) is specifically designed for AI systems that may impact +children and youth as potential users. AIAs in Healthcare +(I15) is intended to assess risks associated with designing and +developing AI systems that require access to the UK National +Medical Imaging Platform. +Finding to RQ2.3: The current RAI risk assessment +frameworks generally consider the entire lifecycle of +AI systems rather than focusing only on the AI model +pipeline. However, these frameworks do not provide +clear guidance on extending or adapting them to fit +diverse contexts. This limitation restricts the effective- +ness of RAI risk assessment frameworks as the risks +and mitigation may vary depending on the context (e.g. +different organizations, sectors, or regions) in which AI +systems are used. +C. RQ3: How are risks assessed? +This section presents the assessment processes of the col- +lection frameworks. +The frameworks are categorized into two types: procedural +and descriptive. The descriptive frameworks are less concrete +by providing general non-prescriptive assessment and mitiga- +tion and not referring to more specific and concrete solutions. +In contrast, procedural frameworks are more structured and in- +clude more detailed steps (e.g., inputs, processes, outputs) for +conducting AI risk assessments. The procedural frameworks +can also contain suggested mitigation solutions, assessment +templates, or checklists. +The collected frameworks examine underlying risks and/or +corresponding mitigation plans. To better present the results, +we summarize the different types of risks (i.e., risk factors) +the frameworks take into account. We adapted the risk cate- +gorization from a traditional risk management framework [26] +and added AI-specific context. The adapted risk factors are +categorized as follows: +• Hazard: A hazard refers to any dangerous situation or +condition arising from AI systems or related activities/ar- +tifacts that can result in harm to HSE wellbeing. Hazards +are sources of harm or exploit external to AI systems. +• Exposure: Exposure refers to individuals, property, sys- +tems, or other elements located within zones affected by +AI-related hazards that are therefore at risk of potential +losses. +• Vulnerability: Vulnerability pertains to the characteris- +tics and circumstances of an AI system or related artifacts +that make it susceptible to the detrimental effects of a +hazard. Compared to hazards, vulnerabilities are internal +weaknesses/issues of AI systems. +• Risks by/after mitigation (Mitigation risk): Mitigation +risks refer to the potential newly introduced risks brought +about by the implementation of specific mitigation, re- +silience, or control measures, or residual risks that persist +even after the implementation of mitigation measures. +For each of the collected frameworks, we summarized their +types (i.e., descriptive or procedural) and examined mitigation +measures and risk factors in Table I. +We only articulate the answers to RQ3.1 (framework inputs) +and RQ3.3 (framework outputs) for the procedural framework, +as the descriptive frameworks do not have direct inputs or +outputs. RQ3.2 (assessment processes) fits all frameworks, and +the answer is thus presented based on all frameworks collected. + +Fig. 6: AI system lifecycle (adapted from [1], [7], [12], [25]). +Fig. 7: Stages covered by collected frameworks. +1) RQ3.1: What are the inputs?: This RQ investigate the +inputs and the forms of inputs of the procedural frameworks. +The procedural frameworks are all based on certain forms of +questionnaires (e.g., self-assessment template, checklist etc.). +The inputs to these frameworks are answers to predefined +questions provided by relevant stakeholders (e.g., development +teams including system developers, data scientists etc.). +2 frameworks (EU ALTAI, I2 and CA AIA, I3) are de- +signed as interactive online tools. Users can input the required +information about their AI systems and get instant feedback +based on their inputs. Similarly, I5 and I9 are based on +excel sheets where users can fill in system details or check +if the recommended practices for minimizing potentials are +met. I13 and I14 provide self-assessment templates where +predefined questions regarding the AI system (e.g., intended +use, stakeholders, benefits/harms) need to be answered. The +other seven procedural frameworks (I4, I7, I8, I10, I11, I15, +I16) are available as published reports, where more detailed +descriptions of the contexts are given. In these reports, AI +risk/impact assessment questionnaires/checklists are given. It +is important to note that in Q&A-style assessments, both the +quality of the answers and the underpinning methodology used +to generate them are crucial factors, rather than relying solely +on subjective inputs from the assessors. +Finding to RQ3.1: The current RAI risk assessment +frameworks primarily rely on subjective evaluation +from the assessors via a series of questions or check- +lists without the support of more objective tools and +techniques, leading to potentially biased results. +2) RQ3.2: What are the processes?: This section discusses +how risk assessments are conducted in both descriptive and +procedural frameworks. +The descriptive industrial frameworks include AI RMF (I1) +by US NIST, Model AI governance framework (I6) by Sin- +gapore, and recommended practices for assessing the impact +of autonomous and intelligent systems on human well-being +(I12) by IEEE. +AI RMF (I1) is a framework with four components (map, +measure, manage, and govern) that gives organizations rec- +ommendations to adopt and adapt to their specific needs. AI +RMF (I1) is a non-prescriptive framework that aims to identify, +assess, and manage context-related risks by presenting desired +outcomes and general approaches for risk management. It pro- +motes the development of a culture of active risk management +through its recommendations and non-exhaustive solutions +presented in its companion playbook8. AI RMF (I1) is a +non-prescriptive framework that aims to identify, assess, and +manage context-related risks by presenting desired outcomes +and general approaches for risk management. It promotes the +development of a culture of active risk management through +its recommendations and non-exhaustive solutions presented +in its companion playbook. Similarly, Singapore’s Model +Framework (I6) and IEEE’s standard on AI impact assessment +(I12) are designed to be flexible by providing higher-level +guidance on the assessment processes. +8https://pages.nist.gov/AIRMF/ + +Questions +not +Stage not +Stage +classified, 8 +specified, 7 +specified, 9Planning & +Business and ethical requirements analysis: identification of the system's concept and objectives, stakeholders (and possible impacts to +Requirements analysis +stakeholders), intended uses (application domain) etc. Ethical considerations (ethics application) etc +Architectural/structural design (e.g., software architecture design, AI/ML paradigm design (e.g., centralized/distributed/decentralized), +Design +detailed design of desired behavior of AI/non-AI components (e.g., UI design, data source identification, model/algorithm selection) +System construction of both AI (e.g., data collection and processing, existing/new model/algorithm creation/selection) and non-Al +Implementation +components, including unit testing and integration testing. +Implemented system tested against a finite set of test cases. AI/ML model(s) verified & validated on test data, model output calibrated +Testing +and interpreted +Deployment +Deploying (e.g., canary/blue-green/shadow deployment) the tested system, and verifying regulatory/ethical compliance +Operation & +Operating, continuously monitoring (assess both intended and unintended system/model output and impacts), feedback gathering, and +Monitoring +maintenance of the deployed systemAs for the procedural frameworks, the assessment pro- +cesses are based on the input answers, where potential risks +are identified through the Q&A processes. The assessment +and evaluation processes of various procedural frameworks +can be grouped into four categories: risk/principle-based (I2, +I5, I7, I8, I11, I16), system development process-based (I4, +I5), essential system component-based (I3, I9), and sys- +tem description- and requirements-based (I10, I13, I14, I15). +The risk/principle-based assessments include questions de- +signed for each of the different risks/principles. The process- +based assessments include questions throughout different AI- +SDLC stages, from planning to monitoring & operations. +The component-based assessments are formulated based on +essential components (e.g., algorithms, data). The system +description- and requirements-based solutions offer mecha- +nisms for the assessee to provide information about their AI +systems and reflect on compliance with specific requirements. +For the more developed tools and frameworks, such as +I2 and I3, the risk scores and potential risks are calculated +automatically based on the selections/inputs. As for other +procedural frameworks, such as report- or excel-based ones, +they identify and assess risks by the assessment conductors +through a more manual process. The assessors evaluate the +system’s details, such as intended and unintended uses, stake- +holders, data integrity, algorithmic explainability, and consult +with external or internal stakeholders if necessary. This process +enables a seemingly systematic analysis of an AI system to +evaluate its impact and risks. +A valuable part of the AI risk assessment is the mitigation +plans suggested by some frameworks. In Table I, we summa- +rize whether clear mitigation considerations are included in the +frameworks by examining the questions/recommendations in- +cluded in each of the 16 frameworks (Yes: Mitigation specified. +*Yes: Mitigation included but not specified. No: Mitigation not +included). Only 5 (I2, I3, I6, I10, I11) out of 16 frameworks +have specified mitigation-related aspects. 7 frameworks (I1, +I4, I7, I12, I14-I16) have more or less included risk mitigation +measures without clearly specifying them. 4 frameworks (I5, +I8, I9, I13) do not cover mitigation. +As for the risk factors, none of the frameworks specified +the different risk factors they considered. However, given the +potential value of such categorization in helping organizations +better triage and prioritize risks, we examined the frameworks +and their questions/recommendations and extracted the risk +factors each framework takes into account (see Table I and Fig. +8). Despite their respective focus (e.g., I14 focuses on hazards +while touching vulnerability and exposure), all 16 frameworks +consider potential vulnerability, and 15 frameworks cover haz- +ard and exposure. However, mitigation risks are significantly +underemphasized and only covered by AU NSW AI Assurance +Framework (I7) and ECP’s AI impact assessment framework +(I15). Even for these two frameworks that consider mitigation +risks, they do not provide a comprehensive assessment rather +briefly mention such risks. For example, in I7: “Are there any +residual risks?”, and in I15: “Considering planned mitigations, +could the AI system cause significant or irreversible harms?”. +Fig. 8: Risk factors considered by collected frameworks. +Finding 1 to RQ3.2: RAI risk assessment frameworks +need to distinguish among risk factors (i.e., hazard, +exposure, vulnerability, and mitigation risk). Although +collected frameworks categorically encompass these +factors to some degree, they may focus on particular +factors while briefly touching on others. Further, mit- +igation risks are significantly neglected. This can lead +to potential failure to identify and mitigate crucial RAI +risks. +Finding 2 to RQ3.2: Existing RAI risk assessment +frameworks provide some information on assessment +procedures but fail to clearly specify inputs/outputs, +stakeholders, and resources needed at each step. +Finding 3 to RQ3.2: Many RAI risk assessment frame- +works plainly list assessment measures (e.g., questions, +checklists) without considering their interconnections +or dependencies, leading to an inefficient assessment +process. +3) RQ3.3: What are the outputs?: This section discuss the +outputs of the procedural frameworks. +Whether the output of a documented report is specified or +not, the outputs of the procedural frameworks are, or at least +should be, risk/impact assessment reports. Some frameworks, +such as I2, which creates a visualization of the risk level +correlated to the RAI principles, and I3, which calculates +risk and mitigation scores for various risk areas and gener- +ates the level of impact, generate reports automatically. For +I10, assessors must manually generate a report based on the +questionnaire and their answers to the questions. The other +procedural frameworks (I4, I5, I7-I9, I11, I13-I16) serve as +(self-)assessment tools to guide assessors in identifying risks +and do not require the preparation of a report. However, since +assessors should clearly document all the answers and the +related questions when using the procedural frameworks, the +processes result in documented assessment reports. + +15 +15 +16 +2Finding to RQ3: While organizations may rely on RAI +risk assessment frameworks for potential mitigation +solutions, current frameworks either fail to provide +concrete mitigation solutions, or lack a structured way +to present the solutions. This makes it challenging for +organizations to address identified risks effectively. +IV. DISCUSSION +A. On the concreteness of RAI risk assessment frameworks +1) Relative concreteness: A risk assessment framework +may appear concrete at one level but too abstract for the +next. For example, management teams may consider certain +assessment questions concrete, while development teams may +find them not doable. Additionally, even seemingly concrete +checklists or templates for RAI risk assessment may only +be effective if assessors have a standardized and trustworthy +approach to completing each item. Therefore, it is essential +to have well-structured, concrete, and reusable solutions (e.g., +design patterns [24], [27]) in the lower level that align/connect +with higher-level practices such as governance guidelines to +ensure a comprehensive and effective risk assessment. +2) Trivialized concreteness: Our mapping study reveals that +many frameworks trivialize the concept of “concreteness” by: +• Applying existing assessment concepts to new AI-specific +artifacts/processes without further specifying potential +solutions. Examples include acknowledging the existence +of RAI risks and broadly mentioning that they need to +be identified, documented, and mitigated. +• Identifying new concepts in AI and providing some +sub-categorization without providing potential solutions. +Examples include acknowledging bias as a common issue +in AI systems and listing different sources of bias (e.g., +data, algorithm), but not providing specified solutions to +different biases. +• Identifying important RAI risks and referring to poten- +tially stale non-AI frameworks, which may not be suitable +for addressing RAI risks. +It’s important to note that while higher-level frameworks +may seem abstract, they do not always trivialize concreteness. +These frameworks are generally more abstract because they +need to be widely applicable and less prone to obsolescence. +They can be helpful, particularly for management teams, +as they point out areas where organizations can uplift their +practices. The key criteria for determining if a higher-level +framework is concrete or not include: +• Whether high-level abstractions of potential assessment +and/or mitigation measures are underpinned (specified or +reasonably inferable) by lower-level concrete assessment +techniques. +• Whether there is a clear understanding among higher- +level stakeholders (e.g., management) about the inputs, +processes, outputs, as well as required personnel and +resources to complete the assessment. This understanding +may not necessarily require technical expertise but rather +an understanding of the trust placed in the lower-level +concrete assessment techniques utilized. +B. Essentials to “Concreteness” +We summarize the essential qualities, elements, and pro- +cesses that a concrete RAI risk assessment framework should +process in Fig. 9. +A concrete RAI risk assessment framework should have the +following characteristics: 1) The assessment and/or mitigation +(e.g., questions/checklists/recommendations) proposed at one +level/stage are reasonably underpinned/aligned/connected to +other level/stage even if the measure itself is narrow-scoped +and not directly covering different levels/stages. 2) The or- +ganization of assessment and mitigation should be layered, +considering their dependencies on each other. This allows +for a clearer assessment logic and more efficient assessment +processes. Currently, only EU’s ALTAI (I2) achieves a certain +level of interconnectivity by providing an interactive online +assessment tool where the following questions may vary +depending on the previous answers. 3) The framework should +be extensible, dynamic, and adaptive in that it can be adapted +and extended to more specific contexts. All the qualities above +result in enhanced assessment efficiency. +The elements to be covered by a comprehensive RAI +risk assessment framework should include different dimen- +sions, contexts, measurements, and mitigations. Assessment +and mitigation should be organized based on different RAI +principles, RAI stakeholders, and AI-SDLC stages. Existing +organizational governance structures and measures should also +be considered. Even if a framework focuses on a specific +aspect (e.g., stage/principle-specific, designed for assessment +instead of mitigation), it needs to be well connected (i.e., in- +terconnected) with other aspects. Additionally, the framework +should consider and specify contextual elements such as appli- +cable regions, sectors, and compatibility with an organization’s +existing risk management processes and structures. Moreover, +different risk factors should be considered, and corresponding +assessment and mitigation be presented. Especially, mitiga- +tion risks are significantly neglected by existing frameworks. +Reusable mitigation plans should be suggested in a structured +way, along with their pros and cons considered. +Specifications on the procedures required to conduct the +RAI risk assessment can help assessors and assessees from +different levels better understand the inputs/processes/outputs +and required stakeholders and resources (e.g., data, tools, +funds) for each step. However, only half (I1, I3, I4, I7, I10, I12, +I14, I15) of the 16 frameworks provided such specifications +to a certain extent. Furthermore, 7 out of the 8 frameworks +merely stated the steps needed to conduct the assessment, with +only I4 specifying stakeholders involved in each step. None +of the frameworks provides details on the resources required +to complete each step. +C. Threats to validity +External. The term “AI risk assessment” along with a set +of other terms such as “AI risk management” and “AI impact + +Fig. 9: Essentials to building a concrete RAI risk assessment framework. +assessment” has been used to mean largely the same topic: +identification, assessment/measurement, and mitigation of RAI +risks. We extended our search terms with a set of supportive +terms that are being used interchangeably in the search string +to ensure that all the relevant work were covered to mitigate +this issue. Another issue is that we only include the publicly +accessible RAI risk assessment frameworks, although some +organizations have their own frameworks for internal use. +Internal. To mitigate the threat of not finding all rele- +vant studies, we conducted a rigorous search using defined +keywords with support terms and conducted snowballing to +recover the missing studies from the literature. To address +the bias from data collection and synthesis, one researcher +performed the tasks and the other researcher reviewed and +double-checked the results. The two researchers discussed the +inconsistency and reached a common ground. +V. RELATED WORK +The pressing need to manage RAI risks has attracted +significant attention in both industry and academia. Many +studies on RAI risks have been published in recent years. +However, they heavily focus on AI risk conceptualization and +taxonomy (e.g., [28]–[31]) and provide no concrete solutions +(i.e., assessment/mitigation techniques) to RAI risks. With the +increasing interest in managing RAI risks, more actionable +solutions to managing RAI risks have been proposed. Zhang +et al. [32] proposed to evaluate model risks by inspecting their +behavior on counterfactuals. Schwee et al. [33] introduced a +toolchain for assessing privacy risks. The toolchain takes in +a model trained from the dataset to be shared and creates a +privacy risk report. Yajima et al. [34] showcased their work in +progress on assessing machine learning security risks. Failure +mode and effect analysis (FMEA) has been adopted/extended +for assessing RAI risks in [35]–[37]. +Notably, EY and Trilateral Research published a survey of +AI risk assessment methodologies in January 2022 [38]. With +the objective of providing RAI governors with noteworthy +practices and regulations in the field, this survey presents +a high-level overview of the global landscape of AI risk +assessment. The report discusses: 1) regulations and legislation +worldwide containing AI risk assessment related elements; +2) solutions to RAI risk assessment by several international +organizations; 3) standards related to AI risk management +and governance; 4) a brief overview of part of the proposed +approaches in industry and academia. While this report is cate- +gorically comprehensive, it mainly aims to help RAI governors +grasp the worldwide outline of the field. Furthermore, it lacks +detailed and systematic analysis of the existing frameworks. +In contrast, the objective of this study is to provide RAI +practitioners with a systematic summary of the existing RAI +risk assessment frameworks, and shed light on the future + +Interconnected +Layered +Industry-level +Efficient +Organization-level +Different levels +Qualities +Extensible +Team-level +Dynamic +RAI principles +Assessors +Adaptive +RAI Stakeholders +Different Roles +Assessees +Dimension +AI software +lifecycle stages +Existing (other) risk +assessment frameworks +Region +Governance structure +Essentials to +Elements +Context +concreteness +Sector +Hazard +Organization +Exposure +Measurement +Vulnerability +Risk factors +Tools and techniques +Mitigation risk +for assessment +Control +Mitigation +Reusable solutions +Steps +Stakeholders +Processes +Inputs & outputs +Resourcesdevelopment of concrete RAI risk assessment frameworks. +VI. CONCLUSION AND FUTURE WORK +This paper conducts a systematic mapping study to evaluate +the capabilities and limitations of existing RAI risk assessment +frameworks. We examine key characteristics of a concrete +framework, including specified RAI principles, stakeholders, +AI system lifecycle stages, applicable regions and sectors, risk +factors, and reusable mitigations. We provide insights to help +facilitating the development of concrete RAI risk assessment +frameworks. These includes presenting the assessment and +mitigation measures in an interconnected and layered way and +specifying the assessment procedures as well as associated +inputs/outputs, stakeholders, and resources. For future work, +we are developing a question bank with questions clearly +labelled with respect to different characteristics, mitigations, +and risk factors etc. Based on the question bank, we plan to +develop a concrete RAI risk assessment framework. +REFERENCES +[1] Q. Lu, L. Zhu, X. Xu, J. 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Kjærgaard, +“Tool-chain for supporting privacy risk assessments,” in Proceedings of +the 7th ACM International Conference on Systems for Energy-Efficient +Buildings, Cities, and Transportation, 2020, pp. 140–149. +[34] J. Yajima, M. Inui, T. Oikawa, F. Kasahara, I. Morikawa, and +N. Yoshioka, “A new approach for machine learning security risk +assessment: Work in progress,” in Proceedings of the 1st International +Conference on AI Engineering: Software Engineering for AI, ser. CAIN +’22. New York, NY, USA: Association for Computing Machinery, 2022, +p. 52–53. [Online]. Available: https://doi.org/10.1145/3522664.3528613 +[35] G. A. Dominguez, K. Kawaai, and H. Maruyama, “Fails: a tool for +assessing risk in ml systems,” in 2021 28th Asia-Pacific Software +Engineering Conference Workshops (APSEC Workshops). +IEEE, 2021, +pp. 1–4. +[36] S. Rismani and A. Moon, “How do ai systems fail socially?: an engi- +neering risk analysis approach,” in 2021 IEEE International Symposium +on Ethics in Engineering, Science and Technology (ETHICS). +IEEE, +2021, pp. 1–8. +[37] J. Li and M. Chignell, “Fmea-ai: Ai fairness impact assessment using +failure mode and effects analysis,” AI and Ethics, pp. 1–14, 2022. +[38] EY, +Trilateral +Research, +“A +survey +of +artificial +intelligence +risk +assessment +methodologies: +The +global +state +of +play +and +leading +practices +identified.” +[Online]. +Available: +https: +//www.trilateralresearch.com/wp-content/uploads/2022/01/A-survey-of- +AI-Risk-Assessment-Methodologies-full-report.pdf + diff --git a/CdFJT4oBgHgl3EQftC3b/content/tmp_files/load_file.txt b/CdFJT4oBgHgl3EQftC3b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e16891cd1baa0fce6f2d565e62642b26415cc625 --- /dev/null +++ b/CdFJT4oBgHgl3EQftC3b/content/tmp_files/load_file.txt @@ -0,0 +1,847 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf,len=846 +page_content='A Systematic Mapping Study on Responsible AI Risk Assessment Boming Xia∗†, Qinghua Lu∗†, Harsha Perera∗, Liming Zhu∗†, Zhenchang Xing∗, Yue Liu∗†, Jon Whittle∗ ∗CSIRO’s Data61, Sydney, Australia †University of New South Wales, Sydney, Australia Abstract—The rapid development of artificial intelligence (AI) has led to increasing concerns about the capability of AI systems to make decisions and behave responsibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Responsible AI (RAI) refers to the development and use of AI systems that benefit humans, society, and the environment while minimising the risk of negative consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' To ensure responsible AI, the risks as- sociated with AI systems’ development and use must be identified, assessed and mitigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Various AI risk assessment frameworks have been released recently by governments, organisations, and companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' However, it can be challenging for AI stakeholders to have a clear picture of the available frameworks and determine the most suitable ones for a specific context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Additionally, there is a need to identify areas that require further research or development of new frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' To fill the gap, we present a mapping study of 16 existing RAI risk assessment frameworks from the industry, governments, and non-government organiza- tions (NGOs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' We identify key characteristics of each framework and analyse them in terms of RAI principles, stakeholders, system lifecycle stages, geographical locations, targeted domains, and assessment methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Our study provides a comprehensive analysis of the current state of the frameworks and highlights areas of convergence and divergence among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' We also identify the deficiencies in existing frameworks and outlines the essential characteristics a concrete framework should possess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Our findings and insights can help relevant stakeholders choose suitable RAI risk assessment frameworks and guide the design of future frameworks towards concreteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Index Terms—artificial intelligence, machine learning, risk assessment, impact assessment, responsible AI, risk mitigation, pattern I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' INTRODUCTION The adoption of artificial intelligence (AI) in various appli- cation domains has led to numerous advantages, such as im- proved efficiency and reduced cost in manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' However, the risks associated with AI systems have also attracted signif- icant attention from both industry and academia [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' For example, an AI system may make biased decisions that lead to unintended discrimination [4]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Also, the AI system’s dataset may contain sensitive information, risking violation of laws such as EU General Data Protection Regulation (GDPR)1 and EU AI Act (proposed)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The AI incident database3 has collected over 2200 (as of January 2023) reported real-world incidents caused by AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Responsible AI (RAI) is developing and applying AI sys- tems that benefit humans, society, and the environment (HSE) 1https://gdprinfo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='eu/ 2https://artificialintelligenceact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='eu/ 3https://incidentdatabase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='ai/ while minimizing the associated risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' A number of RAI principle frameworks that AI systems and stakeholders should adhere to have been released recently, such as Australia’s AI Ethics Principles4 and European Commission’s Ethics guidelines for trustworthy AI5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Many organizations have de- veloped principle-driven RAI risk assessment frameworks to implement RAI based on the RAI principles, m (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', US NIST AI risk management framework [7], EU Assessment List for Trust AI framework [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' These frameworks are designed to help organizations and individuals systematically assess and mitigate potential risks associated with AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Despite the availability of these RAI risk assessment frameworks, AI system stakeholders need to gain a holistic view of the existing frameworks to choose the most appropriate one for their context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Also, it is unclear how effective these frameworks are at assessing and mitigating RAI risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' To bridge the gaps, we have performed a systematic map- ping study on the existing RAI risk assessment frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The main objectives of this study are: 1) to provide a summary of the current available higher-quality AI risk frameworks to which researchers and practitioners can refer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 2) to investigate the capabilities and limitations of the RAI risk assessment frameworks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' and 3) to provide insights for future research and development on concrete AI risk assessment frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The main contributions of this study are: We present a comprehensive qualitative and quantitative analysis and synthesis of 16 state-of-practice RAI risk assessment frameworks selected from the grey literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' We provide empirical findings and insights on the capa- bilities and limitations of the existing frameworks and highlight the essentials for developing concrete RAI risk assessment frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The remainder of this paper is organized as follows: Sec- tion II presents the methodology and research questions (RQ) followed by the results and findings for each RQ in section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Section IV discusses RAI risk assessment framework “concreteness” and threats to validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Then, section V lists related work while section VI concludes the paper with a summary and the future work of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='au/publications/australias-artificial-intelligence- ethics-framework/australias-ai-ethics-principles 5https://ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='eu/digital-single-market/en/news/ethics-guidelines- trustworthy-ai arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='11616v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='SE] 27 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 1: Methodology overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' METHODOLOGY We perform the systematic mapping study following Kitchenham’s guideline [9] on conducting literature reviews in software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The overall methodology is presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' To investigate the capabilities of the existing AI risk assessment frameworks, we derived the following RQs: RQ1: Who have published RAI risk assessment frame- works?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' RQ2: What are the characteristics of the existing RAI risk assessment frameworks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' – RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='1 What RAI principles are addressed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' – RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='2 Who are the stakeholders?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' ∗ RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='1 Who conducts the assessment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' ∗ RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='2 Whose activities are assessed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' – RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='3 What is the scope of the frameworks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' ∗ RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='1 Which development stages are covered by the frameworks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' ∗ RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='2 Where can the frameworks be applied?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' ∗ RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='3 Which domains/sectors are the frame- works designed for?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' RQ3: How are the RAI risks assessed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' – RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='1: What are the inputs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' – RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='2: What is the assessment process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' – RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='3: What are the outputs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The data sources include ACM, IEEE, Science Direct, Springer, Google scholar for academic papers and Google Search for industrial frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The search was conducted in November 2022 and the search term is (“artificial in- telligence” OR “machine learning” OR AI OR ML) AND (impact OR risk) AND (assess OR assessment OR assessing OR evaluate OR evaluation OR evaluating OR measure OR measurement OR measuring OR mitigate OR mitigation OR mitigating OR manage OR managing OR management).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The study only includes frameworks with relatively concrete AI risk assessment solutions and excludes frameworks discussing high-level AI risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Although the literature search included academic databases, academic papers and frameworks are excluded considering the following reasons: a) one of our inclusion criteria is to include frameworks that are currently being used in practice (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', governmental/industrial/international organizations consulting extensively with practitioners and extracting proven uses);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' b) based on the inclusion and exclusion criteria, the academic pa- pers we collected are either discussing the identified industrial frameworks, or lack of details on concrete AI risk assessment solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' In the end, we selected 16 industrial frameworks to be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The complete research protocol is available online6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' We adopt Australia’s AI ethics principles in this study: Human, societal and environmental (HSE) wellbeing, Human-centered values, Fairness, Privacy protection and se- curity, Reliability and safety, Transparency and explainability, Contestability, and Accountability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' RESEARCH RESULTS This section presents the results and findings of each RQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' RQ1: Who have published RAI risk assessment frame- works?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' As illustrated in Table I and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 2, we identified 16 frame- works to be included in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' These frameworks have been published by organizations based in the United States (US), the United Kingdom (UK), the European Union (EU), Canada (CA), Australia (AU), Singapore (SA), the Netherlands (NL), and Germany (DE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Additionally, one framework has been released by the World Economic Forum, an international (INT) organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Although we did not set a specific time limit for the literature search, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 2b, the majority of the frameworks (10 out of 16, 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='5%) were published or last updated (some frameworks tend to be updated over time) in 2022, followed by 1 framework (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='25%) published in 2021, 3 frameworks (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='75%) published in 2020, 0 framework published in 2019, and 2 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='5%) frameworks published in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Also, a significant proportion of the frameworks (9 out of 16, 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='25%) were published by government agencies worldwide, with 6 of them last updated in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' This suggests that the issue of RAI risks has been gaining significant attention worldwide, particularly among government agencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' In terms of the number of published frameworks (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 2a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 2c), the US leads in research and development on RAI risk assessment and published 6 related frameworks, including 3 frameworks from US government agencies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', I1 by National Institute of Standards and Technology (NIST), and I8 6https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='com/document/d/1F sAmRI7zvJBYyiF96cn5oNtG3O8 psyB/edit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='usp=sharing&ouid=111846093034327217492&rtpof=true&sd=true Retrieved Academic: 10988 Grey: 3160 Tentative Academic: 104 Grey: 198 Tentative Academic: 104+39=143 Grey: 198+24=222 Selected Research protocol Academic: 0 Grey: 16 ReportingTABLE I: Industrial AI risk assessment frameworks (collected in November 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Demographics (RQ1) Characteristics (RQ2) Processes (RQ3) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Frameworks Region Affiliation Affiliation type First release Last update RAI Princples Stakeholders Stages Region Sector Type Mitigation Risk factors I1 AI risk management framework (AI RMF) [7] US National Institute of Standards and technology (NIST) Government 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='05 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='08 All principles Specified All stages Region- agnostic Sector- agnostic Descriptive Yes Hazard, exposure, vulnerability I2 Assessment list for trustworthy AI (ALTAI) [8] EU European Commission High- Level Expert Group on AI Government 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='06 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='07 All principles Specified All stages Region- agnostic Sector- agnostic Procedural Yes Hazard, exposure, vulnerability I3 Algorithm Impact Assessment tool (AIA) [10] CA Government of Canada Government 2019 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='11 Not specified Not specified Planning & requirements analysis, design, testing Region- agnostic Sector- agnostic Procedural Yes Hazard, exposure, vulnerability I4 Fundamental rights and algorithm impact assessment (FRAIA) [11] NL Ministry of the Interior and Kingdom Relations (BZK) Government 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='03 N/A Not specified Specified All stages Region- agnostic Public sectors Procedural Yes Hazard, exposure, vulnerability I5 AI and data protection risk toolkit [12] UK Information Commissioner’s Office (ICO) Government 2021 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='05 HSE wellbeing, human- centered values, fairness, privacy protection & security, reliability & safety, transparency & explainability, accountability Specified All stages Reusable anywhere with adjustments Sector- agnostic Procedural No Hazard, exposure, vulnerability I6 Model AI governance framework [13] SA Personal Data Protection Commission (PDPC) Government 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='01 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='01 HSE wellbeing, human- centered values, fairness, transparency & explainability, reliability & safety Not specified Not specified Region- agnostic Sector- agnostic Descriptive Yes Hazard, exposure, vulnerability I7 NSW artificial intelligence assurance framework [14] AU NSW Government Government 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='03 N/A HSE wellbeing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' human- centered values,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' fairness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' privacy protection & security,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' reliability & safety,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' transparency & explainability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' accountability Specified All stages Reusable anywhere with adjustments Sector- agnostic Procedural Yes Hazard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' exposure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' vulnerability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' mitigation risk I8 Ethics & algorithms toolkit [15] US GovEX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' the City and County of San Francisco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Harvard DataSmart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' and Data Community DC Government involved 2018 N/A HSE wellbeing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' human- centered values,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' fairness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' privacy protection & security,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' reliability & safety,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' transparency & explainability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' accountability Specified Not specified Region- agnostic Sector- agnostic Procedural No Hazard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' exposure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' vulnerability I9 RFD-BUS012A artificial intelligence assessment tool [16] US Pennsylvania Office of Administration Government 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='09 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='08 Not specified Not specified Planning & requirements analysis, design Region- agnostic Sector- agnostic Procedural No Vulnerability I10 Model rules on impact assessment of algorithmic decision-making systems used by public administration [17] EU European Law Institute (ELI) NGO 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='01 N/A Not specified Not specified Not specified Region- agnostic Public sectors Procedural Yes Hazard, exposure, vulnerability I11 Artificial intelligence for children toolkit [18] INT World Economic Forum (WEF) NGO 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='03 N/A HSE wellbeing, human- centered values, fairness, privacy protection & security, reliability & safety, transparency & explainability, contestability Specified Not specified Region- agnostic Children & youth Procedural Yes Hazard, exposure, vulnerability I12 Recommended practices for assessing the impact of autonomous and intelligent systems on human well-being [19] US IEEE NGO 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='05 N/A HSE wellbeing, human- centered values Specified All stages Region- agnostic Sector- agnostic Descriptive Yes Hazard, exposure, vulnerability I13 Responsible AI impact assessment template [20] US Microsoft Industry 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='06 N/A All principles Not specified Not specified Region- agnostic Sector- agnostic Procedural No Hazard, exposure, vulnerability I14 Algorithmic impact assessments (AIAs) in healthcare [21] UK Ada Lovelace Institute NGO 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='01 N/A HSE wellbeing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' human centered values,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' fairness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' privacy protection & security,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' reliability & safety,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' transparency & explainability Specified Planning & requirements analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' design,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' tetsing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' deployment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' monitoring UK only UK healthcare Procedural Yes Hazard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' exposure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' vulnerability I15 Artificial intelligence impact assessment [22] NL ECP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Platform for the Information Society NGO 2018 N/A Not specified Specified Not specified Region- agnostic Sector- agnostic Procedural Yes Hazard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' exposure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' vulnerability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' mitigation risk I16 Automated decision-making systems in the public sector: an impact assessment tool for public authorities [23] DE Algorithm Watch NGO 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='06 N/A All principles Not specified Not specified Region- agnostic Public sectors Procedural Yes Hazard, exposure, vulnerability (a) Percentage by region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' (b) Number by year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' (c) Number by year and region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 2: Demographics of collected frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' and I9 by two state/city government agencies) and 2 from US- based organizations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', I12 by IEEE and I13 by Microsoft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The UK, EU, and Netherlands rank second with 2 frameworks developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' In the UK, one framework was developed by a government agency (Information Commissioner’s Office, ICO) and Ada Lovelace Institute published the other specifically for the proposed National Medical Imaging Platform of the National Health Service (NHS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The EU published 2 frame- works on RAI risk assessment in 2020 (last updated) and 2022, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Notably, the EU has drafted its AI Act, which marks a significant step towards operationalizing RAI by legislation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The Netherlands published 2 frameworks, one in 2018 by a non-government organization (ECP) and the other in 2022 by its Ministry of the Interior and Kingdom Relations (BZK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Australia’s New South Wales government published the nation’s first AI Assurance Framework in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Singapore had its AI governance framework published in 2019, while it launched AI Verify in May 2022 to objectively assess AI systems in a verifiable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The Canadian government released its Algorithm impact assessment tool in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The World Economic Forum (WEF) published a toolkit for managing RAI risks to children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Lastly, one framework published by a German-based organization, Algorithm Watch, is identified in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Finding to RQ1: The growing number of RAI risk assessment frameworks worldwide indicates increasing global concern about the risks associated with the development and use of AI systems and a growing recognition of RAI approaches to assess and mitigate RAI risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' RQ2: What are the characteristics of the existing AI risk assessment frameworks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' This subsection discusses the characteristics of the collected frameworks based on the following aspects: RAI principles, stakeholders, software development lifecycle stages, geograph- ical locations, and targeted sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' To improve presentation, we first classify the frameworks based on whether they have clear specifications on differ- ent characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', whether RAI principles/stakeholder- s/stages are specified).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Then, we further categorize them to see whether they have formulated the assessment and mitigation based on different sub-categories of those characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', different RAI principles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 1) RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='1 What RAI principles are addressed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' : This sub- RQ aims to investigate the RAI principles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', the correspond- ing risk category) addressed by the identified frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' We have mapped the various principles from different frameworks to Australia’s AI ethics principles (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 3a, among the 16 identified frame- works, 11 frameworks (I1, I2, I5-I8, I11-I14, I16) have speci- fied their guiding principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 5 frameworks (I3, I4, I9, I10, I15) do not explicitly state their corresponding principles, although they may implicitly encompass these principles through their framework description and introduction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', I3) or references to other existing frameworks, standards, and guidelines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', I4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Among the 11 frameworks with specified guiding princi- ples, only 5 frameworks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', I1, I2, I11, I13, I16) organize their sets of RAI risk assessment questions or checklists based on different RAI principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' All the 11 frameworks that explicitly specify the guid- ing principles or targeted risks consider HSE wellbeing and human-centred values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Out of these 11 frameworks, 10 frame- works cover fairness, reliability & safety, transparency & explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The only exception is framework I12, which focuses mainly on HSE wellbeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Privacy protection & se- curity is covered by 9 (I1, I2, I5, I7, I8, I11, I13, I14, I16) and accountability is covered by 8 frameworks (I1, I2, I5, I7, I8, I11, I13, I16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Only 5 frameworks (I1, I2, I11, I13, I16) include contestability (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 2) RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='2: Who are the stakeholders?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' : This subsection examines the stakeholders involved in the frameworks from two perspectives: the framework user(s) who are responsible for conducting the risk assessment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', assessor), and those whose activities are being assessed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', assessee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The stake- holders classification is based on [24], where the stakeholders are categorized into three levels: industry-level, organization- level, and team-level (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' INT, 1 DE,1 SA, 1 US, 5 CA, 1 AU,1 NL, 2 EU,2 UK, 21 3 6 1 1 2 11 1 1 1 2 1 1 1 1 3 1 1 1(a) Principle overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' (b) Principle coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 3: Ethical principles covered by the identified frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 4: Stakeholders classification [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 5 shows that 10 of the collected frameworks (I1, I2, I4, I5, I7, I8, I11, I12, I14, I15) have mentioned their targeted stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' For example, NIST’s AI RMF (I1) specifies the framework is intended for “AI actors” defined by the Organisation for Economic Co-operation and Development (OECD), while EU’s ALTAI (I2) has listed the example stakeholders in its guide on “How to complete ALTAI7”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' However, only the Netherlands BZK’s FRAIA (I4) has clearly specified the different stakeholders associated with different assessment stages to answer stage-specific questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The data synthesis results show that all 16 frameworks involve the participation of RAI governors as the assessors and development teams (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', data scientists, system developers) as the assessee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' RAI governors are those who set and enforce RAI 7https://altai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='insight-centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='org/Home/HowToComplete Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 5: Stakeholders of collected frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' policies within an organization or community, and they can be internal or external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' One issue identified through the data synthesis process is the lack of consideration of more diverse and inclusive (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', comprehensive) roles of stakeholders from different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' For example, industry-level procurers are largely neglected, with only I1, I2 and I7 considering this aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Team-level speaking, all 10 frameworks with identified stakeholders require input from AI system development teams (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', assessees) on infor- mation such as intended use, data source, data privacy, and algorithm transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The assessees typically include prod- uct managers, project managers, team leaders, data scientists, and system developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' However, 9 out of 10 frameworks fail to consider more diverse roles of assessees (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', architects, UI/UX designers [7], [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The US NIST’s AI RMF (I1) is distinguished by its inclusion of a broader range of stake- holders involved in various stages of AI system development and post-development, such as procurement, deployment, and operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' However, I1 does not explicitly present categorized assessments and mitigations based on different stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Finding to RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='1 & RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='2: The current RAI risk as- sessment frameworks are developed with ad-hoc scope and focus, making them difficult for organizations to use effectively in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' This can be seen in the lack of consideration for certain key stakeholders, lifecycle stages, or ethical principles in their assessment and mitigation, failing to identify and mitigate important risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 3) RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='3 What is the scope of the frameworks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' : With the RQ, we aim to explore the scope of the existing AI risk assessment frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' a) RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='1: Which development stages are covered by the frameworks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' This RQ aims to investigate the stages covered by the collected frameworks in the AI system development lifecycle (AI-SDLC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' By referencing several existing sources with AI- SDLC [1], [7], [12], [25], we first summarized the typical stages included in (AI) SDLC (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', planning & requirement analysis, design, implementation, testing, deployment, opera- tion & monitoring).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' we adapted the tasks in each stage uestions tlassified,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Principles Principles 5 not specified,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 11 Questions specified,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='not ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='classified ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='611 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='8Industry-level stakeholders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='AI technology producers/procurers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Al impacted subjects ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='AI solution producers/procurers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='RAI governors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='AI users/consumers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='RAI tool producers/procurers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Organization-level stakeholders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Employees ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Board members ·Executives ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Managers· ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Team-level stakeholders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Product managers · Project managers · Team leaders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Business analysts · Architects ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='UX/UI designers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Data scientists · Developers · Testers · OperatorsQuestions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='not ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Stakeholder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='classified,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 9 Stakeholder specified,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 10 not specified,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 6with additional AI-specific context and derived an AI-SDLC (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The detailed results of the AI-SDLC stages covered by the collected frameworks are presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 7 shows that 7 of the collected frameworks do not specify AI system lifecycle stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Although the other 9 frameworks (I1-I5, I7, I9, I12, I14) have clarified when they can be applied during the AI system lifecycle, The UK ICO’s AI and Data Protection Risk toolkit (I5) is the only one that has categorized AI risk assessment and evaluation processes based on different stages of the AI system lifecycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Netherlands BZK’s FRAIA (I4) is similarly structured in a more coarse- grained way in that the assessment is conducted based on three stages: input, throughput, and output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 6 (I1, I2, I4, I5, I7, I12) out of 9 frameworks with specified AI system lifecycle stages can be used to evaluate potential risks throughout the entire AI system lifecycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The other 3 frameworks (I3, I9, I14) focus on the initial stages of ideation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', planning & requirements analysis, design).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' In addition, I3 covers the testing stage, while I14 covers the testing, deployment and follow-up monitoring stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' b) RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='2: Where can the frameworks be applied?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' With the RQ, we aim to explore whether there are geograph- ical constraints to applying the existing AI risk assessment frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The government-developed frameworks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', I1-I9) can be applied anywhere, although some of them may require adjust- ments considering region-specific elements in the frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' For example, The UK ICO’s AI and Data Protection Risk toolkit (I5) is aligned with the UK’s General Data Protection Regulation (GDPR), and the AU NSW’s AI Assurance Frame- work (I7) references relevant policies in the Australian state of New South Wales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 6 out of 7 frameworks developed by NGOs and industrial companies (I10-I13, I15-I16) are region- agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' At the same time, I14 is specially designed for UK NHS’s planned National Medical Imaging platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' c) RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='2 Which domains/sectors are the frameworks designed for?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' This RQ intends to investigate the domains and sectors where the frameworks can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Most of the collected frameworks are generally designed across various domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' However, 5 frameworks have been designed for specific purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' FRAIA (I4), Model Rules (I10) and Impact Assessment Tool for Public Authorities (I16) are intended for evaluating the development and deployment of AI systems in the public sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The AI for Children toolkit (I11) is specifically designed for AI systems that may impact children and youth as potential users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' AIAs in Healthcare (I15) is intended to assess risks associated with designing and developing AI systems that require access to the UK National Medical Imaging Platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Finding to RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='3: The current RAI risk assessment frameworks generally consider the entire lifecycle of AI systems rather than focusing only on the AI model pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' However, these frameworks do not provide clear guidance on extending or adapting them to fit diverse contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' This limitation restricts the effective- ness of RAI risk assessment frameworks as the risks and mitigation may vary depending on the context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' different organizations, sectors, or regions) in which AI systems are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' RQ3: How are risks assessed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' This section presents the assessment processes of the col- lection frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The frameworks are categorized into two types: procedural and descriptive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The descriptive frameworks are less concrete by providing general non-prescriptive assessment and mitiga- tion and not referring to more specific and concrete solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' In contrast, procedural frameworks are more structured and in- clude more detailed steps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', inputs, processes, outputs) for conducting AI risk assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The procedural frameworks can also contain suggested mitigation solutions, assessment templates, or checklists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The collected frameworks examine underlying risks and/or corresponding mitigation plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' To better present the results, we summarize the different types of risks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', risk factors) the frameworks take into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' We adapted the risk cate- gorization from a traditional risk management framework [26] and added AI-specific context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The adapted risk factors are categorized as follows: Hazard: A hazard refers to any dangerous situation or condition arising from AI systems or related activities/ar- tifacts that can result in harm to HSE wellbeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Hazards are sources of harm or exploit external to AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Exposure: Exposure refers to individuals, property, sys- tems, or other elements located within zones affected by AI-related hazards that are therefore at risk of potential losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Vulnerability: Vulnerability pertains to the characteris- tics and circumstances of an AI system or related artifacts that make it susceptible to the detrimental effects of a hazard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Compared to hazards, vulnerabilities are internal weaknesses/issues of AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Risks by/after mitigation (Mitigation risk): Mitigation risks refer to the potential newly introduced risks brought about by the implementation of specific mitigation, re- silience, or control measures, or residual risks that persist even after the implementation of mitigation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' For each of the collected frameworks, we summarized their types (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', descriptive or procedural) and examined mitigation measures and risk factors in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' We only articulate the answers to RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='1 (framework inputs) and RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='3 (framework outputs) for the procedural framework, as the descriptive frameworks do not have direct inputs or outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='2 (assessment processes) fits all frameworks, and the answer is thus presented based on all frameworks collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 6: AI system lifecycle (adapted from [1], [7], [12], [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 7: Stages covered by collected frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 1) RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='1: What are the inputs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' : This RQ investigate the inputs and the forms of inputs of the procedural frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The procedural frameworks are all based on certain forms of questionnaires (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', self-assessment template, checklist etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The inputs to these frameworks are answers to predefined questions provided by relevant stakeholders (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', development teams including system developers, data scientists etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 2 frameworks (EU ALTAI, I2 and CA AIA, I3) are de- signed as interactive online tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Users can input the required information about their AI systems and get instant feedback based on their inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Similarly, I5 and I9 are based on excel sheets where users can fill in system details or check if the recommended practices for minimizing potentials are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' I13 and I14 provide self-assessment templates where predefined questions regarding the AI system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', intended use, stakeholders, benefits/harms) need to be answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The other seven procedural frameworks (I4, I7, I8, I10, I11, I15, I16) are available as published reports, where more detailed descriptions of the contexts are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' In these reports, AI risk/impact assessment questionnaires/checklists are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' It is important to note that in Q&A-style assessments, both the quality of the answers and the underpinning methodology used to generate them are crucial factors, rather than relying solely on subjective inputs from the assessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Finding to RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='1: The current RAI risk assessment frameworks primarily rely on subjective evaluation from the assessors via a series of questions or check- lists without the support of more objective tools and techniques, leading to potentially biased results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 2) RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='2: What are the processes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' : This section discusses how risk assessments are conducted in both descriptive and procedural frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The descriptive industrial frameworks include AI RMF (I1) by US NIST, Model AI governance framework (I6) by Sin- gapore, and recommended practices for assessing the impact of autonomous and intelligent systems on human well-being (I12) by IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' AI RMF (I1) is a framework with four components (map, measure, manage, and govern) that gives organizations rec- ommendations to adopt and adapt to their specific needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' AI RMF (I1) is a non-prescriptive framework that aims to identify, assess, and manage context-related risks by presenting desired outcomes and general approaches for risk management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' It pro- motes the development of a culture of active risk management through its recommendations and non-exhaustive solutions presented in its companion playbook8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' AI RMF (I1) is a non-prescriptive framework that aims to identify, assess, and manage context-related risks by presenting desired outcomes and general approaches for risk management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' It promotes the development of a culture of active risk management through its recommendations and non-exhaustive solutions presented in its companion playbook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Similarly, Singapore’s Model Framework (I6) and IEEE’s standard on AI impact assessment (I12) are designed to be flexible by providing higher-level guidance on the assessment processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 8https://pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content="gov/AIRMF/ Questions not Stage not Stage classified, 8 specified, 7 specified, 9Planning & Business and ethical requirements analysis: identification of the system's concept and objectives, stakeholders (and possible impacts to Requirements analysis stakeholders), intended uses (application domain) etc." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Ethical considerations (ethics application) etc Architectural/structural design (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', software architecture design, AI/ML paradigm design (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', centralized/distributed/decentralized), Design detailed design of desired behavior of AI/non-AI components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', UI design, data source identification, model/algorithm selection) System construction of both AI (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', data collection and processing, existing/new model/algorithm creation/selection) and non-Al Implementation components, including unit testing and integration testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Implemented system tested against a finite set of test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' AI/ML model(s) verified & validated on test data, model output calibrated Testing and interpreted Deployment Deploying (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', canary/blue-green/shadow deployment) the tested system, and verifying regulatory/ethical compliance Operation & Operating, continuously monitoring (assess both intended and unintended system/model output and impacts), feedback gathering, and Monitoring maintenance of the deployed systemAs for the procedural frameworks, the assessment pro- cesses are based on the input answers, where potential risks are identified through the Q&A processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The assessment and evaluation processes of various procedural frameworks can be grouped into four categories: risk/principle-based (I2, I5, I7, I8, I11, I16), system development process-based (I4, I5), essential system component-based (I3, I9), and sys- tem description- and requirements-based (I10, I13, I14, I15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The risk/principle-based assessments include questions de- signed for each of the different risks/principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The process- based assessments include questions throughout different AI- SDLC stages, from planning to monitoring & operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The component-based assessments are formulated based on essential components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', algorithms, data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The system description- and requirements-based solutions offer mecha- nisms for the assessee to provide information about their AI systems and reflect on compliance with specific requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' For the more developed tools and frameworks, such as I2 and I3, the risk scores and potential risks are calculated automatically based on the selections/inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' As for other procedural frameworks, such as report- or excel-based ones, they identify and assess risks by the assessment conductors through a more manual process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The assessors evaluate the system’s details, such as intended and unintended uses, stake- holders, data integrity, algorithmic explainability, and consult with external or internal stakeholders if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' This process enables a seemingly systematic analysis of an AI system to evaluate its impact and risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' A valuable part of the AI risk assessment is the mitigation plans suggested by some frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' In Table I, we summa- rize whether clear mitigation considerations are included in the frameworks by examining the questions/recommendations in- cluded in each of the 16 frameworks (Yes: Mitigation specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Yes: Mitigation included but not specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' No: Mitigation not included).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Only 5 (I2, I3, I6, I10, I11) out of 16 frameworks have specified mitigation-related aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 7 frameworks (I1, I4, I7, I12, I14-I16) have more or less included risk mitigation measures without clearly specifying them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 4 frameworks (I5, I8, I9, I13) do not cover mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' As for the risk factors, none of the frameworks specified the different risk factors they considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' However, given the potential value of such categorization in helping organizations better triage and prioritize risks, we examined the frameworks and their questions/recommendations and extracted the risk factors each framework takes into account (see Table I and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Despite their respective focus (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', I14 focuses on hazards while touching vulnerability and exposure), all 16 frameworks consider potential vulnerability, and 15 frameworks cover haz- ard and exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' However, mitigation risks are significantly underemphasized and only covered by AU NSW AI Assurance Framework (I7) and ECP’s AI impact assessment framework (I15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Even for these two frameworks that consider mitigation risks, they do not provide a comprehensive assessment rather briefly mention such risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' For example, in I7: “Are there any residual risks?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', and in I15: “Considering planned mitigations, could the AI system cause significant or irreversible harms?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 8: Risk factors considered by collected frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Finding 1 to RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='2: RAI risk assessment frameworks need to distinguish among risk factors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', hazard, exposure, vulnerability, and mitigation risk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Although collected frameworks categorically encompass these factors to some degree, they may focus on particular factors while briefly touching on others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Further, mit- igation risks are significantly neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' This can lead to potential failure to identify and mitigate crucial RAI risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Finding 2 to RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='2: Existing RAI risk assessment frameworks provide some information on assessment procedures but fail to clearly specify inputs/outputs, stakeholders, and resources needed at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Finding 3 to RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='2: Many RAI risk assessment frame- works plainly list assessment measures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', questions, checklists) without considering their interconnections or dependencies, leading to an inefficient assessment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 3) RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='3: What are the outputs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' : This section discuss the outputs of the procedural frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Whether the output of a documented report is specified or not, the outputs of the procedural frameworks are, or at least should be, risk/impact assessment reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Some frameworks, such as I2, which creates a visualization of the risk level correlated to the RAI principles, and I3, which calculates risk and mitigation scores for various risk areas and gener- ates the level of impact, generate reports automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' For I10, assessors must manually generate a report based on the questionnaire and their answers to the questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The other procedural frameworks (I4, I5, I7-I9, I11, I13-I16) serve as (self-)assessment tools to guide assessors in identifying risks and do not require the preparation of a report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' However, since assessors should clearly document all the answers and the related questions when using the procedural frameworks, the processes result in documented assessment reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 15 15 16 2Finding to RQ3: While organizations may rely on RAI risk assessment frameworks for potential mitigation solutions, current frameworks either fail to provide concrete mitigation solutions, or lack a structured way to present the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' This makes it challenging for organizations to address identified risks effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' On the concreteness of RAI risk assessment frameworks 1) Relative concreteness: A risk assessment framework may appear concrete at one level but too abstract for the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' For example, management teams may consider certain assessment questions concrete, while development teams may find them not doable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Additionally, even seemingly concrete checklists or templates for RAI risk assessment may only be effective if assessors have a standardized and trustworthy approach to completing each item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Therefore, it is essential to have well-structured, concrete, and reusable solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', design patterns [24], [27]) in the lower level that align/connect with higher-level practices such as governance guidelines to ensure a comprehensive and effective risk assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 2) Trivialized concreteness: Our mapping study reveals that many frameworks trivialize the concept of “concreteness” by: Applying existing assessment concepts to new AI-specific artifacts/processes without further specifying potential solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Examples include acknowledging the existence of RAI risks and broadly mentioning that they need to be identified, documented, and mitigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Identifying new concepts in AI and providing some sub-categorization without providing potential solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Examples include acknowledging bias as a common issue in AI systems and listing different sources of bias (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', data, algorithm), but not providing specified solutions to different biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Identifying important RAI risks and referring to poten- tially stale non-AI frameworks, which may not be suitable for addressing RAI risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' It’s important to note that while higher-level frameworks may seem abstract, they do not always trivialize concreteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' These frameworks are generally more abstract because they need to be widely applicable and less prone to obsolescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' They can be helpful, particularly for management teams, as they point out areas where organizations can uplift their practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The key criteria for determining if a higher-level framework is concrete or not include: Whether high-level abstractions of potential assessment and/or mitigation measures are underpinned (specified or reasonably inferable) by lower-level concrete assessment techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Whether there is a clear understanding among higher- level stakeholders (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', management) about the inputs, processes, outputs, as well as required personnel and resources to complete the assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' This understanding may not necessarily require technical expertise but rather an understanding of the trust placed in the lower-level concrete assessment techniques utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Essentials to “Concreteness” We summarize the essential qualities, elements, and pro- cesses that a concrete RAI risk assessment framework should process in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' A concrete RAI risk assessment framework should have the following characteristics: 1) The assessment and/or mitigation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', questions/checklists/recommendations) proposed at one level/stage are reasonably underpinned/aligned/connected to other level/stage even if the measure itself is narrow-scoped and not directly covering different levels/stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 2) The or- ganization of assessment and mitigation should be layered, considering their dependencies on each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' This allows for a clearer assessment logic and more efficient assessment processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Currently, only EU’s ALTAI (I2) achieves a certain level of interconnectivity by providing an interactive online assessment tool where the following questions may vary depending on the previous answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 3) The framework should be extensible, dynamic, and adaptive in that it can be adapted and extended to more specific contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' All the qualities above result in enhanced assessment efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The elements to be covered by a comprehensive RAI risk assessment framework should include different dimen- sions, contexts, measurements, and mitigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Assessment and mitigation should be organized based on different RAI principles, RAI stakeholders, and AI-SDLC stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Existing organizational governance structures and measures should also be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Even if a framework focuses on a specific aspect (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', stage/principle-specific, designed for assessment instead of mitigation), it needs to be well connected (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', in- terconnected) with other aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Additionally, the framework should consider and specify contextual elements such as appli- cable regions, sectors, and compatibility with an organization’s existing risk management processes and structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Moreover, different risk factors should be considered, and corresponding assessment and mitigation be presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Especially, mitiga- tion risks are significantly neglected by existing frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Reusable mitigation plans should be suggested in a structured way, along with their pros and cons considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Specifications on the procedures required to conduct the RAI risk assessment can help assessors and assessees from different levels better understand the inputs/processes/outputs and required stakeholders and resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', data, tools, funds) for each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' However, only half (I1, I3, I4, I7, I10, I12, I14, I15) of the 16 frameworks provided such specifications to a certain extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Furthermore, 7 out of the 8 frameworks merely stated the steps needed to conduct the assessment, with only I4 specifying stakeholders involved in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' None of the frameworks provides details on the resources required to complete each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Threats to validity External.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The term “AI risk assessment” along with a set of other terms such as “AI risk management” and “AI impact Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 9: Essentials to building a concrete RAI risk assessment framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' assessment” has been used to mean largely the same topic: identification, assessment/measurement, and mitigation of RAI risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' We extended our search terms with a set of supportive terms that are being used interchangeably in the search string to ensure that all the relevant work were covered to mitigate this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Another issue is that we only include the publicly accessible RAI risk assessment frameworks, although some organizations have their own frameworks for internal use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Internal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' To mitigate the threat of not finding all rele- vant studies, we conducted a rigorous search using defined keywords with support terms and conducted snowballing to recover the missing studies from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' To address the bias from data collection and synthesis, one researcher performed the tasks and the other researcher reviewed and double-checked the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The two researchers discussed the inconsistency and reached a common ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' RELATED WORK The pressing need to manage RAI risks has attracted significant attention in both industry and academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Many studies on RAI risks have been published in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' However, they heavily focus on AI risk conceptualization and taxonomy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', [28]–[31]) and provide no concrete solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=', assessment/mitigation techniques) to RAI risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' With the increasing interest in managing RAI risks, more actionable solutions to managing RAI risks have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' [32] proposed to evaluate model risks by inspecting their behavior on counterfactuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Schwee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' [33] introduced a toolchain for assessing privacy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The toolchain takes in a model trained from the dataset to be shared and creates a privacy risk report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Yajima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' [34] showcased their work in progress on assessing machine learning security risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Failure mode and effect analysis (FMEA) has been adopted/extended for assessing RAI risks in [35]–[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Notably, EY and Trilateral Research published a survey of AI risk assessment methodologies in January 2022 [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' With the objective of providing RAI governors with noteworthy practices and regulations in the field, this survey presents a high-level overview of the global landscape of AI risk assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' The report discusses: 1) regulations and legislation worldwide containing AI risk assessment related elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 2) solutions to RAI risk assessment by several international organizations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 3) standards related to AI risk management and governance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' 4) a brief overview of part of the proposed approaches in industry and academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' While this report is cate- gorically comprehensive, it mainly aims to help RAI governors grasp the worldwide outline of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Furthermore, it lacks detailed and systematic analysis of the existing frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' In contrast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' the objective of this study is to provide RAI practitioners with a systematic summary of the existing RAI risk assessment frameworks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' and shed light on the future ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Interconnected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Layered ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Industry-level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Efficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Organization-level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Different levels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Qualities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Extensible ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Team-level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Dynamic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='RAI principles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Assessors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Adaptive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='RAI Stakeholders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Different Roles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Assessees ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Dimension ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='AI software ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='lifecycle stages ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Existing (other) risk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='assessment frameworks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Region ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Governance structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Essentials to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Elements ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Context ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='concreteness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Sector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Hazard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Organization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Exposure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Measurement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Vulnerability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Risk factors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Tools and techniques ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Mitigation risk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='for assessment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Mitigation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Reusable solutions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Steps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Stakeholders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Processes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Inputs & outputs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='Resourcesdevelopment of concrete RAI risk assessment frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK This paper conducts a systematic mapping study to evaluate the capabilities and limitations of existing RAI risk assessment frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' We examine key characteristics of a concrete framework, including specified RAI principles, stakeholders, AI system lifecycle stages, applicable regions and sectors, risk factors, and reusable mitigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' We provide insights to help facilitating the development of concrete RAI risk assessment frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' These includes presenting the assessment and mitigation measures in an interconnected and layered way and specifying the assessment procedures as well as associated inputs/outputs, stakeholders, and resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' For future work, we are developing a question bank with questions clearly labelled with respect to different characteristics, mitigations, and risk factors etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Based on the question bank, we plan to develop a concrete RAI risk assessment framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' REFERENCES [1] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Lu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content=' Zhu, X.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='com/wp-content/uploads/2022/01/A-survey-of- AI-Risk-Assessment-Methodologies-full-report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} +page_content='pdf' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFJT4oBgHgl3EQftC3b/content/2301.11616v1.pdf'} diff --git a/GNFLT4oBgHgl3EQfGi_B/content/tmp_files/2301.11993v1.pdf.txt b/GNFLT4oBgHgl3EQfGi_B/content/tmp_files/2301.11993v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2952b3f3c63beb1d2334d6e73063dd9e72e59b1c --- /dev/null +++ b/GNFLT4oBgHgl3EQfGi_B/content/tmp_files/2301.11993v1.pdf.txt @@ -0,0 +1,3951 @@ +Quantum Langevin theory for two coupled phase-conjugated electromagnetic waves +Yue Jiang,1, 2, ∗ Yefeng Mei,3, † and Shengwang Du4, ‡ +1JILA, National Institute of Standards and Technology and the University of Colorado, Boulder, Colorado 80309, USA +2Department of Physics, University of Colorado, Boulder, Colorado 80309, USA +3Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA +4Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA +(Dated: January 31, 2023) +While loss-gain-induced Langevin noises have been intensively studied in quantum optics, the ef- +fect of a complex-valued nonlinear coupling coefficient on the noises of two coupled phase-conjugated +optical fields has never been questioned before. Here, we provide a general macroscopic phenomeno- +logical formula of quantum Langevin equations for two coupled phase-conjugated fields with linear +loss (gain) and complex nonlinear coupling coefficient. The macroscopic phenomenological formula +is obtained from the coupling matrix to preserve the field commutation relations and correlations, +which does not require knowing the microscopic details of light-matter interaction and internal +atomic structures. To validate this phenomenological formula, we take spontaneous four-wave mix- +ing in a double-Λ four-level atomic system as an example to numerically confirm that our macroscopic +phenomenological result is consistent with that obtained from the microscopic Heisenberg-Langevin +theory. Finally, we apply the quantum Langevin equations to study the effects of linear gain and +loss, complex phase mismatching, as well as complex nonlinear coupling coefficient in entangled +photon pair (biphoton) generation, particularly to their temporal quantum correlations. +I. +INTRODUCTION +Quantum Langevin equations is a common approach +to studying an open quantum system involving loss or +gain, where the stochastic coupling between the system +and its environment is molded as a set of Langevin noise +operators [1–5]. For example, in the parametric down- +conversion (PDC) process, a pump laser beam passes +through a χ(2) nonlinear crystal and is down-converted +into a pair of phase-conjugated electromagnetic (EM) +waves. +In the simplest case with the perfect phase- +matching condition and an undepleted pump beam, with- +out linear loss or gain, the two phase-conjugated single- +mode fields are governed by the following coupled equa- +tions [6] +∂ +∂z +�ˆa1 +ˆa† +2 +� += M +�ˆa1 +ˆa† +2 +� += +� +0 +iκ +−iκ +0 +� �ˆa1 +ˆa† +2 +� +, +(1) +where ˆam and ˆa† +m (m = 1, 2) are the field annihilation +and creation operators, M is the 2 × 2 coupling matrix, +and κ is the (real) nonlinear coupling coefficient. Here +we consider only the forward-wave case with both fields +propagating along the same +z direction. If losses are +presented during the propagation of the two fields, the +coupling matrix is +M = +� +−α1 +iκ +−iκ −α2 +� +, +(2) +∗ yue.jiang@jila.colorado.edu +† meiyf@umich.edu +‡ dusw@utdallas.edu +and their coupled equations become [3, 7] +∂ +∂z +�ˆa1 +ˆa† +2 +� += +� +−α1 +iκ +−iκ −α2 +� �ˆa1 +ˆa† +2 +� ++ +�√2α1 ˆf1 +√2α2 ˆf † +2 +� +, +(3) +where αm > 0 are the loss (absorption) coefficients, +and ˆfm are the associated Langevin noise operators sat- +isfying [ ˆfm(ω, z), ˆf † +n(ω′, z′)] = δmnδ(ω − ω′)δ(z − z′). +If there is linear gain instead of loss, for example in +channel 1, i.e., α1 < 0, equation (3) can be modi- +fied by taking √2α1 ˆf1 → √−2α1 ˆf † +1. +One can show +that these Langevin noise operators are necessary to pre- +serve the commutation relations during propagation, i.e. +[ˆam(ω, z), ˆa† +n(ω′, z)] = [ˆam(ω, 0), ˆa† +n(ω′, 0)] = δmnδ(ω − +ω′). +Equation (3) has been widely applied for PDC pro- +cesses where the nonlinear coupling coefficient κ is real +[3, 7–9]. +However, in a more general case of cou- +pled phase-conjugated fields, such as four-wave mixing +(FWM) near atomic resonances [10–12], the nonlinear +coupling coefficient κ can take a complex value involving +complicated atomic transitions. +In this case, equation +(3) is not valid and its solution does not preserve com- +mutation relations of the fields. What are the general +quantum Langevin coupled equations accounting for the +complex nonlinear coupling coefficient? +To answer the question, the common approach is to +derive quantum Langevin equations by solving the light- +matter coupled Heisenberg equations, which requires +knowing microscopic details of light-matter interaction +such as atomic populations and transitions [11–13]. The +complexity of this approach increases dramatically as +more atomic transitions are involved and it is extremely +difficult for experimentalists to follow, particularly in +some situations where it is impossible to obtain full mi- +croscopic details. Then our reduced question becomes: +arXiv:2301.11993v1 [quant-ph] 27 Jan 2023 + +2 +Is it possible to obtain self-consistent quantum Langevin +coupled equations from the general expression of the cou- +pling matrix? We call this the macroscopic phenomeno- +logical approach. To our best knowledge, there has been +no published work in investigating Langevin noises in- +duced by a complex nonlinear coupling coefficient κ. +In this article, for the first time, we provide a gen- +eral macroscopic phenomenological formula of quantum +Langevin equations for two coupled phase-conjugated +fields with linear loss (gain) and complex nonlinear cou- +pling coefficient, in both forward- and backward-wave +configurations. The macroscopic phenomenological for- +mula is obtained from the coupling matrix by preserv- +ing commutation relations and correlations of the fields, +which does not require knowing the microscopic details of +light-matter interaction and internal atomic structures. +We aim to make it readable and accessible for experi- +mental researchers in the quantum optics community. +This article is structured as follows. In Sec. II, to ful- +fill the requirement of preserving commutation relations, +we formulate the general macroscopic phenomenologi- +cal quantum Langevin coupled equations and their solu- +tions from the coupling matrix taking into account linear +loss (gain) and complex nonlinear coupling coefficient, +in both forward- and backward-wave configurations. In +Sec. III, taking spontaneous four-wave mixing (SFWM) +in a double-Λ four-level atomic system as an example, +we derive the coupled Langevin equations from micro- +scopic light-atom Heisenberg interaction for this special +case. We numerically confirm that the macroscopic phe- +nomenological solution in Sec. II agrees well with the +microscopic approach. In Sec. IV, we apply the quan- +tum Langevin theory to study effects of linear gain and +loss, complex phase mismatching, and complex nonlinear +coupling coefficient in entangled photon pair (biphoton) +generation, particularly to their temporal quantum cor- +relations. We conclude in the last section V. +II. +QUANTUM LANGEVIN EQUATIONS +Here we consider the two coupled single-mode phase- +conjugated fields in either forward-wave or backward- +wave configuration, as illustrated in Fig. 1. +In the +forward-wave configuration [Fig. 1(a)], both fields prop- +agate along +z direction through a nonlinear medium +with a length L. +In the backward-wave configuration +[Fig. 1(b)], the two fields propagate in opposing direc- +tions. The field annihilation operators ˆam(t, z) can be +expressed as +ˆa1(t, z) = +1 +√ +2π +� +dωˆa1(ω, z)ei( ω +c z−ωt), +ˆa2(t, z) = +1 +√ +2π +� +dωˆa2(ω, z)ei(± ω +c z−ωt), +(4) +where ± represents that field 2 propagates along +z or +−z direction, for the forward-wave or backward-wave +configuration, respectively. +The filed operators satisfy +the following commutation relations +� +ˆam (t, z) , ˆa† +n (t′, z) +� += δmnδ(t − t′), +� +ˆam (ω, z) , ˆa† +n (ω′, z) +� += δmnδ(ω − ω′). +(5) +In the forward-wave configuration, both fields are input +at z = 0, or ˆa1(0) and ˆa2(0) are the “initial” boundary +conditions. The general coupling matrix is [14] +MF = +� +−α1 + i ∆k +2 +iκ +−iκ +−α∗ +2 − i ∆k +2 +� +, +(6) +where αm = −i ωm +2c χm with χm being linear suscepti- +bility, and ∆k (real) is the phase mismatching in vac- +uum. In general, αm is complex valued, whose real part +Re{αm} > 0 represents loss (or gain for Re{αm} < 0) +and imaginary part represents phase velocity dispersion. +The nonlinear coupling coefficient κ can also be complex- +valued. In the backward-wave configuration, the general +coupling matrix becomes [12, 15] +MB = +� +−α1 + i ∆k +2 +iκ +iκ +α∗ +2 − i ∆k +2 +� +, +(7) +and the “initial” boundary conditions are ˆa1(0) and +ˆa2(L): field 1 is input at z = 0 and field 2 is input at +z = L. +One can show that, under the following unitary gauge +transformation +�ˆa1 +ˆa† +2 +� += +� +eiθ/2 +0 +0 +e−iθ/2 +� �ˆa1 +ˆa† +2 +� += U +�ˆa1 +ˆa† +2 +� += +� ˆa1eiθ/2 +ˆa† +2e−iθ/2 +� +, +(8) +the corresponding coupling matrix become +MF(θ) = UMFU† = +� +−α1 + i ∆k +2 +iκeiθ +−iκe−iθ +−α∗ +2 − i ∆k +2 +� +, +(9) +and +MB(θ) = UMBU† = +� +−α1 + i ∆k +2 +iκeiθ +iκe−iθ +α∗ +2 − i ∆k +2 +� +. +(10) +As physics is preserved and unchanged under the above +gauge transformation, we take θ = 0 throughout this +article for convenience and simplification. +In presence of linear loss or gain, i.e., Re{αm} ̸= 0, or +complex nonlinear coupling coefficient, κ ̸= κ∗, the two- +mode coupled equations must include Langevin noise op- +erators to preserve the commutation relations of the field +operators in Eq. (5). The noise operators should only +be related to Re{αm} and Im{κ}. As κ is real, the cou- +pled equations in forward-wave configuration should be +reduced to the known Eq. (3). For both forward- and +backward-wave configurations in the same nonlinear ma- +terial, the noise origin is the same except field 2 prop- +agates along ±z direction for different configurations. +With these guidelines, we provide quantum Langevin +equations for the two phase-conjugated fields from their +coupling matrix in the following subsections. + +3 +𝑧 +𝑎�� +𝑎�� +� +Medium +0 +𝐿 +𝜅 +𝑧 +𝑎�� +𝑎�� +� +Medium +0 +𝐿 +𝜅 +(b) +(a) +Figure 1. +Schematics of two coupled phase-conjugated electromagnetic waves: +(a) forward-wave configuration, and (b) +backward-wave configuration. κ is the nonlinear coupling coefficient between the two modes. +A. +Forward-Wave Configuration +In +the +forward-wave +configuration +as +shown +in +Fig. 1(a), we find that its quantum Langevin coupled +equations can be expressed in the following general form +∂ +∂z +�ˆa1 +ˆa† +2 +� += MF +�ˆa1 +ˆa† +2 +� ++ NFR +� ˆf1 +ˆf † +2 +� ++ NFI +� ˆf † +1ˆf2 +� +(11) +with the “initial” condition at z = 0: +� +ˆam(ω, 0), ˆa† +n(ω′, 0) +� += δmnδ(ω − ω′). +(12) +The Langevin noise operators satisfy +� +ˆfm(ω, z), ˆf † +n(ω′, z′) +� += δmnδ(ω − ω′)δ(z − z′) +(13) +and have the following correlations +� +ˆf † +m(ω, z) ˆfn(ω′, z′) +� += 0, +� +ˆfm(ω, z) ˆf † +n(ω′, z′) +� += δmnδ(ω − ω′)δ(z − z′), +� +ˆfm(ω, z) ˆfn(ω′, z′) +� += +� +ˆf † +m(ω, z) ˆf † +n(ω′, z′) +� += 0. +(14) +The Langevin noise matrix is given by +NF ≡ +� +−(MF + MF +∗) = NFR + iNFI, +(15) +where NFR and NFI are the real and imaginary parts of +the matrix NF (i.e., NFmn = NFRmn + iNFImn), respec- +tively. +We obtain the solution of Eq. (11) at the output sur- +face z = L as the following +�ˆa1 (L) +ˆa† +2 (L) +� += eMFL +�ˆa1 (0) +ˆa† +2 (0) +� ++ +� L +0 +eMF(L−z) +� +NFR +� ˆf1 (z) +ˆf † +2 (z) +� ++ NFI +� ˆf † +1 (z) +ˆf2 (z) +�� +dz. +(16) +Defining +eMFL ≡ +� +A B +C D +� +, +(17) +eMF(L−z) ≡ +� +A1 (z) B1 (z) +C1 (z) D1 (z) +� +, +(18) +we rewrite Eq. (16) as +�ˆa1 (L) +ˆa† +2 (L) +� += +� +A B +C D +� �ˆa1 (0) +ˆa† +2 (0) +� ++ +� L +0 +� +A1 (z) B1 (z) +C1 (z) D1 (z) +� � +NFR +� ˆf1 (z) +ˆf † +2 (z) +� ++ NFI +� ˆf † +1 (z) +ˆf2 (z) +�� +dz. +(19) +We numerically confirm that the solution preserves the +commutation relations +� +ˆam(ω, L), ˆa† +n(ω′, L) +� += +� +ˆam(ω, 0), ˆa† +n(ω′, 0) +� += δmnδ(ω − ω′). +(20) +Now we examine some special cases. +Case 1: We first consider the coupling matrix MF in Eq. +(6) where the nonlinear coupling coefficient κ is real and +both modes have losses (Re{αm} ≥ 0) . This works for +most PDC processes [3, 7]. Under such a condition, we +have the following diagonalized noise matrix +NF = NFR = +�� +2Re{α1} +0 +0 +� +2Re{α2} +� +, +(21) +and the coupled Langevin equations +∂ +∂z +�ˆa1 +ˆa† +2 +� += MF +�ˆa1 +ˆa† +2 +� ++ +�� +2Re{α1} ˆf1 +� +2Re{α2} ˆf † +2 +� +, +(22) +which is the well-known result in literature [3, 7]. +Case 2: κ is real, the mode 1 has linear loss (Re{α1} = +α ≥ 0), and the mode 2 has linear gain (Re{α2} = −g ≤ +0). The noise matrix becomes +NF = +�√ +2α +0 +0 +i√2g +� +. +(23) +We have the following coupled Langevin equations +∂ +∂z +�ˆa1 +ˆa† +2 +� += MF +�ˆa1 +ˆa† +2 +� ++ +�√ +2α ˆf1 +√2g ˆf2 +� +. +(24) + +4 +Case 3: The two modes are perfectly phase-matched +without linear gain or loss: ∆k = 0, α1 = α2 = 0, but +the nonlinear coupling coefficient is complex-valued κ = +η + iζ. In this case, the coupled matrix is +MF = +� +0 +−ζ + iη +ζ − iη +0 +� +. +(25) +The noise matrix becomes +NF = Θ(ζ) +� +ζ +� +1 +1 +−1 1 +� ++ iΘ(−ζ) +� +−ζ +� +1 +1 +−1 1 +� +, +(26) +where Θ(ζ) is Heaviside step function, Θ(ζ) = 1 if ζ > 0, +Θ(ζ) = 0 if ζ ≤ 0. The Langevin coupled equations are +∂ +∂z +�ˆa1 +ˆa† +2 +� +=MF +�ˆa1 +ˆa† +2 +� ++ Θ(ζ) +� +ζ +� +1 +1 +−1 1 +� � ˆf1 +ˆf † +2 +� ++ Θ(−ζ) +� +−ζ +� +1 +1 +−1 1 +� � ˆf † +1ˆf2 +� +. +(27) +Eq. (27) shows that a complex-valued nonlinear coupling +coefficient also leads to Langevin noises even when there +is no linear gain or loss. This is revealed by this article +for the first time. +Case 4: As κ is real and there is no linear loss or gain +(α1 = α2 = 0), the coupled equations can be written as +i ∂ +∂z +�ˆa1 +ˆa† +2 +� += +� +− ∆k +2 +−κ +κ +∆k +2 +� �ˆa1 +ˆa† +2 +� += ˆH +�ˆa1 +ˆa† +2 +� +. +(28) +The effective Hamiltonian ˆH has anti-parity-time (APT) +symmetry, which has been demonstrated in FWM in cold +atoms [14, 16]. +B. +Backward-Wave Configuration +In the back-wave configuration as shown in Fig. 1(b), +the quantum Langevin coupled equations can be ex- +pressed in the following general form +∂ +∂z +�ˆa1 +ˆa† +2 +� += MB +�ˆa1 +ˆa† +2 +� ++ NBR +� ˆf1 +ˆf † +2 +� ++ NBI +� ˆf † +1ˆf2 +� +. +(29) +Different +from +the +forward-wave +configuration, +the +“boundary” condition is +� +ˆa1(ω, 0), ˆa† +1(ω′, 0) +� += +� +ˆa2(ω, L), ˆa† +2(ω′, L) +� += δ(ω − ω′). +(30) +The Langevin noise operators satisfy the same commu- +tation relations and correlations in Eqs. (13) and (14). +The Langevin noise matrix is given by +NB ≡ +� +1 +0 +0 −1 +� �� +−MB11 −MB12 +MB21 +MB22 +� ++ +� +−MB11 −MB12 +MB21 +MB22 +�∗ += NBR + iNBI, +(31) +where NBR and NBI are the real and imaginary parts of +the matrix NB, respectively. One can show that the noise +matrix defined in Eq. (31) has the same origin as that +in the forward-wave configuration in the same nonlinear +material: +NB = +� +1 +0 +0 −1 +� +NF. +(32) +We note that the choice of noise matrix is not unique. +For example, transformation ˆf1 → − ˆf1 or/and ˆf2 → − ˆf2 +does not affect computing any physical observable. We +elaborate on this more in Appendix A. +We obtain the solution of Eq. (29) at z = L as follow- +ing +�ˆa1 (L) +ˆa† +2 (L) +� += eMBL +�ˆa1 (0) +ˆa† +2 (0) +� ++ +� L +0 +eMB(L−z) +� +NBR +� ˆf1 (z) +ˆf † +2 (z) +� ++ NBI +� ˆf † +1 (z) +ˆf2 (z) +�� +dz. +(33) +We define +eMBL ≡ +� ¯A +¯B +¯C +¯D +� +, +(34) +eMB(L−z) ≡ +� ¯A1 (z) +¯B1 (z) +¯C1 (z) +¯D1 (z) +� +. +(35) +Different from the forward-wave case, in the backward- +wave configuration, the mode 1 input is at z = 0 and the +mode 2 input is at z = L. With known ˆa1(0) and ˆa2(L), +we rearrange Eq. (33) and obtain solutions for ˆa1(L) and +ˆa2(0): +�ˆa1 (L) +ˆa† +2 (0) +� += +� +A B +C D +� �ˆa1 (0) +ˆa† +2 (L) +� ++ +� +1 −B +0 −D +� � L +0 +� ¯A1 (z) +¯B1 (z) +¯C1 (z) +¯D1 (z) +� � +NBR +� ˆf1 (z) +ˆf † +2 (z) +� ++ NBI +� ˆf † +1 (z) +ˆf2 (z) +�� +dz, +(36) + +5 +where +A = ¯A − +¯B ¯C +¯D , +B = +¯B +¯D, +C = − +¯C +¯D, +D = 1 +¯D. +(37) +We numerically confirm that Eq. (36) preserves the com- +mutation relations +� +ˆa1(ω, L), ˆa† +1(ω′, L) +� += +� +ˆa1(ω, 0), ˆa† +1(ω′, 0) +� +, +� +ˆa2(ω, 0), ˆa† +2(ω′, 0) +� += +� +ˆa2(ω, L), ˆa† +2(ω′, L) +� +. +(38) +Similarly to the forward-wave configuration, we exam- +ine the following four special cases. +Case 1: We assume the nonlinear coupling coefficient κ +is real and both modes have losses (Re{αm} ≥ 0). Under +such a condition, we have the following diagonalized noise +matrix +NB = +�� +2Re{α1} +0 +0 +− +� +2Re{α2} +� +, +(39) +and the coupled Langevin equations +∂ +∂z +�ˆa1 +ˆa† +2 +� += MB +�ˆa1 +ˆa† +2 +� ++ +� � +2Re{α1} ˆf1 +− +� +2Re{α2} ˆf † +2 +� +. +(40) +Case 2: κ is real, mode 1 has linear loss (Re{α1} = α ≥ +0), and mode 2 has linear gain (Re{α2} = −g ≤ 0). The +noise matrix becomes +NF = +�√ +2α +0 +0 +−i√2g +� +. +(41) +We have the following coupled Langevin equations +∂ +∂z +�ˆa1 +ˆa† +2 +� += MB +�ˆa1 +ˆa† +2 +� ++ +� √ +2α ˆf1 +−√2g ˆf2 +� +. +(42) +Case 3: The two modes are perfectly phase-matched +without linear gain and loss: ∆k = 0, α1 = α2 = 0, +but the nonlinear coupling coefficient is complex-valued +κ = η + iζ. In this case, the coupled matrix is +MB = +� +0 +−ζ + iη +−ζ + iη +0 +� +. +(43) +The noise matrix becomes +NB = Θ(ζ) +� +ζ +� +1 +1 +1 −1 +� ++ iΘ(−ζ) +� +−ζ +� +1 +1 +1 −1 +� +. +(44) +The Langevin coupled equations are +∂ +∂z +�ˆa1 +ˆa† +2 +� +=MB +�ˆa1 +ˆa† +2 +� ++ Θ(ζ) +� +ζ +� +1 +1 +1 −1 +� � ˆf1 +ˆf † +2 +� ++ Θ(−ζ) +� +−ζ +� +1 +1 +1 −1 +� � ˆf † +1ˆf2 +� +. +(45) +Eq. (45) shows that in the backward-wave configuration, +a complex-valued nonlinear coupling coefficient also leads +to Langevin noises even though there is no linear gain or +loss. +Case 4: As κ is real and there are equal losses in both +modes (α1 = α2 = α > 0) with perfect phase matching +(∆k = 0), the coupled equations can be written as +i ∂ +∂z +�ˆa1 +ˆa† +2 +� += +� +−iα −κ +−κ +iα +� �ˆa1 +ˆa† +2 +� += ˆH +�ˆa1 +ˆa† +2 +� +. +(46) +Interestingly, the effective Hamiltonian ˆH here follows +parity-time (PT) symmetry [17, 18]. +III. +MICROSCOPIC ORIGIN OF LANGEVIN +NOISES: SFWM +One could validate the above phenomenological ap- +proach of quantum Langevin coupled equations by con- +firming the microscopic origin of the Langevin noises. +However, +for two systems with the same quantum +Langevin equations, their microscopic structures may be +quite different. Therefore it is impossible to sort all mi- +croscopic systems. In this section, we focus on SFWM in +a double-Λ four-level atomic system [10–12, 19, 20] with +electromagnetically induced transparency (EIT) [21, 22], +and show that the phenomenological approach in the +above section agrees with the numerical results from the +microscopic quantum theory of light-atom interaction. +We start from a single-atom picture, considering an +EM wave couples the atomic transition |j⟩ and |k⟩. The +induced single atom polarization ˆpjk ∝ µjkˆσjk, where +µjk is the electric dipole moment matrix element, ˆσjk = +|j ⟩⟨ k| is single atom transition operator from state |k⟩ to +|j⟩. In the Heisenberg-Langevin picture, the single-atom +transition operator can be expressed as +ˆσjk = ˆσ(0) +jk + +� +µν +βµν ˆf (σ) +µν , +(47) +where ˆσ(0) +jk = ⟨ˆσjk⟩ is the zeroth-order steady state so- +lution. The single atom noise operator between atomic +transition |ν⟩ → |µ⟩ is represented by ˆf (σ) +µν , which satisfies +the following correlations: +⟨ ˆf (σ) +µν (ω) ˆf (σ)† +µ′ν′ (ω′)⟩ = ⟨ ˆf (σ) +µν (ω) ˆf (σ) +ν′µ′(ω′)⟩ += Dµν,ν′µ′δ(ω − ω′), +⟨ ˆf (σ)† +µν (ω) ˆf (σ) +µ′ν′(ω′)⟩ = ⟨ ˆf (σ) +νµ (ω) ˆf (σ) +µ′ν′(ω′)⟩ += Dνµ,µ′ν′δ(ω − ω′), +(48) +where Dµν,ν′µ′ and Dνµ,µ′ν′ are diffusion coefficients. +In a continuous medium with atomic number density +n, the noises from different atoms are uncorrelated. We +have the spatially averaged atomic operator +ˆ¯σjk ≡ ˆσ(0) +jk + +1 +√ +nA +� +µν +βµν ˆ¯f (σ) +µν , +(49) + +6 +|1⟩ +|2⟩ +Δ� +𝑎��� +𝑧 +0 +𝐿 +(a) +(b) +|3⟩ +𝜔�� +𝜔� +𝐸� +𝐸� +𝑎�� +|4⟩ +𝜔� +𝜔� +𝜛 +𝜛 +Figure 2. Spontaneous four-wave mixing (SFWM) in a double-Λ four-level cold atomic medium. (a) Backward-wave geometry +of SFWM optical configuration. Driven by counter-propagating pump (Ep) and coupling (Ec) beams, phase-matched backward +Stokes (ˆas) and anti-Stokes (ˆaas) are spontaneously generated from a laser-cooled atomic medium. (b) Atomic energy-level +diagram. The pump (ωp) laser is detuned with ∆p from transition |1⟩ → |4⟩, and the coupling (ωc) laser is on-resonant with +transition |2⟩ → |3⟩. Stokes (ωs) photons are spontaneously generated from transition |4⟩ → |2⟩, and anti-Stokes (ωas) photons +from transition |3⟩ → |1⟩. ϖ = ωas − ω13 is the anti-Stokes photon frequency detuning from transition |1⟩ → |3⟩. +where A is the single-mode cross-section area, and the +spatially averaged atomic noise operators ˆ¯f (σ) +µν satisfy the +following modified correlations +⟨ ˆ¯f (σ) +µν (ω, z) ˆ¯f (σ)† +µ′ν′ (ω′, z′)⟩ = ⟨ ˆ¯f (σ) +µν (ω, z) ˆ¯f (σ) +ν′µ′(ω′, z′)⟩ += Dµν,ν′µ′δ(ω − ω′)δ(z − z′), +⟨ ˆ¯f (σ)† +µν (ω, z) ˆ¯f (σ) +µ′ν′(ω′, z′)⟩ = ⟨ ˆ¯f (σ) +νµ (ω, z) ˆ¯f (σ) +µ′ν′(ω′, z′)⟩ += Dνµ,µ′ν′δ(ω − ω′)δ(z − z′), +(50) +where the diffusion coefficients are the same as those from +the single-atom picture. +The electric field and polarization are described as +ˆE(t, z) = 1 +2 +� +ˆE(+)(t, z) + ˆE(−)(t, z) +� +, +ˆP(t, z) = 1 +2 +� +ˆP (+)(t, z) + ˆP (−)(t, z) +� +, +(51) +Where ˆE(+), ˆP (+) and ˆE(−), ˆP (−) are positive and nega- +tive frequency parts. We take the following Fourier trans- +form +ˆE(+)(t, z) = +1 +√ +2π +� +dω ˆE(ω, z)ei(± ω +c z−ωt), +ˆP (+)(t, z) = +1 +√ +2π +� +dω ˆP(ω, z)ei(± ω +c z−ωt), +(52) +where ˆE(ω, z), ˆP(ω, z) are complex amplitudes in fre- +quency domain. +The Maxwell equation under slowly +varying envelope approximation (SVEA) can be written +as +±∂ ˆE(ω, z) +∂z += i +2ωη ˆP(ω, z), +(53) +where ± represents for propagation direction along ±z, +and free space impedance η = 1/(cε0) = 377 Ohm, with +c being the speed of light in vacuum, and ε0 the vacuum +permittivity. With quantized electric field +ˆE(ω, z) = +� +2ℏω +cε0Aˆa(ω, z), +(54) +and +ˆP(ω, z) = 2nµjkˆ¯σjk(ω, z), +(55) +we obtain the Langevin equation for the EM field in the +atomic medium +±∂ˆa(ω, z) +∂z += i nAgjkˆ¯σjk(ω, z) += i nAgjkˆσ(0) +jk (ω, z) + ˆ¯F(ω, z), +(56) +where +gjk = µjk +� +ωjk +2cε0ℏA, +ˆ¯F(ω, z) = i +√ +nAgjk +� +µν +βµν ˆ¯f (σ) +µν (ω, z) += iµjk +� nωjk +2cε0ℏ +� +µν +βµν ˆ¯f (σ) +µν (ω, z). +(57) +Here gjk = g∗ +kj is single photon-atom coupling strength. +Now we turn to the backward-wave SFWM in a double- +Λ four-level atomic system as illustrated in Fig. 2. In +presence of counter-propagating pump (Ep, ωp) and cou- +pling (Ec, ωc) laser beams, phase-matched Stokes (ωs) +and anti-Stokes (ωas) are spontaneously generated and +propagate through the medium in opposing directions. +In the rotating reference frame, the interaction Hamilto- +nian for a single atom is +ˆV = − ℏ +� +g31ˆaasˆσ31 + g13ˆa† +asˆσ13 +� +− ℏ +� +g42ˆasˆσ42 + g24ˆa† +sˆσ24 +� +− 1 +2ℏ (Ωcˆσ32 + Ω∗ +c ˆσ23) − 1 +2ℏ +� +Ωpˆσ41 + Ω∗ +pˆσ14 +� +− ℏ∆pˆσ44 − ℏϖˆσ33 − ℏϖˆσ22, +(58) +where Ωc = µ32Ec/ℏ is coupling Rabi frequency. The +coupling laser is on-resonant with transition |2⟩ → |3⟩. +Ωp = µ41Ep/ℏ is pump Rabi frequency. +The pump +laser is far detuned from the transition |1⟩ → |4⟩ with +∆p = ωp − ω14 so that the atomic population mainly oc- +cupies the ground state |1⟩. We take this ground-state + +7 +approximation through this section. +With continuous- +wave pump and coupling driving fields, the energy con- +servation leads to ωas+ωs = ωc+ωp. Here ϖ = ωas−ω13 +is the anti-Stokes frequency detuning and thus the Stokes +frequency detuning is ωs − ωs0 = −ϖ. +The atomic evolution is governed by the following +Heisenberg-Langevin equation [11] +∂ +∂t ˆσjk = i +ℏ[ ˆV , ˆσjk] − γjkˆσjk + rA +jk + ˆf (σ) +jk , +(59) +where γjk = γkj (nonzero only as j ̸= k) are dephasing +rates, rA +jk (nonzero only as j = k) are the population +transfer resulting from spontaneous emission decay. The +full equation of motion can be found in Appendix B. The +diffusion coefficients Djk,j′k′ can be obtained through the +Einstein relation +Djk,j′k′ = ∂ +∂t ⟨ˆσjkˆσj′k′⟩ +− +� +ˆAjkˆσj′k′ +� +− +� +ˆσjk ˆAj′k′ +� +, +(60) +where ˆAjk = +∂ +∂t ˆσjk − ˆf (σ) +jk . For the SFWM governed by +Eq. (59), we have [11, 12] +� +�� +D12,21 D12,24 +D42,21 D42,24 +D12,31 D12,34 +D42,31 D42,34 +D13,21 D13,24 +D43,21 D43,24 +D13,31 D13,34 +D43,31 D43,34 +� +�� += +� +�� +2γ12 ⟨ˆσ11⟩ + Γ31 ⟨ˆσ33⟩ + Γ41 ⟨ˆσ44⟩ γ12 ⟨ˆσ14⟩ +0 +0 +γ12 ⟨ˆσ41⟩ +0 +0 +0 +0 +0 +Γ3 ⟨ˆσ11⟩ + Γ31 ⟨ˆσ33⟩ + Γ41 ⟨ˆσ44⟩ Γ3 ⟨ˆσ14⟩ +0 +0 +Γ3 ⟨ˆσ41⟩ +Γ3 ⟨ˆσ44⟩ +� +�� , +(61) +� +�� +D21,12 D21,42 +D24,12 D24,42 +D21,13 D21,43 +D24,13 D24,43 +D31,12 D31,42 +D34,12 D34,42 +D31,13 D31,43 +D34,13 D34,43 +� +�� += +� +�� +2γ12 ⟨ˆσ22⟩ + Γ32 ⟨ˆσ33⟩ + Γ42 ⟨ˆσ44⟩ +0 +γ12 ⟨ˆσ23⟩ +0 +0 +Γ4 ⟨ˆσ22⟩ + Γ32 ⟨ˆσ33⟩ + Γ42 ⟨ˆσ44⟩ +0 +Γ4 ⟨ˆσ23⟩ +γ12 ⟨ˆσ32⟩ +0 +0 +0 +0 +Γ4 ⟨ˆσ32⟩ +0 +Γ4 ⟨ˆσ33⟩ +� +�� . +(62) +Solving Eq. (59) under the ground-state approximation +⟨ˆσ11⟩ ∼= 1 with weak pump excitation ∆p ≫ {Ωp, Γ4}, we +get the single-atom steady-state solutions (with µν = +12, 13, 42, 43) +ˆσ13 = ˆσ(0) +13 + +� +µν +βas +µν ˆf (σ) +µν , +ˆσ42 = ˆσ(0) +42 + +� +µν +βs +µν ˆf (σ) +µν , +(63) +where +ˆσ(0) +13 =4 (ϖ + iγ12) +T (ϖ) +g31ˆaas ++ +ΩcΩp +T (ϖ) (∆p + iγ14)g24ˆa† +s, +ˆσ(0) +42 =(ϖ + iγ13) +T (ϖ) +|Ωp|2 +(∆p − iγ24) +1 +(∆p + iγ14)g24ˆa† +s ++ +Ω∗ +pΩ∗ +c +T (ϖ) (∆p − iγ24)g31ˆaas, +(64) +βas +12 = i2Ωc +T (ϖ), +βas +13 = −i4 (ϖ + iγ12) +T (ϖ) +, +βas +42 = − +iΩcΩp +T (ϖ) (∆p − iγ24), +βas +43 = +i2Ωp (ϖ + iγ12) +T (ϖ) (∆p − iγ34), +βs +12 = i2 (ϖ + iγ13) +T (ϖ) +Ω∗ +p +(∆p − iγ24), +βs +13 = − +iΩ∗ +pΩ∗ +c +T (ϖ) (∆p − iγ24), +βs +42 = − +i +(∆p − iγ24), +βs +43 = − +iΩ∗ +c +2 (∆p − iγ24) (∆p − iγ34), +(65) +where T(ϖ) ≡ |Ωc|2 − 4 (ϖ + iγ13) (ϖ + iγ12). We then +obtain the ensemble spatially averaged atomic operators + +8 +-50 +0 +50 +0 +0.2 +0.4 +0.6 +0.8 +1 +-50 +0 +50 +-10 +-8 +-6 +-4 +-2 +0 +10-7 +-50 +0 +50 +0 +0.2 +0.4 +0.6 +0.8 +1 +Macro +Micro +NLN +-50 +0 +50 +-2 +-1 +0 +1 +2 +3 +10-8 +( +- +) +( +- +) +( +- +) +( +- +) +(a) +(b) +(c) +(d) +Figure 3. Comparison of commutation relations between the macroscopic (“Macro”, blue solid lines) and microscopic (“Micro”, +red dashed lines) approaches in the group delay regime: (a) [ˆaas(L), ˆa† +as(L)], (b) [ˆaas(L), ˆa† +as(L)] − δ(ϖ − ϖ′), (c) [ˆas(0), ˆa† +s(0)], +and (d)[ˆas(0), ˆa† +s(0)] − δ(ϖ − ϖ′). The results with no Langevin noise operators (“NLN”) are shown as black dotted lines in +(a) and (c). +for generating anti-Stokes and Stokes fields from Eq. (49) +ˆ¯σ13 = ˆσ(0) +13 + +1 +√ +nA +� +µν +βas +µν ˆ¯f +(σ) +µν , +ˆ¯σ42 = ˆσ(0) +42 + +1 +√ +nA +� +µν +βs +µν ˆ¯f +(σ) +µν . +(66) +Following the procedures in Eqs. (56) and (57), +∂ˆaas(ω, z) +∂z += i nAg13ˆ¯σ13(ω, z), +∂ˆa† +s(ω, z) +∂z += i nAg42ˆ¯σ42(ω, z), +(67) +we get coupled equations for counter-propagating anti- +Stokes (propagating along +z) and Stokes (propagating +along −z) fields in the backward-wave configuration +∂ +∂z +�ˆaas +ˆa† +s +� += +� +−αas + i ∆k +2 +iκas +iκs +α∗ +s − i ∆k +2 +� �ˆaas +ˆa† +s +� ++ +� ˆ¯Fas +− ˆ¯F † +s +� +, +(68) +where +ˆ¯Fas = ig13 +√ +nA +� +βas +12 ˆ¯f (σ) +12 + βas +13 ˆ¯f (σ) +13 + βas +42 ˆ¯f (σ) +42 + βas +43 ˆ¯f (σ) +43 +� +, +ˆ¯F † +s = −ig42 +√ +nA +� +βs +12 ˆ¯f (σ) +12 + βs +13 ˆ¯f (σ) +13 + βs +42 ˆ¯f (σ) +42 + βs +43 ˆ¯f (σ) +43 +� +, +(69) +and +αas = −iωas +2c χas, +αs = −iωs +2c χs, +κas = +√ωasωs +2c +χ(3) +as EpEc, +κs = +√ωsωas +2c +χ(3)∗ +s +E∗ +pE∗ +c , +χas = 4n |µ13|2 +ε0ℏ +(ϖ + iγ12) +T (ϖ) +, +χs = n |µ24|2 +ε0ℏ +(ϖ − iγ13) +T ∗ (ϖ) +|Ωp|2 +∆2p + γ2 +14 +, +χ(3) +as = nµ13µ32µ24µ41 +ε0ℏ3 +1 +T (ϖ) +1 +(∆p + iγ14), +χ(3) +s += nµ13µ32µ24µ41 +ε0ℏ3 +1 +T ∗ (ϖ) +1 +(∆p + iγ14), +(70) + +9 +-10 +-5 +0 +5 +10 +1 +1.0001 +1.0002 +1.0003 +1.0004 +Macro +Micro +-10 +-5 +0 +5 +10 +0 +1 +2 +3 +4 +10-4 +-10 +-5 +0 +5 +10 +1 +1.0001 +1.0002 +1.0003 +1.0004 +-10 +-5 +0 +5 +10 +0 +1 +2 +3 +4 +10-4 +( +- +) +( +- +) +( +- +) +( +- +) +(a) +(b) +(c) +(d) +Figure 4. +Four real correlations of Stokes and anti-Stokes fields in the group delay regime: +(a) ⟨ˆaas(L)ˆa† +as(L)⟩, (b) +⟨ˆa† +as(L)ˆaas(L)⟩, (c) ⟨ˆas(0)ˆa† +s(0)⟩, and (d) ⟨ˆa† +s(0)ˆas(0)⟩. +The macroscopic (“Macro”) and microscopic (“Micro”) approaches +are shown as blue solid and red dashed lines, respectively. +The expressions for βas +µν and βs +µν are listed in Eqs. (65). +∆k = (ωas−ωs)/c−(⃗kc+⃗kp)· ˆz is the phase mismatching +in vacuum. +Here the complex αas represents the EIT +loss and phase dispersion. +α∗ +s is the Raman gain and +dispersion along −z propagation direction. One can show +that the nonlinear coupling coefficients can be expressed +as κas = κeiθ and κs = κe−iθ, where +κ = +√ωasωs +2c +nµ13µ24 +ε0ℏ +���� +ΩpΩc +∆p + iγ14 +���� +1 +T(ϖ), +(71) +and θ is the phase of ΩpΩc/(∆p + iγ14). As a result, κas +and κs fulfill the gauge transformation discussed in Sec. +II. Therefore, to be consistent with the treatment in Sec. +II, we rewrite Eq. (68) to +∂ +∂z +�ˆaas +ˆa† +s +� += MB +�ˆaas +ˆa† +s +� ++ +� ˆFas +− ˆF † +s +� +, +(72) +where +MB = +� +−αas + i ∆k +2 +iκ +iκ +α∗ +s − i ∆k +2 +� +, +ˆFas = ˆ¯Fase−iθ/2, +ˆF † +s = ˆ¯F † +s eiθ/2. +(73) +Similarly, we rewrite the SFWM quantum Langevin +equations in the forward-wave configuration in Ap- +pendix C. +We now turn to compare Eq. +(72) with Eq. +(29) +from the phenomenological approach in Sec. II, where +we take mode 1 as anti-Stokes and mode 2 as Stokes in +the backward-wave configuration. +From Eq. +(29), we +have +ˆFas = NBR11 ˆf1 + NBI11 ˆf † +1 + NBI12 ˆf2 + NBR12 ˆf † +2, +ˆF † +s = −NBR21 ˆf1 − NBI21 ˆf † +1 − NBI22 ˆf2 − NBR22 ˆf † +2. +(74) +Therefore, we obtain ˆFas and ˆF † +s from two different ap- +proaches: Eq. (69) from the microscopic photon-atom +interaction, and Eq. (74) from the macroscopic phe- +nomenological approach. +Although we remark that +the atomic noise operators ˆ¯f (σ) +µν +are different from the +field noise operators ˆfm, the correlations of ˆFas and ˆFs +uniquely determine the system performance. While we +find it difficult to analytically prove the two approaches +are equivalent, we could numerically compute and com- +pare the commutation relations and correlations of ˆaas, +ˆa† +as, ˆas, and ˆa† +s. +We consider here the backward-wave SFWM in laser- +cooled +85Rb atoms with relevant atomic energy lev- +els being |1⟩ = +��52S1/2, F = 2 +� +, |2⟩ = +��52S1/2, F = 3 +� +, + +10 +-2 +-1 +0 +1 +2 +Macro +Micro +-2 +-1 +0 +1 +2 +-2 +-1 +0 +1 +2 +-10 +-5 +0 +5 +10 +-2 +-1 +0 +1 +2 +-10 +0 +10 +-2 +-1 +0 +1 +2 +-10 +-5 +0 +5 +10 +-2 +-1 +0 +1 +2 +-2 +-1 +0 +1 +2 +-2 +-1 +0 +1 +2 +-2 +-1 +0 +1 +2 +-10 +-5 +0 +5 +10 +-2 +-1 +0 +1 +2 +-10 +0 +10 +-2 +-1 +0 +1 +2 +-10 +-5 +0 +5 +10 +-2 +-1 +0 +1 +2 +10-2 ( +- +) +10-2 ( +- +) +10-2 ( +- +) +10-2 ( +- +) +10-2 ( +- +) +10-2 ( +- +) +(a) +(b) +(c) +(d) +(e) +(f) +Figure 5. +Twelve complex correlations of Stokes and anti-Stokes fields in the group delay regime: (a) ⟨ˆaas(L)ˆaas(L)⟩ = +⟨ˆa† +as(L)ˆa† +as(L)⟩∗, (b) ⟨ˆaas(L)ˆas(0)⟩ = ⟨ˆa† +s(0)ˆa† +as(L)⟩∗, (c) ⟨ˆaas(L)ˆa† +s(0)⟩ = ⟨ˆas(0)ˆa† +as(L)⟩∗, (d) ⟨ˆa† +as(L)ˆas(0)⟩ = ⟨ˆa† +s(0)ˆaas(L)⟩∗, +(e) ⟨ˆas(0)ˆaas(L)⟩ = ⟨ˆa† +as(L)ˆa† +s(0)⟩∗, and (f) ⟨ˆas(0)ˆas(0)⟩ = ⟨ˆa† +s(0)ˆa† +s(0)⟩∗. The macroscopic (“Macro”) and microscopic (“Mi- +cro”) approaches are shown as blue solid and red dashed lines, respectively. +|3⟩ = +��52P1/2, F = 3 +� +, |4⟩ = +��52P3/2, F = 3 +� +. The decay +and dephasing rates for corresponding energy levels are +Γ3 = Γ4 = 2π × 6 MHz, Γ31 = 5 +9Γ3, Γ32 = 4 +9Γ3, Γ41 = +4 +9Γ4, Γ42 = 5 +9Γ4, γ13 = γ23 = γ14 = γ24 = 2π × 3 MHz, +and γ12 = 2π × 0.03 MHz. With vacuum inputs in both +Stokes (z = L) and anti-Stokes (z = 0) modes, we have +⟨ˆaas(ϖ, 0)ˆa† +as(ϖ′, 0)⟩ = ⟨ˆas(ϖ, L)ˆa† +s(ϖ′, L)⟩ = δ(ϖ − ϖ′) +and ⟨ˆa† +as(ϖ, 0)ˆaas(ϖ′, 0)⟩ = ⟨ˆa† +s(ϖ, L)ˆas(ϖ′, L)⟩ = 0. +There is also no correlation between Stokes and anti- +Stokes fields at their inputs. +We numerically compute SFWM in two different +regimes to confirm the consistency between the macro- +scopic and microscopic theories. i) The first is the group +delay regime, where the SFWM spectrum bandwidth is +determined by the EIT slow-light induced phase mis- +matching [10]. +The working parameters are: +Ωp = +2π × 1.2 MHz, Ωc = 2π × 12 MHz, ∆p = 2π × 500 MHz. +The cold atomic medium with length L = 2 cm has den- +sity n = 5.1 × 1016 m−3, corresponding to an atomic +optical depth OD = 80 on the anti-Stokes resonance +transition. ii) The second is the Rabi oscillation regime, +where biphoton correlation reveals single-atom dynamics +[10]. The working parameters are: Ωp = 2π × 1.2 MHz, +Ωc = 2π ×24 MHz, ∆p = ωp −ω14 = 2π ×500 MHz. The +cold atomic medium with length L = 0.2 cm has density +n = 6.4×1014 m−3, corresponding to OD = 0.1. In both +cases, we take ∆k = 127 rad/m. +The numerical results in the group delay regime are + +11 +-50 +0 +50 +0 +0.2 +0.4 +0.6 +0.8 +1 +-50 +0 +50 +-10 +-5 +0 +10-9 +-50 +0 +50 +0 +0.2 +0.4 +0.6 +0.8 +1 +Macro +Micro +NLN +-50 +0 +50 +-1 +0 +1 +2 +10-11 +( +- +) +( +- +) +( +- +) +( +- +) +(a) +(b) +(c) +(d) +Figure 6. Comparison of commutation relations between the macroscopic (“Macro”, blue solid lines) and microscopic (“Micro”, +red dashed lines) approaches in the damped Rabi oscillation regime: (a) [ˆaas(L), ˆa† +as(L)], (b) [ˆaas(L), ˆa† +as(L)] − δ(ϖ − ϖ′), (c) +[ˆas(0), ˆa† +s(0)], and (d)[ˆas(0), ˆa† +s(0)] − δ(ϖ − ϖ′). The results with no Langevin noise operators (“NLN”) are shown as black +dotted lines in (a) and (c). +plotted in Figs. 3, 4, and 5. +The commutation re- +lations [ˆaas(L), ˆa† +as(L)] and [ˆas(0), ˆa† +s(0)] are shown in +Fig. 3. +Both macroscopic and microscopic approaches +agree well with each other [Figs. 3(a) and (c)], with neg- +ligible relative small difference < 1.0 × 10−6 [Figs. 3(b) +and (d)]. As expected, the macroscopic phenomenologi- +cal results give perfect flat lines at [ˆaas(L,ϖ),ˆa† +as(L,ϖ′)] +δ(ϖ−ϖ′) += +[ˆas(0,ϖ),ˆa† +s(0,ϖ′)] +δ(ϖ−ϖ′) += 1 which is the starting point of Sec. +II. The microscopic results of field commutations are +consistent with the macroscopic approach, but with < +1.0 × 10−6 deviation at some spectra points. +As we +understand, these small spectra discrepancies may be +caused by the ground-state and zeroth-order approxi- +mations we take for solving the microscopic Heisenberg- +Langevin equations (59). If the Langevin noise operators +are not taken into account, as shown in the black dotted +curves in Figs. 3(a) and (c), the anti-Stokes commuta- +tion relation is not preserved and displays EIT transmis- +sion spectrum, while Stokes commutation relation still +approximately holds due to the negligible gain or loss in +Stokes channel under the ground-state approximation. +Figure 4 displays four real-valued correlations of +Stokes and anti-Stokes fields: +(a )⟨ˆaas(L)ˆa† +as(L)⟩, (b) +⟨ˆa† +as(L)ˆaas(L)⟩, (c) ⟨ˆas(0)ˆa† +s(0)⟩, and (d) ⟨ˆa† +s(0)ˆas(0)⟩. +Figure 5 shows the twelve (six pairs) complex-valued +correlations +of +Stokes +and +anti-Stokes +fields: +(a) +⟨ˆaas(L)ˆaas(L)⟩ = ⟨ˆa† +as(L)ˆa† +as(L)⟩∗, (b) ⟨ˆaas(L)ˆas(0)⟩ = +⟨ˆa† +s(0)ˆa† +as(L)⟩∗, (c) ⟨ˆaas(L)ˆa† +s(0)⟩ = ⟨ˆas(0)ˆa† +as(L)⟩∗, (d) +⟨ˆa† +as(L)ˆas(0)⟩ = ⟨ˆa† +s(0)ˆaas(L)⟩∗, (e) ⟨ˆas(0)ˆaas(L)⟩ = +⟨ˆa† +as(L)ˆa† +s(0)⟩∗, and (f) ⟨ˆas(0)ˆas(0)⟩ = ⟨ˆa† +s(0)ˆa† +s(0)⟩∗. +The macroscopic solutions agree well with those obtained +from the microscopic approach. +The numerical results in the Rabi oscillation regime are +plotted in Figs. 6, 7, and 8. The macroscopic phenomeno- +logical results also agree remarkably well with those from +the microscopic theory. +IV. +BIPHOTON GENERATION +We now turn to apply the quantum Langevin the- +ory to study time-frequency entangled photon pair +(biphoton) generation through spontaneous four-wave +mixing +process, +especially +in +a +variety +of +situa- +tions involving gain, loss, and/or complex nonlinear +coupling +coefficient. +We +consider +continuous-wave +pumping +whose +time +translation +symmetry +leads +to +frequency +anti-correlation +ω1 + ω2 +=constant +between the paired photons. +In the spontaneous + +12 +-50 +0 +50 +1 +1+0.5E-7 +1+1.0E-7 +1+1.5E-7 +Macro +Micro +-50 +0 +50 +0 +5 +10 +15 +10-8 +-50 +0 +50 +1 +1+0.2E-7 +1+0.4E-7 +1+0.6E-7 +1+0.8E-7 +1+1.0E-7 +-50 +0 +50 +0 +2 +4 +6 +8 +10 +10-8 +( +- +) +( +- +) +( +- +) +( +- +) +(a) +(b) +(c) +(d) +Figure 7. Four real correlations of Stokes and anti-Stokes fields in the damped Rabi oscillation regime: (a) ⟨ˆaas(L)ˆa† +as(L)⟩, (b) +⟨ˆa† +as(L)ˆaas(L)⟩, (c) ⟨ˆas(0)ˆa† +s(0)⟩, and (d) ⟨ˆa† +s(0)ˆas(0)⟩. The macroscopic (“Macro”) and microscopic (“Micro”) approaches are +shown as blue solid and red dashed lines, respectively. +four-wave mixing process, both input states are vac- +uum: +⟨ˆa† +1(ϖ, 0)ˆa1(ϖ′, 0)⟩ = ⟨ˆa† +2(ϖ, 0)ˆa2(ϖ′, 0)⟩ = 0, +⟨ˆa1(ϖ′, 0)ˆa† +1(ϖ, 0)⟩ += +⟨ˆa2(ϖ′, 0)ˆa† +2(ϖ, 0)⟩ += +δ(ϖ +− ϖ′) +for +the +forward-wave +configuration, +and ⟨ˆa† +1(ϖ, 0)ˆa1(ϖ′, 0)⟩ += +⟨ˆa† +2(ϖ, L)ˆa2(ϖ′, L)⟩ += +0, +⟨ˆa1(ϖ, 0)ˆa† +1(ϖ′, 0)⟩ = ⟨ˆa2(ϖ, L)ˆa† +2(ϖ′, L)⟩ = δ(ϖ − ϖ′) +for the backward-wave configuration. From Eq. (4), with +ω1 = ω10 + ϖ and ω2 = ω20 − ϖ, we have +ˆa1(t, z1) = eiω10( z1 +c −t) +√ +2π +� +dϖˆa1(ϖ, z1)eiϖ( z1 +c −t)e−i ∆k +2 z1, +ˆa2(t, z2) = eiω20(± z2 +c −t) +√ +2π +� +dϖˆa2(ϖ, z2)eiϖ(± z2 +c −t)e−i ∆k +2 z2, +(75) +where ± represents the forward-wave (+) or backward- +wave (−) configuration, z = z1 and z = z2 are the +output positions of channels 1 and 2, respectively. For +the forward-wave configuration, z1 = z2 = L. For the +backward-wave configuration, z1 = L and z2 = 0. The +phase mismatching in vacuum ∆k = (ωas ±ωs)/c−(⃗kc + +⃗kp)· ˆz ≃ (ωas0 ±ωs0)/c−(⃗kc +⃗kp)· ˆz is nearly a constant. +The vacuum time delay zi/c constants are usually very +small in usual experimental conditions, from now on we +ignore these constants for simplification and rewrite the +above equations to (otherwise one just needs to make a +time translation t → t − zi/c) +ˆa1(t, z1) = e−iω10t +√ +2π +� +dϖˆa1(ϖ, z1)e−iϖt, +ˆa2(t, z2) = e−iω20t +√ +2π +� +dϖˆa2(ϖ, z2)eiϖt. +(76) +The photon rate in channel m can be computed from +Rm ≡ +� +ˆa† +m (t, zm) ˆam (t, zm) +� += 1 +2π +�� ∞ +−∞ +dϖdϖ′e−iϖteiϖ′t � +ˆa† +m (ϖ′, zm) ˆam (ϖ, zm) +� +. +(77) +Here we are particularly interested in the two-photon +Glauber correlation in the time domain, which can be +computed from the following two different orders +G(2) +2,1 (t2, t1) +≡⟨ˆa† +1 (t1, z1) ˆa† +2 (t2, z2) ˆa2 (t2, z2) ˆa1 (t1, z1)⟩ +=|⟨ˆa2 (t2, z2) ˆa1 (t1, z1)⟩|2 ++ |⟨ˆa† +2 (t2, z2) ˆa1 (t1, z1)⟩|2 + R1R2, +(78) + +13 +-5 +0 +5 +Macro +Micro +-5 +0 +5 +-5 +0 +5 +-50 +0 +50 +-5 +0 +5 +-50 +0 +50 +-5 +0 +5 +-50 +0 +50 +-5 +0 +5 +-5 +0 +5 +-5 +0 +5 +-5 +0 +5 +-50 +0 +50 +-5 +0 +5 +-50 +0 +50 +-5 +0 +5 +-50 +0 +50 +-5 +0 +5 +10-5 ( +- +) +10-5 ( +- +) +10-5 ( +- +) +10-5 ( +- +) +10-5 ( +- +) +10-5 ( +- +) +(a) +(b) +(c) +(d) +(e) +(f) +Figure 8. +Twelve complex correlations of Stokes and anti-Stokes fields in the damped Rabi oscillation regime: +(a) +⟨ˆaas(L)ˆaas(L)⟩ = ⟨ˆa† +as(L)ˆa† +as(L)⟩∗, (b) ⟨ˆaas(L)ˆas(0)⟩ = ⟨ˆa† +s(0)ˆa† +as(L)⟩∗, (c) ⟨ˆaas(L)ˆa† +s(0)⟩ = ⟨ˆas(0)ˆa† +as(L)⟩∗, (d) ⟨ˆa† +as(L)ˆas(0)⟩ = +⟨ˆa† +s(0)ˆaas(L)⟩∗, (e) ⟨ˆas(0)ˆaas(L)⟩ = ⟨ˆa† +as(L)ˆa† +s(0)⟩∗, and (f) ⟨ˆas(0)ˆas(0)⟩ = ⟨ˆa† +s(0)ˆa† +s(0)⟩∗. The macroscopic (“Macro”) and mi- +croscopic (“Micro”) approaches are shown as blue solid and red dashed lines, respectively. +G(2) +1,2 (t1, t2) +≡⟨ˆa† +2 (t2, z2) ˆa† +1 (t1, z1) ˆa1 (t1, z1) ˆa2 (t2, z2)⟩ +=|⟨ˆa1 (t1, z1) ˆa2 (t2, z2)⟩|2 ++ |⟨ˆa† +2 (t2, z2) ˆa1 (t1, z1)⟩|2 + R1R2, +(79) +where we have applied the Gaussian moment theorem +[23, 24] to decompose the fourth-order field correlations +to the sum of the products of second-order field corre- +lations. +The first term in Eqs. +(78) and (79) can be +expressed as |Ψ2,1(t2, t1)|2 and |Ψ1,2(t1, t2)|2, where +Ψ2,1(t2, t1) = ⟨ˆa2 (t2, z2) ˆa1 (t1, z1)⟩ += e−iω20t2e−iω10t1ψ2,1(t1 − t2), +(80) +Ψ1,2(t1, t2) = ⟨ˆa1 (t1, z1) ˆa2 (t2, z2)⟩ += e−iω20t2e−iω10t1ψ1,2(t1 − t2), +(81) +are the two-photon wavefunctions with the relative parts +ψ2,1(t1 − t2) += 1 +2π +�� +dϖdϖ′⟨ˆa2(ϖ′, z2)ˆa1(ϖ, z1)⟩e−iϖ(t1−t2). (82) +ψ1,2(t1 − t2) += 1 +2π +�� +dϖdϖ′⟨ˆa1(ϖ, z1)ˆa2(ϖ′, z2)⟩e−iϖ(t1−t2). (83) +One can show that the second term in Eqs. (78) and (79) +is zero if the nonlinear coupling coefficient is real-valued, + +14 +and it is usually very small as compared to other terms. +The third term in Eqs. (78) and (79) is the accidental +coincidence counts. The two-photon wavefunction and +Glauber correlation satisfy the following exchange sym- +metry +ψ21(t1 − t2) = ψ2,1(t1 − t2) = ψ1,2(t1 − t2), +Ψ21(t2, t1) = Ψ2,1(t2, t1) = Ψ1,2(t1, t2), +G(2) +21 (t2, t1) = G(2) +2,1 (t2, t1) = G(2) +1,2 (t1, t2) . +(84) +The normalized two-photon correlation is defined as +g(2) +21 (t2, t1) ≡ G(2) +21 (t2, t1) +R1R2 +. +(85) +As the system has time translation symmetry with +continuous-wave pumping, G(2) +21 (t2, t1) = G(2) +21 (t1 − t2) +depends only on the relative time t1 − t2. +A. +Loss and Gain +To simplify and unify the descriptions for account- +ing both forward- and backward-wave cases, we define +“input-output” fields: +ˆa1,in ≡ ˆa1(0), ˆa2,in ≡ ˆa2(0), +ˆa1,out ≡ ˆa1(L), and ˆa2,out ≡ ˆa2(L) for the forward-wave +case; ˆa1,in ≡ ˆa1(0), ˆa2,in ≡ ˆa2(L), ˆa1,out ≡ ˆa1(L), and +ˆa2,out ≡ ˆa2(0) for the backward-wave case. In this sub- +section, we aim to investigate the roles of loss and gain +in biphoton generation, considering linear loss in mode 1 +(Re{α1} = α ≥ 0) and linear gain (Re{α2} = −g ≤ 0) +in mode 2. We also assume κ is real, or its contribution +to Langevin noises is much smaller than the linear gain +and loss, i.e., Im{κ} ≪ {α, g}. In this case, for forward- +and backward-wave configurations, the noise matrix is +reduced to +NF,B = +�√ +2α +0 +0 +±i√2g +� +. +(86) +Hence, the output fields in Eqs. (19) and (36) can be +rewritten as +�ˆa1,out +ˆa† +2,out +� += +� +A B +C D +� �ˆa1,in +ˆa† +2,in +� ++ +� L +0 +� +X11 X12 +X21 X22 +� � ˆf1 (z) +ˆf2 (z) +� +dz. +(87) +where Xmn are combined coefficients. We further rewrite +Eq. (87) as +ˆa1,out = Aˆa1,in + Bˆa† +2,in + +� L +0 +� +X11 ˆf1(z) + X12 ˆf2(z) +� +, +ˆa2,out = C∗ˆa† +1,in + D∗ˆa2,in + +� L +0 +� +X∗ +21 ˆf † +1(z) + X∗ +22 ˆf † +2(z) +� +. +(88) +As shown in Eq. +(84), there are two different orders +[⟨: ˆa2ˆa1 :⟩ or ⟨: ˆa1ˆa2 :⟩] to compute the two-photon wave- +function and Galuber correlation. Although these two +orders are equivalent, the numerical computation com- +plexity may be significantly different. Computing bipho- +ton wavefunction in Eq. (83) in the order ⟨: ˆa1ˆa2 :⟩ in- +volves nonzero noise field correlations ⟨ ˆfm ˆf † +m⟩, while in +the order ⟨: ˆa2ˆa1 :⟩ [Eq. +(82)] these noise field corre- +lations disappear because of ⟨ ˆf † +m ˆfm⟩ = 0. +These field +correlations in the frequency domain can be expressed as +⟨ˆa2out (ϖ′) ˆa1out (ϖ)⟩ = δ(ϖ − ϖ′) [BD∗] , +(89) +⟨ˆa1out (ϖ) ˆa2out (ϖ′)⟩ += δ(ϖ − ϖ′) +� +AC∗ + +� L +0 +dz (X11X∗ +21 + X12X∗ +22) +� +. +(90) +Therefore, we obtain the biphoton wavefunction follow- +ing the order ⟨: ˆa2ˆa1 :⟩ +ψ21(τ) = +�� +dϖdϖ′⟨ˆa2,out(ϖ′)ˆa1,out(ϖ)⟩e−iϖτ += +� +dϖBD∗e−iϖτ. +(91) +where τ = t1 − t2. If following the order ⟨: ˆa1ˆa2 :⟩, we +have +ψ12(τ) = +�� +dϖdϖ′⟨ˆa1,out(ϖ)ˆa2,out(ϖ′)⟩e−iϖτ += +� +dϖ +� +AC∗ + +� L +0 +dz (X11X∗ +21 + X12X∗ +22) +� +e−iϖτ. +(92) +One can show that the second term in Eqs. (78) and (79) +is zero in this loss-gain configuration. The single-channel +photon rates can be obtained as +R1 = 1 +2π +� +|B|2dϖ, +R2 = 1 +2π +� � +|C|2 + +� L +0 +dz +� +|X21|2 + |X22|2� +� +dϖ. +(93) +It is interesting to remark that, in the loss-gain config- +uration, the biphoton field correlation following the order +⟨: ˆagainˆaloss :⟩ does not involve noise field correlations as +shown in Eqs. (89) and (91), which dramatically reduces +the computation complexity. On the other side, taking +the order ⟨: ˆalossˆagain :⟩ must include noise field corre- +lations as shown in Eqs. (90) and (92). This may be +understood in the heralded photon picture [25]: When +a photon in a lossy channel is detected (annihilated) by +a detector, we can always ensure there is its partner (or +paired) photon in another channel; On the other side, +when a photon is detected in a gain channel which pro- +duces multiple photons, we can not always ensure it has +a partner photon in another channel. The exchange sym- +metry can only be preserved by taking into account the +Langevin noises. + +15 +0 +1 +2 +109 +Macro +Micro +NLN +-0.5 +0 +0.5 +1 +0 +1 +2 +(a) +(b) +Figure 9. Two-photon Glauber correlation in time domain in +the group delay regime: (a) G(2) +s,as(τ) and (b) G(2) +as,s(τ). The +simulation conditions are the same as that in Figs. 3, 4, and +5. NLN: no Langevin noise included. +In the SFWM described in Sec. III, the anti-Stokes +photons experience finite EIT loss due to the ground +state dephasing (γ12 ̸= 0), and the Stokes photons prop- +agate with negligible but small Raman gain. +Figure +9 displays the two-photon Glauber correlation in the +group delay regime with the same parameters as those +in Figs. 3, 4 and 5. As shown in Fig. 9(a) and (b), both +macroscopic and microscopic approaches with Langevin +noises give consistent results. As expected, the compu- +tation of G(2) +s,as(τ) (following the order ⟨: ˆasˆaas :⟩) with- +out Langevin noise operators (black dotted line: NLN) +agrees with the exact results obtained from both macro- +scopic (blue solid line) and microscopic (red dashed line) +approaches, shown in Fig. 9(a). +On the contrary, the +computation of G(2) +as,s(τ) (following the order ⟨: ˆaasˆas :⟩) +without Langevin noise operators deviates significantly +from the exact results, as shown in Fig. 9(b). +B. +Complex Phase Mismatching +Different from the Heisenberg picture where the evo- +lution of field operators is governed by their Langevin +coupled equations, reference [10] provides a perturbation +theory to describe biphoton state in the interaction pic- +ture. The solution from Heisenberg-Langevin theory may +contain correlations of more than two photons, while the +perturbation theory focuses only on the two-photon state +by ignoring higher-order terms. These two treatments are +expected to give the same results in the limit of small pa- +rameter gain. Although the perturbation theory in the +interaction picture provides a much clear physics picture +of two-photon state, treating loss and gain requires a +proper justification. In the perturbation theory, linear +loss and gain are included in the complex phase mis- +matching ∆˜k(ϖ) [10]. For the SFWM described in Sec. +III, Ref. [10] derives the biphoton relative wavefunction +with perturbation theory as +ψ(τ) = iL +2π +� +dϖκ(ϖ)Φ(ϖ)e−iϖτ, +(94) +where the longitudinal detuning function is +Φ(ϖ) = sinc +� +∆˜kL +2 +� +ei(kas+ks)L, +(95) +There is a statement in Ref. [10]: “It is found that to be +consistent with the Heisenberg–Langevin theory in the +low-gain limit, the argument in Φ should be replaced by +∆˜k = +� +⃗kas + ⃗k∗ +s − ⃗kc − ⃗kp +� +· ˆz, where ⃗k∗ +s is the conjugate +of ⃗ks.” For the SFWM in the double-Λ four-level atomic +system, there is small Raman gain in the Stokes chan- +nel. What happens if there is loss in the Stokes channel? +Should we take ⃗k∗ +s or ⃗ks in the complex phase mismatch- +ing ∆˜k(ϖ)? Although Ref. [10] takes ⃗k∗ +s for Stokes pho- +tons with gain, it is not clear whether it still holds for the +case with loss. In this subsection, we do not only provide +a justification for the above statement in Ref. [10] from +the quantum Langevin theory by taking small parametric +gain approximation, but also extend the complex phase +mismatching to the case with loss in the Stokes channel. +We take the same backward-wave configuration in +Ref. [10]. We assume anti-Stokes photons in mode 1 are +lossless with EIT and there is gain (or loss) in Stokes +mode 2. The small parametric gain fulfills |κ| ≪ {α, g}. +In the backward-wave configuration, using Eq. +(7), +(34), and (37), we obtain analytical expressions of +A, B, C, and D as +A = +� +q2 − 4κ2e−(α1−α∗ +2)L/2 +qsinh +� +L +2 +� +q2 − 4κ2 +� ++ +� +q2 − 4κ2cosh +� +L +2 +� +q2 − 4κ2 +�, +B = +2iκ +q + +� +q2 − 4κ2coth( L +2 +� +q2 − 4κ2) +, +C = +−2iκ +q + +� +q2 − 4κ2coth( L +2 +� +q2 − 4κ2) +, +D = +� +q2 − 4κ2e(α1−α∗ +2)L/2 +qsinh +� +L +2 +� +q2 − 4κ2 +� ++ +� +q2 − 4κ2cosh +� +L +2 +� +q2 − 4κ2 +�, +(96) +where q ≡ α1 + α∗ +2 − i∆k. In the small parametric gain +approximation, we have +� +q2 − 4κ2 ≈ q += α1 + α∗ +2 − i∆k = −i(∆k1 − ∆k2 +∗ + ∆k), +(97) +and +α1 − α∗ +2 = −i(∆k1 + ∆k2 +∗). +(98) + +16 +where ∆km = ωm +2c χm is the wavenumber difference from +that in vacuum. Hence, we simplify A, B, C, and D to +A =exp [i∆k1L] exp +�i∆kL +2 +� +, +B =iκLsinc +�(∆k1 − ∆k∗ +2 + ∆k)L +2 +� +× exp +�i(∆k1 − ∆k∗ +2 + ∆k)L +2 +� +, +C = − iκLsinc +�(∆k1 − ∆k∗ +2 + ∆k)L +2 +� +× exp +�i(∆k1 − ∆k∗ +2 + ∆k)L +2 +� +, +D =exp [−i∆k∗ +2L] exp +�i∆kL +2 +� +. +(99) +We first look at the case with gain in the Stokes (mode +2). As discussed in Sec. IV A, we take the order ⟨: ˆa2ˆa1 :⟩ +ψ21(τ) = +�� +dϖdϖ′⟨ˆa2,out(ϖ′)ˆa1,out(ϖ)⟩e−iϖτ += +� +dϖBD∗e−iϖτ, +(100) +where +BD∗ = iκLsinc +�(∆k1 − ∆k∗ +2 + ∆k)L +2 +� +× exp +�i(∆k1 − ∆k∗ +2 + 2∆k2)L +2 +� +. +(101) +Comparing Eqs. (100) and (101) with Eqs. (94) and (95), +particularly for the argument in the sinc function, we +have ∆˜k = ∆k1 − ∆k∗ +2 + ∆k = k1 − k∗ +2 − kc + kp = +kas − k∗ +s − kc + kp which is consistent with the statement +in Ref. [10]. +We now look at the case with loss in the Stokes (mode +2). We take the order ⟨: ˆa1ˆa2 :⟩ and have +ψ12(τ) = +�� +dϖdϖ′⟨ˆa1,out(ϖ)ˆa2,out(ϖ′)⟩e−iϖτ += +� +dϖAC∗e−iϖτ, +(102) +where +AC∗ = iκ∗Lsinc +�(∆k∗ +1 − ∆k2 + ∆k)L +2 +� +exp +�i(2∆k1 − ∆k∗ +1 + ∆k2)L +2 +� +. +(103) +Comparing Eqs. (102) and (103) with Eqs. (94) and (95), +we have ∆˜k = ∆k∗ +1 − ∆k2 + ∆k = k1 − k2 − kc + kp = +kas − ks − kc + kp, which is different from the case with +gain. Here we have taken k1 ≃ k∗ +1 for lossless mode 1. +Although our discussion is based on the backward- +wave configuration, the conclusion can be extended to +the forward-wave configuration, which is derived in de- +tail in Appendix D. Therefore, in the case with gain in +the Stokes mode 2, the complex phase mismatching is +∆˜k = +� +⃗kas + ⃗k∗ +s − ⃗kc − ⃗kp +� +· ˆz. In the case with loss in +the Stokes mode 2, the complex phase mismatching be- +comes ∆˜k = +� +⃗kas + ⃗ks − ⃗kc − ⃗kp +� +· ˆz. +C. +Complex Nonlinear Coupling Coefficient and +Rabi Oscillation +As illustrated in Fig. 2, we can understand the SFWM +process in the following picture. After a Stoke and anti- +Stokes photon pair is born from a single atom follow- +ing the atomic transitions [Fig. 2(b)], the paired pho- +tons then propagate through the medium [Fig. 2(a)]. As +the photon pair can be generated at any atom inside the +medium, the overall two-photon wavefunction (or prob- +ability amplitude) is a superposition of all possible such +generation-propagation two-photon Feynman paths. Fol- +lowing this picture, when the propagation effect can be +ignored, the biphoton state reveals the single atom dy- +namics, which is connected to the nonlinear coupling co- +efficient. In the following, we consider SFWM in the limit +of small optical depth (OD) where the linear propaga- +tion effect is small and show how the complex spectrum +of nonlinear coupling coefficient reveals single-atom Rabi +oscillation. +We +rewrite +the +nonlinear +coupling +coefficient +in +Eq. (71) as: +κ(ϖ) = J +� +1 +(ϖ − Ωe/2 + iγe) − +1 +(ϖ + Ωe/2 + iγe) +� +, +(104) +where +J = − +√ωasωsnµ13µ24 +8cε0ℏΩe +���� +ΩpΩc +∆p + iγ14 +���� . +(105) +Here Ωe = +� +|Ωc|2 − (γ13 − γ12)2 is the effective coupling +Rabi frequency, and γe = (γ12 + γ13)/2 is the effective +dephasing rate. Obviously, the nonlinear coupling coeffi- +cient κ(ϖ) has a complex spectrum, with two resonances +separated by the effective coupling Rabi frequency Ωe. In +the ground-state approximation with major atomic pop- +ulation in state |1⟩, the undepleted pump laser beam is +far detuned from the transition |1⟩ → |4⟩ and its exci- +tation is weak such that we can take χs ≃ 0. On the +other side, from Eq. +(70) we have the complex linear +susceptibility for anti-Stokes photons +χas(ϖ) = −n |µ13|2 +ε0ℏ +(ϖ + iγ12) +(ϖ − Ωe/2 + iγe)(ϖ + Ωe/2 + iγe) +(106) +Although the anti-Stokes photon absorption at ϖ = 0 +is suppressed by the EIT effect, there are two absorp- +tion resonances appearing at ϖ = ±Ωe/2 which coin- +cide with the two resonances of nonlinear coupling coef- +ficient in Eq. (104). We take the pump laser with weak + +17 +intensity (∝ |Ωp|2) and large detuning (∆p) such that +Re{αas(ϖ = ±Ωe/2)}>Im{κ(ϖ = ±Ωe/2)}, which are +usually satisfied in the ground state condition. As the +propagation effect is small and the phase matching is not +important, the paired photons are mostly generated from +the two resonances (ϖ = ±Ωe/2) of the nonlinear cou- +pling coefficient. +In the forward-wave configuration, with the coupling +matrix +MF = +� +−αas + i ∆k +2 +iκ +−iκ +−i ∆k +2 +� +, +(107) +and short medium length L satisfying |MFL| ≪ 1, we +have approximately +� +A B +C D +� += eMFL ∼= 1 + MFL += +� +1 − αasL + i ∆k +2 L +iκL +−iκL +1 − i ∆k +2 L +� +. +(108) +As discussed in Sec. IV A, the biphoton field correlation +following the order ⟨: ˆasˆaas :⟩ does not need count the +Langevin noise operators: +⟨ˆas(ϖ′, L)ˆaas(ϖ, L)⟩ = BD∗δ(ϖ − ϖ′) += iκL(1 + i∆k +2 L)δ(ϖ − ϖ′) +∼= iκ(ϖ)Lδ(ϖ − ϖ′), +(109) +where we have neglected higher order terms O(L2). From +Eq. (82), we have the relative biphoton wavefunction +ψs−as(τ) = iL +2π +� +dϖκ(ϖ)e−iϖτ, +(110) +which is the Fourier transform of the nonlinear coupling +coefficient with τ = tas − ts. Substituting Eq. (104) into +Eq. (110) we obtain +ψs−as(τ) = LJe−γeτ[e−iΩeτ/2 − eiΩeτ/2]Θ(τ) += −2iLJe−γeτ sin +�Ωeτ +2 +� +Θ(τ), +(111) +where Θ(τ) is the Heaviside function. +Equation (111) +shows a damped Rabi oscillation, resulting from the beat- +ing between biphotons generated from the two resonances +at ϖ = ±Ωe/2. The Heaviside function shows the anti- +Stokes photon is always generated after its paired Stokes +photon following the time order of atomic transitions +|1⟩ → |4⟩ → |2⟩ → |3⟩ → |1⟩ in an SFWM cycle shown in +Fig. 2(b). +In the backward-wave configuration, the coupling ma- +trix becomes +MB = +� +−αas + i ∆k +2 +iκ +iκ +−i ∆k +2 +� +. +(112) +0 +2 +4 +6 +105 +Macro +Micro +NLN +0 +2 +4 +6 +-0.4 +-0.2 +0 +0.2 +0.4 +0 +2 +4 +6 +| +s-as|2 +(a) +(b) +(c) +Figure 10. Two-photon Glauber correlation in time domain +in the damped Rabi oscillation regime: (a) G(2) +s,as(τ) and (b) +G(2) +as,s(τ). The simulation conditions are the same as that in +Figs. 6, 7, and 8. (c) shows the analytic solution for the bipho- +ton waveform |ψs−as(τ)|2. NLN: no Langevin noise included. +With |MBL| ≪ 1 we have +� ¯A +¯B +¯C +¯D +� += eMBL ∼= 1 + MBL += +� +1 − αasL + i ∆k +2 L +iκL +iκL +1 − i ∆k +2 L +� +, +(113) +and +� +A B +C D +� += +� +1 − αasL + i ∆k +2 L +iκL +−iκL +1 + i ∆k +2 L +� +, +(114) +where we have neglect higher order terms O(L2). Simi- +larly, we have +⟨ˆas(ϖ′, 0)ˆaas(ϖ, L)⟩ ∼= iκ(ϖ)Lδ(ϖ − ϖ′), +(115) +which is the same as Eq. (109) of the forward-wave con- +figuration. Therefore, we obtain Rabi oscillations in both +forward- and backward-wave configurations. +Equation +(111) is identical to the result derived from the pertur- +bation theory in the interaction picture [10]. + +18 +Figure 10 displays the two-photon Glauber correlation +in the damped Rabi oscillation regime with the same pa- +rameters as those in Figs. +6, 7 and 8. +As illustrated +in Fig. 10(a) and (b), both macroscopic and microscopic +approaches with Langevin noises give consistent results. +As expected, the computation of G(2) +s,as(τ) (following the +order ⟨: ˆasˆaas :⟩) without Langevin noise operators (dot +points) agrees with the exact results obtained from both +microscopic (red dashed line) and macroscopic (blue solid +line) approaches, shown in Fig. 10(a). +On the con- +trary, the computation of G(2) +as,s(τ) (following the order +⟨: ˆaasˆas :⟩) without Langevin noise operators (dot points: +NLN) deviates significantly from the exact results and vi- +olates the causality, as shown in Fig. 10(b). Fig. 10(c) +shows the result from the analytic solution in Eq. (111) +which agree well with the exact results in Figs. 10(a) and +(b). +It is interesting to examine a system without gain and +loss whose Langevin noises are purely contributed by the +complex nonlinear coupling coefficient. In this case, the +above approximation and conclusion do not hold. Let’s +now consider the case 3 with the forward-wave config- +uration in Sec. II A, where α1 = α2 = ∆k = 0, and +κ = η + iζ. As shown in Sec. II A, the noise matrix is +different as ζ is positive or negative. We first consider +ζ > 0, the Langevin coupled equations (27) becomes +∂ +∂z +�ˆa1 +ˆa† +2 +� += +� +0 +iκ +−iκ +0 +� �ˆa1 +ˆa† +2 +� ++ +� +ζ +� +1 +1 +−1 1 +� � ˆf1 +ˆf † +2 +� +. +(116) +Under the condition |MFL| ≪ 1, we solve Eq. +(116) to +the first order of L and have +ˆa1(L) ∼= ˆa1(0) + iκLˆa† +2(0) + +� +ζ +� L +0 +dz +� +ˆf1 + ˆf † +2 +� +, +ˆa2(L) ∼= ˆa2(0) + iκ∗Lˆa† +1(0) + +� +ζ +� L +0 +dz +� +− ˆf † +1 + ˆf2 +� +. +(117) +The two-photon field correlations are +⟨ˆa1(L)ˆa2(L)⟩ = ⟨ˆa2(L)ˆa1(L)⟩ ∼= i +2(κ + κ∗)Lδ(ϖ − ϖ′). +(118) +As ζ < 0, the Langevin coupled equations (27) becomes +∂ +∂z +�ˆa1 +ˆa† +2 +� += +� +0 +iκ +−iκ +0 +� �ˆa1 +ˆa† +2 +� ++ +� +−ζ +� +1 +1 +−1 1 +� � ˆf † +1ˆf2 +� +. (119) +Under the condition |MFL| ≪ 1, we solve Eq. +(119) to +the first order of L and have +ˆa1(L) ∼= ˆa1(0) + iκLˆa† +2(0) + +� +−ζ +� L +0 +dz +� +ˆf † +1 + ˆf2 +� +, +ˆa2(L) ∼= ˆa2(0) + iκ∗Lˆa† +1(0) + +� +−ζ +� L +0 +dz +� +− ˆf1 + ˆf † +2 +� +. +(120) +The two-photon field correlations are +⟨ˆa1(L)ˆa2(L)⟩ = ⟨ˆa2(L)ˆa1(L)⟩ ∼= i +2(k + k∗)Lδ(ϖ − ϖ′), +(121) +which is the same as Eq. (118). The biphoton relative +wavefunction is +ψ21(τ) = ψ∗ +21(−τ) = iL +2π +� +dϖ1 +2(k + k∗)e−iϖτ. +(122) +One can prove that under the same limit |MBL| ≪ 1, the +backward-wave configuration gives the same two-photon +field correlation [Eqs. +(118) and (121)] and temporal +wavefunction [Eq. (122)]. Equation (122) suggests the +biphoton temporal wavefunction has time reversal sym- +metry when there is no linear gain and loss. +V. +CONCLUSION +In summary, we provide a macroscopic phenomenolog- +ical formula of quantum Langevin equations for two cou- +pled phase-conjugated fields with linear loss (gain) and +complex nonlinear coupling coefficient, in both forward- +and backward-wave configurations. +The macroscopic +phenomenological formula, obtained from the coupling +matrix and the requirement of preserving commutation +relations of field operators during propagation, does not +require knowing microscopic details of light-matter inter- +action and internal atomic structures. To validate this +phenomenological formula, we take SFWM in a double- +Λ four-level atomic system as an example to numeri- +cally confirm that our macroscopic phenomenological re- +sult is consistent with that obtained from microscopic +Heisenberg-Langevin theory. As compared to the com- +plicated microscopic theory which varies from system +to system, the macroscopic coupled equations are much +more friendly to experimentalists. We apply the quantum +Langevin equations to study the effects of gain and/or +loss as well as complex nonlinear coupling coefficient in +biphoton generation, particularly to the temporal quan- +tum correlations. We show that the computation com- +plexity can be dramatically reduced by taking a proper +order of field operators based on loss and gain. Making +a comparison between the quantum Langevin theory (in +the Heisenberg picture) and the perturbation theory (in +the interaction picture [10]), we extend the expression of +complex phase mismatching to account for loss and gain. +At last, we reveal Rabi oscillation in SFWM biphoton +temporal correlation when the propagation effect is small. +Although in this article we focus on biphoton generation +from the spontaneous parametric process, the quantum +Langevin coupled equations can also be used to study +two-mode squeezing, parametric oscillation, and other +quantum light state generation. +ACKNOWLEDGMENTS +S.D. +acknowledges +support +from +DOE +(DE- +SC0022069), +AFOSR +(FA9550-22-1-0043) +and +NSF +(CNS-2114076, 2228725). + +19 +Appendix A: Noise Matrix in Backward-Wave +Configuration +In the macroscopic quantum Langevin equations, the +requirement of preserving commutation relations allows +multiple choices of the noise matrix. For example, ˆf1 → +− ˆf1 or/and ˆf2 → − ˆf2 do not affect any computation re- +sults of physical observables involving pairs of Langevin +noise operators. As an example, here we provide several +equivalent noise matrices for backward-wave configura- +tion: +NB1 ≡ +� +1 +0 +0 −1 +� �� +−MB11 −MB12 +MB21 +MB22 +� ++ +� +−MB11 −MB12 +MB21 +MB22 +�∗ += +� +1 +0 +0 −1 +� +NF, +NB2 ≡ NB1 +� +1 +0 +0 −1 +� += +�� +−MB11 MB12 +−MB21 MB22 +� ++ +� +−MB11 MB12 +−MB21 MB22 +�∗ +, +NB3 ≡ NB1 +� +−1 0 +0 +1 +� +, +NB4 ≡ NB1 +� +−1 +0 +0 +−1 +� += −NB1. +(A1) +We take the first choice NB1 in the main text [see Eq. (31) +in Sec. II B] so that it is consistent with the microscopic +treatment in Sec. III. +Appendix B: Heisenberg-Langevin Equations of +SFWM +The full Heisenberg equation of motion can be written +as +˙ˆS = i( ˆO ˆS − ˆS ˆO) + ˆG + ˆF, +(B1) +where +ˆS = +� +�� +ˆσ11 ˆσ12 ˆσ13 ˆσ14 +ˆσ21 ˆσ22 ˆσ23 ˆσ24 +ˆσ31 ˆσ32 ˆσ33 ˆσ34 +ˆσ41 ˆσ42 ˆσ43 ˆσ44 +� +�� , +(B2) +ˆO = − +� +�� +0 +0 +g31ˆaas Ωp/2 +0 +ϖ +Ωc/2 +g42ˆas +g13ˆa∗ +as Ω∗ +c/2 +ϖ +0 +Ω∗ +p/2 +g24ˆa∗ +s +0 +∆p +� +�� , +(B3) +ˆG = +� +�� +Γ31ˆσ33 + Γ41ˆσ44 +−γ12ˆσ12 +−γ13ˆσ13 −γ14ˆσ14 +−γ12ˆσ21 +Γ32ˆσ33 + Γ42ˆσ44 −γ23ˆσ23 −γ24ˆσ24 +−γ13ˆσ31 +−γ23ˆσ32 +−Γ3ˆσ33 −γ34ˆσ34 +−γ14ˆσ41 +−γ24ˆσ42 +−γ34ˆσ43 −Γ4ˆσ44 +� +�� , +(B4) +ˆF = +� +���� +ˆf (σ) +11 +ˆf (σ) +12 +ˆf (σ) +13 +ˆf (σ) +14 +ˆf (σ) +21 +ˆf (σ) +22 +ˆf (σ) +23 +ˆf (σ) +24 +ˆf (σ) +31 +ˆf (σ) +32 +ˆf (σ) +33 +ˆf (σ) +34 +ˆf (σ) +41 +ˆf (σ) +42 +ˆf (σ) +43 +ˆf (σ) +44 +� +���� . +(B5) +Γm = Γm1 + Γm2 is the total spontaneous decay rate of +excited state |m⟩, where m = 3, or 4, and Γmj is the decay +rate from state |m⟩ to |j⟩. For the two hyperfine ground +states, there are Γ1 = Γ2 = 0. +For cold atoms with +only spontaneous emisson decay, the dephasing rates γjk +(j ̸= k) between states |k⟩ and |j⟩ are γ13 = γ23 = Γ3/2, +γ14 = γ24 = Γ4/2, γ34 = (Γ3+Γ4)/2. γ12 is the dephasing +rate between two hyperfine ground states |1⟩ and |2⟩. +Appendix C: Microscopic SFWM Quantum +Langevin Equations in Forward-Wave Configuration +Although Sec. III focuses on numerical confirmation +of backward-wave SFWM, we remark that it may be +helpful for general readers to write the SFWM quantum +Langevin equations in the forward-wave configuration as +well. +In the forward-wave configuration with both Stokes +and anti-Stokes fields propagating along +z direction, +the coupled Langevin equations become +∂ +∂z +�ˆaas +ˆa† +s +� += MF +�ˆaas +ˆa† +s +� ++ +� ˆFas +ˆF † +s +� +, +(C1) +where +MF = +� +−αas + i ∆k +2 +iκ +−iκ +−α∗ +s − i ∆k +2 +� +, +(C2) +with ∆k = (ωas+ωs)/c−(⃗kc+⃗kp)·ˆz. The noise operators +ˆFas and ˆF † +s , defined in Eq. (69), originate from micro- +scopic atom-light interaction. To compare Eq. (C1) with +Eq. (11) from the phenomenological approach in Sec. II, +we take mode 1 as anti-Stokes and mode 2 as Stokes in +the forward-wave configuration. From Eq. (11), we can +also obtain ˆFas and ˆF † +s from the noise matrix: +ˆFas = NFR11 ˆf1 + NFI11 ˆf † +1 + NFI12 ˆf2 + NFR12 ˆf † +2, +ˆF † +s = NFR21 ˆf1 + NFI21 ˆf † +1 + NFI22 ˆf2 + NFR22 ˆf † +2. +(C3) +Appendix D: Complex Phase Mismatching in +Forward-Wave Configuration +In the forward-wave configuration, similar to the +backward-wave configuration in Sec. IV B, we assume +anti-Stokes photons in mode 1 are lossless with EIT and +there is gain (or loss) in Stokes mode 2. The small para- +metric gain fulfills |κ| ≪ {α, g}. Using Eq. (6) and (17), + +20 +we obtain analytical expressions of A, B, C, and D as +A = +� +q2 + 4κ2cosh +� +L +2 +� +q2 + 4κ2 +� +− qsinh +� +L +2 +� +q2 + 4κ2 +� +� +q2 + 4κ2e(α1+α∗ +2)L/2 +, +B = +2iκsinh +� +L +2 +� +q2 + 4κ2 +� +� +q2 + 4κ2e(α1+α∗ +2)L/2 , +C = +−2iκsinh +� +L +2 +� +q2 + 4κ2 +� +� +q2 + 4κ2e(α1+α∗ +2)L/2 , +D = +� +q2 + 4κ2cosh +� +L +2 +� +q2 + 4κ2 +� ++ qsinh +� +L +2 +� +q2 + 4κ2 +� +� +q2 + 4κ2e(α1+α∗ +2)L/2 +, +(D1) +where q ≡ α1 − α∗ +2 − i∆k. In the small parametric gain +approximation, we have +� +q2 − 4κ2 ≈ q += α1 − α∗ +2 − i∆k = −i(∆k1 + ∆k∗ +2 + ∆k), +(D2) +and +α1 + α∗ +2 = −i(∆k1 − ∆k∗ +2), +(D3) +where ∆km = ωm +2c χm is the wavenumber difference from +that in vacuum. Hence, we simplify A, B, C, and D to +A =exp [i∆k1L] exp +�i∆kL +2 +� +, +B =iκLsinc +�(∆k1 + ∆k∗ +2 + ∆k)L +2 +� +× exp +�i(∆k1 − ∆k∗ +2)L +2 +� +, +C = − iκLsinc +�(∆k1 + ∆k∗ +2 + ∆k)L +2 +� +× exp +�i(∆k1 − ∆k∗ +2)L +2 +� +, +D =exp [−i∆k∗ +2L] exp +�−i∆kL +2 +� +. +(D4) +We first look at the case with gain in the Stokes (mode +2). As discussed in Sec. IV A, we take the order ⟨: ˆa2ˆa1 :⟩ +ψ21(τ) = +�� +dϖdϖ′⟨ˆa2,out(ϖ′)ˆa1,out(ϖ)⟩e−iϖτ += +� +dϖBD∗e−iϖτ, +(D5) +where +BD∗ = iκLsinc +�(∆k1 + ∆k∗ +2 + ∆k)L +2 +� +× exp +�i(∆k1 − ∆k∗ +2 + 2∆k2 + ∆k)L +2 +� +. +(D6) +Comparing Eqs. (D5) and (D6) with Eqs. (94) and (95), +particularly for the argument in the sinc function, we +have ∆˜k = ∆k1 + ∆k∗ +2 + ∆k = k1 + k∗ +2 − kc − kp = +kas + k∗ +s − kc − kp which is consistent with the statement +in Ref. [10]. +We now look at the case with loss in the Stokes (mode +2). 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A 92, 043836 (2015). + diff --git a/GNFLT4oBgHgl3EQfGi_B/content/tmp_files/load_file.txt b/GNFLT4oBgHgl3EQfGi_B/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e236f27ca345a4fd278123be80c7d10dff58de0b --- /dev/null +++ b/GNFLT4oBgHgl3EQfGi_B/content/tmp_files/load_file.txt @@ -0,0 +1,834 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf,len=833 +page_content='Quantum Langevin theory for two coupled phase-conjugated electromagnetic waves Yue Jiang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' ∗ Yefeng Mei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' † and Shengwang Du4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' ‡ 1JILA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' National Institute of Standards and Technology and the University of Colorado,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Colorado 80309,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' USA 2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' University of Colorado,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Colorado 80309,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' USA 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Michigan 48109,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' USA 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The University of Texas at Dallas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Richardson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Texas 75080,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' USA (Dated: January 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 2023) While loss-gain-induced Langevin noises have been intensively studied in quantum optics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' the ef- fect of a complex-valued nonlinear coupling coefficient on the noises of two coupled phase-conjugated optical fields has never been questioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Here, we provide a general macroscopic phenomeno- logical formula of quantum Langevin equations for two coupled phase-conjugated fields with linear loss (gain) and complex nonlinear coupling coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The macroscopic phenomenological formula is obtained from the coupling matrix to preserve the field commutation relations and correlations, which does not require knowing the microscopic details of light-matter interaction and internal atomic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' To validate this phenomenological formula, we take spontaneous four-wave mix- ing in a double-Λ four-level atomic system as an example to numerically confirm that our macroscopic phenomenological result is consistent with that obtained from the microscopic Heisenberg-Langevin theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Finally, we apply the quantum Langevin equations to study the effects of linear gain and loss, complex phase mismatching, as well as complex nonlinear coupling coefficient in entangled photon pair (biphoton) generation, particularly to their temporal quantum correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' INTRODUCTION Quantum Langevin equations is a common approach to studying an open quantum system involving loss or gain, where the stochastic coupling between the system and its environment is molded as a set of Langevin noise operators [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' For example, in the parametric down- conversion (PDC) process, a pump laser beam passes through a χ(2) nonlinear crystal and is down-converted into a pair of phase-conjugated electromagnetic (EM) waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the simplest case with the perfect phase- matching condition and an undepleted pump beam, with- out linear loss or gain, the two phase-conjugated single- mode fields are governed by the following coupled equa- tions [6] ∂ ∂z �ˆa1 ˆa† 2 � = M �ˆa1 ˆa† 2 � = � 0 iκ −iκ 0 � �ˆa1 ˆa† 2 � , (1) where ˆam and ˆa† m (m = 1, 2) are the field annihilation and creation operators, M is the 2 × 2 coupling matrix, and κ is the (real) nonlinear coupling coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Here we consider only the forward-wave case with both fields propagating along the same +z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' If losses are presented during the propagation of the two fields, the coupling matrix is M = � −α1 iκ −iκ −α2 � , (2) ∗ yue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='jiang@jila.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='edu † meiyf@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='edu ‡ dusw@utdallas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='edu and their coupled equations become [3, 7] ∂ ∂z �ˆa1 ˆa† 2 � = � −α1 iκ −iκ −α2 � �ˆa1 ˆa† 2 � + �√2α1 ˆf1 √2α2 ˆf † 2 � , (3) where αm > 0 are the loss (absorption) coefficients, and ˆfm are the associated Langevin noise operators sat- isfying [ ˆfm(ω, z), ˆf † n(ω′, z′)] = δmnδ(ω − ω′)δ(z − z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' If there is linear gain instead of loss, for example in channel 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=', α1 < 0, equation (3) can be modi- fied by taking √2α1 ˆf1 → √−2α1 ˆf † 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' One can show that these Langevin noise operators are necessary to pre- serve the commutation relations during propagation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' [ˆam(ω, z), ˆa† n(ω′, z)] = [ˆam(ω, 0), ˆa† n(ω′, 0)] = δmnδ(ω − ω′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Equation (3) has been widely applied for PDC pro- cesses where the nonlinear coupling coefficient κ is real [3, 7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' However, in a more general case of cou- pled phase-conjugated fields, such as four-wave mixing (FWM) near atomic resonances [10–12], the nonlinear coupling coefficient κ can take a complex value involving complicated atomic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In this case, equation (3) is not valid and its solution does not preserve com- mutation relations of the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' What are the general quantum Langevin coupled equations accounting for the complex nonlinear coupling coefficient?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' To answer the question, the common approach is to derive quantum Langevin equations by solving the light- matter coupled Heisenberg equations, which requires knowing microscopic details of light-matter interaction such as atomic populations and transitions [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The complexity of this approach increases dramatically as more atomic transitions are involved and it is extremely difficult for experimentalists to follow, particularly in some situations where it is impossible to obtain full mi- croscopic details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Then our reduced question becomes: arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='11993v1 [quant-ph] 27 Jan 2023 2 Is it possible to obtain self-consistent quantum Langevin coupled equations from the general expression of the cou- pling matrix?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We call this the macroscopic phenomeno- logical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' To our best knowledge, there has been no published work in investigating Langevin noises in- duced by a complex nonlinear coupling coefficient κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In this article, for the first time, we provide a gen- eral macroscopic phenomenological formula of quantum Langevin equations for two coupled phase-conjugated fields with linear loss (gain) and complex nonlinear cou- pling coefficient, in both forward- and backward-wave configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The macroscopic phenomenological for- mula is obtained from the coupling matrix by preserv- ing commutation relations and correlations of the fields, which does not require knowing the microscopic details of light-matter interaction and internal atomic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We aim to make it readable and accessible for experi- mental researchers in the quantum optics community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' This article is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' II, to ful- fill the requirement of preserving commutation relations, we formulate the general macroscopic phenomenologi- cal quantum Langevin coupled equations and their solu- tions from the coupling matrix taking into account linear loss (gain) and complex nonlinear coupling coefficient, in both forward- and backward-wave configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' III, taking spontaneous four-wave mixing (SFWM) in a double-Λ four-level atomic system as an example, we derive the coupled Langevin equations from micro- scopic light-atom Heisenberg interaction for this special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We numerically confirm that the macroscopic phe- nomenological solution in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' II agrees well with the microscopic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' IV, we apply the quan- tum Langevin theory to study effects of linear gain and loss, complex phase mismatching, and complex nonlinear coupling coefficient in entangled photon pair (biphoton) generation, particularly to their temporal quantum cor- relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We conclude in the last section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' QUANTUM LANGEVIN EQUATIONS Here we consider the two coupled single-mode phase- conjugated fields in either forward-wave or backward- wave configuration, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the forward-wave configuration [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 1(a)], both fields prop- agate along +z direction through a nonlinear medium with a length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the backward-wave configuration [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 1(b)], the two fields propagate in opposing direc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The field annihilation operators ˆam(t, z) can be expressed as ˆa1(t, z) = 1 √ 2π � dωˆa1(ω, z)ei( ω c z−ωt), ˆa2(t, z) = 1 √ 2π � dωˆa2(ω, z)ei(± ω c z−ωt), (4) where ± represents that field 2 propagates along +z or −z direction, for the forward-wave or backward-wave configuration, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The filed operators satisfy the following commutation relations � ˆam (t, z) , ˆa† n (t′, z) � = δmnδ(t − t′), � ˆam (ω, z) , ˆa† n (ω′, z) � = δmnδ(ω − ω′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (5) In the forward-wave configuration, both fields are input at z = 0, or ˆa1(0) and ˆa2(0) are the “initial” boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The general coupling matrix is [14] MF = � −α1 + i ∆k 2 iκ −iκ −α∗ 2 − i ∆k 2 � , (6) where αm = −i ωm 2c χm with χm being linear suscepti- bility, and ∆k (real) is the phase mismatching in vac- uum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In general, αm is complex valued, whose real part Re{αm} > 0 represents loss (or gain for Re{αm} < 0) and imaginary part represents phase velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The nonlinear coupling coefficient κ can also be complex- valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the backward-wave configuration, the general coupling matrix becomes [12, 15] MB = � −α1 + i ∆k 2 iκ iκ α∗ 2 − i ∆k 2 � , (7) and the “initial” boundary conditions are ˆa1(0) and ˆa2(L): field 1 is input at z = 0 and field 2 is input at z = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' One can show that, under the following unitary gauge transformation �ˆa1 ˆa† 2 � = � eiθ/2 0 0 e−iθ/2 � �ˆa1 ˆa† 2 � = U �ˆa1 ˆa† 2 � = � ˆa1eiθ/2 ˆa† 2e−iθ/2 � , (8) the corresponding coupling matrix become MF(θ) = UMFU† = � −α1 + i ∆k 2 iκeiθ −iκe−iθ −α∗ 2 − i ∆k 2 � , (9) and MB(θ) = UMBU† = � −α1 + i ∆k 2 iκeiθ iκe−iθ α∗ 2 − i ∆k 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (10) As physics is preserved and unchanged under the above gauge transformation, we take θ = 0 throughout this article for convenience and simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In presence of linear loss or gain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=', Re{αm} ̸= 0, or complex nonlinear coupling coefficient, κ ̸= κ∗, the two- mode coupled equations must include Langevin noise op- erators to preserve the commutation relations of the field operators in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The noise operators should only be related to Re{αm} and Im{κ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' As κ is real, the cou- pled equations in forward-wave configuration should be reduced to the known Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' For both forward- and backward-wave configurations in the same nonlinear ma- terial, the noise origin is the same except field 2 prop- agates along ±z direction for different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' With these guidelines, we provide quantum Langevin equations for the two phase-conjugated fields from their coupling matrix in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 3 𝑧 𝑎�� 𝑎�� � Medium 0 𝐿 𝜅 𝑧 𝑎�� 𝑎�� � Medium 0 𝐿 𝜅 (b) (a) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Schematics of two coupled phase-conjugated electromagnetic waves: (a) forward-wave configuration, and (b) backward-wave configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' κ is the nonlinear coupling coefficient between the two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Forward-Wave Configuration In the forward-wave configuration as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 1(a), we find that its quantum Langevin coupled equations can be expressed in the following general form ∂ ∂z �ˆa1 ˆa† 2 � = MF �ˆa1 ˆa† 2 � + NFR � ˆf1 ˆf † 2 � + NFI � ˆf † 1ˆf2 � (11) with the “initial” condition at z = 0: � ˆam(ω, 0), ˆa† n(ω′, 0) � = δmnδ(ω − ω′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (12) The Langevin noise operators satisfy � ˆfm(ω, z), ˆf † n(ω′, z′) � = δmnδ(ω − ω′)δ(z − z′) (13) and have the following correlations � ˆf † m(ω, z) ˆfn(ω′, z′) � = 0, � ˆfm(ω, z) ˆf † n(ω′, z′) � = δmnδ(ω − ω′)δ(z − z′), � ˆfm(ω, z) ˆfn(ω′, z′) � = � ˆf † m(ω, z) ˆf † n(ω′, z′) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (14) The Langevin noise matrix is given by NF ≡ � −(MF + MF ∗) = NFR + iNFI, (15) where NFR and NFI are the real and imaginary parts of the matrix NF (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=', NFmn = NFRmn + iNFImn), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We obtain the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (11) at the output sur- face z = L as the following �ˆa1 (L) ˆa† 2 (L) � = eMFL �ˆa1 (0) ˆa† 2 (0) � + � L 0 eMF(L−z) � NFR � ˆf1 (z) ˆf † 2 (z) � + NFI � ˆf † 1 (z) ˆf2 (z) �� dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (16) Defining eMFL ≡ � A B C D � , (17) eMF(L−z) ≡ � A1 (z) B1 (z) C1 (z) D1 (z) � , (18) we rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (16) as �ˆa1 (L) ˆa† 2 (L) � = � A B C D � �ˆa1 (0) ˆa† 2 (0) � + � L 0 � A1 (z) B1 (z) C1 (z) D1 (z) � � NFR � ˆf1 (z) ˆf † 2 (z) � + NFI � ˆf † 1 (z) ˆf2 (z) �� dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (19) We numerically confirm that the solution preserves the commutation relations � ˆam(ω, L), ˆa† n(ω′, L) � = � ˆam(ω, 0), ˆa† n(ω′, 0) � = δmnδ(ω − ω′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (20) Now we examine some special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Case 1: We first consider the coupling matrix MF in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (6) where the nonlinear coupling coefficient κ is real and both modes have losses (Re{αm} ≥ 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' This works for most PDC processes [3, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Under such a condition, we have the following diagonalized noise matrix NF = NFR = �� 2Re{α1} 0 0 � 2Re{α2} � , (21) and the coupled Langevin equations ∂ ∂z �ˆa1 ˆa† 2 � = MF �ˆa1 ˆa† 2 � + �� 2Re{α1} ˆf1 � 2Re{α2} ˆf † 2 � , (22) which is the well-known result in literature [3, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Case 2: κ is real, the mode 1 has linear loss (Re{α1} = α ≥ 0), and the mode 2 has linear gain (Re{α2} = −g ≤ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The noise matrix becomes NF = �√ 2α 0 0 i√2g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (23) We have the following coupled Langevin equations ∂ ∂z �ˆa1 ˆa† 2 � = MF �ˆa1 ˆa† 2 � + �√ 2α ˆf1 √2g ˆf2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (24) 4 Case 3: The two modes are perfectly phase-matched without linear gain or loss: ∆k = 0, α1 = α2 = 0, but the nonlinear coupling coefficient is complex-valued κ = η + iζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In this case, the coupled matrix is MF = � 0 −ζ + iη ζ − iη 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (25) The noise matrix becomes NF = Θ(ζ) � ζ � 1 1 −1 1 � + iΘ(−ζ) � −ζ � 1 1 −1 1 � , (26) where Θ(ζ) is Heaviside step function, Θ(ζ) = 1 if ζ > 0, Θ(ζ) = 0 if ζ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The Langevin coupled equations are ∂ ∂z �ˆa1 ˆa† 2 � =MF �ˆa1 ˆa† 2 � + Θ(ζ) � ζ � 1 1 −1 1 � � ˆf1 ˆf † 2 � + Θ(−ζ) � −ζ � 1 1 −1 1 � � ˆf † 1ˆf2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (27) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (27) shows that a complex-valued nonlinear coupling coefficient also leads to Langevin noises even when there is no linear gain or loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' This is revealed by this article for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Case 4: As κ is real and there is no linear loss or gain (α1 = α2 = 0), the coupled equations can be written as i ∂ ∂z �ˆa1 ˆa† 2 � = � − ∆k 2 −κ κ ∆k 2 � �ˆa1 ˆa† 2 � = ˆH �ˆa1 ˆa† 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (28) The effective Hamiltonian ˆH has anti-parity-time (APT) symmetry, which has been demonstrated in FWM in cold atoms [14, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Backward-Wave Configuration In the back-wave configuration as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 1(b), the quantum Langevin coupled equations can be ex- pressed in the following general form ∂ ∂z �ˆa1 ˆa† 2 � = MB �ˆa1 ˆa† 2 � + NBR � ˆf1 ˆf † 2 � + NBI � ˆf † 1ˆf2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (29) Different from the forward-wave configuration, the “boundary” condition is � ˆa1(ω, 0), ˆa† 1(ω′, 0) � = � ˆa2(ω, L), ˆa† 2(ω′, L) � = δ(ω − ω′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (30) The Langevin noise operators satisfy the same commu- tation relations and correlations in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (13) and (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The Langevin noise matrix is given by NB ≡ � 1 0 0 −1 � �� −MB11 −MB12 MB21 MB22 � + � −MB11 −MB12 MB21 MB22 �∗ = NBR + iNBI, (31) where NBR and NBI are the real and imaginary parts of the matrix NB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' One can show that the noise matrix defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (31) has the same origin as that in the forward-wave configuration in the same nonlinear material: NB = � 1 0 0 −1 � NF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (32) We note that the choice of noise matrix is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' For example, transformation ˆf1 → − ˆf1 or/and ˆf2 → − ˆf2 does not affect computing any physical observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We elaborate on this more in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We obtain the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (29) at z = L as follow- ing �ˆa1 (L) ˆa† 2 (L) � = eMBL �ˆa1 (0) ˆa† 2 (0) � + � L 0 eMB(L−z) � NBR � ˆf1 (z) ˆf † 2 (z) � + NBI � ˆf † 1 (z) ˆf2 (z) �� dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (33) We define eMBL ≡ � ¯A ¯B ¯C ¯D � , (34) eMB(L−z) ≡ � ¯A1 (z) ¯B1 (z) ¯C1 (z) ¯D1 (z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (35) Different from the forward-wave case, in the backward- wave configuration, the mode 1 input is at z = 0 and the mode 2 input is at z = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' With known ˆa1(0) and ˆa2(L), we rearrange Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (33) and obtain solutions for ˆa1(L) and ˆa2(0): �ˆa1 (L) ˆa† 2 (0) � = � A B C D � �ˆa1 (0) ˆa† 2 (L) � + � 1 −B 0 −D � � L 0 � ¯A1 (z) ¯B1 (z) ¯C1 (z) ¯D1 (z) � � NBR � ˆf1 (z) ˆf † 2 (z) � + NBI � ˆf † 1 (z) ˆf2 (z) �� dz, (36) 5 where A = ¯A − ¯B ¯C ¯D , B = ¯B ¯D, C = − ¯C ¯D, D = 1 ¯D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (37) We numerically confirm that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (36) preserves the com- mutation relations � ˆa1(ω, L), ˆa† 1(ω′, L) � = � ˆa1(ω, 0), ˆa† 1(ω′, 0) � , � ˆa2(ω, 0), ˆa† 2(ω′, 0) � = � ˆa2(ω, L), ˆa† 2(ω′, L) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (38) Similarly to the forward-wave configuration, we exam- ine the following four special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Case 1: We assume the nonlinear coupling coefficient κ is real and both modes have losses (Re{αm} ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Under such a condition, we have the following diagonalized noise matrix NB = �� 2Re{α1} 0 0 − � 2Re{α2} � , (39) and the coupled Langevin equations ∂ ∂z �ˆa1 ˆa† 2 � = MB �ˆa1 ˆa† 2 � + � � 2Re{α1} ˆf1 − � 2Re{α2} ˆf † 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (40) Case 2: κ is real, mode 1 has linear loss (Re{α1} = α ≥ 0), and mode 2 has linear gain (Re{α2} = −g ≤ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The noise matrix becomes NF = �√ 2α 0 0 −i√2g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (41) We have the following coupled Langevin equations ∂ ∂z �ˆa1 ˆa† 2 � = MB �ˆa1 ˆa† 2 � + � √ 2α ˆf1 −√2g ˆf2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (42) Case 3: The two modes are perfectly phase-matched without linear gain and loss: ∆k = 0, α1 = α2 = 0, but the nonlinear coupling coefficient is complex-valued κ = η + iζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In this case, the coupled matrix is MB = � 0 −ζ + iη −ζ + iη 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (43) The noise matrix becomes NB = Θ(ζ) � ζ � 1 1 1 −1 � + iΘ(−ζ) � −ζ � 1 1 1 −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (44) The Langevin coupled equations are ∂ ∂z �ˆa1 ˆa† 2 � =MB �ˆa1 ˆa† 2 � + Θ(ζ) � ζ � 1 1 1 −1 � � ˆf1 ˆf † 2 � + Θ(−ζ) � −ζ � 1 1 1 −1 � � ˆf † 1ˆf2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (45) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (45) shows that in the backward-wave configuration, a complex-valued nonlinear coupling coefficient also leads to Langevin noises even though there is no linear gain or loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Case 4: As κ is real and there are equal losses in both modes (α1 = α2 = α > 0) with perfect phase matching (∆k = 0), the coupled equations can be written as i ∂ ∂z �ˆa1 ˆa† 2 � = � −iα −κ −κ iα � �ˆa1 ˆa† 2 � = ˆH �ˆa1 ˆa† 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (46) Interestingly, the effective Hamiltonian ˆH here follows parity-time (PT) symmetry [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' MICROSCOPIC ORIGIN OF LANGEVIN NOISES: SFWM One could validate the above phenomenological ap- proach of quantum Langevin coupled equations by con- firming the microscopic origin of the Langevin noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' However, for two systems with the same quantum Langevin equations, their microscopic structures may be quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Therefore it is impossible to sort all mi- croscopic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In this section, we focus on SFWM in a double-Λ four-level atomic system [10–12, 19, 20] with electromagnetically induced transparency (EIT) [21, 22], and show that the phenomenological approach in the above section agrees with the numerical results from the microscopic quantum theory of light-atom interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We start from a single-atom picture, considering an EM wave couples the atomic transition |j⟩ and |k⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The induced single atom polarization ˆpjk ∝ µjkˆσjk, where µjk is the electric dipole moment matrix element, ˆσjk = |j ⟩⟨ k| is single atom transition operator from state |k⟩ to |j⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the Heisenberg-Langevin picture, the single-atom transition operator can be expressed as ˆσjk = ˆσ(0) jk + � µν βµν ˆf (σ) µν , (47) where ˆσ(0) jk = ⟨ˆσjk⟩ is the zeroth-order steady state so- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The single atom noise operator between atomic transition |ν⟩ → |µ⟩ is represented by ˆf (σ) µν , which satisfies the following correlations: ⟨ ˆf (σ) µν (ω) ˆf (σ)† µ′ν′ (ω′)⟩ = ⟨ ˆf (σ) µν (ω) ˆf (σ) ν′µ′(ω′)⟩ = Dµν,ν′µ′δ(ω − ω′), ⟨ ˆf (σ)† µν (ω) ˆf (σ) µ′ν′(ω′)⟩ = ⟨ ˆf (σ) νµ (ω) ˆf (σ) µ′ν′(ω′)⟩ = Dνµ,µ′ν′δ(ω − ω′), (48) where Dµν,ν′µ′ and Dνµ,µ′ν′ are diffusion coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In a continuous medium with atomic number density n, the noises from different atoms are uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We have the spatially averaged atomic operator ˆ¯σjk ≡ ˆσ(0) jk + 1 √ nA � µν βµν ˆ¯f (σ) µν , (49) 6 |1⟩ |2⟩ Δ� 𝑎��� 𝑧 0 𝐿 (a) (b) |3⟩ 𝜔�� 𝜔� 𝐸� 𝐸� 𝑎�� |4⟩ 𝜔� 𝜔� 𝜛 𝜛 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Spontaneous four-wave mixing (SFWM) in a double-Λ four-level cold atomic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (a) Backward-wave geometry of SFWM optical configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Driven by counter-propagating pump (Ep) and coupling (Ec) beams, phase-matched backward Stokes (ˆas) and anti-Stokes (ˆaas) are spontaneously generated from a laser-cooled atomic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (b) Atomic energy-level diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The pump (ωp) laser is detuned with ∆p from transition |1⟩ → |4⟩, and the coupling (ωc) laser is on-resonant with transition |2⟩ → |3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Stokes (ωs) photons are spontaneously generated from transition |4⟩ → |2⟩, and anti-Stokes (ωas) photons from transition |3⟩ → |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' ϖ = ωas − ω13 is the anti-Stokes photon frequency detuning from transition |1⟩ → |3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' where A is the single-mode cross-section area, and the spatially averaged atomic noise operators ˆ¯f (σ) µν satisfy the following modified correlations ⟨ ˆ¯f (σ) µν (ω, z) ˆ¯f (σ)† µ′ν′ (ω′, z′)⟩ = ⟨ ˆ¯f (σ) µν (ω, z) ˆ¯f (σ) ν′µ′(ω′, z′)⟩ = Dµν,ν′µ′δ(ω − ω′)δ(z − z′), ⟨ ˆ¯f (σ)† µν (ω, z) ˆ¯f (σ) µ′ν′(ω′, z′)⟩ = ⟨ ˆ¯f (σ) νµ (ω, z) ˆ¯f (σ) µ′ν′(ω′, z′)⟩ = Dνµ,µ′ν′δ(ω − ω′)δ(z − z′), (50) where the diffusion coefficients are the same as those from the single-atom picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The electric field and polarization are described as ˆE(t, z) = 1 2 � ˆE(+)(t, z) + ˆE(−)(t, z) � , ˆP(t, z) = 1 2 � ˆP (+)(t, z) + ˆP (−)(t, z) � , (51) Where ˆE(+), ˆP (+) and ˆE(−), ˆP (−) are positive and nega- tive frequency parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We take the following Fourier trans- form ˆE(+)(t, z) = 1 √ 2π � dω ˆE(ω, z)ei(± ω c z−ωt), ˆP (+)(t, z) = 1 √ 2π � dω ˆP(ω, z)ei(± ω c z−ωt), (52) where ˆE(ω, z), ˆP(ω, z) are complex amplitudes in fre- quency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The Maxwell equation under slowly varying envelope approximation (SVEA) can be written as ±∂ ˆE(ω, z) ∂z = i 2ωη ˆP(ω, z), (53) where ± represents for propagation direction along ±z, and free space impedance η = 1/(cε0) = 377 Ohm, with c being the speed of light in vacuum, and ε0 the vacuum permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' With quantized electric field ˆE(ω, z) = � 2ℏω cε0Aˆa(ω, z), (54) and ˆP(ω, z) = 2nµjkˆ¯σjk(ω, z), (55) we obtain the Langevin equation for the EM field in the atomic medium ±∂ˆa(ω, z) ∂z = i nAgjkˆ¯σjk(ω, z) = i nAgjkˆσ(0) jk (ω, z) + ˆ¯F(ω, z), (56) where gjk = µjk � ωjk 2cε0ℏA, ˆ¯F(ω, z) = i √ nAgjk � µν βµν ˆ¯f (σ) µν (ω, z) = iµjk � nωjk 2cε0ℏ � µν βµν ˆ¯f (σ) µν (ω, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (57) Here gjk = g∗ kj is single photon-atom coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Now we turn to the backward-wave SFWM in a double- Λ four-level atomic system as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In presence of counter-propagating pump (Ep, ωp) and cou- pling (Ec, ωc) laser beams, phase-matched Stokes (ωs) and anti-Stokes (ωas) are spontaneously generated and propagate through the medium in opposing directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the rotating reference frame, the interaction Hamilto- nian for a single atom is ˆV = − ℏ � g31ˆaasˆσ31 + g13ˆa† asˆσ13 � − ℏ � g42ˆasˆσ42 + g24ˆa† sˆσ24 � − 1 2ℏ (Ωcˆσ32 + Ω∗ c ˆσ23) − 1 2ℏ � Ωpˆσ41 + Ω∗ pˆσ14 � − ℏ∆pˆσ44 − ℏϖˆσ33 − ℏϖˆσ22, (58) where Ωc = µ32Ec/ℏ is coupling Rabi frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The coupling laser is on-resonant with transition |2⟩ → |3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Ωp = µ41Ep/ℏ is pump Rabi frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The pump laser is far detuned from the transition |1⟩ → |4⟩ with ∆p = ωp − ω14 so that the atomic population mainly oc- cupies the ground state |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We take this ground-state 7 approximation through this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' With continuous- wave pump and coupling driving fields, the energy con- servation leads to ωas+ωs = ωc+ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Here ϖ = ωas−ω13 is the anti-Stokes frequency detuning and thus the Stokes frequency detuning is ωs − ωs0 = −ϖ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The atomic evolution is governed by the following Heisenberg-Langevin equation [11] ∂ ∂t ˆσjk = i ℏ[ ˆV , ˆσjk] − γjkˆσjk + rA jk + ˆf (σ) jk , (59) where γjk = γkj (nonzero only as j ̸= k) are dephasing rates, rA jk (nonzero only as j = k) are the population transfer resulting from spontaneous emission decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The full equation of motion can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The diffusion coefficients Djk,j′k′ can be obtained through the Einstein relation Djk,j′k′ = ∂ ∂t ⟨ˆσjkˆσj′k′⟩ − � ˆAjkˆσj′k′ � − � ˆσjk ˆAj′k′ � , (60) where ˆAjk = ∂ ∂t ˆσjk − ˆf (σ) jk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' For the SFWM governed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (59),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' we have [11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 12] � �� D12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='21 D12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='24 D42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='21 D42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='24 D12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='31 D12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='34 D42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='31 D42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='34 D13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='21 D13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='24 D43,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='21 D43,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='24 D13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='31 D13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='34 D43,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='31 D43,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='34 � �� = � �� 2γ12 ⟨ˆσ11⟩ + Γ31 ⟨ˆσ33⟩ + Γ41 ⟨ˆσ44⟩ γ12 ⟨ˆσ14⟩ 0 0 γ12 ⟨ˆσ41⟩ 0 0 0 0 0 Γ3 ⟨ˆσ11⟩ + Γ31 ⟨ˆσ33⟩ + Γ41 ⟨ˆσ44⟩ Γ3 ⟨ˆσ14⟩ 0 0 Γ3 ⟨ˆσ41⟩ Γ3 ⟨ˆσ44⟩ � �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (61) � �� D21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='12 D21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='42 D24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='12 D24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='42 D21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='13 D21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='43 D24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='13 D24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='43 D31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='12 D31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='42 D34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='12 D34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='42 D31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='13 D31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='43 D34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='13 D34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='43 � �� = � �� 2γ12 ⟨ˆσ22⟩ + Γ32 ⟨ˆσ33⟩ + Γ42 ⟨ˆσ44⟩ 0 γ12 ⟨ˆσ23⟩ 0 0 Γ4 ⟨ˆσ22⟩ + Γ32 ⟨ˆσ33⟩ + Γ42 ⟨ˆσ44⟩ 0 Γ4 ⟨ˆσ23⟩ γ12 ⟨ˆσ32⟩ 0 0 0 0 Γ4 ⟨ˆσ32⟩ 0 Γ4 ⟨ˆσ33⟩ � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (62) Solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (59) under the ground-state approximation ⟨ˆσ11⟩ ∼= 1 with weak pump excitation ∆p ≫ {Ωp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Γ4},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' we get the single-atom steady-state solutions (with µν = 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 43) ˆσ13 = ˆσ(0) 13 + � µν βas µν ˆf (σ) µν ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' ˆσ42 = ˆσ(0) 42 + � µν βs µν ˆf (σ) µν ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (63) where ˆσ(0) 13 =4 (ϖ + iγ12) T (ϖ) g31ˆaas + ΩcΩp T (ϖ) (∆p + iγ14)g24ˆa† s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' ˆσ(0) 42 =(ϖ + iγ13) T (ϖ) |Ωp|2 (∆p − iγ24) 1 (∆p + iγ14)g24ˆa† s + Ω∗ pΩ∗ c T (ϖ) (∆p − iγ24)g31ˆaas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (64) βas 12 = i2Ωc T (ϖ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' βas 13 = −i4 (ϖ + iγ12) T (ϖ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' βas 42 = − iΩcΩp T (ϖ) (∆p − iγ24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' βas 43 = i2Ωp (ϖ + iγ12) T (ϖ) (∆p − iγ34),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' βs 12 = i2 (ϖ + iγ13) T (ϖ) Ω∗ p (∆p − iγ24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' βs 13 = − iΩ∗ pΩ∗ c T (ϖ) (∆p − iγ24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' βs 42 = − i (∆p − iγ24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' βs 43 = − iΩ∗ c 2 (∆p − iγ24) (∆p − iγ34),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (65) where T(ϖ) ≡ |Ωc|2 − 4 (ϖ + iγ13) (ϖ + iγ12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We then obtain the ensemble spatially averaged atomic operators 8 50 0 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='8 1 50 0 50 10 8 6 4 2 0 10-7 50 0 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='8 1 Macro Micro NLN 50 0 50 2 1 0 1 2 3 10-8 ( ) ( ) ( ) ( ) (a) (b) (c) (d) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Comparison of commutation relations between the macroscopic (“Macro”, blue solid lines) and microscopic (“Micro”, red dashed lines) approaches in the group delay regime: (a) [ˆaas(L), ˆa† as(L)], (b) [ˆaas(L), ˆa† as(L)] − δ(ϖ − ϖ′), (c) [ˆas(0), ˆa† s(0)], and (d)[ˆas(0), ˆa† s(0)] − δ(ϖ − ϖ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The results with no Langevin noise operators (“NLN”) are shown as black dotted lines in (a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' for generating anti-Stokes and Stokes fields from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (49) ˆ¯σ13 = ˆσ(0) 13 + 1 √ nA � µν βas µν ˆ¯f (σ) µν , ˆ¯σ42 = ˆσ(0) 42 + 1 √ nA � µν βs µν ˆ¯f (σ) µν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (66) Following the procedures in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (56) and (57),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' ∂ˆaas(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' z) ∂z = i nAg13ˆ¯σ13(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' ∂ˆa† s(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' z) ∂z = i nAg42ˆ¯σ42(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (67) we get coupled equations for counter-propagating anti- Stokes (propagating along +z) and Stokes (propagating along −z) fields in the backward-wave configuration ∂ ∂z �ˆaas ˆa† s � = � −αas + i ∆k 2 iκas iκs α∗ s − i ∆k 2 � �ˆaas ˆa† s � + � ˆ¯Fas − ˆ¯F † s � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (68) where ˆ¯Fas = ig13 √ nA � βas 12 ˆ¯f (σ) 12 + βas 13 ˆ¯f (σ) 13 + βas 42 ˆ¯f (σ) 42 + βas 43 ˆ¯f (σ) 43 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' ˆ¯F † s = −ig42 √ nA � βs 12 ˆ¯f (σ) 12 + βs 13 ˆ¯f (σ) 13 + βs 42 ˆ¯f (σ) 42 + βs 43 ˆ¯f (σ) 43 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (69) and αas = −iωas 2c χas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' αs = −iωs 2c χs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' κas = √ωasωs 2c χ(3) as EpEc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' κs = √ωsωas 2c χ(3)∗ s E∗ pE∗ c ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' χas = 4n |µ13|2 ε0ℏ (ϖ + iγ12) T (ϖ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' χs = n |µ24|2 ε0ℏ (ϖ − iγ13) T ∗ (ϖ) |Ωp|2 ∆2p + γ2 14 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' χ(3) as = nµ13µ32µ24µ41 ε0ℏ3 1 T (ϖ) 1 (∆p + iγ14),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' χ(3) s = nµ13µ32µ24µ41 ε0ℏ3 1 T ∗ (ϖ) 1 (∆p + iγ14),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (70) 9 10 5 0 5 10 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='0001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='0002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='0003 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='0004 Macro Micro 10 5 0 5 10 0 1 2 3 4 10-4 10 5 0 5 10 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='0001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='0002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='0003 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='0004 10 5 0 5 10 0 1 2 3 4 10-4 ( ) ( ) ( ) ( ) (a) (b) (c) (d) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Four real correlations of Stokes and anti-Stokes fields in the group delay regime: (a) ⟨ˆaas(L)ˆa† as(L)⟩, (b) ⟨ˆa† as(L)ˆaas(L)⟩, (c) ⟨ˆas(0)ˆa† s(0)⟩, and (d) ⟨ˆa† s(0)ˆas(0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The macroscopic (“Macro”) and microscopic (“Micro”) approaches are shown as blue solid and red dashed lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The expressions for βas µν and βs µν are listed in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' ∆k = (ωas−ωs)/c−(⃗kc+⃗kp)· ˆz is the phase mismatching in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Here the complex αas represents the EIT loss and phase dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' α∗ s is the Raman gain and dispersion along −z propagation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' One can show that the nonlinear coupling coefficients can be expressed as κas = κeiθ and κs = κe−iθ, where κ = √ωasωs 2c nµ13µ24 ε0ℏ ���� ΩpΩc ∆p + iγ14 ���� 1 T(ϖ), (71) and θ is the phase of ΩpΩc/(∆p + iγ14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' As a result, κas and κs fulfill the gauge transformation discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Therefore, to be consistent with the treatment in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' II, we rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (68) to ∂ ∂z �ˆaas ˆa† s � = MB �ˆaas ˆa† s � + � ˆFas − ˆF † s � , (72) where MB = � −αas + i ∆k 2 iκ iκ α∗ s − i ∆k 2 � , ˆFas = ˆ¯Fase−iθ/2, ˆF † s = ˆ¯F † s eiθ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (73) Similarly, we rewrite the SFWM quantum Langevin equations in the forward-wave configuration in Ap- pendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We now turn to compare Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (72) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (29) from the phenomenological approach in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' II, where we take mode 1 as anti-Stokes and mode 2 as Stokes in the backward-wave configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (29), we have ˆFas = NBR11 ˆf1 + NBI11 ˆf † 1 + NBI12 ˆf2 + NBR12 ˆf † 2, ˆF † s = −NBR21 ˆf1 − NBI21 ˆf † 1 − NBI22 ˆf2 − NBR22 ˆf † 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (74) Therefore, we obtain ˆFas and ˆF † s from two different ap- proaches: Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (69) from the microscopic photon-atom interaction, and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (74) from the macroscopic phe- nomenological approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Although we remark that the atomic noise operators ˆ¯f (σ) µν are different from the field noise operators ˆfm, the correlations of ˆFas and ˆFs uniquely determine the system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' While we find it difficult to analytically prove the two approaches are equivalent, we could numerically compute and com- pare the commutation relations and correlations of ˆaas, ˆa† as, ˆas, and ˆa† s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We consider here the backward-wave SFWM in laser- cooled 85Rb atoms with relevant atomic energy lev- els being |1⟩ = ��52S1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' F = 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' |2⟩ = ��52S1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' F = 3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 10 2 1 0 1 2 Macro Micro 2 1 0 1 2 2 1 0 1 2 10 5 0 5 10 2 1 0 1 2 10 0 10 2 1 0 1 2 10 5 0 5 10 2 1 0 1 2 2 1 0 1 2 2 1 0 1 2 2 1 0 1 2 10 5 0 5 10 2 1 0 1 2 10 0 10 2 1 0 1 2 10 5 0 5 10 2 1 0 1 2 10-2 ( ) 10-2 ( ) 10-2 ( ) 10-2 ( ) 10-2 ( ) 10-2 ( ) (a) (b) (c) (d) (e) (f) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Twelve complex correlations of Stokes and anti-Stokes fields in the group delay regime: (a) ⟨ˆaas(L)ˆaas(L)⟩ = ⟨ˆa† as(L)ˆa† as(L)⟩∗, (b) ⟨ˆaas(L)ˆas(0)⟩ = ⟨ˆa† s(0)ˆa† as(L)⟩∗, (c) ⟨ˆaas(L)ˆa† s(0)⟩ = ⟨ˆas(0)ˆa† as(L)⟩∗, (d) ⟨ˆa† as(L)ˆas(0)⟩ = ⟨ˆa† s(0)ˆaas(L)⟩∗, (e) ⟨ˆas(0)ˆaas(L)⟩ = ⟨ˆa† as(L)ˆa† s(0)⟩∗, and (f) ⟨ˆas(0)ˆas(0)⟩ = ⟨ˆa† s(0)ˆa† s(0)⟩∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The macroscopic (“Macro”) and microscopic (“Mi- cro”) approaches are shown as blue solid and red dashed lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' |3⟩ = ��52P1/2, F = 3 � , |4⟩ = ��52P3/2, F = 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The decay and dephasing rates for corresponding energy levels are Γ3 = Γ4 = 2π × 6 MHz, Γ31 = 5 9Γ3, Γ32 = 4 9Γ3, Γ41 = 4 9Γ4, Γ42 = 5 9Γ4, γ13 = γ23 = γ14 = γ24 = 2π × 3 MHz, and γ12 = 2π × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='03 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' With vacuum inputs in both Stokes (z = L) and anti-Stokes (z = 0) modes, we have ⟨ˆaas(ϖ, 0)ˆa† as(ϖ′, 0)⟩ = ⟨ˆas(ϖ, L)ˆa† s(ϖ′, L)⟩ = δ(ϖ − ϖ′) and ⟨ˆa† as(ϖ, 0)ˆaas(ϖ′, 0)⟩ = ⟨ˆa† s(ϖ, L)ˆas(ϖ′, L)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' There is also no correlation between Stokes and anti- Stokes fields at their inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We numerically compute SFWM in two different regimes to confirm the consistency between the macro- scopic and microscopic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' i) The first is the group delay regime, where the SFWM spectrum bandwidth is determined by the EIT slow-light induced phase mis- matching [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The working parameters are: Ωp = 2π × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='2 MHz, Ωc = 2π × 12 MHz, ∆p = 2π × 500 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The cold atomic medium with length L = 2 cm has den- sity n = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='1 × 1016 m−3, corresponding to an atomic optical depth OD = 80 on the anti-Stokes resonance transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' ii) The second is the Rabi oscillation regime, where biphoton correlation reveals single-atom dynamics [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The working parameters are: Ωp = 2π × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='2 MHz, Ωc = 2π ×24 MHz, ∆p = ωp −ω14 = 2π ×500 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The cold atomic medium with length L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='2 cm has density n = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='4×1014 m−3, corresponding to OD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In both cases, we take ∆k = 127 rad/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The numerical results in the group delay regime are 11 50 0 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='8 1 50 0 50 10 5 0 10-9 50 0 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='8 1 Macro Micro NLN 50 0 50 1 0 1 2 10-11 ( ) ( ) ( ) ( ) (a) (b) (c) (d) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Comparison of commutation relations between the macroscopic (“Macro”, blue solid lines) and microscopic (“Micro”, red dashed lines) approaches in the damped Rabi oscillation regime: (a) [ˆaas(L), ˆa† as(L)], (b) [ˆaas(L), ˆa† as(L)] − δ(ϖ − ϖ′), (c) [ˆas(0), ˆa† s(0)], and (d)[ˆas(0), ˆa† s(0)] − δ(ϖ − ϖ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The results with no Langevin noise operators (“NLN”) are shown as black dotted lines in (a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 3, 4, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The commutation re- lations [ˆaas(L), ˆa† as(L)] and [ˆas(0), ˆa† s(0)] are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Both macroscopic and microscopic approaches agree well with each other [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 3(a) and (c)], with neg- ligible relative small difference < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='0 × 10−6 [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 3(b) and (d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' As expected, the macroscopic phenomenologi- cal results give perfect flat lines at [ˆaas(L,ϖ),ˆa† as(L,ϖ′)] δ(ϖ−ϖ′) = [ˆas(0,ϖ),ˆa† s(0,ϖ′)] δ(ϖ−ϖ′) = 1 which is the starting point of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The microscopic results of field commutations are consistent with the macroscopic approach, but with < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='0 × 10−6 deviation at some spectra points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' As we understand, these small spectra discrepancies may be caused by the ground-state and zeroth-order approxi- mations we take for solving the microscopic Heisenberg- Langevin equations (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' If the Langevin noise operators are not taken into account, as shown in the black dotted curves in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 3(a) and (c), the anti-Stokes commuta- tion relation is not preserved and displays EIT transmis- sion spectrum, while Stokes commutation relation still approximately holds due to the negligible gain or loss in Stokes channel under the ground-state approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Figure 4 displays four real-valued correlations of Stokes and anti-Stokes fields: (a )⟨ˆaas(L)ˆa† as(L)⟩, (b) ⟨ˆa† as(L)ˆaas(L)⟩, (c) ⟨ˆas(0)ˆa† s(0)⟩, and (d) ⟨ˆa† s(0)ˆas(0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Figure 5 shows the twelve (six pairs) complex-valued correlations of Stokes and anti-Stokes fields: (a) ⟨ˆaas(L)ˆaas(L)⟩ = ⟨ˆa† as(L)ˆa† as(L)⟩∗, (b) ⟨ˆaas(L)ˆas(0)⟩ = ⟨ˆa† s(0)ˆa† as(L)⟩∗, (c) ⟨ˆaas(L)ˆa† s(0)⟩ = ⟨ˆas(0)ˆa† as(L)⟩∗, (d) ⟨ˆa† as(L)ˆas(0)⟩ = ⟨ˆa† s(0)ˆaas(L)⟩∗, (e) ⟨ˆas(0)ˆaas(L)⟩ = ⟨ˆa† as(L)ˆa† s(0)⟩∗, and (f) ⟨ˆas(0)ˆas(0)⟩ = ⟨ˆa† s(0)ˆa† s(0)⟩∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The macroscopic solutions agree well with those obtained from the microscopic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The numerical results in the Rabi oscillation regime are plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 6, 7, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The macroscopic phenomeno- logical results also agree remarkably well with those from the microscopic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' BIPHOTON GENERATION We now turn to apply the quantum Langevin the- ory to study time-frequency entangled photon pair (biphoton) generation through spontaneous four-wave mixing process, especially in a variety of situa- tions involving gain, loss, and/or complex nonlinear coupling coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We consider continuous-wave pumping whose time translation symmetry leads to frequency anti-correlation ω1 + ω2 =constant between the paired photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the spontaneous 12 50 0 50 1 1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='5E-7 1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='0E-7 1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='5E-7 Macro Micro 50 0 50 0 5 10 15 10-8 50 0 50 1 1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='2E-7 1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='4E-7 1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='6E-7 1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='8E-7 1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='0E-7 50 0 50 0 2 4 6 8 10 10-8 ( ) ( ) ( ) ( ) (a) (b) (c) (d) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Four real correlations of Stokes and anti-Stokes fields in the damped Rabi oscillation regime: (a) ⟨ˆaas(L)ˆa† as(L)⟩, (b) ⟨ˆa† as(L)ˆaas(L)⟩, (c) ⟨ˆas(0)ˆa† s(0)⟩, and (d) ⟨ˆa† s(0)ˆas(0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The macroscopic (“Macro”) and microscopic (“Micro”) approaches are shown as blue solid and red dashed lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' four-wave mixing process, both input states are vac- uum: ⟨ˆa† 1(ϖ, 0)ˆa1(ϖ′, 0)⟩ = ⟨ˆa† 2(ϖ, 0)ˆa2(ϖ′, 0)⟩ = 0, ⟨ˆa1(ϖ′, 0)ˆa† 1(ϖ, 0)⟩ = ⟨ˆa2(ϖ′, 0)ˆa† 2(ϖ, 0)⟩ = δ(ϖ − ϖ′) for the forward-wave configuration, and ⟨ˆa† 1(ϖ, 0)ˆa1(ϖ′, 0)⟩ = ⟨ˆa† 2(ϖ, L)ˆa2(ϖ′, L)⟩ = 0, ⟨ˆa1(ϖ, 0)ˆa† 1(ϖ′, 0)⟩ = ⟨ˆa2(ϖ, L)ˆa† 2(ϖ′, L)⟩ = δ(ϖ − ϖ′) for the backward-wave configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (4), with ω1 = ω10 + ϖ and ω2 = ω20 − ϖ, we have ˆa1(t, z1) = eiω10( z1 c −t) √ 2π � dϖˆa1(ϖ, z1)eiϖ( z1 c −t)e−i ∆k 2 z1, ˆa2(t, z2) = eiω20(± z2 c −t) √ 2π � dϖˆa2(ϖ, z2)eiϖ(± z2 c −t)e−i ∆k 2 z2, (75) where ± represents the forward-wave (+) or backward- wave (−) configuration, z = z1 and z = z2 are the output positions of channels 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' For the forward-wave configuration, z1 = z2 = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' For the backward-wave configuration, z1 = L and z2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The phase mismatching in vacuum ∆k = (ωas ±ωs)/c−(⃗kc + ⃗kp)· ˆz ≃ (ωas0 ±ωs0)/c−(⃗kc +⃗kp)· ˆz is nearly a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The vacuum time delay zi/c constants are usually very small in usual experimental conditions, from now on we ignore these constants for simplification and rewrite the above equations to (otherwise one just needs to make a time translation t → t − zi/c) ˆa1(t, z1) = e−iω10t √ 2π � dϖˆa1(ϖ, z1)e−iϖt, ˆa2(t, z2) = e−iω20t √ 2π � dϖˆa2(ϖ, z2)eiϖt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (76) The photon rate in channel m can be computed from Rm ≡ � ˆa† m (t, zm) ˆam (t, zm) � = 1 2π �� ∞ −∞ dϖdϖ′e−iϖteiϖ′t � ˆa† m (ϖ′, zm) ˆam (ϖ, zm) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (77) Here we are particularly interested in the two-photon Glauber correlation in the time domain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' which can be computed from the following two different orders G(2) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='1 (t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' t1) ≡⟨ˆa† 1 (t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' z1) ˆa† 2 (t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' z2) ˆa2 (t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' z2) ˆa1 (t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' z1)⟩ =|⟨ˆa2 (t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' z2) ˆa1 (t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' z1)⟩|2 + |⟨ˆa† 2 (t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' z2) ˆa1 (t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' z1)⟩|2 + R1R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (78) 13 5 0 5 Macro Micro 5 0 5 5 0 5 50 0 50 5 0 5 50 0 50 5 0 5 50 0 50 5 0 5 5 0 5 5 0 5 5 0 5 50 0 50 5 0 5 50 0 50 5 0 5 50 0 50 5 0 5 10-5 ( ) 10-5 ( ) 10-5 ( ) 10-5 ( ) 10-5 ( ) 10-5 ( ) (a) (b) (c) (d) (e) (f) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Twelve complex correlations of Stokes and anti-Stokes fields in the damped Rabi oscillation regime: (a) ⟨ˆaas(L)ˆaas(L)⟩ = ⟨ˆa† as(L)ˆa† as(L)⟩∗, (b) ⟨ˆaas(L)ˆas(0)⟩ = ⟨ˆa† s(0)ˆa† as(L)⟩∗, (c) ⟨ˆaas(L)ˆa† s(0)⟩ = ⟨ˆas(0)ˆa† as(L)⟩∗, (d) ⟨ˆa† as(L)ˆas(0)⟩ = ⟨ˆa† s(0)ˆaas(L)⟩∗, (e) ⟨ˆas(0)ˆaas(L)⟩ = ⟨ˆa† as(L)ˆa† s(0)⟩∗, and (f) ⟨ˆas(0)ˆas(0)⟩ = ⟨ˆa† s(0)ˆa† s(0)⟩∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The macroscopic (“Macro”) and mi- croscopic (“Micro”) approaches are shown as blue solid and red dashed lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' G(2) 1,2 (t1, t2) ≡⟨ˆa† 2 (t2, z2) ˆa† 1 (t1, z1) ˆa1 (t1, z1) ˆa2 (t2, z2)⟩ =|⟨ˆa1 (t1, z1) ˆa2 (t2, z2)⟩|2 + |⟨ˆa† 2 (t2, z2) ˆa1 (t1, z1)⟩|2 + R1R2, (79) where we have applied the Gaussian moment theorem [23, 24] to decompose the fourth-order field correlations to the sum of the products of second-order field corre- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The first term in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (78) and (79) can be expressed as |Ψ2,1(t2, t1)|2 and |Ψ1,2(t1, t2)|2, where Ψ2,1(t2, t1) = ⟨ˆa2 (t2, z2) ˆa1 (t1, z1)⟩ = e−iω20t2e−iω10t1ψ2,1(t1 − t2), (80) Ψ1,2(t1, t2) = ⟨ˆa1 (t1, z1) ˆa2 (t2, z2)⟩ = e−iω20t2e−iω10t1ψ1,2(t1 − t2), (81) are the two-photon wavefunctions with the relative parts ψ2,1(t1 − t2) = 1 2π �� dϖdϖ′⟨ˆa2(ϖ′, z2)ˆa1(ϖ, z1)⟩e−iϖ(t1−t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (82) ψ1,2(t1 − t2) = 1 2π �� dϖdϖ′⟨ˆa1(ϖ, z1)ˆa2(ϖ′, z2)⟩e−iϖ(t1−t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (83) One can show that the second term in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (78) and (79) is zero if the nonlinear coupling coefficient is real-valued, 14 and it is usually very small as compared to other terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The third term in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (78) and (79) is the accidental coincidence counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The two-photon wavefunction and Glauber correlation satisfy the following exchange sym- metry ψ21(t1 − t2) = ψ2,1(t1 − t2) = ψ1,2(t1 − t2), Ψ21(t2, t1) = Ψ2,1(t2, t1) = Ψ1,2(t1, t2), G(2) 21 (t2, t1) = G(2) 2,1 (t2, t1) = G(2) 1,2 (t1, t2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (84) The normalized two-photon correlation is defined as g(2) 21 (t2, t1) ≡ G(2) 21 (t2, t1) R1R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (85) As the system has time translation symmetry with continuous-wave pumping, G(2) 21 (t2, t1) = G(2) 21 (t1 − t2) depends only on the relative time t1 − t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Loss and Gain To simplify and unify the descriptions for account- ing both forward- and backward-wave cases, we define “input-output” fields: ˆa1,in ≡ ˆa1(0), ˆa2,in ≡ ˆa2(0), ˆa1,out ≡ ˆa1(L), and ˆa2,out ≡ ˆa2(L) for the forward-wave case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' ˆa1,in ≡ ˆa1(0), ˆa2,in ≡ ˆa2(L), ˆa1,out ≡ ˆa1(L), and ˆa2,out ≡ ˆa2(0) for the backward-wave case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In this sub- section, we aim to investigate the roles of loss and gain in biphoton generation, considering linear loss in mode 1 (Re{α1} = α ≥ 0) and linear gain (Re{α2} = −g ≤ 0) in mode 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We also assume κ is real, or its contribution to Langevin noises is much smaller than the linear gain and loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=', Im{κ} ≪ {α, g}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In this case, for forward- and backward-wave configurations, the noise matrix is reduced to NF,B = �√ 2α 0 0 ±i√2g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (86) Hence, the output fields in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (19) and (36) can be rewritten as �ˆa1,out ˆa† 2,out � = � A B C D � �ˆa1,in ˆa† 2,in � + � L 0 � X11 X12 X21 X22 � � ˆf1 (z) ˆf2 (z) � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (87) where Xmn are combined coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We further rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (87) as ˆa1,out = Aˆa1,in + Bˆa† 2,in + � L 0 � X11 ˆf1(z) + X12 ˆf2(z) � , ˆa2,out = C∗ˆa† 1,in + D∗ˆa2,in + � L 0 � X∗ 21 ˆf † 1(z) + X∗ 22 ˆf † 2(z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (88) As shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (84), there are two different orders [⟨: ˆa2ˆa1 :⟩ or ⟨: ˆa1ˆa2 :⟩] to compute the two-photon wave- function and Galuber correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Although these two orders are equivalent, the numerical computation com- plexity may be significantly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Computing bipho- ton wavefunction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (83) in the order ⟨: ˆa1ˆa2 :⟩ in- volves nonzero noise field correlations ⟨ ˆfm ˆf † m⟩, while in the order ⟨: ˆa2ˆa1 :⟩ [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (82)] these noise field corre- lations disappear because of ⟨ ˆf † m ˆfm⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' These field correlations in the frequency domain can be expressed as ⟨ˆa2out (ϖ′) ˆa1out (ϖ)⟩ = δ(ϖ − ϖ′) [BD∗] , (89) ⟨ˆa1out (ϖ) ˆa2out (ϖ′)⟩ = δ(ϖ − ϖ′) � AC∗ + � L 0 dz (X11X∗ 21 + X12X∗ 22) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (90) Therefore, we obtain the biphoton wavefunction follow- ing the order ⟨: ˆa2ˆa1 :⟩ ψ21(τ) = �� dϖdϖ′⟨ˆa2,out(ϖ′)ˆa1,out(ϖ)⟩e−iϖτ = � dϖBD∗e−iϖτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (91) where τ = t1 − t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' If following the order ⟨: ˆa1ˆa2 :⟩, we have ψ12(τ) = �� dϖdϖ′⟨ˆa1,out(ϖ)ˆa2,out(ϖ′)⟩e−iϖτ = � dϖ � AC∗ + � L 0 dz (X11X∗ 21 + X12X∗ 22) � e−iϖτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (92) One can show that the second term in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (78) and (79) is zero in this loss-gain configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The single-channel photon rates can be obtained as R1 = 1 2π � |B|2dϖ, R2 = 1 2π � � |C|2 + � L 0 dz � |X21|2 + |X22|2� � dϖ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (93) It is interesting to remark that, in the loss-gain config- uration, the biphoton field correlation following the order ⟨: ˆagainˆaloss :⟩ does not involve noise field correlations as shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (89) and (91), which dramatically reduces the computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' On the other side, taking the order ⟨: ˆalossˆagain :⟩ must include noise field corre- lations as shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (90) and (92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' This may be understood in the heralded photon picture [25]: When a photon in a lossy channel is detected (annihilated) by a detector, we can always ensure there is its partner (or paired) photon in another channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' On the other side, when a photon is detected in a gain channel which pro- duces multiple photons, we can not always ensure it has a partner photon in another channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The exchange sym- metry can only be preserved by taking into account the Langevin noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 15 0 1 2 109 Macro Micro NLN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='5 1 0 1 2 (a) (b) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Two-photon Glauber correlation in time domain in the group delay regime: (a) G(2) s,as(τ) and (b) G(2) as,s(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The simulation conditions are the same as that in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 3, 4, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' NLN: no Langevin noise included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the SFWM described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' III, the anti-Stokes photons experience finite EIT loss due to the ground state dephasing (γ12 ̸= 0), and the Stokes photons prop- agate with negligible but small Raman gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Figure 9 displays the two-photon Glauber correlation in the group delay regime with the same parameters as those in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 3, 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 9(a) and (b), both macroscopic and microscopic approaches with Langevin noises give consistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' As expected, the compu- tation of G(2) s,as(τ) (following the order ⟨: ˆasˆaas :⟩) with- out Langevin noise operators (black dotted line: NLN) agrees with the exact results obtained from both macro- scopic (blue solid line) and microscopic (red dashed line) approaches, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 9(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' On the contrary, the computation of G(2) as,s(τ) (following the order ⟨: ˆaasˆas :⟩) without Langevin noise operators deviates significantly from the exact results, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 9(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Complex Phase Mismatching Different from the Heisenberg picture where the evo- lution of field operators is governed by their Langevin coupled equations, reference [10] provides a perturbation theory to describe biphoton state in the interaction pic- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The solution from Heisenberg-Langevin theory may contain correlations of more than two photons, while the perturbation theory focuses only on the two-photon state by ignoring higher-order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' These two treatments are expected to give the same results in the limit of small pa- rameter gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Although the perturbation theory in the interaction picture provides a much clear physics picture of two-photon state, treating loss and gain requires a proper justification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the perturbation theory, linear loss and gain are included in the complex phase mis- matching ∆˜k(ϖ) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' For the SFWM described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' III, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' [10] derives the biphoton relative wavefunction with perturbation theory as ψ(τ) = iL 2π � dϖκ(ϖ)Φ(ϖ)e−iϖτ, (94) where the longitudinal detuning function is Φ(ϖ) = sinc � ∆˜kL 2 � ei(kas+ks)L, (95) There is a statement in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' [10]: “It is found that to be consistent with the Heisenberg–Langevin theory in the low-gain limit, the argument in Φ should be replaced by ∆˜k = � ⃗kas + ⃗k∗ s − ⃗kc − ⃗kp � ˆz, where ⃗k∗ s is the conjugate of ⃗ks.” For the SFWM in the double-Λ four-level atomic system, there is small Raman gain in the Stokes chan- nel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' What happens if there is loss in the Stokes channel?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Should we take ⃗k∗ s or ⃗ks in the complex phase mismatch- ing ∆˜k(ϖ)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Although Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' [10] takes ⃗k∗ s for Stokes pho- tons with gain, it is not clear whether it still holds for the case with loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In this subsection, we do not only provide a justification for the above statement in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' [10] from the quantum Langevin theory by taking small parametric gain approximation, but also extend the complex phase mismatching to the case with loss in the Stokes channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We take the same backward-wave configuration in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We assume anti-Stokes photons in mode 1 are lossless with EIT and there is gain (or loss) in Stokes mode 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The small parametric gain fulfills |κ| ≪ {α, g}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the backward-wave configuration, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (7), (34), and (37), we obtain analytical expressions of A, B, C, and D as A = � q2 − 4κ2e−(α1−α∗ 2)L/2 qsinh � L 2 � q2 − 4κ2 � + � q2 − 4κ2cosh � L 2 � q2 − 4κ2 �, B = 2iκ q + � q2 − 4κ2coth( L 2 � q2 − 4κ2) , C = −2iκ q + � q2 − 4κ2coth( L 2 � q2 − 4κ2) , D = � q2 − 4κ2e(α1−α∗ 2)L/2 qsinh � L 2 � q2 − 4κ2 � + � q2 − 4κ2cosh � L 2 � q2 − 4κ2 �, (96) where q ≡ α1 + α∗ 2 − i∆k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the small parametric gain approximation, we have � q2 − 4κ2 ≈ q = α1 + α∗ 2 − i∆k = −i(∆k1 − ∆k2 ∗ + ∆k), (97) and α1 − α∗ 2 = −i(∆k1 + ∆k2 ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (98) 16 where ∆km = ωm 2c χm is the wavenumber difference from that in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Hence, we simplify A, B, C, and D to A =exp [i∆k1L] exp �i∆kL 2 � , B =iκLsinc �(∆k1 − ∆k∗ 2 + ∆k)L 2 � × exp �i(∆k1 − ∆k∗ 2 + ∆k)L 2 � , C = − iκLsinc �(∆k1 − ∆k∗ 2 + ∆k)L 2 � × exp �i(∆k1 − ∆k∗ 2 + ∆k)L 2 � , D =exp [−i∆k∗ 2L] exp �i∆kL 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (99) We first look at the case with gain in the Stokes (mode 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' IV A, we take the order ⟨: ˆa2ˆa1 :⟩ ψ21(τ) = �� dϖdϖ′⟨ˆa2,out(ϖ′)ˆa1,out(ϖ)⟩e−iϖτ = � dϖBD∗e−iϖτ, (100) where BD∗ = iκLsinc �(∆k1 − ∆k∗ 2 + ∆k)L 2 � × exp �i(∆k1 − ∆k∗ 2 + 2∆k2)L 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (101) Comparing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (100) and (101) with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (94) and (95), particularly for the argument in the sinc function, we have ∆˜k = ∆k1 − ∆k∗ 2 + ∆k = k1 − k∗ 2 − kc + kp = kas − k∗ s − kc + kp which is consistent with the statement in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We now look at the case with loss in the Stokes (mode 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We take the order ⟨: ˆa1ˆa2 :⟩ and have ψ12(τ) = �� dϖdϖ′⟨ˆa1,out(ϖ)ˆa2,out(ϖ′)⟩e−iϖτ = � dϖAC∗e−iϖτ, (102) where AC∗ = iκ∗Lsinc �(∆k∗ 1 − ∆k2 + ∆k)L 2 � exp �i(2∆k1 − ∆k∗ 1 + ∆k2)L 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (103) Comparing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (102) and (103) with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (94) and (95), we have ∆˜k = ∆k∗ 1 − ∆k2 + ∆k = k1 − k2 − kc + kp = kas − ks − kc + kp, which is different from the case with gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Here we have taken k1 ≃ k∗ 1 for lossless mode 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Although our discussion is based on the backward- wave configuration, the conclusion can be extended to the forward-wave configuration, which is derived in de- tail in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Therefore, in the case with gain in the Stokes mode 2, the complex phase mismatching is ∆˜k = � ⃗kas + ⃗k∗ s − ⃗kc − ⃗kp � ˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the case with loss in the Stokes mode 2, the complex phase mismatching be- comes ∆˜k = � ⃗kas + ⃗ks − ⃗kc − ⃗kp � ˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Complex Nonlinear Coupling Coefficient and Rabi Oscillation As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 2, we can understand the SFWM process in the following picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' After a Stoke and anti- Stokes photon pair is born from a single atom follow- ing the atomic transitions [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 2(b)], the paired pho- tons then propagate through the medium [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 2(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' As the photon pair can be generated at any atom inside the medium, the overall two-photon wavefunction (or prob- ability amplitude) is a superposition of all possible such generation-propagation two-photon Feynman paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Fol- lowing this picture, when the propagation effect can be ignored, the biphoton state reveals the single atom dy- namics, which is connected to the nonlinear coupling co- efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the following, we consider SFWM in the limit of small optical depth (OD) where the linear propaga- tion effect is small and show how the complex spectrum of nonlinear coupling coefficient reveals single-atom Rabi oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We rewrite the nonlinear coupling coefficient in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (71) as: κ(ϖ) = J � 1 (ϖ − Ωe/2 + iγe) − 1 (ϖ + Ωe/2 + iγe) � , (104) where J = − √ωasωsnµ13µ24 8cε0ℏΩe ���� ΩpΩc ∆p + iγ14 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (105) Here Ωe = � |Ωc|2 − (γ13 − γ12)2 is the effective coupling Rabi frequency, and γe = (γ12 + γ13)/2 is the effective dephasing rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Obviously, the nonlinear coupling coeffi- cient κ(ϖ) has a complex spectrum, with two resonances separated by the effective coupling Rabi frequency Ωe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the ground-state approximation with major atomic pop- ulation in state |1⟩, the undepleted pump laser beam is far detuned from the transition |1⟩ → |4⟩ and its exci- tation is weak such that we can take χs ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' On the other side, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (70) we have the complex linear susceptibility for anti-Stokes photons χas(ϖ) = −n |µ13|2 ε0ℏ (ϖ + iγ12) (ϖ − Ωe/2 + iγe)(ϖ + Ωe/2 + iγe) (106) Although the anti-Stokes photon absorption at ϖ = 0 is suppressed by the EIT effect, there are two absorp- tion resonances appearing at ϖ = ±Ωe/2 which coin- cide with the two resonances of nonlinear coupling coef- ficient in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (104).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We take the pump laser with weak 17 intensity (∝ |Ωp|2) and large detuning (∆p) such that Re{αas(ϖ = ±Ωe/2)}>Im{κ(ϖ = ±Ωe/2)}, which are usually satisfied in the ground state condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' As the propagation effect is small and the phase matching is not important, the paired photons are mostly generated from the two resonances (ϖ = ±Ωe/2) of the nonlinear cou- pling coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the forward-wave configuration, with the coupling matrix MF = � −αas + i ∆k 2 iκ −iκ −i ∆k 2 � , (107) and short medium length L satisfying |MFL| ≪ 1, we have approximately � A B C D � = eMFL ∼= 1 + MFL = � 1 − αasL + i ∆k 2 L iκL −iκL 1 − i ∆k 2 L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (108) As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' IV A, the biphoton field correlation following the order ⟨: ˆasˆaas :⟩ does not need count the Langevin noise operators: ⟨ˆas(ϖ′, L)ˆaas(ϖ, L)⟩ = BD∗δ(ϖ − ϖ′) = iκL(1 + i∆k 2 L)δ(ϖ − ϖ′) ∼= iκ(ϖ)Lδ(ϖ − ϖ′), (109) where we have neglected higher order terms O(L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (82), we have the relative biphoton wavefunction ψs−as(τ) = iL 2π � dϖκ(ϖ)e−iϖτ, (110) which is the Fourier transform of the nonlinear coupling coefficient with τ = tas − ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (104) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (110) we obtain ψs−as(τ) = LJe−γeτ[e−iΩeτ/2 − eiΩeτ/2]Θ(τ) = −2iLJe−γeτ sin �Ωeτ 2 � Θ(τ), (111) where Θ(τ) is the Heaviside function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Equation (111) shows a damped Rabi oscillation, resulting from the beat- ing between biphotons generated from the two resonances at ϖ = ±Ωe/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The Heaviside function shows the anti- Stokes photon is always generated after its paired Stokes photon following the time order of atomic transitions |1⟩ → |4⟩ → |2⟩ → |3⟩ → |1⟩ in an SFWM cycle shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the backward-wave configuration, the coupling ma- trix becomes MB = � −αas + i ∆k 2 iκ iκ −i ∆k 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (112) 0 2 4 6 105 Macro Micro NLN 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='4 0 2 4 6 | s-as|2 (a) (b) (c) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Two-photon Glauber correlation in time domain in the damped Rabi oscillation regime: (a) G(2) s,as(τ) and (b) G(2) as,s(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The simulation conditions are the same as that in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 6, 7, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (c) shows the analytic solution for the bipho- ton waveform |ψs−as(τ)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' NLN: no Langevin noise included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' With |MBL| ≪ 1 we have � ¯A ¯B ¯C ¯D � = eMBL ∼= 1 + MBL = � 1 − αasL + i ∆k 2 L iκL iκL 1 − i ∆k 2 L � , (113) and � A B C D � = � 1 − αasL + i ∆k 2 L iκL −iκL 1 + i ∆k 2 L � , (114) where we have neglect higher order terms O(L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Simi- larly, we have ⟨ˆas(ϖ′, 0)ˆaas(ϖ, L)⟩ ∼= iκ(ϖ)Lδ(ϖ − ϖ′), (115) which is the same as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (109) of the forward-wave con- figuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Therefore, we obtain Rabi oscillations in both forward- and backward-wave configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Equation (111) is identical to the result derived from the pertur- bation theory in the interaction picture [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 18 Figure 10 displays the two-photon Glauber correlation in the damped Rabi oscillation regime with the same pa- rameters as those in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 6, 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 10(a) and (b), both macroscopic and microscopic approaches with Langevin noises give consistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' As expected, the computation of G(2) s,as(τ) (following the order ⟨: ˆasˆaas :⟩) without Langevin noise operators (dot points) agrees with the exact results obtained from both microscopic (red dashed line) and macroscopic (blue solid line) approaches, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 10(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' On the con- trary, the computation of G(2) as,s(τ) (following the order ⟨: ˆaasˆas :⟩) without Langevin noise operators (dot points: NLN) deviates significantly from the exact results and vi- olates the causality, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 10(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 10(c) shows the result from the analytic solution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (111) which agree well with the exact results in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 10(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' It is interesting to examine a system without gain and loss whose Langevin noises are purely contributed by the complex nonlinear coupling coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In this case, the above approximation and conclusion do not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Let’s now consider the case 3 with the forward-wave config- uration in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' II A, where α1 = α2 = ∆k = 0, and κ = η + iζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' As shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' II A, the noise matrix is different as ζ is positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We first consider ζ > 0, the Langevin coupled equations (27) becomes ∂ ∂z �ˆa1 ˆa† 2 � = � 0 iκ −iκ 0 � �ˆa1 ˆa† 2 � + � ζ � 1 1 −1 1 � � ˆf1 ˆf † 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (116) Under the condition |MFL| ≪ 1, we solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (116) to the first order of L and have ˆa1(L) ∼= ˆa1(0) + iκLˆa† 2(0) + � ζ � L 0 dz � ˆf1 + ˆf † 2 � , ˆa2(L) ∼= ˆa2(0) + iκ∗Lˆa† 1(0) + � ζ � L 0 dz � − ˆf † 1 + ˆf2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (117) The two-photon field correlations are ⟨ˆa1(L)ˆa2(L)⟩ = ⟨ˆa2(L)ˆa1(L)⟩ ∼= i 2(κ + κ∗)Lδ(ϖ − ϖ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (118) As ζ < 0, the Langevin coupled equations (27) becomes ∂ ∂z �ˆa1 ˆa† 2 � = � 0 iκ −iκ 0 � �ˆa1 ˆa† 2 � + � −ζ � 1 1 −1 1 � � ˆf † 1ˆf2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (119) Under the condition |MFL| ≪ 1, we solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (119) to the first order of L and have ˆa1(L) ∼= ˆa1(0) + iκLˆa† 2(0) + � −ζ � L 0 dz � ˆf † 1 + ˆf2 � , ˆa2(L) ∼= ˆa2(0) + iκ∗Lˆa† 1(0) + � −ζ � L 0 dz � − ˆf1 + ˆf † 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (120) The two-photon field correlations are ⟨ˆa1(L)ˆa2(L)⟩ = ⟨ˆa2(L)ˆa1(L)⟩ ∼= i 2(k + k∗)Lδ(ϖ − ϖ′), (121) which is the same as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (118).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The biphoton relative wavefunction is ψ21(τ) = ψ∗ 21(−τ) = iL 2π � dϖ1 2(k + k∗)e−iϖτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (122) One can prove that under the same limit |MBL| ≪ 1, the backward-wave configuration gives the same two-photon field correlation [Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (118) and (121)] and temporal wavefunction [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (122)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Equation (122) suggests the biphoton temporal wavefunction has time reversal sym- metry when there is no linear gain and loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' CONCLUSION In summary, we provide a macroscopic phenomenolog- ical formula of quantum Langevin equations for two cou- pled phase-conjugated fields with linear loss (gain) and complex nonlinear coupling coefficient, in both forward- and backward-wave configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The macroscopic phenomenological formula, obtained from the coupling matrix and the requirement of preserving commutation relations of field operators during propagation, does not require knowing microscopic details of light-matter inter- action and internal atomic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' To validate this phenomenological formula, we take SFWM in a double- Λ four-level atomic system as an example to numeri- cally confirm that our macroscopic phenomenological re- sult is consistent with that obtained from microscopic Heisenberg-Langevin theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' As compared to the com- plicated microscopic theory which varies from system to system, the macroscopic coupled equations are much more friendly to experimentalists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We apply the quantum Langevin equations to study the effects of gain and/or loss as well as complex nonlinear coupling coefficient in biphoton generation, particularly to the temporal quan- tum correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We show that the computation com- plexity can be dramatically reduced by taking a proper order of field operators based on loss and gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Making a comparison between the quantum Langevin theory (in the Heisenberg picture) and the perturbation theory (in the interaction picture [10]), we extend the expression of complex phase mismatching to account for loss and gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' At last, we reveal Rabi oscillation in SFWM biphoton temporal correlation when the propagation effect is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Although in this article we focus on biphoton generation from the spontaneous parametric process, the quantum Langevin coupled equations can also be used to study two-mode squeezing, parametric oscillation, and other quantum light state generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' ACKNOWLEDGMENTS S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' acknowledges support from DOE (DE- SC0022069), AFOSR (FA9550-22-1-0043) and NSF (CNS-2114076, 2228725).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' 19 Appendix A: Noise Matrix in Backward-Wave Configuration In the macroscopic quantum Langevin equations, the requirement of preserving commutation relations allows multiple choices of the noise matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' For example, ˆf1 → − ˆf1 or/and ˆf2 → − ˆf2 do not affect any computation re- sults of physical observables involving pairs of Langevin noise operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' As an example, here we provide several equivalent noise matrices for backward-wave configura- tion: NB1 ≡ � 1 0 0 −1 � �� −MB11 −MB12 MB21 MB22 � + � −MB11 −MB12 MB21 MB22 �∗ = � 1 0 0 −1 � NF, NB2 ≡ NB1 � 1 0 0 −1 � = �� −MB11 MB12 −MB21 MB22 � + � −MB11 MB12 −MB21 MB22 �∗ , NB3 ≡ NB1 � −1 0 0 1 � , NB4 ≡ NB1 � −1 0 0 −1 � = −NB1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (A1) We take the first choice NB1 in the main text [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (31) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' II B] so that it is consistent with the microscopic treatment in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Appendix B: Heisenberg-Langevin Equations of SFWM The full Heisenberg equation of motion can be written as ˙ˆS = i( ˆO ˆS − ˆS ˆO) + ˆG + ˆF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (B1) where ˆS = � �� ˆσ11 ˆσ12 ˆσ13 ˆσ14 ˆσ21 ˆσ22 ˆσ23 ˆσ24 ˆσ31 ˆσ32 ˆσ33 ˆσ34 ˆσ41 ˆσ42 ˆσ43 ˆσ44 � �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (B2) ˆO = − � �� 0 0 g31ˆaas Ωp/2 0 ϖ Ωc/2 g42ˆas g13ˆa∗ as Ω∗ c/2 ϖ 0 Ω∗ p/2 g24ˆa∗ s 0 ∆p � �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (B3) ˆG = � �� Γ31ˆσ33 + Γ41ˆσ44 −γ12ˆσ12 −γ13ˆσ13 −γ14ˆσ14 −γ12ˆσ21 Γ32ˆσ33 + Γ42ˆσ44 −γ23ˆσ23 −γ24ˆσ24 −γ13ˆσ31 −γ23ˆσ32 −Γ3ˆσ33 −γ34ˆσ34 −γ14ˆσ41 −γ24ˆσ42 −γ34ˆσ43 −Γ4ˆσ44 � �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (B4) ˆF = � ���� ˆf (σ) 11 ˆf (σ) 12 ˆf (σ) 13 ˆf (σ) 14 ˆf (σ) 21 ˆf (σ) 22 ˆf (σ) 23 ˆf (σ) 24 ˆf (σ) 31 ˆf (σ) 32 ˆf (σ) 33 ˆf (σ) 34 ˆf (σ) 41 ˆf (σ) 42 ˆf (σ) 43 ˆf (σ) 44 � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (B5) Γm = Γm1 + Γm2 is the total spontaneous decay rate of excited state |m⟩, where m = 3, or 4, and Γmj is the decay rate from state |m⟩ to |j⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' For the two hyperfine ground states, there are Γ1 = Γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' For cold atoms with only spontaneous emisson decay, the dephasing rates γjk (j ̸= k) between states |k⟩ and |j⟩ are γ13 = γ23 = Γ3/2, γ14 = γ24 = Γ4/2, γ34 = (Γ3+Γ4)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' γ12 is the dephasing rate between two hyperfine ground states |1⟩ and |2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Appendix C: Microscopic SFWM Quantum Langevin Equations in Forward-Wave Configuration Although Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' III focuses on numerical confirmation of backward-wave SFWM, we remark that it may be helpful for general readers to write the SFWM quantum Langevin equations in the forward-wave configuration as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the forward-wave configuration with both Stokes and anti-Stokes fields propagating along +z direction, the coupled Langevin equations become ∂ ∂z �ˆaas ˆa† s � = MF �ˆaas ˆa† s � + � ˆFas ˆF † s � , (C1) where MF = � −αas + i ∆k 2 iκ −iκ −α∗ s − i ∆k 2 � , (C2) with ∆k = (ωas+ωs)/c−(⃗kc+⃗kp)·ˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The noise operators ˆFas and ˆF † s , defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (69), originate from micro- scopic atom-light interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' To compare Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (C1) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (11) from the phenomenological approach in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' II, we take mode 1 as anti-Stokes and mode 2 as Stokes in the forward-wave configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (11), we can also obtain ˆFas and ˆF † s from the noise matrix: ˆFas = NFR11 ˆf1 + NFI11 ˆf † 1 + NFI12 ˆf2 + NFR12 ˆf † 2, ˆF † s = NFR21 ˆf1 + NFI21 ˆf † 1 + NFI22 ˆf2 + NFR22 ˆf † 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (C3) Appendix D: Complex Phase Mismatching in Forward-Wave Configuration In the forward-wave configuration, similar to the backward-wave configuration in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' IV B, we assume anti-Stokes photons in mode 1 are lossless with EIT and there is gain (or loss) in Stokes mode 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' The small para- metric gain fulfills |κ| ≪ {α, g}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (6) and (17), 20 we obtain analytical expressions of A, B, C, and D as A = � q2 + 4κ2cosh � L 2 � q2 + 4κ2 � − qsinh � L 2 � q2 + 4κ2 � � q2 + 4κ2e(α1+α∗ 2)L/2 , B = 2iκsinh � L 2 � q2 + 4κ2 � � q2 + 4κ2e(α1+α∗ 2)L/2 , C = −2iκsinh � L 2 � q2 + 4κ2 � � q2 + 4κ2e(α1+α∗ 2)L/2 , D = � q2 + 4κ2cosh � L 2 � q2 + 4κ2 � + qsinh � L 2 � q2 + 4κ2 � � q2 + 4κ2e(α1+α∗ 2)L/2 , (D1) where q ≡ α1 − α∗ 2 − i∆k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' In the small parametric gain approximation, we have � q2 − 4κ2 ≈ q = α1 − α∗ 2 − i∆k = −i(∆k1 + ∆k∗ 2 + ∆k), (D2) and α1 + α∗ 2 = −i(∆k1 − ∆k∗ 2), (D3) where ∆km = ωm 2c χm is the wavenumber difference from that in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Hence, we simplify A, B, C, and D to A =exp [i∆k1L] exp �i∆kL 2 � , B =iκLsinc �(∆k1 + ∆k∗ 2 + ∆k)L 2 � × exp �i(∆k1 − ∆k∗ 2)L 2 � , C = − iκLsinc �(∆k1 + ∆k∗ 2 + ∆k)L 2 � × exp �i(∆k1 − ∆k∗ 2)L 2 � , D =exp [−i∆k∗ 2L] exp �−i∆kL 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (D4) We first look at the case with gain in the Stokes (mode 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' IV A, we take the order ⟨: ˆa2ˆa1 :⟩ ψ21(τ) = �� dϖdϖ′⟨ˆa2,out(ϖ′)ˆa1,out(ϖ)⟩e−iϖτ = � dϖBD∗e−iϖτ, (D5) where BD∗ = iκLsinc �(∆k1 + ∆k∗ 2 + ∆k)L 2 � × exp �i(∆k1 − ∆k∗ 2 + 2∆k2 + ∆k)L 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (D6) Comparing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (D5) and (D6) with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (94) and (95), particularly for the argument in the sinc function, we have ∆˜k = ∆k1 + ∆k∗ 2 + ∆k = k1 + k∗ 2 − kc − kp = kas + k∗ s − kc − kp which is consistent with the statement in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We now look at the case with loss in the Stokes (mode 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' We take the order ⟨: ˆa1ˆa2 :⟩ and have ψ12(τ) = �� dϖdϖ′⟨ˆa1,out(ϖ)ˆa2,out(ϖ′)⟩e−iϖτ = � dϖAC∗e−iϖτ, (D7) where AC∗ = iκ∗Lsinc �(∆k∗ 1 + ∆k2 + ∆k)L 2 � × exp �i(2∆k1 − ∆k∗ 1 + ∆k2 + ∆k)L 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (D8) Comparing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (D7) and (D8) with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' (94) and (95), we have ∆˜k = ∆k∗ 1 + ∆k2 + ∆k = k1 + k2 − kc + kp = kas + ks − kc − kp, which is different from the case with gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Here we have taken k1 ≃ k∗ 1 for loss- less mode 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Therefore, in the case with loss in the Stokes mode 2, the complex phase mismatching becomes ∆˜k = � ⃗kas + ⃗ks − ⃗kc − ⃗kp � ˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Gardiner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} +page_content=' Collett, Input and output in damped quantum systems: 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFLT4oBgHgl3EQfGi_B/content/2301.11993v1.pdf'} diff --git a/I9AyT4oBgHgl3EQfTfcL/content/tmp_files/2301.00104v1.pdf.txt b/I9AyT4oBgHgl3EQfTfcL/content/tmp_files/2301.00104v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7ff0901be1f19bd6daf70b9f14174e5b949fb9e4 --- /dev/null +++ b/I9AyT4oBgHgl3EQfTfcL/content/tmp_files/2301.00104v1.pdf.txt @@ -0,0 +1,1865 @@ +arXiv:2301.00104v1 [cs.CR] 31 Dec 2022 +Separating Computational and Statistical Differential Privacy +(Under Plausible Assumptions) +Badih Ghazi∗ +Rahul Ilango† +Pritish Kamath‡ +Ravi Kumar§ +Pasin Manurangsi¶ +Abstract +Computational differential privacy (CDP) is a natural relaxation of the standard notion of +(statistical) differential privacy (SDP) proposed by Beimel, Nissim, and Omri (CRYPTO 2008) +and Mironov, Pandey, Reingold, and Vadhan (CRYPTO 2009). In contrast to SDP, CDP only +requires privacy guarantees to hold against computationally-bounded adversaries rather than +computationally-unbounded statistical adversaries. Despite the question being raised explicitly +in several works (e.g., Bun, Chen, and Vadhan, TCC 2016), it has remained tantalizingly open +whether there is any task achievable with the CDP notion but not the SDP notion. Even a +candidate such task is unknown. Indeed, it is even unclear what the truth could be! +In this work, we give the first construction of a task achievable with the CDP notion but not +the SDP notion. More specifically, under strong but plausible cryptographic assumptions, we +construct a task for which there exists an ε-CDP mechanism with ε = O(1) achieving 1 − o(1) +utility, but any (ε, δ)-SDP mechanism, including computationally unbounded ones, that achieves +a constant utility must use either a super-constant ε or a non-negligible δ. To prove this, we +introduce a new approach for showing that a mechanism satisfies CDP: first we show that a +mechanism is “private” against a certain class of decision tree adversaries, and then we use +cryptographic constructions to “lift” this into privacy against computational adversaries. We +believe this approach could be useful to devise further tasks separating CDP from SDP. +∗Google Research, Mountain View. badihghazi@gmail.com. +†MIT. Part of this work was done during an internship at Google Research. rilango@mit.edu. +‡Google Research, Mountain View. pritish@alum.mit.edu. +§Google Research, Mountain View. ravi.k53@gmail.com. +¶Google Research, Thailand. pasin@google.com. + +Contents +1 +Introduction +1 +2 +Overview of the Results +3 +2.1 +The d-Distance Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +2.2 +SDP Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.3 +A CDP Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.4 +Final Steps +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.5 +On the Plausiblility of the Cryptographic Assumptions . . . . . . . . . . . . . . . . . +6 +3 +Preliminaries +7 +3.1 +Dataset and Adjacency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3.2 +Mechanism, Utility Function, and Usefulness +. . . . . . . . . . . . . . . . . . . . . . +8 +3.3 +Notions of Differential Privacy +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +4 +Low Diameter Set Problem and Nearby Point Problem +9 +4.1 +Simplification of Input Representation . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +4.2 +Nearby Point Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +4.3 +Verifiable Low Diameter Set Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +4.4 +From Low Diameter Set Problem to Nearby Point Problem +. . . . . . . . . . . . . . +11 +5 +CDP Mechanism for Verifiable Low Diameter Set Problem +11 +5.1 +CDP Mechanism without Verifiability +. . . . . . . . . . . . . . . . . . . . . . . . . . +12 +5.1.1 +Additional Preliminaries: Cryptographic Primitives . . . . . . . . . . . . . . . +12 +5.1.2 +Public-Coin Differing-Inputs Circuits from CRKHFs . . . . . . . . . . . . . . +13 +5.1.3 +From Differing-Inputs Circuits to CDP . . . . . . . . . . . . . . . . . . . . . . +15 +5.2 +CDP Mechanism for VLDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +5.2.1 +Witness-Indistinguishable Proofs . . . . . . . . . . . . . . . . . . . . . . . . . +17 +5.2.2 +Making Utility Function Efficient Using Witness-Indistinguishable Proofs +. . +17 +6 +SDP Lower Bounds for the Nearby Point Problem +19 +6.1 +Additional Preliminaries: Tools from Differential Privacy +. . . . . . . . . . . . . . . +20 +6.2 +Weak Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +6.3 +Boosting the Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +6.4 +Boosting the Failure Probability +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +6.5 +Putting Things Together: Proof of Theorem 30 . . . . . . . . . . . . . . . . . . . . . +24 +7 +Putting Things Together: Proof of Theorem 5 +24 +8 +Conclusion and Discussion +24 +A Comparison of various diO assumptions +29 + +1 +Introduction +The framework of differential privacy (DP) [DMNS06, DKM+06] gives formal privacy guarantees +on the outputs of randomized algorithms. +It has been the subject of a significant body of re- +search, leading to numerous practical deployments including the US census [Abo18], and industrial +applications [EPK14, Sha14, Gre16, App17, DKY17, KT18, RSP+21]. +The definition of DP requires privacy against computationally unbounded, i.e., statistical, adver- +saries. A natural modification is to instead only require privacy against computationally bounded +adversaries. In cryptography, considering computationally bounded adversaries instead of statisti- +cal ones enables a vast array of applications, like public-key cryptography. Could the same be true +for DP? A good survey of the area can be found in Vadhan’s monograph [Vad17, Section 10]. De- +spite Beimel, Nissim, and Omri [BNO08] defining computational differential privacy (CDP) in 2008 +(definitions that were further extended by Mironov, Pandey, Reingold, and Vadhan [MPRV09]), +the central question of separating it from statistical differential privacy (SDP)1, in the standard +client-server model, remains open: +Question 1. [Vad17, Open Problem 10.7] +Is there a computational task solvable by a single cura- +tor with computational differential privacy but is impossible to achieve with information-theoretic +differential privacy?2 +There have been several positive and negative results towards resolving this question. In the +positive direction, it is known that in the multi-party setting, CDP is stronger than SDP [MMP+10, +MPRV09]. Roughly speaking, this is because secure multi-party computation enables many data cu- +rators to simulate acting as a single central curator, without compromising privacy. Still, the multi- +party setting seems very different than the single-curator (aka central) setting. Indeed, [MMP+10] +remark3 that their “strong separation between (information-theoretic) differential privacy and com- +putational differential privacy ... stands in sharp contrast with the client-server setting where ... +there are not even candidates for a separation.” +In the central setting, Bun, Chen, and Vadhan [BCV16] show there is a task for which there +is a CDP mechanism, but any SDP mechanism for this task must be inefficient (modulo certain +cryptographic assumptions). We stress that the task they consider does have an inefficient SDP +mechanism (with parameters that match their CDP mechanism), so it does not resolve Question 1. +While this may seem like a minor technical point, we emphasize that it is of crucial importance. +Perhaps the main practical motivation behind studying CDP is the hope that there are CDP mech- +anisms for natural tasks with parameters that beat the lower bounds against SDP mechanisms. But +if, as in the case of the result in [BCV16], there exists (even an inefficient) SDP mechanism match- +ing the parameters of the CDP mechanism, then clearly there is no hope of the CDP mechanism’s +parameters beating SDP lower bounds. +In the negative direction, Mironov, Pandey, Reingold, and Vadhan [MPRV09] (building on +Green and Tao [GT08], Tao and Ziegler [TZ08], and Reingold, Trevisan, Tulsiani, and Vad- +han [RTTV08]) show a “dense model theorem” for pairs of random variables with “pseudodensity” +with each other. [MPRV09] note that (roughly speaking) extending this dense model theorem to +handle multiple pairs of random variables would prove that any CDP mechanism could be converted +into an SDP mechanism; such an extension is still open [Vad17, Open Problem 10.8]. +1For the formal definitions of CDP and SDP, we refer the reader to Section 3. +2We state this verbatim from [Vad17]. +3This remark is also quoted by Groce, Katz, and Yerukhimovich [GKY11]. +1 + +Groce, Katz, and Yerukhimovich [GKY11] show that CDP mechanisms for certain tasks where +the output is low-dimensional actually do imply SDP mechanisms. Many natural statistical tasks +fall into this category, and consequently, such tasks cannot separate CDP from SDP. This result +was further strengthened by [BCV16]. Furthermore, [GKY11] show that CDP mechanisms con- +structed in a black-box way from a variety of cryptographic objects, such as one-way functions, +random oracles, trapdoor permutations, and cryptographic hash functions, cannot separate CDP +from SDP. +In summary, there are at least two barriers to separate CDP from SDP: +1. High-dimensionality: One needs to consider (perhaps non-natural) tasks with high dimen- +sional outputs; +2. Exotic cryptography: One needs to use cryptography somewhat specially (perhaps either +an exotic primitive or in a non-black-box manner). +In light of these both positive and negative results as well as the lack of a candidate separation, it +was not even clear what the truth could be: is there any task for which there is a CDP mechanism +but not an SDP one? +Our Contributions. +We show, under plausible cryptographic hypotheses, that there are indeed +tasks for which there exist CDP mechanisms but no SDP mechanisms. This not only positively +answers Question 1 but also negatively answers the dense model extension question [Vad17, Open +Problem 10.8]. We state this result now informally and formalize it later in Section 2. We also delay +discussing our precise cryptographic assumptions to Section 2.5, where we discuss their plausibility +in detail. +Theorem 2. [Informal version of Theorem 5] Under cryptographic assumptions, there exists a task +for which there is a CDP mechanism but no SDP mechanism. +Let us take a step back to discuss the implications of Theorem 2. Although (as we will see in +a moment) our task is specifically constructed for the purpose of separating CDP and SDP, the +fact that we can separate them at all opens up a possibility that such a separation even holds for +some “natural” tasks. Indeed, some of the current lower bound techniques for SDP—such as the +ubiquitous “packing lower bounds”4 (see [HT10])—do not necessarily rule out CDP mechanisms. +It seems prudent to carefully reexamine the current lower bound techniques to see whether they +also apply to CDP. The ultimate hope for this program would be to employ CDP to overcome the +known SDP lower bounds for some more “natural” tasks. (Of course, such tasks would also give a +more “natural” separation of CDP and SDP.) +In fact, the technical approach we use in our construction already suggests a general approach +for constructing non-trivial CDP mechanisms that could apply to more tasks. We discuss this in +more detail in Section 2, but the idea is as follows. In order to show a task has a CDP mechanism, +first show there is a mechanism for that task that is “private” against a certain class of decision +tree adversaries. Then, second, use cryptographic assumptions to “lift” this into privacy against +computational adversaries. +4Specifically, when the packing lower bound requires the use of super-polynomially many datasets, the correspond- +ing adversary does not necessarily run in polynomial time. +2 + +Organization. +The rest of the paper is organized as follows. +Section 2 provides a high-level +overview of our techniques as well as a discussion of our cryptographic assumptions and their plau- +sibility. Section 3 contains the background material and Section 4 formally defines the problems. +We provide our CDP mechanism in Section 5, and prove lower bounds against SDP mechanisms in +Section 6. These two components are put together to prove the main result in Section 7. Finally, +we discuss the open problems and future directions in Section 8. +2 +Overview of the Results +We will next discuss the high-level overview of our techniques. +We will sometimes have to be +informal here, but all details are formalized later in the paper. +Let us quickly recall how the +“task” is defined5. +Following [GKY11, BCV16], a task is defined by an efficiently computable +utility function u that takes in an input dataset D and a response y such that u(D, y) = 1 if y is +considered “useful” for D and u(D, y) = 0 otherwise. A mechanism M is said to be α-useful for u +iff E[u(D, M(D))] ≥ α for all input datasets D. We remark that many well-studied problems—such +as linear queries with various error metrics—can be written in this form. We will refer to α as the +usefulness of the mechanism. +One of our main conceptual contributions is to define a class of tasks that seems to naturally +circumvent the two earlier-mentioned barriers—tasks where one needs to output a circuit. +2.1 +The d-Distance Problem +Before we detail why tasks that output a circuit might evade the two barriers, let us describe a +concrete example. We call the following the d-distance problem (where d ∈ N is a parameter): +◮ Given: dataset D that consists of n bits +◮ Output: circuit C mapping n bits to 1 bit +◮ Utility: C is considered useful6 if it outputs +⊲ 1 on D, and +⊲ 0 on all points at distance greater than d from D. +Informally, this problem asks to output a circuit that checks if its input is “close” to D. Looking +ahead, we will ultimately separate CDP from SDP under cryptographic assumptions by considering +a version of this problem where we only care about datasets in a cryptographically special set. +We now revisit the two barriers and discuss how the distance problem might circumvent them. +1. High-dimensionality: The output of this task is a circuit, which is high-dimensional. +2. Exotic cryptography: Because the output of the task is a circuit, it lends itself to a powerful +class of cryptographic objects: circuit obfuscators [BGI+12]. Roughly speaking, circuit obfusca- +tors take as input a circuit C and output a scrambled, obfuscated circuit C′ that computes the +same function as C but which, ideally, has the property that “anything you could do with access +to the circuit C′, you could do with only black-box access to the function the circuit computes.” +Importantly, obfuscation is not in the list of primitives ruled out by the barrier in [GKY11]. +5Please refer to Section 3 for a more formal definition. +6One might be concerned about whether this utility function is actually efficiently computable. We will address +this in Section 2.3 after we describe our final construction. +3 + +2.2 +SDP Lower Bound +Our starting point for separating CDP from SDP is the d-distance problem described above. Indeed, +we show that there is no SDP mechanism for this problem for most settings of d. +Lemma 3. If 0 < d ≤ n.99, then there is no (ε, δ)-SDP mechanism for d-distance that is 0.01-useful +for ε = O(1) and δ negligible in n. +In fact, this lower bound is straightforward (Lemma 15) from the well-known blatant non-privacy +notion (see, e.g., [De12]): no DP algorithm can output a dataset that is (with large probability) +close to the input dataset. Crucially, our lower bounds are non-constructive, and do not yield an +efficient adversary (which would imply a similar lower bound against CDP mechanisms). Thus, to +separate CDP from SDP it suffices to come up with a CDP mechanism M for, say, n.99-distance. +2.3 +A CDP Mechanism +One of our main ideas to help construct a CDP mechanism M is to use obfuscation. In particular, +we will consider mechanisms where the returned circuit is obfuscated. Recall that in order to prove +a mechanism M that outputs a circuit C is CDP, one needs to argue that no efficient adversary +that gets C as input can break the privacy guarantee. By considering mechanisms that return +obfuscated circuits, we can drastically simplify the type of adversaries we need to prove privacy +against. +Instead of proving privacy against adversaries that see the circuit C (i.e., white-box +setting), sufficiently strong obfuscation means we only need to prove privacy against decision tree +adversaries that can query the function computed by the circuit (i.e., black-box setting). In other +words, if we have a mechanism that satisfies DP against black-box adversaries (decision trees) with +a polynomial number of queries, we can then hope to use sufficiently strong obfuscation to “lift” +this into a mechanism that is secure against (white-box) computational adversaries with polynomial +running time. +Of course, one needs to be careful about whether such “sufficiently strong obfuscation” is even +possible, but, putting that aside for the moment, the question of whether there is a CDP mechanism +for n.99-distance (Question 4 below) appears to reduce to whether there is a mechanism for n.99- +distance that is DP against query (a.k.a. decision-tree) adversaries. +Question 4. Let ε = O(1) and 0 ≤ d ≤ n.99. Does there exist an ε-CDP mechanism for d-distance +with constant usefulness? +While we do not resolve Question 4, we (roughly speaking) show that there is a mechanism that +is DP against non-adaptive decision tree adversaries, whose queries are fixed a priori. It turns out +a relatively simple mechanism based on randomized response [War65] works for these less powerful +adversaries. +From Non-Adaptive Lower Bound to Computational Lower Bound. +This switch from +the usual adaptive query adversaries to non-adaptive query adversaries comes at a price however. It +is not clear how to use obfuscation to lift a mechanism that is private against non-adaptive queries +into one that is private against computational adversaries. Indeed, a polynomial-time algorithm +with even black-box access to a function seems to be an inherently adaptive adversary! +Surprisingly, we manage to get around this by using another cryptographic object introduced +by Bitansky, Kalai, and Paneth [BKP18]: collision-resistant keyless hash functions. Informally +4 + +speaking, a hash function being collision-resistant and keyless means that “any efficient adversary +can only generate a number of hash collisions that is at most polynomially larger than the advice +the adversary gets.” +We then modify the d-distance problem to only consider datasets that hash to, say, the all +zeroes string. Formally, zero hash d-distance is the following problem. Let R ⊆ {0, 1}n be the set +of strings that hash to the all zeroes string. +◮ Given: dataset D that consists of n bits +◮ Output: circuit C mapping n bits to 1 bit +◮ Utility: C is considered useful if D /∈ R or both of the following hold: +⊲ it outputs 1 on D +⊲ it outputs 0 on all points in R at distance greater than d from D +In other words, the utility function now completely ignores all points outside of R. +The high-level intuition behind this change is the following: +1. Our CDP mechanism can output a circuit C such that the only inputs where C(x) reveals +information are those x in the set R (i.e., that hash to zero). +2. Any polynomial-time adversary A can only generate fixed polynomial number of elements of +R by the collision-resistance property of the hash function. +3. Combining the above effectively makes the inputs A can query C on “non-adaptive”. +Finally, in order to “lift” the query separation into the computational realm we use another cryp- +tographic tool: differing-inputs obfuscation (diO) [BGI+01, BGI+12, ABG+13]. Roughly speaking, +diO is an obfuscator with the following guarantee: if any efficient adversary can distinguish the +obfuscation of two circuits C1 and C2, then an efficient adversary can find an input x on which +C1(x) ̸= C2(x). In particular, the assumption we use is even weaker than public-coin diO [IPS15], +which is already considered to more plausible than general diO.7 +In summary, diO allows us to reduce computational adversaries to adaptive query adversaries +and collision-resistant keyless hash functions allows us to reduce adaptive query adversaries to +non-adaptive query adversaries. Interestingly, to the best of our knowledge, this is the first time +collision-resistant keyless hash functions are being used together with any obfuscation assumption. +Making the Utility Function Efficiently Computable. +Observant readers may have already +noticed an issue: utility functions that we have considered so far are not necessarily efficiently +computable. Specifically, a trivial way to implement the utility function would be to enumerate all +points at distance at least d, feed it into the circuit, and check that the output is as expected; this +would take 2nΩ(1) time. +To overcome the above problem, we restrict circuits to only those that are relatively simple, so +that there is a small “witness” w that certifies that the circuit outputs zero at all points that are +d-far from D. A naive idea is then to let the CDP mechanism output the circuit C together with +such a witness w. The utility function can then just efficiently check that w is a valid witness (and +that C(D) = 0 or x ∈ R). This makes the utility function efficient but unfortunately compromises +privacy because the witness w itself can leak additional information. To avoid this, we instead use +non-interactive witness indistinguishable (NIWI) proofs (e.g., [BOV07]). Roughly speaking, this +allows us to produce a proof π from w (and C and diO), which does not leak any information about +w (against computationally bounded adversaries), but at the same time still allows us to verify +7See Assumption 22 for formal statement of the assumption and Appendix A for comparison with other diO +assumptions in literature. +5 + +that the underlying witness w is valid. The former is sufficient for CDP, while the latter ensures +that the utility function can be computed efficiently. +This completes the high-level overview of the constructed task and our CDP mechanism. The +cryptographic primitives needed for our mechanism are formalized in Assumptions 18, 22 and 26. +2.4 +Final Steps +Finally, we remark that since our problem is now not exactly the original d-distance problem +anymore, as the utility guarantees are only now meaningful for datasets in R. This means that +we cannot use the lower bound in Lemma 3 for the d-distance problem directly. Fortunately, we +can still adapt its proof—a “packing-style” lower bound on each coordinate—to one which applies +a packing-style argument on each block of coordinates instead. With this, we can prove the lower +bound for zero hash d-distance as long as the set R has sufficiently large density (≈ 1/n−o(log n)). +Putting all the ingredients together, we arrive at the following8: +Theorem 5 (Main Result). Under Assumptions 18, 22 and 26, for any constant εCDP > 0, there +exists an ensemble u = {un}n∈N of polynomial time computable utility functions such that +◮ There is an εCDP-CDP mechanism that is (1 − on(1))-useful for u. +◮ For any constant εSDP > 0, there is no εSDP-SDP mechanism that is 0.01-useful for u. +2.5 +On the Plausiblility of the Cryptographic Assumptions +We now discuss the plausiblility of the three cryptographic assumptions we use for our result: +(i) NIWI: Non-interactive Witness Indistinguishable Proofs (formally, Assumption 26) +(ii) CRKHF: Collision-Resistant Keyless Hash Functions (formally, Assumption 18) +(iii) diO-for-pcS: Differing-Inputs Obfuscation for Public-coin Samplers (formally, Assumption 22) +Regarding (i), NIWI. +Bitansky and Paneth [BP15a] show that NIWIs exist assuming one- +way permutations exist and indistinguishability obfuscation (iO) exists. Recently, Jain, Lin, and +Sahai [JLS21] show that the existence of iO follows from well-founded assumptions; consequently, +NIWIs exist based on widely-believed assumptions. (We note that other previous works have also +constructed NIWIs based on other more specific assumptions [BOV07, GOS12].) +Regarding (ii), CRKHF. +Bitansky, Kalai, and Paneth [BKP18] defined CRKHFs to model the +properties of existing hash functions like SHA-2 used in practice. They suggest several candidates +for CRKHFs, such as hash functions based on AES and Goldreich’s one-way functions. They also +note that CRKHFs exist in the Random Oracle model, as a random function is a CRKHF. Still, +it is an open question to base the security of a CRKHF on a standard cryptographic assumption. +Part of the difficulty of doing this, as [BKP18] describe, is that most cryptographic assumptions +involve some sort of structure that is useful for constructing cryptographic objects. In contrast, +the goal of a CRKHF is to have no structure at all. In summary, given the various CRKHF candi- +dates, the existence in the Random Oracle model, and the fact that CRKHFs exist “in practice,” +this assumption is quite plausible. For our specific construction, we need a different hash length +8We remark that εSDP-SDP mechanism here refers to an ensemble of mechanisms {Mn} which are (εSDP, negl)-SDP. +(See Definition 7.) +6 + +(equivalently, different compression rate) than that used in [BKP18]; please refer to the discussion +preceding Assumption 18 for the parameters and justification. +Finally, we remark that, even though the existence of CRKHFs is not known to reduce to any +“well-founded” assumption, even refuting their existence would answer a longstanding question in +cryptography: giving non-contrived separations between the Random Oracle model [BR93] and the +standard model. In the words of Bitansky, Kalai, and Paneth [BKP18] +“Any attack on the multi-collision resistance of a [keyless] cryptographic hash function +would constitute a strong and natural separation between the hash and random oracles. +For several cryptographic hash functions used in practice, the only known separations +from random oracles are highly contrived [CGH04].” +Regarding (iii), diO-for-pcS. +One can think of diO [BGI+01, BGI+12] as an “extractable” +strengthening of iO. While iO has now become a widely-believed assumption [JLS21], the exis- +tence of diO is controversial. Several papers (e.g., [BP15b, GGHW17, BSW16]) cast doubt on the +existence of diO, especially in the case where an arbitrary auxillary input is allowed; we stress that +all the negative results for diO hold for contrived auxillary inputs and/or distributions. On the +positive side, [BCP14] show that diO reduces to iO in special cases, such as when the number of +differing-inputs is bounded by a polynomial. More related to our result, [IPS15] gives a definition +of public-coin diO that avoids the difficulties presented by earlier negative results regarding auxil- +iary inputs, although [BP15b] presented some evidence against this definition in special cases. Our +specific assumption of diO-for-pcS is in fact weaker than the assumption of public-coin diO. In the +definition of public-coin diO, as in [IPS15], we start with any public-coin sampler (pcS), for which it +is hard to find an input on which two circuits differ, even given the knowledge of all the randomness +that underlies the circuits. The security of the obfuscation is required to hold even against adver- +saries that know all the randomness that underlies the generation of the two circuits. However, +in our definition, the security of the obfuscation is required to hold only against adversaries that +observes a single obfuscated circuit, which makes the assumption weaker. See Appendix A for a +more detailed discussion on comparison of this assumption with other diO assumptions in literature. +Finally, we only use the existence of diO-for-pcS for a simple circuit family for our result, so even if +general purpose diO-for-pcS does not exist, we think it is plausible that diO-for-pcS exists for the +specific family of circuits we need for our result. (See Assumption 22 for the exact pcS family for +which we require a diO.) +Final thoughts on our assumptions. +In conclusion, we view each of our three assumptions +as plausible. Moreover, each of assumptions has at least some evidence that is hard to refute: +NIWIs exist based on a widely-believed assumption, refuting CRKHFs would require giving the +first non-contrived separation between the standard and the Random Oracle model, and despite +many attempts (e.g., [BP15b, GGHW17, BSW16]) to refute diO, the question is still open, especially +for the particular diO-for-pcS version. +3 +Preliminaries +A function g : N → R≥0 is said to be negligible if g(n) = n−ω(1). Let PPT be an abbreviation for +probabilistic polynomial-time Turing machine. +7 + +For x ∈ {0, 1}n and r ∈ N, we use Br(x) to denote the (Hamming) ball of radius r around x, +i.e., {z ∈ {0, 1}n | ∥x − z∥1 ≤ r}. Furthermore, we use diam(S) for a set S ⊆ {0, 1}n to denote the +(Hamming) diameter of S, i.e., maxx,x′ ∈S ∥x − x′∥1. +3.1 +Dataset and Adjacency +For a domain X, we view a dataset D as a histogram over the domain X, i.e., D ∈ ZX +≥0 where Dx +denotes the number of times x ∈ X appears in the dataset. The size of the dataset is defined as +∥D∥1 := � +x∈X Dx. We write X m as a shorthand for the set of all datasets of size m, and X ∗ for +the set of all datasets over domain X. Two datasets are adjacent iff ∥D − D′∥1 = 1, i.e., one of the +datasets is a result of adding or removing a single row from the other dataset. +3.2 +Mechanism, Utility Function, and Usefulness +A mechanism M is a randomized algorithm that takes in a dataset D ∈ X ∗ and outputs an element +from a set Y. The utility of a mechanism is measured by a utility function u, which is a polynomial- +time deterministic algorithm that takes in a dataset D ∈ X ∗ together with a response y ∈ Y and +outputs 0 or 1 (whether the response is good for the dataset). We say that the mechanism M is +α-useful for utility u iff Pr[u(D, M(D)) = 1] ≥ α. +Below, we will often discuss an ensemble M = {Mn}n∈N of mechanisms where9 Mn : X ∗ +n → Yn. +We say that an ensemble of mechanisms is efficient if Mn on input D ∈ X m +n runs in time poly(n, m). +For an ensemble u = {un}n∈N of utility functions and α = {αn ∈ [0, 1]}n∈N, we say that M is α- +useful with respect to u iff Mn is αn-useful with respect to un for all n ∈ N. +For brevity, we will sometimes refer to “ensemble of mechanisms” simply as “mechanism” and +“ensemble of utility functions” simply as “utility function” when there is no ambiguity. +3.3 +Notions of Differential Privacy +We now define the notions of DP that will be used throughout the paper. +(Statistical) Differential Privacy. +The standard (statistical) notion of DP can be defined in +terms of the following notion of indistinguishability. +Definition 6 (Statistical Indistinguishability). Distributions P, Q are said to be (ε, δ)-indistinguishable, +denoted P ≈ε,δ Q, if for all events (measurable sets) E, it holds that +Pr +X∼P[X ∈ E] ≤ eε · Pr +X∼Q[X ∈ E] + δ, +and +Pr +X∼Q[X ∈ E] ≤ eε · Pr +X∼P[X ∈ E] + δ. +For simplicity, we use ≈ε to denote ≈ε,0. +Definition 7 (Statistical Differential Privacy (SDP) [DMNS06, DKM+06]). For ε, δ > 0, a mecha- +nism M is said to be (ε, δ)-SDP if and only if for every pair D, D′ of adjacent datasets, we have that +M(D) ≈ε,δ M(D′). We say that an ensemble M = {Mn}n∈N is ε-SDP for a sequence ε = {εn}n∈N +if there exists a negligible sequence {δn}n∈N such that Mn is (εn, δn)-SDP for all n ∈ N. +We note that the above notation, which omits explicit δ for an ensemble of mechanisms, was also +used by [BCV16]. +9It is always implicitly assumed that Xn, Yn are of size poly(n). +8 + +Computational Differential Privacy. +The notion of computational DP relaxes the notion +of indistinguishability to a computational version, where the privacy holds only with respect to +computationally bounded adversaries. +Definition 8 (Computational Indistinguishability). Two ensembles of distributions P = {Pn}n∈N +and Q = {Qn}n∈N, where Pn and Qn are supported over {0, 1}p(n) for some polynomial p(·), are +said to be ε-computationally-indistinguishable for a sequence ε = {εn}n∈N, denoted P ≈c +ε Q, if there +exists a negligible function negl(·) such that for any PPT adversary A, it holds that +Pr +X∼Pn[A(X) = 1] ≤ eεn · +Pr +X∼Qn[A(X) = 1] + negl(n), and +Pr +X∼Qn[A(X) = 1] ≤ eεn · +Pr +X∼Pn[A(X) = 1] + negl(n). +In the special case of ε = 0, we suppress the subscript and simply write P ≈c Q. +Throughout, when we refer to a sequence {(Dn, D′ +n)}n∈N of adjacent datasets, it is always assumed +that Dn ∈ X mn +n +, D′ +n ∈ X m′ +n +n +are of sizes mn, m′ +n = poly(n). +Definition 9 (Computational Differential Privacy (CDP) [MPRV09]). An ensemble M = {Mn}n∈N +of mechanisms is said to be ε-CDP for a sequence ε = {εn}n∈N, if for any sequence {(Dn, D′ +n)}n∈N +of adjacent datasets, it holds that {Mn(Dn)}n∈N ≈c +εn {Mn(D′ +n)}n∈N. +This definition is often referred to as indistinguishability-based CDP (IND-CDP) in previous +works [MPRV09, GKY11, BCV16]. Since we only use this notion for our main result, we refer to +it simply as CDP. The other definition of CDP used in previous works is simulation-based: +Definition 10 (SIM-CDP [MPRV09]). An ensemble M = (Mn)n∈N of mechanisms is said to be ε- +SIM-CDP if there exists an (εn, 0)-SDP ensemble {M′ +n}n∈N of mechanisms such that for any sequence +{Dn ∈ X ∗ +n}n∈N of datasets, with size of Dn being at most poly(n), it holds that Mn(Dn) ≈c M′ +n(Dn). +It should be noted that SIM-CDP cannot be used for the separation we are looking for. Specif- +ically, if {Mn}n∈N is ε-SIM-CDP, we may use {M′ +n}n∈N as our ε-SDP mechanism. Since the utility +function runs in polynomial time, it follows immediately that, if {Mn}n∈N is α-useful, then {M′ +n}n∈N +is also (α − o(1))-useful. Due to this, we will not consider SIM-CDP again in this paper. +Calculus of ≈ and ≈c. +The following properties are well-known. +Fact 11. The notions of (ε, δ)-indistinguishability and ε-computational-indistinguishability satisfy: +◮ Basic Composition: If P0 ≈ε,δ P1 and P1 ≈ε′,δ′ P2, then P0 ≈ε+ε′,δ+δ′ P2. +Similarly, if +P0 ≈c +ε P1 and P1 ≈c +ε′ P2, then P0 ≈c +ε+ε′ P2. +◮ Post-processing: If P ≈ε,δ Q, then for all (randomized) functions f, it holds that f(P) ≈ε,δ +f(Q). Similarly, if P ≈c +ε Q, then for all PPT algorithms A, it holds that A(P) ≈c +ε A(Q). +4 +Low Diameter Set Problem and Nearby Point Problem +In this section, we introduce the problems that we will use in our separation. Before that, we will +describe a simplifying assumption that we can make about the inputs. +9 + +4.1 +Simplification of Input Representation +Recall that so far a dataset may contain multiple copies of an element. Below, however, it will be +more convenient to only discuss the case where each element appears only once, i.e., D ∈ {0, 1}X . +This is sufficient since if we have a utility function u : {0, 1}X × Y → {0, 1} defined only on +D ∈ {0, 1}X , we can easily define the utility function u : NX × Y → {0, 1} by +u(D, r) = +� +u(D, r) +if D ∈ {0, 1}X , +1 +otherwise. +In other words, the utility function considers any response good for datasets with repetition. Clearly, +if u is efficiently computable, then so is u. Furthermore, suppose that we have an ε-CDP mechanism +M = {Mn}n∈N for u = {un}n∈N. For every dataset D, let D be defined by Di = min +� +Di, 1 +� +. +Then, we may define M = +� +Mn +� +n∈N by M(D) = M(D). It is simple to check that M remains +ε-CDP. Furthermore, if M is α-useful for u, then M remains α-useful for u. +Finally, note that a lower bound for DP algorithms restricted to non-repeated datasets trivially +implies a lower bound against all datasets. +Due to this, we will henceforth focus our attention only on the datasets D ∈ {0, 1}X . Further- +more, throughout the remainder of this paper, we will always pick Xn = [n]. This further simplifies +the input representation to be just a bit vector x ∈ {0, 1}n. We will define an input of our problem +in this way. Furthermore, we will henceforth use x instead of D to denote the input dataset. +4.2 +Nearby Point Problem +We will start by defining our first problem, which asks to output a point that is close to the input +point if the latter belongs to some set R. As we noted in the introduction, when R is the set +of all points (i.e., Rn = {0, 1}n), this is exactly the same as the problem considered in blatant +non-privacy [DN03, DMT07]. As we will see later, the presence of the set R is due to our use of +hashing, which is required in our proof for the CDP mechanism. +Definition 12 (τ-Nearby R-Point Problem). The nearby point problem parameterized by sequences +{τn ∈ N}n∈N and {Rn ⊆ {0, 1}n}n∈N is denoted by NBPτ,R. +For input x ∈ {0, 1}n and output +y ∈ Yn = {0, 1}n, the utility is defined as: +uNBP +τn,Rn(x, y) := 1 {∥x − y∥1 ≤ τn or x /∈ Rn} +For brevity, we will assume throughout that Rn is efficiently recognizable and henceforth we do +not state this explicitly. Note that this assumption implies that the utility function defined above +is efficiently computable. The nearby point problem will be primarily used for proving the lower +bounds against SDP. +4.3 +Verifiable Low Diameter Set Problem +Next, we define circuit-based tasks for which we will give CDP mechanisms. To do so, we need to +first define a “τ-diameter verifier”. +Definition 13 (τ-Diameter Verifier). For a sequence τ = {τn}n∈N of integers, we say that an +efficiently computable (deterministic) verifier V = {Vn}n∈N is a τ-diameter verifier for circuits of +size s(n) if it takes as input a circuit C : {0, 1}n → {0, 1} of (polynomial) size s(n) and a proof π +of size poly(n), and outputs Vn(C, π) = 1 only if diam(C−1(1)) ≤ τn. +10 + +We can now define the (verifiable) low diameter set problem as follows: +Definition 14 (Verifiable τ-Diameter R-Set Problem). The verifiable low diameter set problem +parameterized by sequences τ = {τn}n∈N, R = {Rn ⊆ {0, 1}n}n∈N, and τ-diameter verifier V = +{Vn}n∈N is denoted by VLDSτ,R,V . The input, output, and utility are defined as follows: +◮ Input: x ∈ {0, 1}n. +◮ Output: circuit C and a proof π, both of size poly(n). +◮ Utility: uVLDS +τn,Rn,Vn(x, (C, π)) := 1 {C(x) = 1 or x /∈ Rn} and 1 {Vn(C, π) = 1}. +For convenience, we also define the following utility function +ueval +R (x, C) := 1 {C(x) = 1 or x /∈ R} . +Note that this does not correspond to a hard task, because a circuit that always outputs one is +1-useful. Nonetheless, it will be convenient to state usefulness of some intermediate algorithms via +this utility function. +4.4 +From Low Diameter Set Problem to Nearby Point Problem +Below we provide a simple observation that reduces the task of proving an SDP lower bound for +the verifiable low diameter set problem to that of the nearby point problem. (Note here that the +SDP mechanisms considered below can be computationally inefficient.) +Lemma 15. If there is an (ε, δ)-SDP α-useful mechanism for the VLDSτ,R,V problem, then there +is an (ε, δ)-SDP α-useful mechanism for the NBPτ,R problem. +Proof. Let M be an (ε, δ)-SDP α-useful mechansim for the VLDSτ,R,V problem. We will construct +an (ε, δ)-SDP α-useful mechanism M′ for the NBPτ,R problem. +The mechanism M′ +n on input dataset x ∈ {0, 1}n works as follows. First, let (C, π) ← Mn(x). +If Vn(C, π) = 1, then output the lexicographically first element of C−1(1) (else, output 0n). This +completes our description of M′. +Since M is (ε, δ)-SDP, we have that M′ is also (ε, δ)-SDP by post-processing. It remains to +show that M′ is α-useful. Fix some input x ∈ {0, 1}n. If x /∈ Rn, then any output satisfies utility. +Thus, it suffices to consider the case where x ∈ Rn. With probability α, we have that Vn(C, π) = 1 +(which implies that C−1(1) has diameter at most τn), and x ∈ C−1(1). Consequently, the distance +between x and the lexicographically first element of C−1(1) is at most τn. So with probability at +least α, the output of M′ is useful for x, as desired. +5 +CDP Mechanism for Verifiable Low Diameter Set Problem +In this section we build a CDP mechanism for the verifiable low diameter set problem. We establish +the following result: +Theorem 16. Suppose that Assumptions 18, 22 and 26 hold. Then, for all constant εCDP > 0 and +τ = +� +τn = n0.9� +n∈N, there exists a τ-diameter verifier V and a sequence R = {Rn}n∈N of sets of +sizes |Rn| ≥ 2n/no(log n), such that there exists an εCDP-CDP mechanism that is (1 − on(1))-useful +for uVLDS +τ,R,V . +As discussed in the overview, we first build a mechanism that is CDP but without verifiability +using collision-resistant keyless hash functions and differing-inputs obfuscators (Section 5.1). We +then turn it into a verifiable one using non-interactive witness indistinguishable proofs (Section 5.2). +11 + +5.1 +CDP Mechanism without Verifiability +In this section, we construct our first CDP mechanism (Algorithm 3). We depart from the overview +in Section 2 slightly and do not prove a non-adaptive query lower bound explicitly. Instead, we +directly show in Section 5.1.2 how to sample the appropriate differing-inputs circuit family. This +can be then easily turned into our CDP mechanism via diO in Section 5.1.3. +5.1.1 +Additional Preliminaries: Cryptographic Primitives +Throughout this section, we will repeatedly use the so-called randomized response (RR) mecha- +nism [War65]. Specifically, RRε is an algorithm that takes in x ∈ {0, 1}n and outputs ˜x ∈ {0, 1}n, +where ˜xi = xi with probability +eε +1+eε independently for each i ∈ [n]. It is well-known (and very +simple to verify) that RRε is ε-SDP. +Collision-Resistant Keyless Hash Functions. +In our construction, we will use the Collision- +Resistant Keyless Hash Functions (CRKHFs) [BKP18]. The formal definition is as given below. +Definition 17 (Collision-Resistant Keyless Hash Functions [BKP18]). A sequence of hash func- +tions +� +Hn : {0, 1}n → {0, 1}γ(n)� +n∈N is K-collision resistant for advice length ζ for sequences K = +{Kn}n∈N, ζ = {ζn}n∈N if, for any PPT A and a sequence {zn}n∈N of advices where |zn| = ζn, we +must have +Pr +(Y1,...,YKn)←A(1n;zn) [Y1, . . . , YKn are distinct and Hn(Y1) = · · · = Hn(YKn)] ≤ negl(n). +We will skip the subscript n whenever it is clear from context. +In [BKP18], the hash value length γ(n) is assumed to be either linear, i.e., γ(n) = Ω(n), or +polynomial, i.e., γ(n) = nΘ(1). However, we need a collision-resistant hash function with a much +smaller γ(n), namely O(log2 n). +We remark that this is still very much plausible: as long as +γ(n) is ω(log n), the “guess-and-check” algorithm will only produce a collision with only negligible +probability. A more precise statement of our assumption is stated below. +Assumption 18. There is an efficiently computable sequence H = {Hn}n∈N of hash functions with +hash value length γ(n) = o(log2 n) such that, for any constant c1 > 0, there exists a constant c2 > 0 +such that the hash function sequence is K-collision resistant for advice length ζ where K(n) = nc2 +and ζ(n) = nc1. +We remark that, for the existence of CDP mechanism (shown in this section), we will only use +the multi-collision-resistance without relying on the assumption on the value of γ. The latter is +only used to show that no SDP mechanism exists for the problem (Section 7). +Differing-Inputs Obfuscators for Public-Coin Samplers. +For any two circuits C0 and C1, +a differing-inputs obfuscator diO [BGI+12] guarantees that the non-existence of an efficient ad- +versary that can find an input on which C0 and C1 differ implies that diO(C0) and diO(C1) are +computationally indistinguishable. For our application, it even suffices to assume a weaker notion, +namely that of differing-inputs obfuscator for public-coin samplers, as defined below. +12 + +Definition 19 (Public-Coin Differing-Inputs Circuit Sampler). An efficient non-uniform sampling +algorithm Sampler = {Samplern} is a public-coin differing-inputs sampler for the parameterized +collection C = {Cn} of circuits if the output of Samplern is distributed over Cn × Cn and for every +efficient non-uniform algorithm A = {An}, there exists a negligible function negl(·) such that for +all n ∈ N: +Pr +θ [C0(y) ̸= C1(y) : (C0, C1) ← Samplern(θ), y ← An(θ)] ≤ negl(n). +Here, Samplern is a deterministic algorithm and the only source of randomness is the seed θ. +Definition 20 (Differing-Inputs Obfuscator for Public-Coin Samplers (cf. [IPS15])). A uniform +PPT diO is a differing-inputs obfuscator for public-coin samplers for the parameterized circuit +family C = {Cn} if the following conditions are satisfied: +◮ Correctness: For all n ∈ N, for all C ∈ Cn, for all inputs y, we have that +Pr[C′(y) = C(y) : C′ ← diO(1n, C)] = 1. +◮ Polynomial slowdown: There exists a universal polynomial p(·) such that for all C ∈ Cn, it +holds that +Pr[|C′| ≤ p(|C|) : C′ ← diO(1n, C)] = 1. +◮ Differing-inputs: For every public-coin differing inputs sampler Sampler = {Samplern} for +C, and every (not necessarily uniform) PPT distinguisher D = {Dn}, there exists a negligible +function negl such that the following holds for all n ∈ N: For (C0, C1) ← Samplern(θ) +| Pr +θ [Dn(diO(1n, C0)) = 1] − Pr +θ [Dn(diO(1n, C1)) = 1]| ≤ negl(n). +We note that the notion of diO-for-pcS is in fact weaker than the notion of general public-coin diO +as given by [IPS15]. We elaborate on this comparison in Appendix A. Whenever n is clear from +context, we use diO(C) to denote diO(1n, C) for simplicity. When we want to be explicit about the +randomness ρ (of poly(n) bit length) used by diO we will denote it as diOρ(C). +We only need the existence of a differing-inputs obfuscator for a specific family of circuits. This cir- +cuit family will be defined later and therefore we defer formalizing our assumption to Section 5.1.3. +5.1.2 +Public-Coin Differing-Inputs Circuits from CRKHFs +The first step of our proof is to construct a differing-inputs circuit family based on CRKHFs. Our +sampler is described in Algorithm 1. +We next prove that the above sampler is a public-coin differing-inputs sampler, which means +that any efficient adversary, even with the knowledge of ˜x (which is the only source of randomness), +cannot find an input on which C0 and C1 differ. The proof starts by noticing that any input that +differentiates C0, C1 must, by definition of the circuits, have hash value υn. Therefore, if there were +an adversary that can find a differing input, then we could run it multiple times to get Y1, . . . , YK +that have the same hash value. (See Algorithm 2 below.) However, our proof is not finished yet, +since it is possible that Y1, . . . , YK are not distinct. Indeed, the crux of the construction is that, +due to how we select ˜x and define the circuits, a fixed Y will be a differing input with negligible +probability10. It follows that Y1, . . . , YK must be distinct w.h.p. This is formalized below. +10It is also simple to see that this property suffices to prove a non-adaptive query lower bound as discussed in +Section 2. +13 + +Algorithm 1 Differing-Inputs Circuit Family Sampler LDS-Samplern. +Parameters: +Adjacent datasets x, x′ ∈ {0, 1}n, hash value υn ∈ {0, 1}γ(n), privacy parameter +ε > 0, radius r, ˜r > 0. +Randomness: θ ∼ RRε(0n). +Output: Circuits C0, C1. +˜x ← x ⊕ θ (bit-wise XOR; equivalent to RRε(x)) +C0 ← circuit that takes in z and computes 1 +� +z ∈ Br(x) ∩ B˜r(˜x) ∩ H−1 +n (υn) +� +C1 ← circuit that takes in z and computes 1 +� +z ∈ Br(x′) ∩ B˜r(˜x) ∩ H−1 +n (υn) +� +return (C0, C1) +Lemma 21. Let H be as in Assumption 18. For any constant ε > 0, choosing r = 0.5n0.9 and +˜r = +1 +1+eε n + n0.6 makes LDS-Samplern (Algorithm 1) a public-coin differing-inputs sampler. +Proof. Suppose for the sake of contradiction that for some adjacent x, x′ ∈ {0, 1}n, there exists a +PPT ADI such that +Pr +θ [C0(y) ̸= C1(y) : (C0, C1) ← LDS-Samplern(θ), y ← ADI +n (θ)] ≥ n−c, +(1) +for some constant c > 0. Furthermore, let c1 be such that the total size of the descriptions of +ADI +n , LDS-Samplern is at most nc1. Finally, let c2 > 0 be as in Assumption 18 and K = nc2. +Algorithm 2 Collision-Resistant Hash Function Adversary ACRH +n +. +Parameter: +The target number of collisions K ∈ N, constant c > 0. +Advice: +Descriptions of ADI +n , LDS-Samplern. +Output: Y1, . . . , YK ∈ {0, 1}n or ⊥. +i ← 0 +for j = 1, . . . , K · nc+1 do +θj ← RRε(0n) +(Cj +0, Cj +1) ← LDS-Samplern(θj) +yj ← ADI +n (θj) +if Cj +0(yj) ̸= Cj +1(yj) then +i ← i + 1 +Yi ← yj +if i ≥ K then +break +if i < K then +return ⊥ +else +return Y1, . . . , YK +Consider the adversary ACRH +n +for collision-resistant hash function described in Algorithm 2. +First, note that by (1) and a standard concentration inequality, the probability that ACRH +n +outputs +⊥ is on(1). Furthermore, notice that C0, C1 can differ on y only if Hn(y) = υn, meaning that +Hn(Yi) = υn always. Therefore, it suffices for us to show that the probability that Y1, . . . , YK are +14 + +distinct is 1 − on(1). By a union bound, we have that ACRH +n +violates the collision-resistance of H +as desired. +Thus, we are only left to show that Y1, . . . , YK are not distinct with probability o(1). To see +that this is the case, notice that +Pr[Y1, . . . , YK are not distinct] ≤ +� +1≤i1 j1, yj = Yi1] +≤ Pr[∃j > j1, Cj +0(Yi1) ̸= Cj +1(Yi1)] +≤ +� +j>j1 +Pr[Cj +0(Yi1) ̸= Cj +1(Yi1)]. +(3) +Now, let us bound the RHS probability for a fixed j > j1. To see this, first observe that Yi1 must +belong to the symmetric difference Br(x)△Br(x′); otherwise, we must have Cj1 +0 (Yi1) = Cj1 +1 (Yi1), a +contradiction to our definition of Yi1. +Now, let ˜xj denote the ˜x selected by LDS-Sampler when constructing Cj +0, Cj +1. We have +Pr[Cj +0(Yi1) ̸= Cj +1(Yi1)] ≤ Pr[Yi1 ∈ B˜r(˜xj)]. +(4) +Let d := ∥Yi1 − x∥1 and ˜d := ∥Yi1 − ˜xj∥1. Since Yi1 ∈ Br(x)△Br(x′), it holds that d ∈ {r, r + 1}. +Thus, ˜d is distributed as Bin(d, +eε +1+eε ) + Bin(n − d, +1 +1+eε ). We have E˜xj∼RRε(x) ˜d = +1 +1+eε n + eε−1 +eε+1d. +By Bernstein’s inequality, +Pr[ ˜d ≤ ˜r] ≤ exp +� +− +t2 +eε +(1+eε)2 n + 2 +3t +� +≤ exp(−Ω(n0.8)), +where t = E˜xj∼RRε(x) ˜d − ˜r ≥ eε−1 +eε+1(0.5n0.9 − 1) − n0.6. Plugging into (4), we have +Pr[Cj +0(Yi1) ̸= Cj +1(Yi1)] ≤ exp(−Ω(n0.8)). +(5) +Combing (2), (3), (5), we have +Pr[Y1, . . . , YK are not distinct] ≤ K3nc+1 · exp(−Ω(n0.8)) ≤ exp(−Ω(n0.8)), +where the last inequality follows from K = nO(1). +5.1.3 +From Differing-Inputs Circuits to CDP +We will next construct CDP mechanism from the previously constructed differing-inputs circuit +family. First, let us state the assumption we need here: +Assumption 22. For H as in Assumption 18, any constant ε > 0 and r = 0.5n0.9, ˜r = +1 +1+eε n+n0.6, +there exists a differing-inputs obfuscator diO for the sampler LDS-Sampler. +15 + +Algorithm 3 CDP mechanism MdiO. +Parameter: +Differing-inputs obfuscator diO, hash function H, parameters ε, r, ˜r (as in +Assumption 22), and a hash value υn ∈ {0, 1}γ(n). +Input: Dataset x ∈ {0, 1}n. +Output: Circuit : {0, 1}n → {0, 1}. +˜x ← RRε(x). +C ← circuit that takes in z and compute 1 +� +z ∈ Br(x) ∩ B˜r(˜x) ∩ H−1 +n (υn) +� +�C ← diOρ(C) for randomness ρ +return +�C +Distribution H0: +˜x ← RRε(x) +C(z) := 1 +� +z ∈ Br(x) ∩ B˜r(˜x) ∩ H−1 +n (υn) +� +return diOρ(C) +Distribution H1: +˜x ← RRε(x) +C(z) := 1 +� +z ∈ Br(x′) ∩ B˜r(˜x) ∩ H−1 +n (υn) +� +return diOρ(C) +Distribution H2: +˜x ← RRε(x′) +C(z) := 1 +� +z ∈ Br(x′) ∩ B˜r(˜x) ∩ H−1 +n (υn) +� +return diOρ(C) +Figure 1: Hybrids in proof of Theorem 23. H0 is precisely MdiO(x) and H2 is precisely MdiO(x′). +Our mechanism can then be defined by simply applying obfuscator to the circuit generated +in the same way as C1 in LDS-Samplern. This mechanism MdiO is described more formally in +Algorithm 3. The CDP property of the mechanism follows rather simply from the definition of diO +and the fact that RRε is ε-SDP. +Theorem 23. Under Assumptions 18 and 22, MdiO is ε-CDP. +Proof. For any adjacent datasets x, x′, we want to show that MdiO(x) ≈c +ε MdiO(x′). We show this +using an intermediate hybrid, as shown in Figure 1, where changes from one hybrid to next are +highlighted in red. +◮ Distribution H0 is precisely MdiO(x). +◮ Distribution H1 is a variant of H0, where we change x to x′ in the definition of C, but continue +to sample ˜x ∼ RRε(x). +◮ Distribution H2 is a variant of H1, where we sample ˜x ∼ RRε(x′). Note that this is exactly +MdiO(x′). +We show that H0 ≈c +ε H2 by showing that H0 ≈c H1 and H1 ≈ε,0 H2 and using basic +composition (Fact 11). +We have from Lemma 21, that under Assumption 18, the joint distri- +bution of ˜x ∼ RRε(x), and circuits C in H0 and H1 is precisely the output of LDS-Sampler. +Thus, from Assumption 22, it follows that H0 ≈c H1 by post-processing (Fact 11). +Next, we +have that H1 ≈(ε,0) H2, since the only difference between the two is the distribution of ˜x, and +RRε(x) ≈(ε,0) RRε(x′) (again by post-processing). +Finally, its utility also follows simply from a standard concentration inequality. +16 + +Theorem 24. When choosing ˜r = +1 +1+eε n + n0.6, MdiO is (1 − o(1))-useful for ueval +H−1 +n (υn). +Proof. Consider any dataset x. If x /∈ H−1 +n (υn), then, by definition of uLDS +H−1 +n (υn), the utility always +evaluates to one. Therefore, we may only consider the case where x ∈ H−1 +n (υn). +In this case, observe that Pr +� +ueval +H−1 +n (υn)(x, MdiO(x)) = 1 +� += Pr˜x∼RRε(x)[x ∈ B˜r(˜x)]. Notice that +∥x − ˜x∥1 is distributed as Bin(n, +1 +1+eε ). Therefore, applying Bernstein’s inequality, we have +Pr +˜x∼RRε(x)[x /∈ B˜r(˜x)] ≤ exp +� +− +t2 +eε +(1+eε)2 n + 2 +3t +� +≤ exp(−Ω(n0.2)), +where t = ˜r− +n +1+eε = n0.6. This means that Pr +� +ueval +H−1 +n (υn)(x, MdiO(x)) = 1 +� += 1−o(1) as desired. +5.2 +CDP Mechanism for VLDS +5.2.1 +Witness-Indistinguishable Proofs +For any NP language L with associated verifier VL, let RL denote the corresponding relation +{(x, w) : x ∈ L and VL(x, w) = 1}. Let RL(x) := {w : (x, w) ∈ RL}. +Definition 25 (NIWI Proof System). A pair (P, V ) of PPT algorithms is a non-interactive witness +indistinguishable (NIWI) proof system for an NP relation RL if it satisfies: +◮ Correctness: for every (x, w) ∈ RL +Pr[V (x, π) = 1 : π ← P(x, w)] = 1. +◮ Soundness: there exists a negligible function negl such that for all x /∈ L and π ∈ {0, 1}∗: +Pr[V (x, π) = 1] ≤ negl(|x|). +◮ Witness Indistinguishability: There exists a polynomial ζ(·) and a negligible function negl(·), +such that for any sequence I = {(x, w0, w1) : w0, w1 ∈ RL(x)} and for all circuits C of size at +most ζ(|x|): +���� +Pr +π0←P (x,w0)[C(x, π0) = 1] − +Pr +π1←P (x,w1)[C(x, π1) = 1] +���� ≤ negl(|x|). +Assumption 26 ([BOV07, GOS12, BP15a]). There exists a NIWI proof system for any language +in NP. +5.2.2 +Making Utility Function Efficient Using Witness-Indistinguishable Proofs +We consider the NP language �L defined below, and use the corresponding NIWI verifier to define +the utility for VLDS. +Definition 27. Language �L consists of all circuits �C with a top AND gate, namely of the form +�C0 ∧ �C1 such that there exists some x, ˜x and ρ, such that at least one of �C0 or �C1 can be obtained +as diOρ(C) where C is a circuit that takes in z and computes 1 +� +z ∈ Br(x) ∩ B˜r(˜x) ∩ H−1(υ) +� +. +A “witness” for �C ∈ �L is given by w = (b, x, ˜x, ρ), where b ∈ {0, 1} indicates whether the witness +is provided for �C0 or for �C1. Let ( �P, �V ) denote the NIWI proof system for L (guaranteed to exist +by Assumption 26). +17 + +Algorithm 4 Sub-routine Maux +diO. +Parameter: +Differing-inputs obfuscator diO, hash function H, parameters ε, r, ˜r (as in +Assumption 22), and a hash value υ ∈ {0, 1}γ(n). +Input: Dataset x ∈ {0, 1}n. +Output: Circuit : {0, 1}n → {0, 1}. +˜x ← RRε(x). +C ← circuit that takes in z and compute 1 +� +z ∈ Br(x) ∩ B˜r(˜x) ∩ H−1 +n (υ) +� +�C ← diOρ(C) for randomness ρ +return +�C, ˜x, ρ +Algorithm 5 CDP mechanism Mcdp. +Input: Dataset x ∈ {0, 1}n, radius parameters r, ˜r > 0 and privacy parameter ε. +Output: Circuit C and a proof string π. +�C0, ˜x0, ρ0 ← Maux +diO(x) +�C1, ˜x1, ρ1 ← Maux +diO(x) +�C = �C0 ∧ �C1 +π ← �P( �C, (0, x, ˜x0, ρ0)) (NIWI proof for �C ∈ �L using witness (0, x, ˜x0, ρ0)). +return +�C, π +We consider the verifiable low diameter set problem VLDSτ,H−1(υ),�V . Note that �C ∈ �L auto- +matically implies that �C encodes a τ-diameter set (since �C = �C0 ∧ �C1, it suffices to certify that at +least one of �C0 or �C1 encodes a τ-diameter set) where τ = 2r = n0.9. +Theorem 28. Mcdp (described in Algorithm 5) is 2ε-CDP. +Proof. For any adjacent datasets x, x′, we want to show that Mcdp(x) ≈c +2ε Mcdp(x′). We show this +through the means of intermediate hybrids, as shown in Figure 2, where changes from one hybrid +to next are highlighted in red. +◮ Distribution H0 is precisely Mcdp(x). +◮ Distribution H1 is a variant of H0, where �C1 is generated through x′ instead of x. +◮ Distribution H2 is a variant of H1, where we switch π from corresponding to witness (0, x, ˜x0, ρ0) +to the witness (1, x′, ˜x1, ρ1). +◮ Distribution H3 is a variant of H2, where �C0 is also generated through x′ instead of x. +◮ Distribution H4 is a variant of H3, where we switch π from corresponding to witness (1, x′, ˜x1, ρ1) +to the witness (0, x′, ˜x0, ρ0). Note that this is exactly Mcdp(x′). +From Assumption 26 and post-processing (Fact 11), we have that H1 ≈c H2, and similarly H3 ≈c +H4. +Next, we show that H0 ≈c +ε H1. Note that the output of H0 and H1 do not depend on ˜x1 +and ρ1. Thus the only material change between H0 and H1 is that �C1 ∼ MdiO(x) in H0 versus +�C1 ∼ MdiO(x′) in H1. From Theorem 23, we have that MdiO(x) ≈c +ε MdiO(x′). Thus, it follows +that H0 ≈c +ε H1 by post-processing (Fact 11). Similarly, it follows that H2 ≈c +ε H3 (here we use that +˜x0 and ρ0 are immaterial to the final output of H2 and H3). +Combining these using basic composition (Fact 11), we get that H0 ≈c +2ε H4, thus implying that +Mcdp is 2ε-CDP. +18 + +Distribution H0: +�C0, ˜x0, ρ0 ← Maux +diO(x) +�C1, ˜x1, ρ1 ← Maux +diO(x) +�C = �C0 ∧ �C1 +π ← �P( �C, (0, x, ˜x0, ρ0)) +return +�C, π +Distribution H1: +�C0, ˜x0, ρ0 ← Maux +diO(x) +�C1, ˜x1, ρ1 ← Maux +diO(x′) +�C = �C0 ∧ �C1 +π ← �P( �C, (0, x, ˜x0, ρ0)) +return +�C, π +Distribution H2: +�C0, ˜x0, ρ0 ← Maux +diO(x) +�C1, ˜x1, ρ1 ← Maux +diO(x′) +�C = �C0 ∧ �C1 +π ← �P( �C, (1, x′, ˜x1, ρ1)). +return +�C, π +Distribution H3: +�C0, ˜x0, ρ0 ← Maux +diO(x′) +�C1, ˜x1, ρ1 ← Maux +diO(x′) +�C = �C0 ∧ �C1 +π ← �P( �C, (1, x′, ˜x1, ρ1)). +return +�C, π +Distribution H4: +�C0, ˜x0, ρ0 ← Maux +diO(x′) +�C1, ˜x1, ρ1 ← Maux +diO(x′) +�C = �C0 ∧ �C1 +π ← �P( �C, (0, x′, ˜x0, ρ0)) +return +�C, π +Figure 2: Hybrids in proof of Theorem 28. H0 is precisely Mcdp(x) and H4 is precisely Mcdp(x′). +Corollary 29. Mcdp is (1 − o(1))-useful for uVLDS +τ,H−1(υ),�V . +Proof. The utility for x /∈ H−1(υ) is trivially 1. Consider x ∈ H−1(υ). Suppose the mechanism +MdiO is (1 − η)-useful for ueval +H−1(υ). Since we sample �C0 and �C1 from MdiO independently we have +that �C(x) = 1 with probability at least 1 − 2η. Finally, note that the proof π in the output of +Mcdp is always accepted by �V . From Theorem 24, we have that η = o(1), and hence Mcdp is +1 − 2η = 1 − o(1) useful for uVLDS +τ,H−1(υ),�V . +We end this section by proving Theorem 16. The proof is essentially a straightforward combina- +tion of the previous two results. The only choice left to make is to select the hash value υ; we select +it so that the size of the preimage H−1(υ) is maximized. This ensures that the set R = H−1(υ) +has enough density as required in Theorem 16. (Note: the density requirement in Theorem 16 is +not important for showing the existence of a CDP mechanism, but instead is later used to show the +non-existence of SDP mechanisms.) +Proof of Theorem 16. Let H, τ, �V be as defined above. Furthermore, let υ be such that H−1(υ) is +maximized and ε = εCDP/2. The fact that there exists an εCDP-CDP mechanism that is (1 − o(1))- +useful for uVLDS +τ,R,�V follows immediately from Theorem 28 and Corollary 29. +Furthermore, by our +choice of υ, notice that |R| = |H−1(υ)| ≤ 2n/2γ(n) ≥ 2n/no(log n), where the latter comes from our +assumption on γ in Assumption 18. +6 +SDP Lower Bounds for the Nearby Point Problem +In this section, we will show that there is no O(1)-SDP algorithm for the nearby point problem +with target threshold n0.99 as long as the set Rn is fairly dense, as formalized below. +Theorem 30. Let τ = {τn}n∈N and R = {Rn ⊆ {0, 1}n}n∈N be such that τn ≤ n0.99 and |Rn| ≥ +2n/no(log n). Then, for any constant ε > 0 and any negligible function negl, there exists a sufficiently +large n ∈ N such that there is no (ε, negl(n))-SDP algorithm that is 0.01-useful for unear +τ,R . +19 + +To prove Theorem 30, let us first recall the standard “blatant non-privacy implies non-DP” +proof11, which corresponds to the case Rn = {0, 1}n. At a high-level, these proofs proceed by +showing that the error in each coordinate is large by “matching” each x ∈ {0, 1}n with another +point x′ which is the same as x except with the i-th bit flipped; a basic calculation then shows that +(on average) the i-th bit is predicted incorrectly with large probability. Summing this up over all +the coordinates yield the desired bound. +As we are in the case where Rn ̸= {0, 1}n, we cannot use the proof above directly. Nonetheless, +we can still adapt the above proof. More specifically, instead of looking at each coordinate at a +time, we look at a block of coordinates. For each block, we try to find a matching in the same spirit +as above, but we now allow the x, x′ to have a larger distance; simple calculations give us a lower +bound on being incorrect in this block (Section 6.2). We then “sum up” across all blocks to get +a large distance (Section 6.3). Even though we get a large distance τ via this approach, the error +probability (i.e. one minus usefulness) is small (i.e. o(1)). Fortunately, we can overcome this using +the so-called DP hyperparameter tuning algorithm [LT19, PS21] (Section 6.4). This concludes our +proof overview. +6.1 +Additional Preliminaries: Tools from Differential Privacy +We will require several additional tools from DP literature, which we list below for completeness. +Laplace Mechanism. +The Laplace distribution with scale parameter b > 0, denoted by Lap(b), +is the probability distribution over R with probability mass function z �→ 1 +2b exp(−|z|/b). +Given a function f : X ∗ → R, its sensitivity is defined as ∆(f) := maxD,D′ |f(D) − f(D′)|, +where the maximum is over all pair D, D′ of adjacent datasets. +The Laplace mechanism [DMNS06] is an ε-SDP mechanism that simply outputs f(X)+Lap(∆(f)/ǫ). +Basic Composition. +We will also use the so-called basic composition theorem: an algorithm that +just runs an (ε1, δ1)-SDP and an (ε2, δ2)-SDP algorithms as subroutines, is (ε1 + ε2, δ1 + δ2)-SDP. +Group Privacy. +The following fact is well-known and is often referred to as group privacy. +Fact 31 (Group Privacy (e.g., [SU16])). Let M : X ∗ → Y be an (ε, δ)-SDP mechanism and let +D, D′ ∈ X ∗ be such that ∥D−D′∥ ≤ t, then, for all S ⊆ Y we have Pr[M(D) ∈ S] ≤ eε′·Pr[M(D′) ∈ +S] + δ′, where ε′ = tε and δ′ = etε−1 +eε−1 · δ. +DP Hyperparameter Tuning. +We will also use the following result of Liu and Talwar [LT19] on +DP hyperparameter tuning. We remark that some improvements in the constants has been made in +[PS21], by using a different distribution of the number of repetitions. Nonetheless, since we are only +interested in an asymptotic bound, we choose to work with the slightly simpler hyperparameter +tuning algorithm from [LT19]. +The hyperparameter tuning algorithm from [LT19] allows us to take any DP “base” mechanism +Mbase, which outputs a candidate y and a score q ∈ R, run it multiple times and output a candidate +with score that is above a certain threshold. The precise description is in Algorithm 6. +We will use the following DP guarantee of Mtuning, which was shown in [LT19]12. +11Here we follow the proofs in [Sur19, Man22]. +12Note that this is a simplified version of [LT19, Theorem 3.1] where we simply set ε0 = 1. +20 + +Algorithm 6 DP Hyperparameter Tuning Mtuning. +Parameters: Mechanism Mbase, Threshold s, Number of Steps T, Stopping Probability γ. +Input: Dataset D +for j = 1, . . . , T do +Let (y, q) ← Mbase(D). +if q ≥ s then +return x (and halt) +With probability γ: +return ⊥ (and halt) +Theorem 32 (DP Hyperparameter Tuning [LT19]). Let ε > 0 and δ, γ ∈ [0, 1]. Suppose that Mbase +is (ε, δ)-SDP and T ≥ 2/γ. Then, the DP Hyperparameter Tuning mechanism Mtuning defined in +Algorithm 6 is (2ε + 1, 10e2ε · δ/γ). +6.2 +Weak Hardness +We start with a relatively weak hardness for the case of τ = 0, i.e., the answer is considered correct +iff it is the same as the input. To prove this, we recall a couple of facts. +The first is a simple relation between independent set and maximum matching. Let ind(G) +denote the size of the maximum independent set of G. +Fact 33. For any graph G = (V, E), there exists matching of size at least (|V | − ind(G))/2. +Let Hd denote the distance-d graph on the hypercube, i.e., Hd = ({0, 1}n, E) where (x, x′) ∈ E +iff ∥x − x′∥1 ≤ d. Let +� n +≤d +� += �d +i=0 +�n +i +� +. The following is the “packing” lower bound. +Fact 34. For any d ∈ N, ind(H2d+1) ≤ 2n/ +� n +≤d +� +. +We are now ready to prove a lower bound for the nearby problem. +Theorem 35. For any R ⊆ {0, 1}n, d, ε, δ, let ε′ = (2d + 1)ε and δ′ = eε′−1 +eε−1 δ. Then, for any +(ε, δ)-SDP algorithm M, we have +� +x∈R +Pr[M(x) ̸= x] ≥ 0.5e−ε′(1 − δ′) +� +|R| − 2n +� n +≤d +� +� +. +Proof. Let H2d+1[R] denote the subgraph of H2d+1 induced on R. Notice that ind(H2d+1[R]) ≤ +ind(H2d+1). Therefore, by Fact 33 and Fact 34, we can conclude H2d+1[R] contains a matching of +size at least m ≥ +� +|R| − 2n/ +� n +≤d +�� +/2. Let the matching be (x1, ˜x1), . . . , (xm, ˜xm). +For each i ∈ [m], we have +Pr[M(xi) ̸= xi] + Pr[M(˜xi) ̸= ˜xi] +≥ Pr[M(xi) = ˜xi] + Pr[M(˜xi) ̸= ˜xi] +(Group privacy, Fact 31) +≥ e−ε′(Pr[M(˜xi) = ˜xi] − δ′) + Pr[M(˜xi) ̸= ˜xi] +≥ e−ε′(Pr[M(˜xi) = ˜xi] + Pr[M(˜xi) ̸= ˜xi] − δ′) += e−ε′(1 − δ′). +Adding this over all i ∈ [m] yields the claimed bound. +21 + +6.3 +Boosting the Distance +We can now prove a hardness for larger τ by dividing the coordinates into groups and applying the +previously derived weak hardness result on each group. We note that the “non-usefulness” we get +on the right hand side is still insufficient for Theorem 30; this will be dealt with in Section 6.4. +Theorem 36. Let n = n′ · b′ for some n′, b′ ∈ N. For any R ⊆ {0, 1}n, d, ε, δ, ζ, let ε′ = (2d + 1)ε +and δ′ = eε′−1 +eε−1 δ. Then, for any (ε, δ)-SDP algorithm M, there exists x ∈ R such that +Pr[unear +ζ·b′,R(M(x), x) = 0] ≥ +� +0.5e−ε′(1 − δ′) +� +1 − +2n +|R| · +� n′ +≤d +� +�� +− ζ. +Proof. Let Bi := {(i − 1)n′ + 1, . . . , in′} for all i ∈ [b′]. Furthermore, let R(Bi,z−Bi) denote the set +the set of all x ∈ R such that x−Bi = z−Bi. +First, notice that +� +x∈R +Pr[unear +ζ·b′,R(M(x), x) = 0] = +� +x∈R +Ey←M(x) 1 +�|{i ∈ [n] | yi ̸= xi}| +b′ +> ζ +� +≥ +� +x∈R +Ey←M(x) 1 +�|{i ∈ [b′] | yBi ̸= xBi}| +b′ +> ζ +� +≥ +� +x∈R +Ey←M(x) +� +Pr +i∈[b′][yBi ̸= xBi] − ζ +� += + + 1 +b′ +� +i∈[b′] +� +x∈R +Pr[M(x)Bi ̸= xBi] + + − ζ|R| +≥ + + + 1 +b′ +� +i∈[b′] +� +z−Bi∈{0,1}[n]\Bi +� +x∈R(Bi,z−Bi ) +Pr[M(x)Bi ̸= xBi] + + + − ζ|R|. +For each fixed z−Bi ∈ {0, 1}[n]\Bi, consider the mechanism M′ : {0, 1}Bi → {0, 1}Bi defined by +M′(xBi) := Mi(xBi ◦ z−Bi)|Bi. +It is clear that M′ is (ε, δ)-SDP. +Furthermore, observe that +Pr[M(x)Bi ̸= xBi] = Pr[M′(x) ̸= xBi] for all x ∈ R(Bi,z−Bi). Therefore, by applying Theorem 35 +and plugging it back into the above, we get +� +x∈R +Pr[unear +ζ·b′,R(M(x), x) = 0] +≥ + + + 1 +b′ +� +i∈[b′] +� +z−Bi∈{0,1}[n]\Bi +0.5e−ε′(1 − δ′) +� +|R(Bi,z−Bi)| − 2n′ +� n′ +≤d +� +� + + − ζ|R| += + + 1 +b′ +� +i∈[b′] +0.5e−ε′(1 − δ′) +� +|R| − 2n−n′ · 2n′ +� n′ +≤d +� +� + − ζ|R| += +� +0.5e−ε′(1 − δ′) +� +|R| − 2n +� n′ +≤d +� +�� +− ζ|R|. +Dividing by |R| then gives us the claimed bound. +22 + +6.4 +Boosting the Failure Probability +We will now prove the last part of the lower bound, which is to show that the existence of even +slightly useful mechanism also leads to an existence of a highly useful mechanism, albeit at a slight +increanse in the distance threshold. The formal statement and its proof are given below; the proof +uses the DP hyperparameter tuning algorithm (Theorem 32). +Theorem 37. Suppose that there exists an (ε, δ)-SDP mechanism M : {0, 1}n → {0, 1}n that is +α-useful for unear +τ,R . Then, there exists an (ε′, δ′)-SDP mechanism M′ : {0, 1}n → {0, 1}n that is +(1 − 1/n1000)-useful for unear +τ ′,R where ε′ = 4ε + 1, δ′ = O +� +n11e2ε +α +· δ +� +and τ ′ = τ + O +� ln n +α +� +. +Proof. First, let us construct the mechanism Mbase : {0, 1}n → {0, 1}n × R as follows: +◮ On input x ∈ {0, 1}n, first let y ← M(x). +◮ Then, let q = ∥x − y∥1 + z where z ∼ Lap(1/ε). +◮ Output (x, q). +Since M is (ε, δ)-SDP and the Laplace mechanism is ε-SDP, the basic composition theorem implies +that the entire Mbase mechanism is (2ε, δ)-SDP. +Let �T = ln(5n1000)/α. Let τ ′ = τ − log(10n1000 �T)/ε. We now apply Algorithm 6 with γ = +0.5/(n1000 �T), T = 2/γ and threshold s = τ ′ − log(10n1000 �T)/ε. +Theorem 32 ensures that the +resulting algorithm Mtuning is (4ε + 1, 10e2εδ/γ)-SDP. Our final mechanism � +M is the mechanism +that runs Mtuning. +If the output is not ⊥, � +M returns that output. +Otherwise, � +M returns an +arbritrary element of {0, 1}n. Since � +M is simple a post-processing of Mtuning, we have � +M is also +(4ε + 1, 10e2εδ/γ)-SDP. +We will next show that Mtuning is (1 − 1/n1000)-useful for unear +τ ′,R. By definition of the utility +function, this immediately holds for any x /∈ R. Therefore, we may only consider any x ∈ R. +Consider Mtuning on such an x. Let yi, zi, qi denote the corresponding values of y, z, q in the ith +run of Mbase. +We will consider the following three events: +◮ Let E1 denote the event that |∥xi − yi∥1 − qi| > log(10n1000 �T)/ε for some i ∈ [ �T]. +◮ Let E2 denote the event that uτ,R(yi) = 0 for all i ∈ [ �T]. +◮ Let E3 denote the event that Mtuning halts in the first �T steps. +Before we bound the probability of each events, notice that, if none of E1, E2, E3 occurs, we must +have unear +τ ′,R(y) = 1 (where y denote the output of � +M), since s − τ, τ ′ − s ≥ log(10n1000 �T)/ε. That is, +Pr +y←� +M(x) +[unear +τ ′,R(y) = 0] ≤ Pr[E1 ∨ E2 ∨ E3] ≤ Pr[E1] + Pr[E2] + Pr[E3]. +We will now bound the probability for each event. For E1, it immediately follows from the +Laplace tail bound together with a union bound that +Pr[E1] ≤ �T · 2/(10n1000 �T) = 0.2/n1000. +For E2, the α-usefulness of M implies that +Pr[E2] ≤ (1 − α) +�T ≤ 0.2/n1000. +Finally, for E3, we may simply use a union bound, which gives +Pr[E3] ≤ γ · �T ≤ 0.5/n1000. +23 + +By combining the four inequalities above, we have +Pr +y←� +M(x) +[unear +τ ′,R(y) = 0] < 1/n1000, +as desired. +6.5 +Putting Things Together: Proof of Theorem 30 +Proof of Theorem 30. Suppose for the sake of contradiction that, for some constant ε > 0 and +negligible function negl, there exists an (ε, negl(n))-SDP mechanism Mn that is 0.01-useful for +unear +τn,Rn for every n ∈ N. +By Theorem 37, there is a (4ε + 1, δ′(n)) mechanism M′ +n that is (1 − 1/n1000)-useful for unear +τ ′n,Rn +where δ′(n) is a negligible function and τ ′ +n = τn + O(log n) = O(n0.99). +Plugging this into +Theorem 36 with R = Rn, n′ = n0.005, b′ = n0.995, ζ = τ ′ +n/b′ ≤ O(n−0.005), ε = 4ε + 1, δ = δ′(n), d = +(log n0.004)/3ε (which gives ε′ ≤ log(2n0.004) and δ′ = 1/nω(1) in Theorem 36), we have +1 +n1000 ≥ +� +0.5 · e− log(2n0.004)(1 − n−ω(1)) (1 − o(1)) +� +− O(n−0.005) += O(n−0.004) · (1 − o(1)) − O(n−0.005), +which is a contradiction for any sufficiently large n. +7 +Putting Things Together: Proof of Theorem 5 +Our main theorem follows from trivially combining the main results from the previous two sections. +Proof of Theorem 5. Let u = uVLDS +τ,R,V be as given in Theorem 16, which immediately yields the +existence of an εCDP-CDP mechanism that is (1 − o(1))-useful. Furthermore, by |R| ≥ 2n/no(n), +Theorem 30 implies that there is no εSDP-SDP mechanism that is 0.01-useful for {unear +τ,R }. Finally, +applying Lemma 15, we can conclude that there is no εSDP-SDP mechanism that is 0.01-useful for +{uVLDS +τ,R,V }. This concludes our proof. +8 +Conclusion and Discussion +In this work, we give a first task that, under certain assumptions, admits an efficient CDP algorithm +but does not admit an (even inefficient) SDP algorithm. As mentioned in Section 1, perhaps the +most intriguing next direction would be to see if there are more “natural” tasks for which CDP +algorithms can go beyond known SDP lower bounds. +On the technical front, there are also a few interesting directions. For example, it would be +interesting to see if the three assumptions in our paper can be removed, relaxed, or replaced (by +perhaps more widely believed assumptions). Alternatively, we can ask the opposite question: what +are the (cryptographic) assumptions necessary for separating CDP and SDP? +Such a question +has been extensively studied in the multiparty model [HMST22, GMPS13, GKM+16, HMSS19, +HNO+18]; for example, it is known that key-agreement is necessary and sufficient to get better- +than-local-DP protocol for inner product in the two-party setting [HMST22]. Achieving such a +24 + +result in our setting would significantly deepen our understanding of the CDP-vs-SDP question in +the central model. +Another possible improvement is to strengthen the hardness of the adversary. In this paper, +we only consider polynomial-time adversaries. Indeed, our CDP mechanism does not remain CDP +against quasi-polynomial adversary. The reason is that we choose the hash value length to be only +o(log2 λ) in Assumption 18, so a trivial “guess-and-check” algorithm can break this assumption in +time λO(log λ). However, as far as we are aware, there is no inherent barrier in proving a separation +with CDP that holds even against, e.g., sub-exponential time adversaries. Achieving such a result +(potentially under stronger or different assumptions) would definitely be interesting. +Furthermore, our task (or more precisely the utility function) is non-uniform (through the choice +of υn). It would also be interesting to have a uniform task. +Acknowledgments +We thank Prabhanjan Ananth for helpful discussions about differing-inputs obfuscation, and anony- +mous reviewers for helpful comments. +25 + +References +[ABG+13] +Prabhanjan Ananth, Dan Boneh, Sanjam Garg, Amit Sahai, and Mark Zhandry. +Differing-inputs obfuscation and applications. IACR Cryptol. ePrint Arch., page 689, +2013. +[Abo18] +John M Abowd. 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Confidentiality, 7(2), 2016. +[Sur19] +Ananda Theertha Suresh. Differentially private anonymized histograms. In NeurIPS, +pages 7969–7979, 2019. +[TZ08] +Terence Tao and Tamar Ziegler. The primes contain arbitrarily long polynomial pro- +gressions. Acta Mathematica, 201(2):213 – 305, 2008. +[Vad17] +Salil P. Vadhan. The complexity of differential privacy. In Tutorials on the Foundations +of Cryptography, pages 347–450. Springer International Publishing, 2017. +[War65] +Stanley L Warner. Randomized response: A survey technique for eliminating evasive +answer bias. JASA, 60(309):63–69, 1965. +28 + +A +Comparison of various diO assumptions +We review and compare the various notions of differing inputs obfuscation, showing that the notion +of diO-for-pcS (Definition 20) is in fact weaker (or at least, no stronger) than all notions of differing +inputs obfuscation studied in literature. +The definition of diO as given by [BGI+12] did not include the notion of a sampler. Instead for +any circuits C0 and C1, if an adversary A can distinguish diO(C0) and diO(C1) with non-negligible +advantage, then there exists an adversary A′ that, given any circuits C′ +0 and C′ +1, where C′ +b is +functionally equivalent to Cb for b ∈ {0, 1}, A′(C′ +0, C′ +1) can return x such that C0(x) ̸= C1(x). +This notion is stronger than the corresponding notion involving samplers. Since most applica- +tions of differing-inputs obfuscation in literature are stated using differing-inputs samplers, we will +only refer to diO notions that involve these. +Definition 38 (Differing-Inputs Circuit Sampler [ABG+13]). An efficient non-uniform sampling +algorithm Sampler = {Samplern} is a differing-inputs sampler for the parameterized collection +C = {Cn} of circuits if the output of Samplern is distributed over Cn × Cn × {0, 1}∗ and for every +efficient non-uniform algorithm A = {An}, there exists a negligible function negl(·) such that for +all n ∈ N: +Pr +θ [C0(y) ̸= C1(y) : (C0, C1, aux) ← Samplern(θ), y ← An(C0, C1, aux)] ≤ negl(n). +Plain Sampler. We call a differing-inputs sampler as a Plain Sampler if aux is always ⊥. +Public-Coin Sampler. We call a differing-inputs sampler as Public-Coin Sampler if aux is equal +to θ (precisely Definition 19). +General Sampler. We call a differing-inputs sampler as a General Sampler whenever we want to +emphasize that aux is allowed to be any function of θ. In particular, plain and public-coin +samplers are special cases of general samplers. +Note that, the more information that aux is allowed to contain, the more restricted the distribution +over circuit pairs (C0, C1) gets. In particular, any public-coin Sampler remains a differing-inputs +Sampler if we set aux to be some function of θ (instead of being all of θ), and similarly, any general +differing-inputs Sampler can be converted to a plain-Sampler by simply setting aux = ⊥. +We can consider two notions of security of differing inputs obfuscators, depending on whether or +not the distinguisher has access to aux. Recall that the “differing-inputs” condition in Definition 20 +was +| Pr +θ [Dn(diO(1n, C0)) = 1] − Pr +θ [Dn(diO(1n, C1)) = 1]| ≤ negl(n). +(6) +On the other hand, we could consider a different notion where for any general sampler Sampler, for +(C0, C1, aux) ← Samplern(θ), we replace the “differing-inputs” condition with +| Pr +θ [Dn(diO(1n, C0), aux) = 1] − Pr +θ [Dn(diO(1n, C1), aux) = 1]| ≤ negl(n). +(7) +Depending on the type of sampler (plain or public-coin or general) and the notion of security +for differing inputs obfuscators ((6) or (7)), we get various kinds of diO assumptions, which we list +below. +29 + +plain-diO +pc-diO +gen-diO +diO-for-genS +diO-for-pcS +Figure 3: Comparisons between different diO assumptions, where A → B denotes that existence of +A implies existence of B, or in other words, existence of A is a stronger assumption than existence of +B. Existence of diO-for-pcS (assumption used in this paper) is the weakest among all the notions. +Plain diO. We refer to plain-diO as the notion of diO that holds only against plain samplers. Note, +there is no difference here between the security notions of (6) and (7), since aux = ⊥ anyway. +Public-Coin diO. We refer to pc-diO, as the notion of public-coin diO defined by [IPS15], cor- +responding to the notion of diO that holds only against public-coin samplers, where the +distinguisher also has access to aux = θ, as in (7). +General diO. We refer to gen-diO, as the notion of general diO defined by [ABG+13], correspond- +ing to the notion of diO that holds for general samplers, and where the distinguisher also has +access to aux, as in (7). +diO for General Samplers. We define diO-for-genS as the notion of diO that holds only against +general samplers, but where the distinguisher does not have access to aux = θ, as in (6). +diO for Public-Coin Samplers. This is precisely Definition 20, where the security of diO holds +only for public-coin samplers, where the distinguisher does not have access to aux, as in (6). +Comparison between different diO assumptions. +The comparison between the assumptions +asserting existence of each type of diO is illustrated in Figure 3, with justification for each arrow +given as follows: +◮ Existence of gen-diO implies existence of plain-diO and pc-diO, since both are special cases +corresponding to plain samplers and public-coin samplers respectively. +◮ To the best of knowledge, it is unknown whether the assumptions of existence of plain-diO +and the existence of pc-diO are comparable or not. +◮ Existence of plain-diO implies existence of diO-for-genS since any general sampler can be +converted to a plain sampler by simply setting aux = ⊥; note that the distinguisher (in the +definition of diO) does not have access to aux in either case. +30 + +◮ Existence of diO-for-genS implies existence of plain-diO and diO-for-pcS since both are special +cases corresponding to plain samplers and public-coin samplers respectively. +◮ Existence of pc-diO implies existence of diO-for-pcS, since the distinguisher in the definition +of diO-for-pcS does not have access to θ and hence is less powerful. +Finally, one may wonder, what was special about the application of diO in this paper that only +required diO-for-pcS and not gen-diO or pc-diO as in prior work in cryptography. The main reason is +that, in cryptographic applications, an aux is provided to adversaries to enable certain cryptographic +functionality (such as by revealing some public key parameters), and thus, it is required that the diO +is secure even given knowledge of this aux information. In applications of pc-diO, the distinguisher +typically does not have access to all of θ (such as some secret key parameters may be hidden), but +security given knowledge of entire θ implies security given partial knowledge of θ. In the setting of +this paper, there wasn’t any particular functionality that needed to be enabled, other than basic +circuit evaluation, and the particular circuit samplers of interest were public-coin differing inputs +samplers, which is why it suffices to only assume diO-for-pcS. +31 + diff --git a/I9AyT4oBgHgl3EQfTfcL/content/tmp_files/load_file.txt b/I9AyT4oBgHgl3EQfTfcL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d09da719ed52b458a293fd027cca3f17811c235a --- /dev/null +++ b/I9AyT4oBgHgl3EQfTfcL/content/tmp_files/load_file.txt @@ -0,0 +1,1729 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf,len=1728 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='00104v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='CR] 31 Dec 2022 Separating Computational and Statistical Differential Privacy (Under Plausible Assumptions) Badih Ghazi∗ Rahul Ilango† Pritish Kamath‡ Ravi Kumar§ Pasin Manurangsi¶ Abstract Computational differential privacy (CDP) is a natural relaxation of the standard notion of (statistical) differential privacy (SDP) proposed by Beimel, Nissim, and Omri (CRYPTO 2008) and Mironov, Pandey, Reingold, and Vadhan (CRYPTO 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In contrast to SDP, CDP only requires privacy guarantees to hold against computationally-bounded adversaries rather than computationally-unbounded statistical adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Despite the question being raised explicitly in several works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', Bun, Chen, and Vadhan, TCC 2016), it has remained tantalizingly open whether there is any task achievable with the CDP notion but not the SDP notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Even a candidate such task is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Indeed, it is even unclear what the truth could be!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In this work, we give the first construction of a task achievable with the CDP notion but not the SDP notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' More specifically, under strong but plausible cryptographic assumptions, we construct a task for which there exists an ε-CDP mechanism with ε = O(1) achieving 1 − o(1) utility, but any (ε, δ)-SDP mechanism, including computationally unbounded ones, that achieves a constant utility must use either a super-constant ε or a non-negligible δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' To prove this, we introduce a new approach for showing that a mechanism satisfies CDP: first we show that a mechanism is “private” against a certain class of decision tree adversaries, and then we use cryptographic constructions to “lift” this into privacy against computational adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We believe this approach could be useful to devise further tasks separating CDP from SDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' ∗Google Research, Mountain View.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' badihghazi@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' †MIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Part of this work was done during an internship at Google Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' rilango@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' ‡Google Research, Mountain View.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' pritish@alum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' §Google Research, Mountain View.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' ravi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='k53@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' ¶Google Research, Thailand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' pasin@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Contents 1 Introduction 1 2 Overview of the Results 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='1 The d-Distance Problem .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 24 7 Putting Things Together: Proof of Theorem 5 24 8 Conclusion and Discussion 24 A Comparison of various diO assumptions 29 1 Introduction The framework of differential privacy (DP) [DMNS06, DKM+06] gives formal privacy guarantees on the outputs of randomized algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' It has been the subject of a significant body of re- search, leading to numerous practical deployments including the US census [Abo18], and industrial applications [EPK14, Sha14, Gre16, App17, DKY17, KT18, RSP+21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The definition of DP requires privacy against computationally unbounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', statistical, adver- saries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' A natural modification is to instead only require privacy against computationally bounded adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In cryptography, considering computationally bounded adversaries instead of statisti- cal ones enables a vast array of applications, like public-key cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Could the same be true for DP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' A good survey of the area can be found in Vadhan’s monograph [Vad17, Section 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' De- spite Beimel, Nissim, and Omri [BNO08] defining computational differential privacy (CDP) in 2008 (definitions that were further extended by Mironov, Pandey, Reingold, and Vadhan [MPRV09]), the central question of separating it from statistical differential privacy (SDP)1, in the standard client-server model, remains open: Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' [Vad17, Open Problem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='7] Is there a computational task solvable by a single cura- tor with computational differential privacy but is impossible to achieve with information-theoretic differential privacy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='2 There have been several positive and negative results towards resolving this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In the positive direction, it is known that in the multi-party setting, CDP is stronger than SDP [MMP+10, MPRV09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Roughly speaking, this is because secure multi-party computation enables many data cu- rators to simulate acting as a single central curator, without compromising privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Still, the multi- party setting seems very different than the single-curator (aka central) setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Indeed, [MMP+10] remark3 that their “strong separation between (information-theoretic) differential privacy and com- putational differential privacy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' stands in sharp contrast with the client-server setting where .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' there are not even candidates for a separation.” In the central setting, Bun, Chen, and Vadhan [BCV16] show there is a task for which there is a CDP mechanism, but any SDP mechanism for this task must be inefficient (modulo certain cryptographic assumptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We stress that the task they consider does have an inefficient SDP mechanism (with parameters that match their CDP mechanism), so it does not resolve Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' While this may seem like a minor technical point, we emphasize that it is of crucial importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Perhaps the main practical motivation behind studying CDP is the hope that there are CDP mech- anisms for natural tasks with parameters that beat the lower bounds against SDP mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' But if, as in the case of the result in [BCV16], there exists (even an inefficient) SDP mechanism match- ing the parameters of the CDP mechanism, then clearly there is no hope of the CDP mechanism’s parameters beating SDP lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In the negative direction, Mironov, Pandey, Reingold, and Vadhan [MPRV09] (building on Green and Tao [GT08], Tao and Ziegler [TZ08], and Reingold, Trevisan, Tulsiani, and Vad- han [RTTV08]) show a “dense model theorem” for pairs of random variables with “pseudodensity” with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' [MPRV09] note that (roughly speaking) extending this dense model theorem to handle multiple pairs of random variables would prove that any CDP mechanism could be converted into an SDP mechanism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' such an extension is still open [Vad17, Open Problem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 1For the formal definitions of CDP and SDP, we refer the reader to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 2We state this verbatim from [Vad17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 3This remark is also quoted by Groce, Katz, and Yerukhimovich [GKY11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 1 Groce, Katz, and Yerukhimovich [GKY11] show that CDP mechanisms for certain tasks where the output is low-dimensional actually do imply SDP mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Many natural statistical tasks fall into this category, and consequently, such tasks cannot separate CDP from SDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' This result was further strengthened by [BCV16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Furthermore, [GKY11] show that CDP mechanisms con- structed in a black-box way from a variety of cryptographic objects, such as one-way functions, random oracles, trapdoor permutations, and cryptographic hash functions, cannot separate CDP from SDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In summary, there are at least two barriers to separate CDP from SDP: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' High-dimensionality: One needs to consider (perhaps non-natural) tasks with high dimen- sional outputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Exotic cryptography: One needs to use cryptography somewhat specially (perhaps either an exotic primitive or in a non-black-box manner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In light of these both positive and negative results as well as the lack of a candidate separation, it was not even clear what the truth could be: is there any task for which there is a CDP mechanism but not an SDP one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Our Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We show, under plausible cryptographic hypotheses, that there are indeed tasks for which there exist CDP mechanisms but no SDP mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' This not only positively answers Question 1 but also negatively answers the dense model extension question [Vad17, Open Problem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We state this result now informally and formalize it later in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We also delay discussing our precise cryptographic assumptions to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='5, where we discuss their plausibility in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' [Informal version of Theorem 5] Under cryptographic assumptions, there exists a task for which there is a CDP mechanism but no SDP mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Let us take a step back to discuss the implications of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Although (as we will see in a moment) our task is specifically constructed for the purpose of separating CDP and SDP, the fact that we can separate them at all opens up a possibility that such a separation even holds for some “natural” tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Indeed, some of the current lower bound techniques for SDP—such as the ubiquitous “packing lower bounds”4 (see [HT10])—do not necessarily rule out CDP mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' It seems prudent to carefully reexamine the current lower bound techniques to see whether they also apply to CDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The ultimate hope for this program would be to employ CDP to overcome the known SDP lower bounds for some more “natural” tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' (Of course, such tasks would also give a more “natural” separation of CDP and SDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=') In fact, the technical approach we use in our construction already suggests a general approach for constructing non-trivial CDP mechanisms that could apply to more tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We discuss this in more detail in Section 2, but the idea is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In order to show a task has a CDP mechanism, first show there is a mechanism for that task that is “private” against a certain class of decision tree adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Then, second, use cryptographic assumptions to “lift” this into privacy against computational adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 4Specifically, when the packing lower bound requires the use of super-polynomially many datasets, the correspond- ing adversary does not necessarily run in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 2 Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Section 2 provides a high-level overview of our techniques as well as a discussion of our cryptographic assumptions and their plau- sibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Section 3 contains the background material and Section 4 formally defines the problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We provide our CDP mechanism in Section 5, and prove lower bounds against SDP mechanisms in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' These two components are put together to prove the main result in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Finally, we discuss the open problems and future directions in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 2 Overview of the Results We will next discuss the high-level overview of our techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We will sometimes have to be informal here, but all details are formalized later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Let us quickly recall how the “task” is defined5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Following [GKY11, BCV16], a task is defined by an efficiently computable utility function u that takes in an input dataset D and a response y such that u(D, y) = 1 if y is considered “useful” for D and u(D, y) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' A mechanism M is said to be α-useful for u iff E[u(D, M(D))] ≥ α for all input datasets D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We remark that many well-studied problems—such as linear queries with various error metrics—can be written in this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We will refer to α as the usefulness of the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' One of our main conceptual contributions is to define a class of tasks that seems to naturally circumvent the two earlier-mentioned barriers—tasks where one needs to output a circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='1 The d-Distance Problem Before we detail why tasks that output a circuit might evade the two barriers, let us describe a concrete example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We call the following the d-distance problem (where d ∈ N is a parameter): ◮ Given: dataset D that consists of n bits ◮ Output: circuit C mapping n bits to 1 bit ◮ Utility: C is considered useful6 if it outputs ⊲ 1 on D, and ⊲ 0 on all points at distance greater than d from D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Informally, this problem asks to output a circuit that checks if its input is “close” to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Looking ahead, we will ultimately separate CDP from SDP under cryptographic assumptions by considering a version of this problem where we only care about datasets in a cryptographically special set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We now revisit the two barriers and discuss how the distance problem might circumvent them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' High-dimensionality: The output of this task is a circuit, which is high-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Exotic cryptography: Because the output of the task is a circuit, it lends itself to a powerful class of cryptographic objects: circuit obfuscators [BGI+12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Roughly speaking, circuit obfusca- tors take as input a circuit C and output a scrambled, obfuscated circuit C′ that computes the same function as C but which, ideally, has the property that “anything you could do with access to the circuit C′, you could do with only black-box access to the function the circuit computes.” Importantly, obfuscation is not in the list of primitives ruled out by the barrier in [GKY11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 5Please refer to Section 3 for a more formal definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 6One might be concerned about whether this utility function is actually efficiently computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We will address this in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='3 after we describe our final construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='2 SDP Lower Bound Our starting point for separating CDP from SDP is the d-distance problem described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Indeed, we show that there is no SDP mechanism for this problem for most settings of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' If 0 < d ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='99, then there is no (ε, δ)-SDP mechanism for d-distance that is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='01-useful for ε = O(1) and δ negligible in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In fact, this lower bound is straightforward (Lemma 15) from the well-known blatant non-privacy notion (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', [De12]): no DP algorithm can output a dataset that is (with large probability) close to the input dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Crucially, our lower bounds are non-constructive, and do not yield an efficient adversary (which would imply a similar lower bound against CDP mechanisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Thus, to separate CDP from SDP it suffices to come up with a CDP mechanism M for, say, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='99-distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='3 A CDP Mechanism One of our main ideas to help construct a CDP mechanism M is to use obfuscation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In particular, we will consider mechanisms where the returned circuit is obfuscated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Recall that in order to prove a mechanism M that outputs a circuit C is CDP, one needs to argue that no efficient adversary that gets C as input can break the privacy guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' By considering mechanisms that return obfuscated circuits, we can drastically simplify the type of adversaries we need to prove privacy against.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Instead of proving privacy against adversaries that see the circuit C (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', white-box setting), sufficiently strong obfuscation means we only need to prove privacy against decision tree adversaries that can query the function computed by the circuit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', black-box setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In other words, if we have a mechanism that satisfies DP against black-box adversaries (decision trees) with a polynomial number of queries, we can then hope to use sufficiently strong obfuscation to “lift” this into a mechanism that is secure against (white-box) computational adversaries with polynomial running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Of course, one needs to be careful about whether such “sufficiently strong obfuscation” is even possible, but, putting that aside for the moment, the question of whether there is a CDP mechanism for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='99-distance (Question 4 below) appears to reduce to whether there is a mechanism for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='99- distance that is DP against query (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' decision-tree) adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Let ε = O(1) and 0 ≤ d ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Does there exist an ε-CDP mechanism for d-distance with constant usefulness?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' While we do not resolve Question 4, we (roughly speaking) show that there is a mechanism that is DP against non-adaptive decision tree adversaries, whose queries are fixed a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' It turns out a relatively simple mechanism based on randomized response [War65] works for these less powerful adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' From Non-Adaptive Lower Bound to Computational Lower Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' This switch from the usual adaptive query adversaries to non-adaptive query adversaries comes at a price however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' It is not clear how to use obfuscation to lift a mechanism that is private against non-adaptive queries into one that is private against computational adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Indeed, a polynomial-time algorithm with even black-box access to a function seems to be an inherently adaptive adversary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Surprisingly, we manage to get around this by using another cryptographic object introduced by Bitansky, Kalai, and Paneth [BKP18]: collision-resistant keyless hash functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Informally 4 speaking, a hash function being collision-resistant and keyless means that “any efficient adversary can only generate a number of hash collisions that is at most polynomially larger than the advice the adversary gets.” We then modify the d-distance problem to only consider datasets that hash to, say, the all zeroes string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Formally, zero hash d-distance is the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Let R ⊆ {0, 1}n be the set of strings that hash to the all zeroes string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' ◮ Given: dataset D that consists of n bits ◮ Output: circuit C mapping n bits to 1 bit ◮ Utility: C is considered useful if D /∈ R or both of the following hold: ⊲ it outputs 1 on D ⊲ it outputs 0 on all points in R at distance greater than d from D In other words, the utility function now completely ignores all points outside of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The high-level intuition behind this change is the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Our CDP mechanism can output a circuit C such that the only inputs where C(x) reveals information are those x in the set R (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', that hash to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Any polynomial-time adversary A can only generate fixed polynomial number of elements of R by the collision-resistance property of the hash function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Combining the above effectively makes the inputs A can query C on “non-adaptive”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Finally, in order to “lift” the query separation into the computational realm we use another cryp- tographic tool: differing-inputs obfuscation (diO) [BGI+01, BGI+12, ABG+13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Roughly speaking, diO is an obfuscator with the following guarantee: if any efficient adversary can distinguish the obfuscation of two circuits C1 and C2, then an efficient adversary can find an input x on which C1(x) ̸= C2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In particular, the assumption we use is even weaker than public-coin diO [IPS15], which is already considered to more plausible than general diO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='7 In summary, diO allows us to reduce computational adversaries to adaptive query adversaries and collision-resistant keyless hash functions allows us to reduce adaptive query adversaries to non-adaptive query adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Interestingly, to the best of our knowledge, this is the first time collision-resistant keyless hash functions are being used together with any obfuscation assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Making the Utility Function Efficiently Computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Observant readers may have already noticed an issue: utility functions that we have considered so far are not necessarily efficiently computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Specifically, a trivial way to implement the utility function would be to enumerate all points at distance at least d, feed it into the circuit, and check that the output is as expected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' this would take 2nΩ(1) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' To overcome the above problem, we restrict circuits to only those that are relatively simple, so that there is a small “witness” w that certifies that the circuit outputs zero at all points that are d-far from D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' A naive idea is then to let the CDP mechanism output the circuit C together with such a witness w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The utility function can then just efficiently check that w is a valid witness (and that C(D) = 0 or x ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' This makes the utility function efficient but unfortunately compromises privacy because the witness w itself can leak additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' To avoid this, we instead use non-interactive witness indistinguishable (NIWI) proofs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', [BOV07]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Roughly speaking, this allows us to produce a proof π from w (and C and diO), which does not leak any information about w (against computationally bounded adversaries), but at the same time still allows us to verify 7See Assumption 22 for formal statement of the assumption and Appendix A for comparison with other diO assumptions in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 5 that the underlying witness w is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The former is sufficient for CDP, while the latter ensures that the utility function can be computed efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' This completes the high-level overview of the constructed task and our CDP mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The cryptographic primitives needed for our mechanism are formalized in Assumptions 18, 22 and 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='4 Final Steps Finally, we remark that since our problem is now not exactly the original d-distance problem anymore, as the utility guarantees are only now meaningful for datasets in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' This means that we cannot use the lower bound in Lemma 3 for the d-distance problem directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Fortunately, we can still adapt its proof—a “packing-style” lower bound on each coordinate—to one which applies a packing-style argument on each block of coordinates instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' With this, we can prove the lower bound for zero hash d-distance as long as the set R has sufficiently large density (≈ 1/n−o(log n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Putting all the ingredients together, we arrive at the following8: Theorem 5 (Main Result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Under Assumptions 18, 22 and 26, for any constant εCDP > 0, there exists an ensemble u = {un}n∈N of polynomial time computable utility functions such that ◮ There is an εCDP-CDP mechanism that is (1 − on(1))-useful for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' ◮ For any constant εSDP > 0, there is no εSDP-SDP mechanism that is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='01-useful for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='5 On the Plausiblility of the Cryptographic Assumptions We now discuss the plausiblility of the three cryptographic assumptions we use for our result: (i) NIWI: Non-interactive Witness Indistinguishable Proofs (formally, Assumption 26) (ii) CRKHF: Collision-Resistant Keyless Hash Functions (formally, Assumption 18) (iii) diO-for-pcS: Differing-Inputs Obfuscation for Public-coin Samplers (formally, Assumption 22) Regarding (i), NIWI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Bitansky and Paneth [BP15a] show that NIWIs exist assuming one- way permutations exist and indistinguishability obfuscation (iO) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Recently, Jain, Lin, and Sahai [JLS21] show that the existence of iO follows from well-founded assumptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' consequently, NIWIs exist based on widely-believed assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' (We note that other previous works have also constructed NIWIs based on other more specific assumptions [BOV07, GOS12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=') Regarding (ii), CRKHF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Bitansky, Kalai, and Paneth [BKP18] defined CRKHFs to model the properties of existing hash functions like SHA-2 used in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' They suggest several candidates for CRKHFs, such as hash functions based on AES and Goldreich’s one-way functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' They also note that CRKHFs exist in the Random Oracle model, as a random function is a CRKHF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Still, it is an open question to base the security of a CRKHF on a standard cryptographic assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Part of the difficulty of doing this, as [BKP18] describe, is that most cryptographic assumptions involve some sort of structure that is useful for constructing cryptographic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In contrast, the goal of a CRKHF is to have no structure at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In summary, given the various CRKHF candi- dates, the existence in the Random Oracle model, and the fact that CRKHFs exist “in practice,” this assumption is quite plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' For our specific construction, we need a different hash length 8We remark that εSDP-SDP mechanism here refers to an ensemble of mechanisms {Mn} which are (εSDP, negl)-SDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' (See Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=') 6 (equivalently, different compression rate) than that used in [BKP18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' please refer to the discussion preceding Assumption 18 for the parameters and justification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Finally, we remark that, even though the existence of CRKHFs is not known to reduce to any “well-founded” assumption, even refuting their existence would answer a longstanding question in cryptography: giving non-contrived separations between the Random Oracle model [BR93] and the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In the words of Bitansky, Kalai, and Paneth [BKP18] “Any attack on the multi-collision resistance of a [keyless] cryptographic hash function would constitute a strong and natural separation between the hash and random oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' For several cryptographic hash functions used in practice, the only known separations from random oracles are highly contrived [CGH04].” Regarding (iii), diO-for-pcS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' One can think of diO [BGI+01, BGI+12] as an “extractable” strengthening of iO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' While iO has now become a widely-believed assumption [JLS21], the exis- tence of diO is controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Several papers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', [BP15b, GGHW17, BSW16]) cast doubt on the existence of diO, especially in the case where an arbitrary auxillary input is allowed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' we stress that all the negative results for diO hold for contrived auxillary inputs and/or distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' On the positive side, [BCP14] show that diO reduces to iO in special cases, such as when the number of differing-inputs is bounded by a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' More related to our result, [IPS15] gives a definition of public-coin diO that avoids the difficulties presented by earlier negative results regarding auxil- iary inputs, although [BP15b] presented some evidence against this definition in special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Our specific assumption of diO-for-pcS is in fact weaker than the assumption of public-coin diO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In the definition of public-coin diO, as in [IPS15], we start with any public-coin sampler (pcS), for which it is hard to find an input on which two circuits differ, even given the knowledge of all the randomness that underlies the circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The security of the obfuscation is required to hold even against adver- saries that know all the randomness that underlies the generation of the two circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' However, in our definition, the security of the obfuscation is required to hold only against adversaries that observes a single obfuscated circuit, which makes the assumption weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' See Appendix A for a more detailed discussion on comparison of this assumption with other diO assumptions in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Finally, we only use the existence of diO-for-pcS for a simple circuit family for our result, so even if general purpose diO-for-pcS does not exist, we think it is plausible that diO-for-pcS exists for the specific family of circuits we need for our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' (See Assumption 22 for the exact pcS family for which we require a diO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=') Final thoughts on our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In conclusion, we view each of our three assumptions as plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Moreover, each of assumptions has at least some evidence that is hard to refute: NIWIs exist based on a widely-believed assumption, refuting CRKHFs would require giving the first non-contrived separation between the standard and the Random Oracle model, and despite many attempts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', [BP15b, GGHW17, BSW16]) to refute diO, the question is still open, especially for the particular diO-for-pcS version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 3 Preliminaries A function g : N → R≥0 is said to be negligible if g(n) = n−ω(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Let PPT be an abbreviation for probabilistic polynomial-time Turing machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 7 For x ∈ {0, 1}n and r ∈ N, we use Br(x) to denote the (Hamming) ball of radius r around x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', {z ∈ {0, 1}n | ∥x − z∥1 ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Furthermore, we use diam(S) for a set S ⊆ {0, 1}n to denote the (Hamming) diameter of S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', maxx,x′ ∈S ∥x − x′∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='1 Dataset and Adjacency For a domain X, we view a dataset D as a histogram over the domain X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', D ∈ ZX ≥0 where Dx denotes the number of times x ∈ X appears in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The size of the dataset is defined as ∥D∥1 := � x∈X Dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We write X m as a shorthand for the set of all datasets of size m, and X ∗ for the set of all datasets over domain X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Two datasets are adjacent iff ∥D − D′∥1 = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', one of the datasets is a result of adding or removing a single row from the other dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='2 Mechanism, Utility Function, and Usefulness A mechanism M is a randomized algorithm that takes in a dataset D ∈ X ∗ and outputs an element from a set Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The utility of a mechanism is measured by a utility function u, which is a polynomial- time deterministic algorithm that takes in a dataset D ∈ X ∗ together with a response y ∈ Y and outputs 0 or 1 (whether the response is good for the dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We say that the mechanism M is α-useful for utility u iff Pr[u(D, M(D)) = 1] ≥ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Below, we will often discuss an ensemble M = {Mn}n∈N of mechanisms where9 Mn : X ∗ n → Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We say that an ensemble of mechanisms is efficient if Mn on input D ∈ X m n runs in time poly(n, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' For an ensemble u = {un}n∈N of utility functions and α = {αn ∈ [0, 1]}n∈N, we say that M is α- useful with respect to u iff Mn is αn-useful with respect to un for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' For brevity, we will sometimes refer to “ensemble of mechanisms” simply as “mechanism” and “ensemble of utility functions” simply as “utility function” when there is no ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='3 Notions of Differential Privacy We now define the notions of DP that will be used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' (Statistical) Differential Privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The standard (statistical) notion of DP can be defined in terms of the following notion of indistinguishability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Definition 6 (Statistical Indistinguishability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Distributions P, Q are said to be (ε, δ)-indistinguishable, denoted P ≈ε,δ Q, if for all events (measurable sets) E, it holds that Pr X∼P[X ∈ E] ≤ eε · Pr X∼Q[X ∈ E] + δ, and Pr X∼Q[X ∈ E] ≤ eε · Pr X∼P[X ∈ E] + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' For simplicity, we use ≈ε to denote ≈ε,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Definition 7 (Statistical Differential Privacy (SDP) [DMNS06, DKM+06]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' For ε, δ > 0, a mecha- nism M is said to be (ε, δ)-SDP if and only if for every pair D, D′ of adjacent datasets, we have that M(D) ≈ε,δ M(D′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We say that an ensemble M = {Mn}n∈N is ε-SDP for a sequence ε = {εn}n∈N if there exists a negligible sequence {δn}n∈N such that Mn is (εn, δn)-SDP for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We note that the above notation, which omits explicit δ for an ensemble of mechanisms, was also used by [BCV16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 9It is always implicitly assumed that Xn, Yn are of size poly(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 8 Computational Differential Privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The notion of computational DP relaxes the notion of indistinguishability to a computational version, where the privacy holds only with respect to computationally bounded adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Definition 8 (Computational Indistinguishability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Two ensembles of distributions P = {Pn}n∈N and Q = {Qn}n∈N, where Pn and Qn are supported over {0, 1}p(n) for some polynomial p(·), are said to be ε-computationally-indistinguishable for a sequence ε = {εn}n∈N, denoted P ≈c ε Q, if there exists a negligible function negl(·) such that for any PPT adversary A, it holds that Pr X∼Pn[A(X) = 1] ≤ eεn · Pr X∼Qn[A(X) = 1] + negl(n), and Pr X∼Qn[A(X) = 1] ≤ eεn · Pr X∼Pn[A(X) = 1] + negl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In the special case of ε = 0, we suppress the subscript and simply write P ≈c Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Throughout, when we refer to a sequence {(Dn, D′ n)}n∈N of adjacent datasets, it is always assumed that Dn ∈ X mn n , D′ n ∈ X m′ n n are of sizes mn, m′ n = poly(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Definition 9 (Computational Differential Privacy (CDP) [MPRV09]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' An ensemble M = {Mn}n∈N of mechanisms is said to be ε-CDP for a sequence ε = {εn}n∈N, if for any sequence {(Dn, D′ n)}n∈N of adjacent datasets, it holds that {Mn(Dn)}n∈N ≈c εn {Mn(D′ n)}n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' This definition is often referred to as indistinguishability-based CDP (IND-CDP) in previous works [MPRV09, GKY11, BCV16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Since we only use this notion for our main result, we refer to it simply as CDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The other definition of CDP used in previous works is simulation-based: Definition 10 (SIM-CDP [MPRV09]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' An ensemble M = (Mn)n∈N of mechanisms is said to be ε- SIM-CDP if there exists an (εn, 0)-SDP ensemble {M′ n}n∈N of mechanisms such that for any sequence {Dn ∈ X ∗ n}n∈N of datasets, with size of Dn being at most poly(n), it holds that Mn(Dn) ≈c M′ n(Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' It should be noted that SIM-CDP cannot be used for the separation we are looking for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Specif- ically, if {Mn}n∈N is ε-SIM-CDP, we may use {M′ n}n∈N as our ε-SDP mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Since the utility function runs in polynomial time, it follows immediately that, if {Mn}n∈N is α-useful, then {M′ n}n∈N is also (α − o(1))-useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Due to this, we will not consider SIM-CDP again in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Calculus of ≈ and ≈c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The following properties are well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Fact 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The notions of (ε, δ)-indistinguishability and ε-computational-indistinguishability satisfy: ◮ Basic Composition: If P0 ≈ε,δ P1 and P1 ≈ε′,δ′ P2, then P0 ≈ε+ε′,δ+δ′ P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Similarly, if P0 ≈c ε P1 and P1 ≈c ε′ P2, then P0 ≈c ε+ε′ P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' ◮ Post-processing: If P ≈ε,δ Q, then for all (randomized) functions f, it holds that f(P) ≈ε,δ f(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Similarly, if P ≈c ε Q, then for all PPT algorithms A, it holds that A(P) ≈c ε A(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 4 Low Diameter Set Problem and Nearby Point Problem In this section, we introduce the problems that we will use in our separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Before that, we will describe a simplifying assumption that we can make about the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='1 Simplification of Input Representation Recall that so far a dataset may contain multiple copies of an element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Below, however, it will be more convenient to only discuss the case where each element appears only once, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', D ∈ {0, 1}X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' This is sufficient since if we have a utility function u : {0, 1}X × Y → {0, 1} defined only on D ∈ {0, 1}X , we can easily define the utility function u : NX × Y → {0, 1} by u(D, r) = � u(D, r) if D ∈ {0, 1}X , 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In other words, the utility function considers any response good for datasets with repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Clearly, if u is efficiently computable, then so is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Furthermore, suppose that we have an ε-CDP mechanism M = {Mn}n∈N for u = {un}n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' For every dataset D, let D be defined by Di = min � Di, 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Then, we may define M = � Mn � n∈N by M(D) = M(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' It is simple to check that M remains ε-CDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Furthermore, if M is α-useful for u, then M remains α-useful for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Finally, note that a lower bound for DP algorithms restricted to non-repeated datasets trivially implies a lower bound against all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Due to this, we will henceforth focus our attention only on the datasets D ∈ {0, 1}X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Further- more, throughout the remainder of this paper, we will always pick Xn = [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' This further simplifies the input representation to be just a bit vector x ∈ {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We will define an input of our problem in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Furthermore, we will henceforth use x instead of D to denote the input dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='2 Nearby Point Problem We will start by defining our first problem, which asks to output a point that is close to the input point if the latter belongs to some set R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' As we noted in the introduction, when R is the set of all points (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', Rn = {0, 1}n), this is exactly the same as the problem considered in blatant non-privacy [DN03, DMT07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' As we will see later, the presence of the set R is due to our use of hashing, which is required in our proof for the CDP mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Definition 12 (τ-Nearby R-Point Problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The nearby point problem parameterized by sequences {τn ∈ N}n∈N and {Rn ⊆ {0, 1}n}n∈N is denoted by NBPτ,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' For input x ∈ {0, 1}n and output y ∈ Yn = {0, 1}n, the utility is defined as: uNBP τn,Rn(x, y) := 1 {∥x − y∥1 ≤ τn or x /∈ Rn} For brevity, we will assume throughout that Rn is efficiently recognizable and henceforth we do not state this explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Note that this assumption implies that the utility function defined above is efficiently computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The nearby point problem will be primarily used for proving the lower bounds against SDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='3 Verifiable Low Diameter Set Problem Next, we define circuit-based tasks for which we will give CDP mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' To do so, we need to first define a “τ-diameter verifier”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Definition 13 (τ-Diameter Verifier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' For a sequence τ = {τn}n∈N of integers, we say that an efficiently computable (deterministic) verifier V = {Vn}n∈N is a τ-diameter verifier for circuits of size s(n) if it takes as input a circuit C : {0, 1}n → {0, 1} of (polynomial) size s(n) and a proof π of size poly(n), and outputs Vn(C, π) = 1 only if diam(C−1(1)) ≤ τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 10 We can now define the (verifiable) low diameter set problem as follows: Definition 14 (Verifiable τ-Diameter R-Set Problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The verifiable low diameter set problem parameterized by sequences τ = {τn}n∈N, R = {Rn ⊆ {0, 1}n}n∈N, and τ-diameter verifier V = {Vn}n∈N is denoted by VLDSτ,R,V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The input, output, and utility are defined as follows: ◮ Input: x ∈ {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' ◮ Output: circuit C and a proof π, both of size poly(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' ◮ Utility: uVLDS τn,Rn,Vn(x, (C, π)) := 1 {C(x) = 1 or x /∈ Rn} and 1 {Vn(C, π) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' For convenience, we also define the following utility function ueval R (x, C) := 1 {C(x) = 1 or x /∈ R} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Note that this does not correspond to a hard task, because a circuit that always outputs one is 1-useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Nonetheless, it will be convenient to state usefulness of some intermediate algorithms via this utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='4 From Low Diameter Set Problem to Nearby Point Problem Below we provide a simple observation that reduces the task of proving an SDP lower bound for the verifiable low diameter set problem to that of the nearby point problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' (Note here that the SDP mechanisms considered below can be computationally inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=') Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' If there is an (ε, δ)-SDP α-useful mechanism for the VLDSτ,R,V problem, then there is an (ε, δ)-SDP α-useful mechanism for the NBPτ,R problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Let M be an (ε, δ)-SDP α-useful mechansim for the VLDSτ,R,V problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We will construct an (ε, δ)-SDP α-useful mechanism M′ for the NBPτ,R problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The mechanism M′ n on input dataset x ∈ {0, 1}n works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' First, let (C, π) ← Mn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' If Vn(C, π) = 1, then output the lexicographically first element of C−1(1) (else, output 0n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' This completes our description of M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Since M is (ε, δ)-SDP, we have that M′ is also (ε, δ)-SDP by post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' It remains to show that M′ is α-useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Fix some input x ∈ {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' If x /∈ Rn, then any output satisfies utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Thus, it suffices to consider the case where x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' With probability α, we have that Vn(C, π) = 1 (which implies that C−1(1) has diameter at most τn), and x ∈ C−1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Consequently, the distance between x and the lexicographically first element of C−1(1) is at most τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' So with probability at least α, the output of M′ is useful for x, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 5 CDP Mechanism for Verifiable Low Diameter Set Problem In this section we build a CDP mechanism for the verifiable low diameter set problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We establish the following result: Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Suppose that Assumptions 18, 22 and 26 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Then, for all constant εCDP > 0 and τ = � τn = n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='9� n∈N, there exists a τ-diameter verifier V and a sequence R = {Rn}n∈N of sets of sizes |Rn| ≥ 2n/no(log n), such that there exists an εCDP-CDP mechanism that is (1 − on(1))-useful for uVLDS τ,R,V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' As discussed in the overview, we first build a mechanism that is CDP but without verifiability using collision-resistant keyless hash functions and differing-inputs obfuscators (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We then turn it into a verifiable one using non-interactive witness indistinguishable proofs (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='1 CDP Mechanism without Verifiability In this section, we construct our first CDP mechanism (Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We depart from the overview in Section 2 slightly and do not prove a non-adaptive query lower bound explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Instead, we directly show in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='2 how to sample the appropriate differing-inputs circuit family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' This can be then easily turned into our CDP mechanism via diO in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='1 Additional Preliminaries: Cryptographic Primitives Throughout this section, we will repeatedly use the so-called randomized response (RR) mecha- nism [War65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Specifically, RRε is an algorithm that takes in x ∈ {0, 1}n and outputs ˜x ∈ {0, 1}n, where ˜xi = xi with probability eε 1+eε independently for each i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' It is well-known (and very simple to verify) that RRε is ε-SDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Collision-Resistant Keyless Hash Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In our construction, we will use the Collision- Resistant Keyless Hash Functions (CRKHFs) [BKP18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The formal definition is as given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Definition 17 (Collision-Resistant Keyless Hash Functions [BKP18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' A sequence of hash func- tions � Hn : {0, 1}n → {0, 1}γ(n)� n∈N is K-collision resistant for advice length ζ for sequences K = {Kn}n∈N, ζ = {ζn}n∈N if, for any PPT A and a sequence {zn}n∈N of advices where |zn| = ζn, we must have Pr (Y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=',YKn)←A(1n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='zn) [Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' , YKn are distinct and Hn(Y1) = · · · = Hn(YKn)] ≤ negl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We will skip the subscript n whenever it is clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' In [BKP18], the hash value length γ(n) is assumed to be either linear, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', γ(n) = Ω(n), or polynomial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=', γ(n) = nΘ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' However, we need a collision-resistant hash function with a much smaller γ(n), namely O(log2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We remark that this is still very much plausible: as long as γ(n) is ω(log n), the “guess-and-check” algorithm will only produce a collision with only negligible probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' A more precise statement of our assumption is stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Assumption 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' There is an efficiently computable sequence H = {Hn}n∈N of hash functions with hash value length γ(n) = o(log2 n) such that, for any constant c1 > 0, there exists a constant c2 > 0 such that the hash function sequence is K-collision resistant for advice length ζ where K(n) = nc2 and ζ(n) = nc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We remark that, for the existence of CDP mechanism (shown in this section), we will only use the multi-collision-resistance without relying on the assumption on the value of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The latter is only used to show that no SDP mechanism exists for the problem (Section 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Differing-Inputs Obfuscators for Public-Coin Samplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' For any two circuits C0 and C1, a differing-inputs obfuscator diO [BGI+12] guarantees that the non-existence of an efficient ad- versary that can find an input on which C0 and C1 differ implies that diO(C0) and diO(C1) are computationally indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' For our application, it even suffices to assume a weaker notion, namely that of differing-inputs obfuscator for public-coin samplers, as defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 12 Definition 19 (Public-Coin Differing-Inputs Circuit Sampler).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' An efficient non-uniform sampling algorithm Sampler = {Samplern} is a public-coin differing-inputs sampler for the parameterized collection C = {Cn} of circuits if the output of Samplern is distributed over Cn × Cn and for every efficient non-uniform algorithm A = {An}, there exists a negligible function negl(·) such that for all n ∈ N: Pr θ [C0(y) ̸= C1(y) : (C0, C1) ← Samplern(θ), y ← An(θ)] ≤ negl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Here, Samplern is a deterministic algorithm and the only source of randomness is the seed θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Definition 20 (Differing-Inputs Obfuscator for Public-Coin Samplers (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' [IPS15])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' A uniform PPT diO is a differing-inputs obfuscator for public-coin samplers for the parameterized circuit family C = {Cn} if the following conditions are satisfied: ◮ Correctness: For all n ∈ N, for all C ∈ Cn, for all inputs y, we have that Pr[C′(y) = C(y) : C′ ← diO(1n, C)] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' ◮ Polynomial slowdown: There exists a universal polynomial p(·) such that for all C ∈ Cn, it holds that Pr[|C′| ≤ p(|C|) : C′ ← diO(1n, C)] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' ◮ Differing-inputs: For every public-coin differing inputs sampler Sampler = {Samplern} for C, and every (not necessarily uniform) PPT distinguisher D = {Dn}, there exists a negligible function negl such that the following holds for all n ∈ N: For (C0, C1) ← Samplern(θ) | Pr θ [Dn(diO(1n, C0)) = 1] − Pr θ [Dn(diO(1n, C1)) = 1]| ≤ negl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We note that the notion of diO-for-pcS is in fact weaker than the notion of general public-coin diO as given by [IPS15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We elaborate on this comparison in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Whenever n is clear from context, we use diO(C) to denote diO(1n, C) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' When we want to be explicit about the randomness ρ (of poly(n) bit length) used by diO we will denote it as diOρ(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We only need the existence of a differing-inputs obfuscator for a specific family of circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' This cir- cuit family will be defined later and therefore we defer formalizing our assumption to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='2 Public-Coin Differing-Inputs Circuits from CRKHFs The first step of our proof is to construct a differing-inputs circuit family based on CRKHFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Our sampler is described in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' We next prove that the above sampler is a public-coin differing-inputs sampler, which means that any efficient adversary, even with the knowledge of ˜x (which is the only source of randomness), cannot find an input on which C0 and C1 differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' The proof starts by noticing that any input that differentiates C0, C1 must, by definition of the circuits, have hash value υn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Therefore, if there were an adversary that can find a differing input, then we could run it multiple times to get Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' , YK that have the same hash value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' (See Algorithm 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=') However, our proof is not finished yet, since it is possible that Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' , YK are not distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Indeed, the crux of the construction is that, due to how we select ˜x and define the circuits, a fixed Y will be a differing input with negligible probability10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' It follows that Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' , YK must be distinct w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' This is formalized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 10It is also simple to see that this property suffices to prove a non-adaptive query lower bound as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' 13 Algorithm 1 Differing-Inputs Circuit Family Sampler LDS-Samplern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Parameters: Adjacent datasets x, x′ ∈ {0, 1}n, hash value υn ∈ {0, 1}γ(n), privacy parameter ε > 0, radius r, ˜r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Randomness: θ ∼ RRε(0n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Output: Circuits C0, C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' ˜x ← x ⊕ θ (bit-wise XOR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' equivalent to RRε(x)) C0 ← circuit that takes in z and computes 1 � z ∈ Br(x) ∩ B˜r(˜x) ∩ H−1 n (υn) � C1 ← circuit that takes in z and computes 1 � z ∈ Br(x′) ∩ B˜r(˜x) ∩ H−1 n (υn) � return (C0, C1) Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Let H be as in Assumption 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' For any constant ε > 0, choosing r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='5n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='9 and ˜r = 1 1+eε n + n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content='6 makes LDS-Samplern (Algorithm 1) a public-coin differing-inputs sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Suppose for the sake of contradiction that for some adjacent x, x′ ∈ {0, 1}n, there exists a PPT ADI such that Pr θ [C0(y) ̸= C1(y) : (C0, C1) ← LDS-Samplern(θ), y ← ADI n (θ)] ≥ n−c, (1) for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Furthermore, let c1 be such that the total size of the descriptions of ADI n , LDS-Samplern is at most nc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Finally, let c2 > 0 be as in Assumption 18 and K = nc2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Algorithm 2 Collision-Resistant Hash Function Adversary ACRH n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Parameter: The target number of collisions K ∈ N, constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Advice: Descriptions of ADI n , LDS-Samplern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Output: Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' , YK ∈ {0, 1}n or ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' i ← 0 for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' , K · nc+1 do θj ← RRε(0n) (Cj 0, Cj 1) ← LDS-Samplern(θj) yj ← ADI n (θj) if Cj 0(yj) ̸= Cj 1(yj) then i ← i + 1 Yi ← yj if i ≥ K then break if i < K then return ⊥ else return Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' , YK Consider the adversary ACRH n for collision-resistant hash function described in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' First, note that by (1) and a standard concentration inequality, the probability that ACRH n outputs ⊥ is on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Furthermore, notice that C0, C1 can differ on y only if Hn(y) = υn, meaning that Hn(Yi) = υn always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Therefore, it suffices for us to show that the probability that Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' , YK are 14 distinct is 1 − on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' By a union bound, we have that ACRH n violates the collision-resistance of H as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' Thus, we are only left to show that Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' , YK are not distinct with probability o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' To see that this is the case, notice that Pr[Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfTfcL/content/2301.00104v1.pdf'} +page_content=' , YK are not distinct] ≤ � 1≤i1 0 if for all t > 0, Pr(|x − E(x)| ≥ t) ≤ 2e−t2/τ 2 +x. +Definition 2 (Well-conditioned covariance matrix). A covariance matrix Σ is well-conditioned +if there is a positive constant d0 such that 0 < d−1 +0 +≤ λmin(Σ) ≤ λmax(Σ) ≤ d0 < ∞. +Definition 3 (Strongly asymptotically unbiased estimate). Let ˆθ be a consistent estimate +of θ with an asymptotic normal distribution √sn(ˆθ − θ) +D +−→ N(µθ, Σθ), where µθ is a +vector with a bounded ℓ2-norm, Σθ is a well-conditioned covariance matrix, and sn is a +sequence of n. Then ˆθ is called a strongly asymptotically unbiased estimate of θ if µθ = 0. +A sub-Gaussian variable is one of the basic concepts in modern statistics (Vershynin, +2018). It generalizes the scope of ordinary Gaussian variables to include all bounded dis- +crete and common continuous variables. The well-conditioned covariance matrix is another +important concept (Bickel and Levina, 2008). A well-conditioned covariance matrix will +ensure that the related statistical optimization is nondegenerate. In addition, we define the +strongly asymptotic unbiasedness to distinguish the consistent estimate whose bias square +vanishes with an equal and a smaller rate than its variance, respectively. If an estimate is +consistent but its bias square and variance vanish at the same rate, the classic confidence +interval cannot cover the true parameter with a probability of 0.95, thus leading to invalid +statistical inference. This problem widely exists in all fields of statistics, especially, in non- +parametric statistics and high-dimensional statistics, and many novel methods are derived +to reduce the bias such that the bias square vanishes faster than the variance (Hall, 1992; +Van de Geer et al., 2014; Jankova and Van De Geer, 2018; Calonico et al., 2018). +Condition 1 (Regularity conditions for multivariable MR). +(C1) For gi = (gi1, . . . , gim)⊤, each entry gij is a bounded sub-Gaussian variable with +E(gij) = 0, var(gij)=1, and sub-Gaussian parameter τg ∈ (0, ∞). For all (i, j) ̸= +(t, s), gij is independent of gts. +(C2) For ui = (ui1, . . . , uip)⊤, each entry uij is a sub-Gaussian variable with E(uij) = 0, +var(uis) ∈ (0, ∞), and sub-Gaussian parameter τu ∈ (0, ∞); vi is a sub-Gaussian +variable with E(vi) = 0, var(vi) ∈ (0, ∞), and sub-Gaussian parameter τv ∈ (0, ∞). +9 + +Besides, (u⊤ +i , vi)⊤ is independent of (u⊤ +t , vt)⊤ for all i ̸= t. Furthermore, Σu×v is a +well-conditioned covariance matrix of (u⊤ +i , vi)⊤. +(C3) For βj = (βj1, . . . , βjp)⊤, √mβjs is a sub-Gaussian variable with E(√mβjs) = 0, +var(√mβjs) ∈ (0, ∞), and sub-Gaussian parameter τβ ∈ (0, ∞). For all j ̸= t, βj +is independent of βt. In addition, Ψββ is a well-conditioned covariance matrix of +√mβj. +(C4) The genetic variant gij, the genetic effect βj, the noise terms ui and vi, are three +mutually independent groups. +Conditions (C1)-(C4) restrict that all variables involved in this paper are sub-Gaussian +distributed. +In practice, gij is standardized from a binomial variable with status 0, 1, +and 2. Hence, it is supposedly a bounded sub-Gaussian variable as long as its minor allele +frequency is not rare. Besides, we assume √mβj to be sub-Gaussian with a well-conditioned +covariance matrix Ψββ because the covariance explained by each variant Σββ decreases as +the number of instrumental variants m increases. +Theorem 1. Denote wαj = ˆαj − αj and ωjs = ˆβjs − βjs, s = 1, . . . , p. Then for all j, +� +� +� +� +� +√n0wαj +√n1wβ1j +... +√npwβ1p +� +� +� +� +� +D +−→ N +� +� +� +� +� +� +� +� +� +� +� +0 +0 +... +0 +� +� +� +� +� , +� +� +� +� +� +� +σyy +n01 +√n0n1σyx1 +· · · +n01 +√n0npσyxp +n01 +√n0n1σyx1 +σx1x1 +· · · +n1p +√n1npσx1xp +... +... +... +... +n0p +√n0npσyxp +n1p +√n1npσx1xp +· · · +σxpxp +� +� +� +� +� +� +� +� +� +� +� +� +, +if n0, . . . , np and m → ∞. +Theorem 1 demonstrates the asymptotic normal distribution of the estimation errors, +based on which we are able to obtain +ΣWβWβ = ∆xx ⊙ Σxx, +σWβwα = δxy ⊙ σxy, +σwαwα = σyy/n0, +(12) +where the (j, s)th element of ∆xx is njs/(njns) and the jth element of δxy is nj0/(n0nj). +As a result, the expectations of SIVW(θ) and HIVW are given by +E(SIVW(θ)) = (∆xx ⊙ Σxx)θ − δxy ⊙ σxy, +(13) +E(HIVW) = Σββ + ∆xx ⊙ Σxx. +(14) +By expressing σxy = Σxxθ + σuv, we obtain an alternative expectation of SIVW(θ)): +E(SIVW(θ)) +� +�� +� +estimation bias += {(∆xx − δxy1⊤) ⊙ Σxx}θ +� +�� +� +measurement error bias +− δxy ⊙ σuv +� +�� +� +confounder bias +. +(15) +From this expectation, it is clear that there are two sources of the estimation error bias: +{(∆xx −δxy1⊤)⊙Σxx}θ comes from the measurement error, while {δxy ⊙σuv} is caused by +the confounder. Here, we call {(∆xx −δxy1⊤)⊙Σxx}θ the measurement error bias because +it has the same statistical impact, i.e., shrinking the coefficient estimate toward zero, as in +10 + +measurement error analysis (Yi, 2017). In contrast, we term {δxy ⊙ σuv} the confounder +bias because σuv ̸= 0 implies that there are underlying confounders simultaneously affecting +both xi and yi. In addition, the overlapping fraction vector δxy trades off these two sources +of biases. Generally, the measurement error bias is dominant when the elements of δxy are +small, while the confounder bias dominates when the elements of δxy are large, and there +may exist a special sample overlap such that δxy⊙σuv = {(∆xx−δxy1⊤)⊙Σxx}θ. In univari- +able MR, this special fraction is n01/n0 = σxxθ/σxy, which guarantees that E(SIVW(θ)) = 0 +and E(ˆθIVW) = θ. This theoretical result explains why in the empirical studies (e.g., Figures +1 and 2 in Sadreev et al. (2021)), ˆθIVW has a negative bias when n01/n0 is small, positive +bias when n01/n0 is large, and is unbiased at this specific point. +Theorem 2. Suppose conditions (C1)-(C4) hold and m, nmin → ∞. Then +(i) if m/√nmin → 0, √nmin(ˆθIVW − θ) +D +−→ N(0, ψθΨ−1 +ββ); +(ii) if m/√nmin → c0, √nmin(ˆθIVW − θ) +D +−→ N(−c0Ψ−1 +ββ(ΨWβWβθ − ψWβwα), ψθΨ−1 +ββ); +(iii) if m/nmin → c0, ˆθIVW − θ +P +−→ −c0(Ψββ + c0ΨWβWβ)−1(ΨWβWβθ − ψWβwα); +(iv) if m/nmin → ∞, ˆθIVW +P +−→ Ψ+ +WβWβψWβwα; +where +ΨWβ×wα = +�ΨWβWβ +ψWβwα +ψ⊤ +Wβwα +ψwαwα +� += +lim +nmin→∞ +�nminΣWβWβ +nminσWβwα +nminσ⊤ +Wβwα +nminσwαwα +� +, +ψθ = ψwαwα + θ⊤ΨWβWβθ − 2θ⊤ψWβwα, and c0 is a positive constant. +Theorem 2 is one of two main theorems in this paper and points out four scenarios. +First, if m goes to infinity with a lower rate than √nmin, ˆθIVW is strongly asymptotically +unbiased. In other words, ˆθIVW is able to reliably infer causality only when the sample +size of GWAS data is quadratically larger than the number of IVs. On the other hand, the +asymptotic covariance matrix of ˆθIVW is the inverse of the cumulative covariance matrix +Ψββ = �m +j=1 cov(βj), therefore, it is optimal to include as many associated variants as +possible in order to have Ψββ large enough. In contrast, using a few top significant variants +to perform MR analysis is not recommended. +Second, if m tends to infinity with the same rate as √nmin, √nmin(ˆθIVW − θ) converges +to an asymptotic normal distribution with a non-zero asymptotic bias {−c0Ψ−1 +ββ(ΨWβWβθ− +ψWβwα)}. In this asymptotic bias, {−c0(ΨWβWβθ−ψWβwα)} is caused by SIVW(θ) and Ψ−1 +ββ +is resulted by H−1 +IVW. Since the asymptotic bias and asymptotic covariance matrix are of +the same order in this scenario, the inference made is invalid although the bias of ˆθIVW +is infinitesimal. Scenario (iii) is more serious than (ii) because the bias of ˆθIVW will not +vanish even when √nmin goes to infinity. In the fourth scenario, ˆθIVW converges to a term +irrelevant to θ. Scenarios (ii) - (iv) indicate that the IVW method is unlikely to make valid +causal inference unless the sample sizes are quadratically larger than the number of IVs. +It is crucial to understand the asymptotic behaviors of ˆθIVW since the IVW method +serves as the foundation for practically all MR techniques. Specifically, IMRP and MR- +PRESSO use hypothesis tests to identify invalid IVs and then apply the IVW method to +11 + +estimate causal effects based on valid IVs only. MR-Robust and MR-Median replace the +quadratic loss function used in IVW by a robust loss function and absolute loss function, +respectively. Although there have been literature studying the bias of ˆθIVW empirically +(Burgess et al., 2011, 2016), they could not explain what causes the bias and how it behaves +asymptotically. +In contrast, Theorem 2 points out the asymptotic properties of ˆθIVW, +representing a significant advance in understanding the IVW method and its extensions. +3 +Bias-corrected Estimating Equation +According to (11), it is possible to remove the bias of SIVW(θ) by subtracting the measure- +ment error bias {ΣWβWβθ−σWβwα}. Motivated by this principle, we propose MRBEE that +estimates the causal effect estimates by solving the new unbiased estimating equation. In +this section, we introduce the estimation of MRBEE, investigate its asymptotic properties, +and discuss three implementation issues including the estimations of the bias-correction +terms, the estimation of sandwich formula of causal effect estimate, and the detection of +potential pleiotropy. +3.1 +Estimation of causal effect +There are many methods that can remove the measurement error bias, including max- +imum likelihood estimation, unbiased estimating functions, and simulation-extrapolation +(SIMEX) methods; see, e.g., Yi (2017). MRBEE is a subtraction correction method belong- +ing to the class of unbiased estimating function methods. Specifically, MRBEE estimates +θ by solving the following unbiased estimating equation: +SBEE(θ) = SIVW(θ) − (ΣWβWβθ − σWβwα), +(16) +where SIVW(θ) = − ˆB⊤( ˆα − ˆBθ)/m. The solution ˆθBEE such that SBEE(ˆθBEE) = 0 is +ˆθBEE = +� ˆB⊤ ˆB +m +− ΣWβWβ +�−1� ˆB⊤ ˆα +m +− σWβwα +� +. +(17) +In practice, ˆθBEE is unreliable when the minimum eigenvalue of ˆB⊤ ˆB/m − ΣWβWβ is nega- +tive, which is also a common problem for subtraction correction methods. In this case, we +recommend first adjusting the negative eigenvalues to be 0 and then using the generalized +inverse of this semi-positive matrix to yield ˆθBEE. +Theorem 3. Suppose conditions (C1)-(C4) hold and m, nmin → ∞. Then +(i) if m/nmin → 0, √nmin(ˆθBEE − θ) +D +−→ N(0, ψθΨ−1 +ββ); +(ii) if m/nmin → c0, √nmin(ˆθBEE − θ) +D +−→ N(0, ψθΨ−1 +ββ + c0Ψ−1 +ββΨBCΨ−1 +ββ); +(iii) if m/nmin → ∞ and m/n2 +min → 0, +� +n2 +min/m(ˆθBEE − θ) +D +−→ N(0, Ψ−1 +ββΨBCΨ−1 +ββ); +12 + +where ψθ is defined in Theorem 2, c0 is a positive constant, and ΨBC is a semi-positive +symmetric matrix whose expression is shown in equation (91). +Theorem 3 indicates the following three scenarios. First, if m/n → 0,√nmin(ˆθBEE − θ) +converges to a normal distribution with a zero mean and the covariance matrix being +exactly the same as ˆθIVW. In other words, ˆθBEE not only enjoys the strongly asymptotic +unbiasedness but also loses no efficiency in comparison to ˆθIVW. Second, if m/nmin → c0 ∈ +(0, ∞), there is an additional covariance matrix c0Ψ−1 +ββΨBCΨ−1 +ββ in the asymptotic normal +distribution, where ΨBC is introduced by the bias-correction terms: +ΨBC = +lim +nmin→∞ var +�nmin +√m +� +(W⊤ +β Wβ − mΣWβWβ)θ − (W⊤ +β wα − mσWβwα) +�� +. +In this scenario, ˆθBEE is again strongly asymptotically unbiased with a convergence rate +√nmin, while ˆθIVW suffers from a bias not vanishing asymptotically. In the third scenario, +ˆθBEE is still strongly asymptotically unbiased with a convergence rate +� +n2 +min/m, and the +asymptotic distribution is dominated by the bias correction term. In contrast, ˆθIVW con- +verges to a term irrelevant to θ. Note that ˆθIVW is not consistent unless m/n → 0 and the +inference made by ˆθIVW is unreliable unless m/√nmin → 0. Therefore, MRBEE is superior +to IVW in terms of both unbiasedness and asymptotic validity. +Most previous works of MR introduced their methods from the perspective of empirical +applications and have not discussed the asymptotic properties; see, e.g., Bowden et al. +(2015, 2016); Verbanck et al. (2018); Morrison et al. (2020). Some works (Zhao et al., +2020; Ye et al., 2021) described the asymptotic behaviors of the causal effect estimates +yielded by their univariate MR methods, but the convergence rates and related conditions +were not straightforward. For example, Zhao et al. (2020) showed that the convergence +rate of their causal effect estimate is O(V1/√V2) where V1 and V2 are two m-concentrations, +which may mislead that this estimate has a O(√m) convergence rate. From Theorem 3, +it is easy to see that ˆθBEE is strongly asymptotically unbiased, the asymptotic covariance +matrix is ψθΨ−1 +ββ, ψθΨ−1 +ββ + c0Ψ−1 +ββΨBCΨ−1 +ββ, and Ψ−1 +ββΨBCΨ−1 +ββ, and the convergence rate +is √nmin, √nmin, and +� +n2 +min/m, with respect to scenarios (i), (ii), and (iii). In addition, +although our method focuses on the multivariable MR model, the theoretical results can +be readily extended to the univariable MR model. To the best of our knowledge, this is +the first theoretical work to demonstrate how the convergence rate and asymptotic normal +distributions vary with the sample sizes of multiple GWAS cohorts and the number of IVs +for univariable and multivariable MR. +3.2 +Estimation of bias-correction terms +In this subsection, we discuss how to estimate the bias-correction terms ΣWβWβ and σWβwα +in practice. Specifically, we apply the method provided by Zhu et al. (2015) to estimate the +covariance matrix ΣWβ×wα of the vector (w⊤ +βj, wαj)⊤ from insignificant GWAS summary +statistics. Let G{0} = (g{0} +ij )n1×M, . . . , G{p} = (g{p} +ij )ns×M be the sample matrices of M +insignificant and independent genetic variants. The insignificance means that the p-value +of the genetic variants are larger than 0.05 for all exposures and outcome, and independence +13 + +means that these variants are in LE. The insignificant GWAS statistics are estimated by +ˆα∗ +j = g{0}⊤ +j +y[0] +n0 +, +ˆβ∗ +js = g{s}⊤ +j +x[s] +ns +, +(18) +for s = 1, . . . , p. With these insignificant effect sizes, ΣWβ×wα can be estimated by +ˆΣWβ×wα = 1 +M +M +� +j=1 +(ˆβ∗ +j1, . . . , ˆβ∗ +jp, ˆα∗ +j)⊤(ˆβ∗ +j1, . . . , ˆβ∗ +jp, ˆα∗ +j), +(19) +because ˆα∗ +j and ˆβ∗ +js follow the same distributions of wαj and wβjs, respectively. Here, ˆΣWβWβ +is the first (p × p) sub-matrix of ˆΣWβ×wα and σWβwα consists of the first p − 1 elements of +the last column of ˆΣWβ×wα. +Theorem 4. Suppose conditions (C1)-(C4) hold. +Let g{s} +ij +satisfy the condition (C1), +E(x[s] +i |g{s} +ij ) = 0 for all 1 ≤ s ≤ p, and E(y[0] +i |g{0} +ij ) = 0. Then +∥Σ +− 1 +2 +Wβ×wα ˆΣWβ×wαΣ +− 1 +2 +Wβ×wα − Ip+1∥2 = OP +� 1 +√ +M +� +, +if nmin and M → ∞. +Theorem 4 shows that ˆΣWβ×wα has a O( +√ +M) convergence rate after adjusting the scale +of ΣWβ×wα. As there may be more than 1 million independent variants in the whole genome, +ˆΣWβ×wα has high precision. In addition, n0, n1, ..., np → ∞ are required such that √n0ˆα∗ +j +and √ns ˆβ∗ +js are asymptotically normally distributed. In addition, many popular GWAS +methods such as cross-phenotype association analysis (CPASSOC, Zhu et al. (2015)) and +multi-trait analysis of GWAS (MTAG, Turley et al. (2018)) need to estimate the covariance +matrix of the estimation errors of GWAS summary statistics. As far as we are concerned, +this theorem is the first one to theoretically guarantee that this covariance matrix can be +consistently estimated from the GWAS insignificant statistics. +3.3 +Estimation of sandwich formula +In this subsection, we illustrate how to estimate the covariance matrix of ˆθBEE, i.e., +cov(ˆθBEE)=ΣBEE(θ), through the famous sandwich formula (Liang and Zeger, 1986): +ΣBEE(θ) = F−1 +BEEVBEE(θ)F−1 +BEE. +(20) +Here, the outer matrix FBEE is the Fisher information matrix, i.e., the expectation of the +Hessian matrix of SBEE(θ): +FBEE = −E +�∂SBEE(θ) +∂θ⊤ +� += Σββ. +(21) +14 + +The inner matrix VBEE(θ) is the covariance matrix of SBEE(θ): +VBEE(θ) = E +� 1 +m +m +� +j=1 +Sj(θ)Sj(θ)⊤ +� +, +(22) +where +Sj(θ) = −(ˆαj − θ⊤ ˆβj) ˆβj − ΣWβWβθ + σWβwα. +(23) +A consistent estimate of ΣBEE(θ) is +ˆΣBEE(ˆθBEE) = ˆF +−1 +BEE ˆVBEE(ˆθBEE)ˆF +−1 +BEE, +(24) +where +ˆFBEE = +ˆB⊤ ˆB +m +− ˆΣWβWβ, +ˆVBEE(ˆθBEE) = 1 +m +m +� +j=1 +ˆSj(ˆθBEE) ˆSj(ˆθBEE)⊤ +ˆSj(ˆθBEE) = −(ˆαj − ˆθ⊤ +BEE ˆβj) ˆβj − ˆΣWβWβ ˆθBEE + ˆσWβwα, +(25) +and ˆΣWβWβ and ˆσWβwα are estimated through (19). +Theorem 5. Under the conditions of Theorem 4, +||Σ +− 1 +2 +BEE(θ) ˆΣBEE(ˆθBEE)Σ +− 1 +2 +BEE(θ) − Ip||2 = OP +� +max +� +1 +√nmin +, +√m +nmin +, +� +log m +m +�� +if nmin, m and M → ∞ and m/n2 +min → 0. +Theorem 5 shows that ˆΣBEE(θ) has a min(√nmin, +� +n2 +min/m, +� +m/ log m) convergence +rate when m/n2 +min → 0. The first two convergence rates are brought by ||ˆFBEE − FBEE||2, +while the third convergence rate is yielded by || ˆVBEE(ˆθBEE) − VBEE(θ)||2, where the non- +asymptotic analysis tool of random matrices are used to derive them (Vershynin, 2018). +Note that the SE estimation should be of the same importance as the causal effect estima- +tion. Although the inference is made based on an unbiased estimate, it could still be invalid +if the SE estimate is not reliable. Our simulations show that the vast majority of current +univariable and multivariable MR approaches are unable to provide accurate SE estimates, +e.g., MR-median consistently overestimates the SE and others have a tendency to under- +estimate it. In contrast, the sandwich formula, whose dependability has been extensively +investigated empirically, is a reliable technique to obtain the SE estimate for MRBEE. This +is yet another advantage of MRBEE over current approaches. +3.4 +Pleiotropy test +Due to the complexity of GWAS data, we cannot completely rule out the possibility of the +existence of UHP and CHP even in the case of modeling multiple exposures. Specifically, +15 + +if UHP and CHP exist, +αj = β⊤ +j θ + γuj + γcj, +(26) +where γuj is a UHP satisfying E(γujβj) = 0 and γcj is a CHP satisfying E(γcjβj) ̸= 0. +Conventional pleiotropy detection methods such as MR-Robust, MR-PRESSO, and IMRP +do not distinguish between UHP and CHP as long as they resemble outliers. Recently, +some novel methods such as CAUSE and MR-CUE have been developed to separate vertical +pleiotropy, UHP and CHP by using a mixture model, allowing slightly larger proportions +of UHP and CHP. However, both the conventional and novel methods only focus on one +exposure, failing to realize that most CHP and UHP may disappear automatically after +specifying all the relevant exposures. +In this paper, we assume that we have excluded all CHP by including all the relevant +exposures and we adopt IMRP (Zhu et al., 2021) to detect UHP. First, we define UHP as +γj = αj − β⊤ +j θ. +(27) +In particular, we assume that γj has a product structure γj = γ∗ +j bj, where γ∗ +j is a fixed +number and bj is a non-random binary indicator. Let O = {j : bj ̸= 0} be the set of UHP. +The number of elements in O (i.e., |O|) should be relatively small, otherwise the UHP +cannot be regarded as outliers. We specify the following variant-specific hypothesis test: +H0 : γj = 0, +v.s. +H1 : γj ̸= 0. +(28) +A natural estimate of γj is +ˆγj = ˆαj − ˆβ⊤ +j ˆθBEE = γj + ϵj. +(29) +where ϵj = wαj − w⊤ +βjθ + w⊤ +βj(ˆθBEE − θ). It is easy to see that E(ϵj) = 0 and var(ϵj) = +θ⊤ΣWβwαθ + σωγωγ − 2θ⊤σWβwα. As a result, tγj = ˆγ2 +j /var(ϵj) can be chosen as a feasible +testing statistic for the hypothesis in (28), which follows a central χ2 +1-distribution under +the null hypothesis. In practice, var(ϵj) can be estimated by +� +var(ϵj) = ˆϑ⊤ +BEESEj ˆRWβ×wαSEj ˆϑBEE, +(30) +where ˆϑBEE = (ˆθ⊤ +BEE, −1)⊤, SEj = diag(se(ˆβj1), . . . , se(ˆβjp), se(ˆαj)), and ˆRWβ×wα is the +correlation matrix of ˆΣWβ×wα. Then γj is considered as an outlier if +Fχ2 +1(ˆtγj) > κ, +(31) +where Fχ2 +1(·) is the CDF of χ2 +1-distribution, ˆtγj = ˆγ2 +j / � +var(ϵj), and κ is a given threshold. +Theorem 6. Assume that |O| is fixed and bounded and γ1∗, . . . , γ∗ +m are a series of non- +random numbers. Then under the conditions of Theorem 5, there exists a threshold κ = +Fχ2 +1(C0 log m) such that +Pr(O = ˆO) → 1 +16 + +where ˆO = {j : Fχ2 +1(ˆtγj) > κ} and C0 is a sufficiently large constant. +Theorem 6 indicates that there is a theoretical threshold κ = Fχ2 +1(C0 log m) to consis- +tently identify all UHP. This threshold increases with a rate O(log m) to reduce the false +discovery rate (FDR) and its concrete value can be chosen by a FDR control method (Ben- +jamini and Hochberg, 1995). In practice, MRBEE will iteratively apply the hypothesis test +(28) to remove the outliers and use the remaining IVs to estimate θ. The stable estimate +is regarded as ˆθBEE. +4 +Simulation +In this section, we conduct numerical comparisons between MRBEE and existing MR +methods. Full details of simulation settings and additional simulation results are shown in +the supplementary material. +4.1 +Univariable MR investigation +We briefly introduce the simulation settings for univariable MR. First, we generate a bi- +nomial variable from Binom(2, bj) where bj ∼ Unif(0.05, 0.5) and standardize it as gij, +the direct effect βj from N(0, 1/m), and ui, vi from a normal distribution with correla- +tion coefficient 0.5. The variances of ui and vi are chosen such that the IV-heritabilities +are σββ/σxx = 0.3 and θ2 × (σββ/σyy) = 0.15, respectively. We specify the causal effect +θ = 0.3/ +√ +2. We compare MRBEE with IVW, DIVW, MR-Egger, MR-Lasso, MR-Median, +IMRP, MR-ConMix, and MR-MiX, where most are implemented by using the R package +MendelianRandomization (Yavorska and Burgess, 2017). +Additionally, the IMRP pro- +cedure is incorporated into MRBEE in which the threshold κ is chosen by R package +FDRestimation (Murray and Blume, 2020). The so-called overlapping fraction is n01/n0, +where the special fraction such that E(SIVW(θ)) = 0 is n01/n0 ≈ 0.77. The number of +independent replications is 1000. +First, we study the influences of overlapping fraction n01/n0 and the number of IVs m, +with the results displayed in Figure 2. Here, we fix n0 = n1 = 20000, specify n01 according +to the overlapping fraction, and assume no UHP or CHP. It is easy to see that in general, +only MRBEE is able to yield an unbiased estimate of θ. For a special overlapping fraction +(placed in the second column of Figure 2), all approaches become unbiased except DIVW. +DIVW performs badly because it will further remove IVs based on their significance levels +and consequently introduces an extra IV selection bias. +In addition, the SE of causal +effect estimate for all methods increases as the overlapping fraction decreases but remains +unchanged by the increase of m. The results are consistent with our theoretical expectation +and asymptotic properties of MRBEE. +As for the standard error, we display the boxplot of ˆse(ˆθ)−se(ˆθ) where se(ˆθ) is approxi- +mated by the empirical SE calculated from the independent replications. It is evident that +the SE estimates produced by all approaches have reduced variances as m grows. However, +only MRBEE and DIVW can provide consistent SE estimates, confirming the accuracy +of MRBEE and DIVW’s SE formulas. Additionally, MR-ConMix is extremely likely to +17 + +0.16 +0.18 +0.20 +0.22 +0.24 +0.26 +0.16 +0.18 +0.20 +0.22 +0.24 +0.26 +0.16 +0.18 +0.20 +0.22 +0.24 +0.26 +θ^ +−0.010 +−0.005 +0.000 +0.005 +0.010 +−0.010 +−0.005 +0.000 +0.005 +0.010 +−0.010 +−0.005 +0.000 +0.005 +0.010 +sd^ (θ^) − sd(θ^) +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +coverage frequency +overlapping fraction = 100% +overlapping fraction = 77% +overlapping fraction = 50% +overlapping fraction = 0% +Causal Effect Estimate +overlapping fraction = 100% +overlapping fraction = 77% +overlapping fraction = 50% +overlapping fraction = 0% +Standard Deviation Estimate +overlapping fraction = 100% +overlapping fraction = 77% +overlapping fraction = 50% +overlapping fraction = 0% +Coverage Frequency +bee +ivw +divw +egger +lasso median +imrp +conmix +mix +bee +ivw +divw +egger +lasso median +imrp +conmix +mix +bee +ivw +divw +egger +lasso median +imrp +conmix +mix +bee +ivw +divw +egger +lasso median +imrp +conmix +mix +bee +ivw +divw +egger +lasso median +imrp +conmix +mix +bee +ivw +divw +egger +lasso median +imrp +conmix +mix +bee +ivw +divw +egger +lasso median +imrp +conmix +mix +bee +ivw +divw +egger +lasso median +imrp +conmix +mix +bee +ivw +divw +egger +lasso median +imrp +conmix +mix +bee +ivw +divw +egger +lasso median +imrp +conmix +mix +bee +ivw +divw +egger +lasso median +imrp +conmix +mix +bee +ivw +divw +egger +lasso median +imrp +conmix +mix +m = 250 +m = 500 +m = 1000 +m = 250 +m = 500 +m = 1000 +m = 250 +m = 500 +m = 1000 +Figure 2: Investigation of MR methods for univarate MR with sample sizes n0 = n1 = 20000, in terms +of overlapping fraction and number of instrumental variants. +18 + +underestimate the standard error, while MR-Egger, MR-Lasso, MR-Median, and MR-Mix +constantly overestimate it. As for IVW, it underestimates the SE when the fraction is large +and overestimates it when the fraction is small. +The coverage frequency refers to the frequency that the confidence interval covers the +true causal effect among simulations. Here, this confidence interval is constructed by dou- +bling ˆse(ˆθ), which means that the coverage frequency corresponding to neither an inflated +type-I error nor an inflated type-II error should be around 0.95. We observed that only MR- +BEE enjoys a coverage frequency around 0.95. When m = 250, MR-Egger, MR-Lasso, and +MR-Median suffer from inflated type-II error rates, likely because these methods cannot +estimate the SE properly. These approaches also result in inflated-type I error rates caused +by weak instrument bias as m increases. Additionally, because MR-Mix overestimates the +SE, it consistently exhibits a substantially inflated type-II error rate. Furthermore, IMRP +and MR-ConMix consistently have inflated type I error rates because they frequently un- +derestimate the SE. +We next verify if the asymptotic normal distributions in Theorem 2 and Theorem 3 +are correct. For a general estimate ˆθ, the asymptotic bias and SE are √sn(ˆθ − θ) and +√snse(ˆθ), respectively, where √sn is the convergence rate of ˆθ. If this estimate is strongly +asymptotically unbiased, the asymptotic bias sn(ˆθ − θ) should also be 0. Besides, if two +estimates have equal asymptotic SEs, they are equally powerful in terms of statistical +efficiency. We select MRBEE, IVW, MR-Median, and MR-Lasso to compare, only consider +two overlapping fractions: 100% and 0%, set n0 = n1 = nmin, and fix the causal effect +θ = 0.5. As for m and nmin, we focus on the following four cases: +(1) m = 2500, 5000, . . . , 50000 and m0.9/n = c0 = 0.1 and 0.2; we examine the direct +bias: ˆθ − θ, asymptotic SE: +� +n2 +min/m se(ˆθ), and coverage frequency; +(2) m = 250, 500, . . . , 5000 and m/n = c0 = 0.1 and 0.2; we examine the direct bias: +ˆθ − θ, asymptotic SE: √nmin se(ˆθ), and coverage frequency; +(3) m = 250, 500, . . . , 5000 and m2/n = c0 = 5 and 10; we examine the asymptotic bias: +√nmin(ˆθ − θ), asymptotic SE: √nmin se(ˆθ), and coverage frequency; +(4) m = 250, 500, . . . , 5000 and m3/n = c0 = 5 and 10; we examine the asymptotic bias: +√nmin(ˆθ − θ), asymptotic SE: √nmin se(ˆθ), and coverage frequency. +Note that we directly generate the estimation errors Wβ and wα according to Theorem 1 +because nmin in cases (3) and (4) can be larger than one million. The calculations involving +individual-data are extremely time-consuming in these cases. +Figure 3 demonstrates the simulation results. +In case (1), ˆθBEE is unbiased while +the other three estimates suffer from non-removable biases. As for the asymptotic SE, +� +n2 +min/m se(ˆθBEE) remains unchanged when nmin and m are sufficiently large (e.g., the +bars colored in blue), verifying conclusion (iii) in Theorem 3. However, the coverage fre- +quency of MRBEE is a little larger than 0.95, meaning that the SE of ˆθBEE is overestimated +in this extreme case. This phenomenon is reasonable because Theorem 4 points out that +the convergence rate of the sandwich formula is min(√nmin, +� +n2 +min/m, +� +m/ log m), which +slows down as m increases. In case (2), the direct bias of ˆθIVW is unchanged as nmin tends +19 + +−0.3 +−0.2 +−0.1 +0.0 +0.1 +−0.3 +−0.2 +−0.1 +0.0 +0.1 +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +−0.2 +−0.1 +0.0 +−0.2 +−0.1 +0.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +−4 +−2 +0 +−4 +−2 +0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +−0.50 +−0.25 +0.00 +0.25 +−0.50 +−0.25 +0.00 +0.25 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0% overlap +100% overlap +direct bias of θ^ ( m(0.9) n is fixed) +0% overlap +100% overlap +asymptotic se of θ^ ( m(0.9) n is fixed) +0% overlap +100% overlap +coverage frequency of θ^ ( m(0.9) n is fixed) +0% overlap +100% overlap +direct bias of θ^ (m/n is fixed) +0% overlap +100% overlap +asymptotic se of θ^ (m/n is fixed) +0% overlap +100% overlap +coverage frequency of θ^ (m/n is fixed) +0% overlap +100% overlap +asymptotic bias of θ^ ( m2 n is fixed) +0% overlap +100% overlap +asymptotic se of θ^ ( m2 n is fixed) +0% overlap +100% overlap +coverage frequency of θ^ ( m2 n is fixed) +0% overlap +100% overlap +asymptotic bias of θ^ ( m3 n is fixed) +0% overlap +100% overlap +asymptotic se of θ^ ( m3 n is fixed) +0% overlap +100% overlap +coverage frequency of θ^ ( m3 n is fixed) +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +MR−BEE +IVW +MR−Median MR−Lasso +m^(0.9)/n = 0.1 +m^(0.9)/n = 0.2 +m^(0.9)/n = 0.1 +m^(0.9)/n = 0.2 +m^(0.9)/n = 0.1 +m^(0.9)/n = 0.2 +n +11.43K +21.33K +30.73K +39.81K +48.67K +57.34K +65.88K +74.29K +82.6K +90.81K +98.95K +107.01K +115K +122.93K +130.81K +138.63K +146.4K +154.13K +161.82K +169.46K +m/n = 0.1 +m/n = 0.2 +m/n = 0.1 +m/n = 0.2 +m/n = 0.1 +m/n = 0.2 +n +2.5K +5K +7.5K +10K +12.5K +15K +17.5K +20K +22.5K +25K +27.5K +30K +32.5K +35K +37.5K +40K +42.5K +45K +47.5K +50K +m^2/n = 10 +m^2/n = 5 +m^2/n = 10 +m^2/n = 5 +m^2/n = 10 +m^2/n = 5 +n +6.25K +25K +56.25K +100K +156.25K +225K +306.25K +400K +506.25K +625K +756.25K +900K +1056.25K +1225K +1406.25K +1600K +1806.25K +2025K +2256.25K +2500K +m^3/n = 0.1 +m^3/n = 0.2 +m^3/n = 0.1 +m^3/n = 0.2 +m^3/n = 0.1 +m^3/n = 0.2 +n +0.025M +0.2M +0.675M +1.6M +3.125M +5.4M +8.575M +12.8M +18.225M +25M +33.275M +43.2M +54.925M +68.6M +84.375M +102.4M +122.825M +145.8M +171.475M +200M +Figure 3: Investigations of MRBEE and IVW in terms of asymptotic bias and covariance matrix. +20 + +to infinity, confirming conclusion (iii) in Theorem 2. As for ˆθBEE, its asymptotic SE is a +little larger than ˆθIVW, verifying item (ii) in Theorem 3. +In case (3), the asymptotic bias of ˆθIVW is constant as nmin goes to infinity, illustrating +that ˆθIVW is not strongly asymptotically unbiased. As a result, the coverage frequencies +of ˆθIVW are significantly smaller than 0.95, confirming our claim that any inference made +based on ˆθIVW is invalid. Besides, the asymptotic SEs of ˆθBEE and ˆθIVW are essentially +the same, indicating that ˆθBEE and ˆθIVW are equally efficient as long as m/nmin → 0. In +case (4), the asymptotic bias of IVW, MR-Median, and MR-Lasso vanish as nmin increases +and their coverage frequencies are around 0.95, which is consistent with conclusion (i) +in Theorem 2. The equal asymptotic SEs also indicate that ˆθBEE and ˆθIVW are equally +efficient in this scenario. In addition, IVW, MR-Median, and MR-Lasso suffer from the +same degree of bias when there is no pleiotropy, while MR-Median not only suffers from a +large asymptotic SE but also is likely to overestimate it. To understand why MR-Median +is always less efficient than IVW when there is no pleiotropy, its asymptotic behavior is +worthy of future investigation. +4.2 +Multivariable MR investigation +For multivariable MR, we consider p = 6 exposures and set the causal effect vector to be +θ = (0.3, 0.3, −0.3, −0.3, 0, 0)⊤. All of the exposures’ IV-heritabilities are 0.3, while the +outcome’s IV-heritability is 0.15. We set an AR(1) structured genetic correlation matrix +with coefficient ρ = −0.5 for the genetic effect βj, while considering a more intricate +correlation structure for the noise terms ui and vi. In order to better mimic real data +analysis, we take into account the scenario of completely overlapping GWAS samples (i.e., +nsk = ns = nk for all s, k). Other cases of sample overlaps and details of the simulation +settings are present in the supplementary materials. +Figure 4 presents the comparison between the multivariable versions of IVW, MR- +Egger, MR-Lasso, MR-Median, and MRBEE. In general, MRBEE is the only method that +can produce unbiased causal effect estimates in all cases. As m increases, the SE of ˆθBEE +remains the same, while the estimation error of the SE estimate becomes smaller. However, +a very large m may conversely reduce the accuracy of the SE estimate in multivariable +MR. For example, the SE estimates of all approaches in the cases of m = 1000 have +larger empirical variances than those in the cases of m = 500. This phenomenon can be +explained by Theorem 5, which indicates that the convergence rate of the sandwich formula +is min(√nmin, +� +n2 +min/m, +� +m/ log m). Hence, a larger m may result in a worse SE estimate +if nmin is not increased as m. +All the multivariable MR methods except MRBEE suffer from larger weak instrument +biases with the increase of m. The SE estimates provided by these methods, in particular +MR-Median, are less reliable than that of MRBEE. Thus, causal inferences based on the +existing multivariable MR methods could be even more unreliable than univariable MR +methods. In addition, ˆθIVW can have a bias toward any direction in multivariable MR. +For example, the bias of ˆθ5,IVW is positive while the bias of ˆθ6,IVW is negative. The actual +directions are jointly determined by the correlations of confounders and genetic effects. +We also examine the impact of omitting some important exposures. We conduct simu- +21 + +0.24 +0.28 +0.32 +0.24 +0.28 +0.32 +0.24 +0.28 +0.32 +θ^ +−0.35 +−0.30 +−0.25 +−0.20 +−0.35 +−0.30 +−0.25 +−0.20 +−0.35 +−0.30 +−0.25 +−0.20 +θ^ +−0.05 +0.00 +0.05 +−0.05 +0.00 +0.05 +−0.05 +0.00 +0.05 +θ^ +−0.01 +0.00 +0.01 +0.02 +−0.01 +0.00 +0.01 +0.02 +−0.01 +0.00 +0.01 +0.02 +sd^ (θ^) − sd(θ^) +−0.01 +0.00 +0.01 +0.02 +−0.01 +0.00 +0.01 +0.02 +−0.01 +0.00 +0.01 +0.02 +sd^ (θ^) − sd(θ^) +−0.01 +0.00 +0.01 +0.02 +−0.01 +0.00 +0.01 +0.02 +−0.01 +0.00 +0.01 +0.02 +sd^ (θ^) − sd(θ^) +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +coverage frequency +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +coverage frequency +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +coverage frequency +θ^ +1 +θ^ +2 +Causal Effect Estimate +θ^ +3 +θ^ +4 +Causal Effect Estimate +θ^ +5 +θ^ +6 +Causal Effect Estimate +θ^ +1 +θ^ +2 +Standard Deviation Estimate +θ^ +3 +θ^ +4 +Standard Deviation Estimate +θ^ +5 +θ^ +6 +Standard Deviation Estimate +θ^ +1 +θ^ +2 +Coverage Frequency +θ^ +3 +θ^ +4 +Coverage Frequency +θ^ +5 +θ^ +6 +Coverage Frequency +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +bee +ivw +egger +lasso median +m = 250 +m = 500 +m = 1000 +m = 250 +m = 500 +m = 1000 +m = 250 +m = 500 +m = 1000 +m = 250 +m = 500 +m = 1000 +m = 250 +m = 500 +m = 1000 +m = 250 +m = 500 +m = 1000 +m = 250 +m = 500 +m = 1000 +m = 250 +m = 500 +m = 1000 +m = 250 +m = 500 +m = 1000 +Figure 4: Investigation of MR methods for multivariable MR with sample sizes n0 = · · · = n6 = 20000 +and overlap-sample sizes n01 = · · · = n65 = 20000, in terms of number of instrumental variants. +22 + +MR−BEE +MR−IVW +MR−Egger +MR−Lasso +MR−Median +0.0 +0.1 +0.2 +0.3 +0.0 +0.1 +0.2 +0.3 +−0.1 +−0.2 +−0.3 +0.0 +−0.1 +−0.2 +−0.3 +0.0 +−0.1 +0.0 +0.1 +−0.1 +0.0 +0.1 +m = 250 +m = 500 +m = 1000 +m = 250 +m = 500 +m = 1000 +m = 250 +m = 500 +m = 1000 +1 exposures +3 exposures +6 exposures +1 exposures +3 exposures +6 exposures +1 exposures +3 exposures +6 exposures +1 exposures +3 exposures +6 exposures +1 exposures +3 exposures +6 exposures +1 exposures +3 exposures +6 exposures +1 exposures +3 exposures +6 exposures +1 exposures +3 exposures +6 exposures +1 exposures +3 exposures +6 exposures +^θ1 +θ^ +2 +θ^ +3 +θ^ +4 +θ^ +5 +θ^ +6 +Figure 5: Investigation of MR methods for multivariable MR with sample sizes n0 = · · · = n6 = 20000 +and overlap-sample sizes n01 = · · · = n65 = 20000, in terms of number of specified exposures. +lations when 1, 3, and all 6 exposures are included in the multivariable MR model, respec- +tively. Figure 5 illustrates the results of the simulations. We observed that if associated +exposures are omitted, the causal effect estimates can suffer severe biases. The degree of +the biases is jointly determined by the genetic covariance matrix and covariance matrix of +confounders. In conclusion, even though MRBEE has eliminated the estimation error bias +and weak instrument bias, OVB still exists if any relevant exposure is not specified in the +multivariable MR model. +4.3 +Other Investigations +For univariable MR, we also investigated the effects of sample sizes, type-I error, winner’s +curse, and outlier detection. Regarding multivariable MR, we investigated the impact of +different sample overlaps. In addition, the precision of estimating ΣWβwα by insignificant +GWAS statistics is also studied. +Only by increasing the sample sizes of the exposure +and outcome cohorts simultaneously, the accuracy of MRBEE can be improved. +The +23 + +traditional MR methods suffer from inflated type-I errors when the overlapping fraction is +large. After accounting for the weak instrument bias and estimate error bias, MRBEE is +almost free of the winner curse’s bias when the overlapping fraction is high. Furthermore, +by applying the iterative method in IMRP, MRBEE can efficiently eliminate pleiotropic +outliers and produce an accurate causal effect estimate. In addition, the estimation error +ΣWβwα decreases with the increase of the number of insignificant variants M. +Finally, +multivariable MRBEE is accurate regardless of sample overlap. We summarized the findings +with the simulation details in the supplementary material. +5 +Real Data Analysis +Cardiovascular disease including coronary artery disease (CAD) is one of the leading causes +of death for both men and women worldwide. There are many epidemiological studies +and MR analyses based on GWAS summary data dedicated to identifying the causal risk +factors for CAD. However, the causal effects of the risk factors on CAD are less clear and +the existing evidence can be contradictory. For example, elevated low-density lipoprotein +cholesterol level (LDL-C) is a well-established causal risk factor for CAD (Group et al., +1994), whereas Wang et al. (2022) concluded by multivariable MR analysis that LDL-C is +not causally related to CAD in Europeans. Additionally, substantial observational analyses +and molecular experiments have suggested that uric acid (UA) and red blood cell counts +(RBC) contribute to the development of CAD (Bujak et al., 2015; Yu and Cheng, 2020). +Nevertheless, Wang et al. (2022) did not observe significant causal effects of the two risk +factors on CAD in Europeans. Furthermore, numerous MR analyses have concluded that +body mass index (BMI) has a positive causal effect on CAD (Zhu, 2020; Wang et al., +2022). However, recent literature indicates that BMI is likely to influence CAD through +the mediation with diseases such as diabetes and hypertension (Gill et al., 2021). These +contradictions may be due to biases in MR methods, including OVB, weak instrument bias, +estimation error bias, etc. +We conducted two data analyses to estimate the causal effects of select risk factors on +CAD. The first analysis uses the 11 exposures in Wang et al. (2022), including BMI, +hemoglobin (HB), hemoglobin a1c (Hba1c), hematocrit (HT), high-density lipoprotein +cholesterol level (HDL-C), height, LDL-C, RBC, systolic blood pressure (SBP), triglyc- +erides (TG), and UA. In Wang et al. (2022), these 11 exposures were divided into two +groups and analyzed separately. In contrast, we analyzed them in one multivariable MR +model to avoid the OVB. In the second analysis, we replace HB, Hba1c, HT, and RBC with +alcohol consumption (alcohol), diabetes, lifetime never smoking status (never.smoking), +and sleeplessness. All the GWAS summary statistics used in our analyses were downloaded +from the Neale lab (http://www.nealelab.is/uk-biobank/). Quality controls (QCs) are +presented in the supplementary material. The total numbers of instrumental variants for +the first and second analyses are 5345 and 5301, respectively. +Figure 6 displays the causal effect estimates with 95% confidence intervals. +MR- +BEE confirms the causal effects of LDL-C, RBC, and UA on CAD. Here, HB, HT, and +RBC have high mutual correlations: +� +cor(xHB, xHT) = 0.89, � +cor(xHB, xRBC) = 0.63, and +24 + +−0.2 +−0.1 +0.0 +0.1 +0.2 +causal effect estimates +−0.2 +−0.1 +0.0 +0.1 +0.2 +causal effect estimates +Multivariable MR Estimation (with 11 exposures in Genome Med.) +Multivariable MR Estimation (with alternative 11 exposures) +BMI +HB +HT +HDL +Height +LDL +Hba1c +RBC +SBP +TG +UA +Alcohol +BMI +Diabetes +HDL +Height +LDL +Never.smoking +SBP +Sleepless +TG +UA +MR−BEE +MR−IVW +MR−Egger +MR−Lasso +MR−Median +Figure 6: +Causal effect estimates of CAD data. +Confidence intervals are yielded by the double SE +estimates. +� +cor(xHT, xRBC) = 0.72, and thus the inferences obtained by the existing methods are not +reliable. For example, the existing MR methods suggest that RBC is not significant, HB +has a significant positive effect, and HT has a significant negative effect, which contradicts +the fact that HT and CAD are positively associated (Sorlie et al., 1981). MRBEE corrects +the estimation error bias and thus leads to a reasonable conclusion – HB and RBC have +positive causal effects on CAD while HT has a positive but insignificant causal effect on +CAD. For the second analysis, MRBEE reveals that BMI is likely to affect CAD through +the mediation of SBP and diabetes. In addition, MRBEE indicates that never.smoking +is protective against CAD, whereas sleeplessness is associated with increasing CAD risk. +Furthermore, due to the weak instrument bias and estimation error bias, the existing meth- +ods overestimate the effects of HDL-C and height and underestimate the causal effects of +diabetes, LDL-C, never.smoking, SBP, and sleeplessness. By using MRBEE, we are able +to obtain reliable causal effect estimates and therefore make valid inferences on the causal +risk factors of CAD. +6 +Discussion +In this paper, we first investigated the asymptotic behavior of the multivariable IVW es- +timate. Since almost all MR methods are based on the IVW method, understanding the +asymptotic behavior of the IVW estimate has very far-reaching implications for the theo- +retical and empirical studies of MR methods. We found that the bias of the multivariable +25 + +IVW estimate is the product of weak instrument bias and estimation error bias. Also, we +revealed that estimation error bias is a linear combination of measurement error bias and +confounder bias, in which the sample overlaps trade off the proportion of these two compo- +nents of estimation error bias. In the literature, although the phenomenon that the IVW +estimate suffers from bias has been observed, a quantitative explanation for its existence is +still absent. Our work fills the gap, which is a significant theoretical contribution to MR. +Subsequently, in this paper, we describe MRBEE that can yield the unbiased causal +effect estimate ˆθBEE. We point out that ˆθBEE is strongly asymptotically unbiased in all +scenarios, indicating that ˆθBEE is asymptotically valid when making causal inferences. We +also discuss how to perform MRBEE in practice, including how to estimate the bias- +correction terms, how to estimate the sandwich formula, and how to identify possible UHP +when multiple exposures are included. +We present corresponding theorems to confirm +that the estimates involved in the implementation of MRBEE are consistent in theory. +In simulations, we show that MRBEE simultaneously estimates causal effects and the SE +unbiasedly, and identifies UHP consistently. In section 5 and also in (Lorincz-Comi et al., +2022), the practical advances of MRBEE are further demonstrated. +It is worth offering guidance on how to properly perform MR analysis from our perspec- +tive. First, we suggest applying the multivariable MR approach instead of the univariable +MR approach because the causal effect estimates obtained by the univariable MR approach +are unreliable due to OVB, regardless of the presence of UHP and CHP in the model. Sec- +ond, rather than selecting the optimal number of instrumental variants such that the F +statistics and conditional F statistics are larger than 10 (Burgess et al., 2011; Sanderson +et al., 2021), we advise including all the independent instrumental variants that are signif- +icantly associated with one or more exposures. Our theory illustrates that the asymptotic +variance of a causal effect estimate is related to the cumulative variance explained by all +specified IVs instead of the average variance explained by each IV. In particular, there +is no need to worry about the issue of weak IVs because MRBEE has demonstrated ef- +ficiency to eliminate weak instrument bias through our simulations and theory. Third, +when performing multivariable MR analysis, it is not necessary to remove variants that are +pleiotropic between the exposures. For example, Wang et al. (2022) observed that LDL-C +was insignificantly associated with CAD in Europeans, which is unlikely to be true because +this risk causality has been well established in randomized clinical trials (Group et al., +1994). The potential reason for this false negative is that Wang et al. (2022) excluded +the IVs associated with RBC, HB, HT, and UA in their multivariable MR analysis. We +believe that the proper way to perform multivariable MR analysis is to simultaneously in- +clude all the relevant exposures, as the multivariable regression can automatically account +for the pleiotropic variants shared by the specified exposures. Fourth, among the existing +multivariable MR approaches including IVW, MR-Egger, and MR-Lasso, we recommend +MRBEE as the primary analysis approach because it has been proven to be the only one +that enjoys strongly asymptotic unbiasedness in the presence of many weak IVs. +26 + +A +Proof +A.1 +Preliminary lemmas +In this subsection, we specify some lemmas that can facilitate the proofs, most of which +can be found in the existing papers. We first discuss the equivalent characterizations of +sub-Gaussian and sub-exponential variables. +Lemma A.1 (Equivalent characterizations of sub-Guassian variables). Given any random +variable X, the following properties are equivalent: +(I) there is a constant K1 ≥ 0 such that +Pr(|X| ≥ t) ≤ 2 exp(−t2/K2 +1), +for all t ≥ 0, +(II) the moments of X satisfy +||X||Lp = (E(|X|p)) +1 +p ≤ K2 +√p, +for all p ≥ 1, +(III) the moment generating function (MGF) of X2 satisfies: +E{exp(λ2X2)} ≤ exp(K2 +3λ2), +for all λ staisfying |λ| ≤ K−1 +3 , +(IV) the MGF of X2 is bounded at some point, namely +E{exp(X2/K2 +4)} ≤ 2, +(V) if E(X) = 0, the MGF of X satisfies +E{exp(λX)} ≤ exp(K2 +5λ2), +for all λ ∈ R, +where K1, . . . , K5 are certain strictly positive constants. +This lemma summarizes some well-known properties of sub-Guassian and can be found +in Vershynin (2018, Proposition 2.5.2). +Lemma A.2 (Equivalent characterizations of sub-exponential variables). Given any ran- +dom variable X, the following properties are equivalent: +(I) there is a constant K1 ≥ 0 such that +Pr(|X| ≥ t) ≤ 2 exp(−t/K1), +for all t ≥ 0, +(II) the moments of X satisfy +||X||Lp = (E(|X|p)) +1 +p ≤ K2p, +for all p ≥ 1, +27 + +(III) the moment generating function (MGF) of |X| satisfies: +E{exp(λ|X|)} ≤ exp(K3λ), +for all λ staisfying 0 ≤ λ ≤ K−1 +3 , +(IV) the MGF of |X| is bounded at some point, namely +E{exp(|X|/K4)} ≤ 2, +(V) if E(X) = 0, the MGF of X satisfies +E{exp(λX)} ≤ exp(K2 +5λ2), +for all λ ≤ K−1 +5 , +where K1, . . . , K5 are certain strictly positive constants. +This lemma summarizes some well-known properties of sub-exponential and can be +found in Vershynin (2018, Proposition 2.7.1). +Lemma A.3 (Product of sub-Gaussian variable is sub-exponential). Suppose that X, Z +are two sub-Gaussian variable, then Y = XZ is a sub-exponential variable. Besides, if X +is a bounded sub-Gaussian variable, then then Y = XZ is a sub-Gaussian variable. +The first claim of this lemma is provided by Vershynin (2018, Proposition 2.7.7). The +second claim of this lemma is a direct inference of Fan et al. (2011, Lemma A.2). +Lemma A.4 (ℓ2-norm of matrices with sub-Gaussian entries). Let X1, . . . , Xn be n (p×1) +independent identically distributed random vector with entries xi1, . . . , xip are sub-Gaussian +with zero-mean. Besides, define the covariance matrix of Xi as +Σ = E(XiX⊤ +i ) +and the related sample covariance matrix +ˆΣ = 1 +n +n +� +i=1 +XiX⊤ +i . +Then for every positive integer n, +E(|| ˆΣ − Σ||2) ≤ C +�p +n + +�p +n +� +||Σ||2, +where C is certain positive constant. +This lemma is provided by Vershynin (2018, Theorem 4.7.1). It shows the convergence +rate of sample covariance matrix is √(n/m). +Lemma A.5 (ℓ2-norm of matrices with sub-exponential entries). Let X1, . . . , Xn be n +(p × 1) independent identically distributed random vector with entries xi1, . . . , xip are sub- +exponential with zero-mean. Besides, define the covariance matrix of Xi as +Σ = E(XiX⊤ +i ) +28 + +and the related sample covariance matrix +ˆΣ = 1 +n +n +� +i=1 +XiX⊤ +i . +Then for ever t ≥ 0, the following inequality holds with probability at least 1 − p exp(−ct2): +|| ˆΣ − Σ||2 ≤ max(||Σ||2δ, δ2), +where c is certain positive constant and δ = t +� +p/n. +This lemma is the direct inference of Vershynin (2010, Theorem 5.44). Besides, by +letting t = √p log n we further obtain +E(|| ˆΣ − Σ||2) = O +�� +p log n +n +� +||Σ||2, +if ˆΣ is the sample covariance matrix of sub-exponential vector. Note that in our method, +the dimension p is fixed and hence we cannot chose t = √p log p such that the estimation +bound becomes +� +(p log p)/n||Σ||2. +Lemma A.6 (Asymptotic normal distribution of Wishart matrix). Suppose X1, X2, . . . , Xn +are n IID relaxation of the p-dimensional variable X ∼ N(0, Σ) with a well-conditioned +covariance matrix Σ. Besides, define the sample covariance matrix of Σ as +ˆΣ = 1 +n +n +� +i=1 +XiX⊤ +i . +If p is a fixed number, then as n → ∞, +√n(vec( ˆΣ) − vec(Σ)) +D +−→ N +� +0, (Ip2 + Kp2)(Σ ⊗ Σ) +� +, +where Kp2 is the so-called commutation matrix, which is able to ensure Kp2vec(A) = +vec(A′) for all (p × p) matrix. +This lemma can be found in Muirhead (2009, equation (5), p90). +A.2 +Specific Lemmas +In this subsection, we specify the following lemmas that are made based on the preliminary +lemmas. +Lemma A.7 (Asymptotic normal distribution of sub-Gaussian and sub-exponential vari- +ables). Suppose X1, . . . , Xn are n independent sub-Gaussian or sub-exponential variables +with mean-zero and variance σ2 +1, . . . , σ2 +n . Then +lim +n→∞ +1 +√n +n +� +i=1 +Xi +D +−→ N(0, σ2 +x), +29 + +where +σ2 +x = lim +n→∞ +1 +n +n +� +i=1 +σ2 +i . +Proof of Lemma A.7. It is easy to verify the Lyapunov’s condition: for all fixed δ > 0, +lim +n→∞ +1 +n1+δ +n +� +i=1 +E(|Xi|2+2δ) ≤ +√2K2 + 2K2δ +2+2δ +nδ +→ 0 +by the (II) of Lemma A.1, if X1, . . . , Xn are sub-Gaussian variables; +lim +n→∞ +1 +n1+δ +n +� +i=1 +E(|Xi|2+2δ) ≤ (2K2 + 2K2δ)2+2δ +nδ +→ 0 +by the (II) of Lemma A.2, if X1, . . . , Xn are sub-exponential variables. And hence the +asymptotic normal distribution holds. +Lemma A.8 (Asymptotic normal distribution of estimation error). Let +ξ[s] +j += +1 +√ns +ns +� +i=1 +g[s] +ij x[s] +i,−j, +where +x[s] +i,−j = x[s] +i − βjsg[s] +i,j, +s = 0, 1, . . . , p, x[0] +i,−j represents y[0] +i,−j and βj0 represent αj. Then +ξ[s] +j +D−→ N(0, σxsxs − σβsβs), +where σx0x0 represents σyy and σβ0β0 represents θ⊤Σββθ. +Proof of Lemma A.8. Note that both g[s] +ij and x[s] +i,−j are sub-Gaussian (x[s] +i,−j is the product of +a sub-Gaussian variable and a bounded sub-Gaussian variable), and it holds E(g[s] +ij x[s] +i,−j) = 0 +and +var(g[s] +ij x[s] +i,−j) = var(g[s] +ij ) × var(x[s] +i,−j) = σxsxs − σβsβs. +(32) +As a result, +ξ[s] +j += +1 +√ns +ns +� +i=1 +g[s] +ij x[s] +i,−j +D−→ N(0, σxsxs − σβsβs), +(33) +according Lemma A.7. +Lemma A.9 (Asymptotic normality of bias-correction terms). Let +ζj = +�nmin +n1 +ξ[1] +j , nmin +n2 +ξ[2] +j , . . . , nmin +np +ξ[p] +j , nmin +n0 +ξ[0] +j +�⊤ +. +30 + +Under the conditions (C1)-(C4), +lim +m→∞ +1 +√m +m +� +j=1 +(vec(ζjζ⊤ +j ) − vec(ΨWβ×wα)) +D−→ N +� +0, (Ip2 + Kp2)(ΨWβ×wα ⊗ ΨWβ×wα) +� +. +as nmin, m → ∞. +Proof of Lemma A.9. By using Lemma A.7, ζj follows N(0, ΨWβ×wα) as nmin → ∞. Then +by using Lemma A.6, this lemma holds. +Lemma A.10 (Asymptotic normality of residual term). Under the conditions (C1)-(C4), +lim +m→∞ +1 +√m +m +� +j=1 +√mβjξ[s] +j +D−→ N(0, σxsxsΣββ), +and +lim +m→∞ +1 +m +m +� +j=1 +√mβj +√mβ⊤ +j ξ[s] +j ξ[k] +j +P−→ +nsk +√nsnk +σxsxkΣββ, +for s = 0, . . . , p, where σx0xk represents σyxk = �p +l=1 θlσxlxk. +Proof of Lemma A.10. By condition (C4), √mβj is independent of ξ[s] +j . By Lemma A.3, +√mβjξ[s] +j +is sub-exponential with mean 0 and covariance matrix +cov(√mβjξ[s] +j ) = cov(√mβj) × var(ξ[s] +j ) += (σxsxs − σβsβs)Σββ. +(34) +Hence, by Lemma A.6, +lim +m→∞ +1 +√m +m +� +j=1 +√mβjξ[s] +j +D−→ N(0, σxsxsΣββ). +On the other hand, βjξ[s] +j +is sub-exponential variable according to Lemma A.3, and +cov(√mβjξ[s] +j , √mβjξ[k] +j ) = cov(ξ[s] +j , ξ[k] +j ) × Σββ += +nsk +√nsnk +(σxsxk − σβsβk)Σββ. +(35) +Hence, by using Lemma A.5 +lim +m→∞ +1 +m +m +� +j=1 +√mβj +√mβ⊤ +j ξ[s] +j ξ[k] +j +P−→ +nsk +√nsnk +σxsxkΣββ. +31 + +A.3 +Proofs of theorems in section 2 +Proof of Theorem 1. +As for the estimation error ωα, we have +wαj = g[0]⊤ +j +y[0] +n0 +− αj = g[0]⊤ +j +y[0] +−j +n0 +, +(36) +where +y[0] +−j = y[0] − αjg[0] +j += +m +� +s̸=j +αtg[0] +t + U[0]θ + v[0], +(37) +and U[0] and v[0] are the corresponding noise terms in the outcome GWAS cohort. Accord- +ing to Lemma A.8, +ξ[0] +j += +1 +√n0 +n0 +� +i=1 +g[0] +ij y[0] +i,−j +D−→ N(0, σyy − θ⊤Σββθ). +(38) +As for the estimation error wβjs, we have +wβjs = g[s]⊤ +j +x[s] +ns +− βjs = g[s]⊤ +j +x[s] +−j +ns +, +(39) +where +x[s] +−j = x[s] − g[s] +j βjs = +� +t̸=j +βtsg[s] +t + u[s]. +(40) +Let +ξ[s] +j += g[s]⊤ +j +x[s] +−j +√ns += +1 +√ns +ns +� +i=1 +g[s] +ij x[s] +i,−j, +(41) +where x[s] +i,−j is the ith element in vector x[s] +−j. According to Lemma A.8, +ξ[s] +j += +1 +√ns +ns +� +i=1 +g[s] +ij x[s] +i,−j +D−→ N(0, σxsxs − σβsβs). +(42) +Now we show the covariance between ξ[s] +j +and ξ[k] +j : +cov(ξ[s] +j , ξ[k] +j ) = E +�x[s]⊤ +−j g[s] +j g[k]⊤ +j +x[k] +−j +√nsnk +� +, +(43) +where x[0] +−j represents y[0] +−j for simplicity. Denote Q[sk] = (Q[sk] +it ) being a (ns × nk) matrix +32 + +whose (i, t)th element is +Q[sk] +it += E(g[s] +ij g[k] +tj ) = +� +1, +(i, t) ∈ Q[sk], +0, +(i, t) /∈ Q[sk], +(44) +where +Q[sk] = {(i, t) : g[s] +ij and g[k] +tj come from the same individual}. +(45) +As a result, +cov(ξ[s] +j , ξ[k] +j ) = E +�x[s]⊤ +−j Q[sk]x[k] +−j +√nsnk +� += +1 +√nsnk +� +(i,t)∈Q[sk] +E(x[s] +i,−jx[k] +t,−j) += +nsk +√nsnk +� +σxsxk − σβsβk +� +, +(46) +where σx0xk represents σyxk for simplicity, and σβ0βk represents +σβ0βk = cov(√mβ⊤ +j θ, √mβjk) = +p +� +l=1 +θlσβlβk. +(47) +Finally, we show ξ[s] +j +is uncorrelated with ξ[s] +t +for all t ̸= j and s = 0, . . . , p. Specifically, +cov(ξ[s] +j , ξ[s] +t ) = E +�x[s]⊤ +−j g[s] +j g[s]⊤ +t +x[s] +−j +ns +� +. +(48) +According the model setting, g[s] +j is independent of g[s] +t for all t ̸= s. Therefore, cov(ξ[s] +j , ξ[s] +t ) = +0. +Note that if m → ∞, Σββ = 1 +mΨββ vanishes. And so Theorem 1 is proved. +Proof of Theorem 2. +The score function of IVW is +− 1 +m +ˆB⊤(ˆa − ˆBˆθIVW) = − 1 +m +ˆB⊤(ˆa − ˆBθ) + 1 +m +ˆB⊤ ˆB(ˆθIVW − θ) +(49) +which leads to +HIVW(ˆθIVW − θ) = −SIVW(θ), +(50) +where +HIVW = 1 +m +ˆB⊤ ˆB, +SIVW(θ) = − 1 +m +ˆB⊤(ˆa − ˆBθ). +(51) +33 + +We first work with the Hessian matrix HIVW: +mHIVW = ˆB⊤ ˆB = B⊤B + B⊤Wβ + W⊤ +β B + W⊤ +β Wβ += J1 + J2 + J3 + J4. +(52) +As for J1, +J1 = +m +� +j=1 +βjβ⊤ +j +P +−→ Ψββ. +(53) +As for J2, +∥√nminJ2∥2 = +���� +1 +√m +m +� +j=1 +(√nminwβj)(√mβj)⊤ +���� +2 +≤ +� +� +� +� +���� +1 +m +m +� +j=1 +(√nminwβj)(√nminwβj)⊤ +���� +2 +× +� +� +� +� +���� +1 +m +m +� +j=1 +(√mβj)(√mβj)⊤ +���� +2 +≤ λ +1 +2max(ΨWβWβ) × λ +1 +2max(Ψββ), +(54) +which means +∥J2∥2 = OP(1/√nmin). +(55) +As for J3, it has the same order as J2. As for J4, +nmin +m J4 = 1 +m +m +� +j=1 +(√nminwβj)(√nminwβj)⊤ +P +−→ ΨWβWβ +(56) +Hence: +(1) If m/nmin → 0, +∥J4∥2 ≤ λmax(ΨWβWβ) × m +nmin +→ 0. +(57) +Therefore, +mHIVW +P +−→ Ψββ. +(58) +(2) If m/nmin → c0 ∈ (0, ∞), then +J4 = +m +nmin +× 1 +m +m +� +j=1 +(√nminwβj)(√nminwβj)⊤ +P +−→ c0ΨWβWβ. +(59) +34 + +Therefore, +mHIVW +P +−→ Ψββ + c0ΨWβWβ. +(60) +(3) If m/nmin → ∞ and m/n1+τ +min → c0 ∈ (0, +∞) with certain constant τ > 0, then +1 +nτ +min +J4 = +m +n1+τ +min +× 1 +m +m +� +j=1 +(√nminwβj)(√nminwβj)⊤ +P +−→ c0ΨWβWβ. +(61) +Therefore, +m +nτ +min +HIVW = c0nminHIVW +P +−→ c0ΨWβWβ. +(62) +We then work with SIVW(θ): +mSIVW(θ) = −B⊤wα − W⊤ +β wα + B⊤Wβθ + W⊤ +β Wβθ += K1 + K2 + K3 + K4. +(63) +As for K1 + K3, +√nmin(K1 + K3) = +1 +√m +m +� +j=1 +(−√nminwαj + √nminw⊤ +βjθ)(√mβj) +D +−→ N(0, ψθΨββ), (64) +where +ψθ = ψwαwα + θ⊤ΨWβWβθ − 2θ⊤ψWβwα. +(65) +As for K2, +nmin +m K2 = − 1 +m +m +� +j=1 +(√nminwαj)(√nminwβj) +P +−→ −ψWβwα. +(66) +As for K4, +nmin +m K4 = +� 1 +m +m +� +j=1 +(√nminwβj +√nminwβj +� +θ +P +−→ ΨWβWβθ, +(67) +Jointing these results, we summary the asymptotic behavior of ˆθIVW: +(1) If m/√nmin → 0, then +√nmin||K2 + K4|| = OP +� +m +√nmin +� += oP(1). +(68) +35 + +Therefore, +√nmin × mSIVW(θ) = √nmin(K1 + K3) + oP(1) +D +−→ N(0, ψθΨββ). +(69) +Note that when m/nmin → 0, mHIVW +P +−→ Ψββ. Therefore, +√nmin(ˆθIVW − θ) = −√nmin(mHIVW)−1(mSIVW(θ)) +D +−→ N(0, ψθΨ−1 +ββ), +(70) +(2) If m/√nmin → c0, then +√nmin(K2 + K4) → −c0ψWβwα + c0ΨWβWβθ, +(71) +and hence +√nmin × mSIVW(θ) +D +−→ N(−c0(ψWβwα + ΨWβWβθ), ψθΨββ). +(72) +Note that when m/nmin → 0, mHIVW +P +−→ Ψββ. Therefore, +√nmin(ˆθIVW − θ) = −√nmin(mHIVW)−1(mSIVW(θ)) +D +−→ N(c0Ψ−1 +ββ(ψWβwα − ΨWβWβθ), ψθΨ−1 +ββ). +(73) +(3) If m/√nmin → ∞ and m/nmin → c0, then ||K1 + K3||2 = OP(1/√nmin), +K2 + K4 +P +−→ −c0ψWβwα + c0ΨWβWβθ, +(74) +and +mHIVW +P +−→ Ψββ + c0ΨWβWβ. +(75) +Hence, +ˆθIVW − θ +P +−→ c0(Ψββ + c0ΨWβWβ)−1(ψWβwα − ΨWβWβθ). +(76) +(4) If m/nmin → ∞ and m/n1+τ +min → c0, then +1 +nτ +min +(K2 + K4) +P +−→ −c0ψWβwα + c0ΨWβWβθ +(77) +and +m +nτ +min +HIVW +P +−→ c0ΨWβWβ. +(78) +Therefore, +ˆθIVW +P +−→ Ψ−1 +WβWβψWβwα. +(79) +36 + +Now Theorem 2 is proved. +A.4 +Proofs of theorems in section 3 +Proofs of Theorem 3. +Note that +0 = SBEE(ˆθBEE) = SBEE(θ) + HBEE(ˆθBEE − θ), +(80) +where +SBEE(θ) = − 1 +m +ˆB⊤( ˆα − ˆBθ) − ΣWβWβθ + σWβwα, +(81) +and +HBEE = 1 +m +ˆB⊤ ˆB − ΣWβWβ. +(82) +As for SBEE(θ), +mSBEE(θ) = −(B + Wβ)⊤(α + wα − Bθ − Wβθ) − mΣWβWβ + mσWβwα += − +� +B⊤(wα − Wβθ) +� ++ +�� +W⊤ +β Wβ − mΣWβWβ +� +θ +� +− +� +W⊤ +β wα − mσWβwα +� += K1 + K2 + K3. +(83) +Here, we define a new vector ϑ = (θ⊤, 1)⊤, an alternative vector +ζj = +�nmin +n1 +ξ[1] +j , nmin +n2 +ξ[2] +j , . . . , nmin +np +ξ[p] +j , nmin +n0 +ξ[0] +j +�⊤ +, +where +ξ[s] +j += +1 +√ns +ns +� +i=1 +g[s] +ij x[s] +is , +s = 0, 1, . . . , p, +and a new covariance matrix +cov(ζj) = ΨWβ×wα = +�ΨWβWβ +ψWβwα +ψ⊤ +Wβwα +ψwαwα +� +. +(84) +As for K1, it can be rewritten as +√nminK1 = − +m +� +j=1 +√nmin(wαj − w⊤ +βjθ)βj = +1 +√m +m +� +j=1 +(√nminζ⊤ +j ϑ)(√mβj) +D +−→ N(0, ψθΨββ), +(85) +where ψθ defined in (65) can be rewritten as +ψθ = ϑ⊤ΨWβ×wαϑ. +(86) +37 + +As for K2 + K3, it can be rewritten as +K2 + K3 = I1:p +p+1 +�W⊤ +β Wβ − mΣWβWβ +W⊤ +β wα − mσWβwα +w⊤ +α Wβ − mσ⊤ +Wβwα +w⊤ +α wα − mσwαwα +� � θ +−1 +� += +√m +nmin +I1:p +p+1 +� 1 +√m +m +� +j=1 +ζjζ⊤ +j − ΨWβ×wα +� +ϑ += +√m +nmin +I1:p +p+1K4ϑ, +(87) +where I1:p +p+1 is a (p × (p + 1)) matrix consisting of the first p row of Ip+1 and +K4 = +1 +√m +m +� +j=1 +ζjζ⊤ +j − ΨWβ×wα. +(88) +According to Lemma A.6, +vec(K4) +D +−→ N +� +0, (I(p+1)2 + K(p+1)2)(ΨWβ×wα ⊗ ΨWβ×wα) +� +. +(89) +As a result, +nmin +√m (K2 + K3) +D +−→ N(0, ΣBC) +(90) +where +ΣBC = +� +ϑ⊤ ⊗ I1:p +p+1 +� +� +�� +� +p×(p+1)2 +� +(I(p+1)2 + K(p+1)2)(ΨWβ×wα ⊗ ΨWβ×wα) +� +� +�� +� +(p+1)2×(p+1)2 +� +ϑ⊤ ⊗ I1:p +p+1 +�⊤ +� +�� +� +(p+1)2×p +. +(91) +So far, we can obtain: +(1) If m/nmin → 0, +√nmin × mSBEE(θ) = √nminK1 + oP(1) +D +−→ N(0, ψθΨββ). +(92) +(2) If m/nmin → c0, +√nmin × mSBEE(θ) = √nminK1 + √nmin(K2 + K3) +D +−→ N(0, ψθΨββ + c0ΣBC). +(93) +(3) If m/nmin → ∞ and √m/nmin → 0, +nmin +√m × mSBEE(θ) = nmin +√m (K2 + K3) + nmin +√m K1 +D +−→ N(0, ΣBC), +(94) +38 + +where +nmin +√m K1 = +�nmin +m × √nminK1 = OP +��nmin +m +� += oP(1). +(95) +Now we move to HBEE: +mHBEE = B⊤B + +� +W⊤ +β Wβ − mΣWβWβ +� ++ B⊤Wβ + W⊤ +β B += J1 + J2 + J3 + J4. +(96) +As for J1 = B⊤B, we have +||J1 − Ψββ||2 = +���� +1 +m +m +� +j=1 +√mβj +√mβ⊤ +j − Ψββ +���� +2 += OP +� 1 +√m +� +. +(97) +As for J2 = W⊤ +β Wβ − mΣWβWβ, we have +J2 = +m +� +j=1 +� +wβjw⊤ +βj − ΣWβWβ +� += +√m +nmin +1 +√m +m +� +j=1 +� +ξjξ⊤ +j − ΨWβWβ +� +. +(98) +As a result, +nmin +√m vec(J2) +D +−→ N(0, (Ip2 + Kp2)(ΨWβWβ ⊗ ΨWβWβ)), +(99) +which means ||J2|| = OP(√m/nmin). As for J3 = B⊤Wβ, +√nmin||J3||2 = +���� +1 +√m +m +� +j=1 +√mβj +√nminω⊤ +βj +���� +2 +≤ +� +� +� +� +���� +1 +m +m +� +j=1 +√mβj +√mβ⊤ +j +���� +2 +� +� +� +� +���� +1 +m +m +� +j=1 +√nminωβj +√nminω⊤ +βj +���� +2 +≤ λ +1 +2max(Ψββ) × λ +1 +2max(ΨWβWβ), +(100) +which means +||J3||2 = OP +� +1 +√nmin +� +(101) +39 + +As for J4, it is easy to see ||J4||2 +2 = ||J3||2 +2. Hence, for all three scenarios in Theorem 3, +||mHBEE − Ψββ||2 = OP +� +max +� 1 +√m, +1 +√nmin +, +√m +nmin +�� +. +(102) +And hence, according to the Slutsky’s theorem, +(1) If m/nmin → 0, +√nmin(ˆθBEE − θ) = −√nminΨ−1 +ββK1 +D +−→ N(0, ψθΨ−1 +ββ). +(103) +(2) If m/nmin → c0, +√nmin(ˆθBEE − θ) = −√nminΨ−1 +ββ(K1 + K2 + K3) +D +−→ N(0, ψθΨ−1 +ββ + c0Ψ−1 +ββΨBCΨ−1 +ββ). +(104) +(2) If m/nmin → ∞ and m/n2 +min → 0, +� +n2 +min/m(ˆθBEE − θ) = −nmin +√m Ψ−1 +ββ(K2 + K3) +D +−→ N(0, Ψ−1 +ββΨBCΨ−1 +ββ). +(105) +Thus, Theorem 3 is proved. +Proof of Theorem 4. +Similar to ξ[s] +j , we define η{s} +j +as +η{s} +j += g{s}⊤ +j +x[s] +√ns += +1 +√ns +ns +� +i=1 +g{s} +ij x[s] +i . +(106) +By using similar deduction as which in the proof of Theorem 1, +η{s} +j +D−→ N(0, σxsxs) +(107) +and +cov(η{s} +j +, η{k} +j +) = +nsk +√nsnk +σxsxk. +(108) +Denote ηj = (η{1} +j +, . . . , η{p} +j +, η{0} +j +) where η{0} +j +represents +1 +√n0g{s}⊤ +j +y[0]. Then we have +cov(ηj) = D−1 +η ΣWβ×wαD−1 +η , +(109) +where +Dη = diag +� 1 +√n1 +, . . . , +1 +√np +, +1 +√n0 +� +. +(110) +40 + +By using Lemma A.4, +���� +1 +M +M +� +j=1 +ηjη⊤ +j − cov(ηj) +���� +2 += OP +� 1 +√ +M +� +, +(111) +and hence +∥Σ +− 1 +2 +Wβ×wα ˆΣWβ×wαΣ +1 +2 +Wβ×wα − Ip+1∥2 ≤ λ−1 +min(cov(ηj)) +���� +1 +M +M +� +j=1 +ηjη⊤ +j − cov(ηj) +���� +2 += OP +� 1 +√ +M +� +. +(112) +Thus, Theorem 4 is proved. +Proof of Theorem 5. +Note that +Sj(θ) = −(ˆαj − θ⊤ ˆβj) ˆβj − ΣWβWβθ + σWβwα += (wαj − θ⊤wβj)βj + +� +(wαj − θ⊤wβj)wβj − ΣWβWβθ + σWβwα +� += J1j + J2j. +(113) +Note that both J1j and J2j are sub-exponential variables with zero mean and covariance +matrix +cov(J1j) = +1 +mnmin +ψθΨββ, +cov(J2j) = +1 +n2 +min +ΣBC. +(114) +Therefore, we obtain +cov(Sj(θ)) = ΣS = +� +� +� +� +� +� +� +1 +mnminψθΨββ, +if m/nmin → 0, +1 +mnminψθΨββ + +c0 +mnminΣBC, +if m/nmin → c0, +1 +n2 +minΣBC, , +if m/nmin → ∞ and √m/nmin → 0. +(115) +Then by using Lemma A.5, +���� +1 +m +m +� +j=1 +Sj(θ)Sj(θ)⊤ − ΣS +���� +2 += OP +�� +log m +m +� +||ΣS||2. +(116) +By using the Slutsky’s theorem, +���� +1 +m +m +� +j=1 +ˆSj(ˆθBEE) ˆSj(ˆθBEE)⊤ − ΣS +���� +2 += OP +�� +log m +m +� +||ΣS||2. +(117) +41 + +where +ˆSj(ˆθBEE) = −(ˆθ⊤ +BEE ˆβj − ˆαj) ˆβj + ˆΣWβWβ ˆθBEE − ˆσWβwα +(118) +On the other hand, according to the proof of Theorem 3, +∥mˆFBEE − Ψββ||2 = OP +� +max +� 1 +√m, +1 +√nmin +, +√m +nmin +�� +. +(119) +Note that ?, A22(p223) illustrates +∥A1A2A3 − B1B2B3∥2 = OP +� +max +� +||A1 − B1||2, ||A2 − B2||2, ||A3 − B3||2 +�� +, +(120) +where A1, A2, A3, B1, B2, B3 are six matrices with non-diverging maximum singular values. +Hence, +|| ˆΣBEE(ˆθBEE) − ΣBEE(θ)||2 = +����(mˆFBEE)−1 +� +m +� +j=1 +ˆSj(ˆθBEE) ˆSj(ˆθBEE)⊤ +� +(mˆFBEE)−1 − mΨ−1 +ββΣSΨ−1 +ββ +���� +2 += OP +� +max +�� +log m +m +, +1 +√nmin +, +√m +nmin +�� +||mΣS||2, +(121) +and consequently +||Σ +− 1 +2 +BEE(θ) ˆΣBEE(θ)Σ +− 1 +2 +BEE(θ) − Ip||2 = OP +� +max +�� +log m +m +, +1 +√nmin +, +√m +nmin +�� +. +(122) +Thus, Theorem 5 is proved. +Proof of Theorem 6. +Note that ||ˆθBEE − θ||2 = OP(n +− 1 +2 +min) and hence ˆαj − ˆβ⊤ +j ˆθBEE and +ˆαj − ˆβ⊤ +j θ have the same distribution. For j ∈ Oc, +ˆγj = εj = ˆαj − ˆβ⊤ +j ˆθBEE = wαj − w⊤ +βjθ + w⊤ +βj(ˆθBEE − θ) +∼ N(0, σεε), +(123) +where +σεε = θ⊤ΣWβwαθ + σωγωγ − 2θ⊤σWβwα. +(124) +As a result, +ˆγ2 +j +σεε +∼ χ2 +1. +(125) +42 + +Denote κ∗ = F −1 +χ2 +1 (κ). Then by using Lemma A.1 of ?, +Pr +� +max +j∈Oc +ˆγ2 +j +σεε +≤ κ∗ +� += 1 − Pr +� +max +j∈Oc +ˆγ2 +j +σεε +> κ∗ +� +≥ 1 − (m − |O|) Pr +� ˆγ2 +j +σεε +> κ∗ +� +≥ 1 − m Pr +� ˆγ2 +j +σεε +> κ∗ +� +≥ 1 − m exp +� +− (√2κ∗ − 1 − 1)2 +4 +� +. (126) +By letting κ∗ = C0 log m with C0 being a sufficiently large constant, +Pr +� +max +j∈Oc +ˆγ2 +j +σεε +≤ κ∗ +� +≥ 1 − exp +� +log m − 2C0 log m − 2√C0 log m − 1 +4 +� +≥ 1 − exp +� +− (2C0 − 4) log m − 2√C0 log m − 1 +4 +� +→ 1, +(127) +if m → ∞. +On the other hand, for j ∈ O, ˆγj = γj + εj, and hence +ˆγ2 +j +σεε +∼ χ2 +1 +� γ2 +j +σεε +� +, +(128) +where χ2 +1(λ) refers to the noncentral chi-squared distribution with degree of freedom 1 +and noncentrality parameter λ. Let Fχ2 +1(λ)(·) be the CDF of this noncentral chi-squared +distribution, which is indeed equal to +Fχ2 +1(λ)(x) = 1 − +� +Q(√x − +√ +λ) + Q(√x + +√ +λ) +� +, +(129) +where Fχ2 +1(λ)(·) be the CDF of χ2 +1(λ) and Q(x) is the Gaussian Q-function, i.e., Q(x) = +1 − Φ(x) and Φ(x) is the CDF of standard normal distribution. +Note that there should exist a constant D0 such that +γ2 +j +σεε +≥ D0nmin +(130) +where D0 is a sufficient large constant. And +Pr +� +min +j∈O +ˆγ2 +j +σεε +≥ κ∗ +� += 1 − Pr +� +min +j∈Oc +ˆγ2 +j +σεε +< κ∗ +� +≥ 1 − Pr +� ˆγ2 +j +σεε +< κ∗ +� +, +j is arbitrary element in O. +(131) +43 + +Hence, +Pr +� +min +j∈O +ˆγ2 +j +σεε +≥ κ∗ +� +≥ Q( +√ +κ∗ − +� +D0nmin) + Q( +√ +κ∗ + +� +D0nmin) +≥ Q( +� +C0 log m − +� +D0nmin) + Q( +� +C0 log m + +� +D0nmin) → 1 +(132) +if m, nmin → ∞. Thus, Theorem 6 is proved. +References +Benjamini, Y. and Y. Hochberg (1995). Controlling the false discovery rate: a practical +and powerful approach to multiple testing. 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Genetic Epidemiology 46(2), 105–121. +48 + diff --git a/KdE4T4oBgHgl3EQfiA17/content/tmp_files/load_file.txt b/KdE4T4oBgHgl3EQfiA17/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0f35fe58d53c994e73403c6f2d96a69d28bb5cc5 --- /dev/null +++ b/KdE4T4oBgHgl3EQfiA17/content/tmp_files/load_file.txt @@ -0,0 +1,2376 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf,len=2375 +page_content='Unbiased estimation and asymptotically valid inference in multivariable Mendelian randomization with many weak instrumental variables Yihe Yang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Noah Lorincz-Comi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Xiaofeng Zhu ∗ Department of Population and Quantitative Health Science Case Western Reserve University Abstract Mendelian randomization (MR) is a popular epidemiological approach that uti- lizes genetic variants as instrumental variables (IVs) to infer the causal relationships between exposures and an outcome in the genome-wide association studies (GWAS) era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' It is well-known that the inverse-variance weighted (IVW) estimate of causal effect suffers from bias caused by the violation of valid IV conditions, however, the quantitative degree of this bias has not been well characterized and understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This paper contributes to the theoretical investigation and practical solution of the causal effect estimation in multivariable MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' First, we prove that the bias of IVW estimate is a product of the weak instrument and estimation error biases, where the estima- tion error bias is caused linearly by measurement error and confounder biases with a trade-off due to the sample overlap among exposure and outcome GWAS cohorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Second, we demonstrate that our novel multivariable MR approach, MR using Bias- corrected Estimating Equation (MRBEE), can estimate the causal effect unbiasedly in the presence of many weak IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Asymptotic behaviors of IVW and MRBEE are investigated under moderate conditions, where MRBEE is shown superior to IVW in terms of unbiasedness and asymptotic validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Simulations exhibit that only MRBEE can provide a strongly asymptotically unbiased estimate of causal effect in compar- ison with existing MR methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Applied to data from the UK Biobank, MRBEE can eliminate weak instrument and estimation error biases and provide valid causal inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' R package MRBEE and supplementary materials are available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Keywords: Causal Inference, Genome-Wide Association Studies, Inverse-Variance Weight- ing, Mendelian Randomization, Weak Instrumental Variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' ∗Email: xxz10@case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This work was supported by grant HG011052 (to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=') from the National Human Genome Research Institute (NHGRI), USA 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='05130v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='ME] 12 Jan 2023 1 Introduction A genome-wide association study (GWAS) refers to the identification of genetic variants statistically associated with complex traits or diseases across the whole genome using large population cohorts (Visscher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' GWAS typically examines associations between single-nucleotide polymorphisms (SNPs) and a trait but can also handle other genetic vari- ants such as insertion and deletions (indels) and structural variations (SVs) (Gresham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The first example of a successful GWAS was the 2005 GWAS which revealed two genetic variants significantly associated with age-related macular degeneration (Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' To date, over 5,000 human GWAS have investigated approximately 2,000 diseases and traits and have identified more than 400,000 genetic associations (Wijmenga and Zher- nakova, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This groundbreaking work has uncovered numerous compelling associations with human complex traits and diseases, shedding light on the disease mechanisms and enhancing clinical care and personalized medicine (Tam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Mendelian randomization (MR) is an epidemiological method that utilizes genetic vari- ants as instrumental variables (IVs) to infer whether an exposure causally influences an outcome (Burgess and Thompson, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Since the genotypes of individuals are ran- domly inherited from their parents and generally do not change during their lifetime, genetic variants are considered to be independent of underlying confounders and hence can be used as IVs to eliminate confounding bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Early MR studies progressed slowly because individual-level data simultaneously including genotypes and phenotypes were rarely available (Ebrahim and Davey Smith, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Recently, many large GWAS have been published and the corresponding summary statistics are available in databases such as the GWAS Catalog (MacArthur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2017) (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='ebi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='uk/gwas/), dbGaP (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='ncbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='nlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='nih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='gov/gap/), and UK biobank (UKBB, Sudlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2015)) (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='ukbiobank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='uk/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The accuracy of causal effect estimation is improved and valuable insights into the causal relationships between common risk factors and diseases are uncovered by utilizing MR with GWAS summary data (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The inverse-variance weighted (IVW) method is the most popular approach used to perform MR with GWAS summary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' A causal effect estimate yielded by the IVW method is supposedly unbiased if three so-called valid IV conditions are satisfied: (IV1) the genetic variants are strongly associated with the exposure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (IV2) the genetic variants are associated with the outcome only through the exposure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' and (IV3) the genetic variants are independent of confounders (Bowden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The directed acyclic graph (DAG) of valid IV conditions is shown in panel (a) in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Due to the complexity of genetic architecture, conditions IV2 and IV3 are often difficult to validate (Zhu, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Meanwhile, it is challenging to quantify the instrument strength and define a universal criterion for concluding that an IV satisfies condition IV1, although the F statistic can be utilized as a rough measure of weak instrument bias (Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Thus, quantifying and eliminating the bias of IVW estimate in MR analysis will lead to valid causal inference and help to understand disease etiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' A genetic variant is termed a pleiotropic variant or pleiotropy if it simultaneously affects multiple traits through different pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' There are two types of pleiotropy: vertical and horizontal pleiotropy, where the former refers to the genetic variant associated with 2 𝑋 𝜃 𝛽 (a) 𝐺 𝑌 C 𝜃 𝛽 (b) 𝐺 𝑌 C uncorrelated horizontal pleiotropy 𝑋 𝐺 𝑋1 𝑋2 𝑋𝑗 𝑋𝑝 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 𝑌 (c) 𝛽1 𝛽2 𝛽𝑗 𝛽𝑝−1 𝛽𝑝 𝜃1 𝜃2 𝜃𝑗 𝜃𝑝−1 𝜃𝑝 C 𝑋𝑝−1 correlated horizontal pleiotropy 𝛾𝑐 𝛾𝑢 Figure 1: DAG of MR and multivariable MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Panel (a): causal path digram with valid genetic IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Panel (b): causal path digram with UHP and CHP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Panel (c): causal path digram for multivariable MR methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' G: genetic IVs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' X: exposure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Y : outcome;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' C: confounders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' β: association between G and X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' θ: causal effect of X on Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' γc: direct correlation between G and C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' γu: direct correlation between G and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' one trait through the mediation of another trait (as described in panel (a) in Figure 1), while the latter refers to the genetic variant independently associated with both traits (as illustrated in panel (b) in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' IVs with evidence of horizontal pleiotropy should be removed before applying IVW because it violates either the (IV2) condition or the (IV3) condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' otherwise, a biased causal effect estimate is likely obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In the literature, there are three strategies to remove the effect of horizontal pleiotropy: 1) Identifying and excluding horizontally pleiotropic IVs by using hypothesis tests, such as the MR pleiotropy residual sum and outlier (MR-PRESSO, Verbanck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2018)) and the iterative MR and pleiotropy (IMRP, Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2021));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 2) Eliminating the effect of horizontal pleiotropy by applying robust tools;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', the MR-Egger (Bowden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2015), MR-Median (Bowden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2016), and MR-Lasso/MR-Robust (Rees et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 3) Automatically separating vertical pleiotropy from horizontal pleiotropy through a mixture mode, among which the representative methods include MR-Mix (Qi and Chatterjee, 2019) and MR contamination mixture (MR-ConMix, Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' It has been gradually realized that horizontal pleiotropy can be divided into uncorrelated horizontal pleiotropy (UHP) and correlated horizontal pleiotropy (CHP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' UHP violates the (IV2) condition and usually refers to a genetic variant that is directly associated with the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In contrast, CHP violates the IV3 condition and may occur when a genetic variant indirectly affects the outcome through the mediation of unspecified exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The DAG of UHP and CHP is shown in panel (b) in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Morrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2020) proposed causal analysis using summary effect (CAUSE), which is the first MR approach accounting for UHP and CHP simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2022) proposed MR-Corr to detect CHP by a Bayesian mixture model and Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2022) extended MR-Corr to MR-CUE (MR with CHP Unraveling shared Etiology and confounding) to detect the UHP and 3 CHP simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Alternatively, Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2021) proposed the constrained maximum likelihood-based MR (cML-MR) method that identifies UHP and CHP through Bayesian information criterion (BIC, Schwarz (1978)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In addition, Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2022) derived MR with automated instrument determination (MRAID) to address UHP and CHP, which allows vertical pleiotropy to be in high linkage disequilibrium (LD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' A significant disadvantage of most existing approaches is that they assume both UHP and CHP to have similar properties as outliers in the traditional regression approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' How- ever, there is substantial evidence that most traits have shared moderate or high genetic correlations (Bulik-Sullivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2015), violating this technical assumption required by most existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Consequently, it is challenging to remove the effect of horizon- tally pleiotropic variants by considering only one exposure in MR analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Multivariable MR, which simultaneously estimates the causal effects of multiple exposures on an outcome, is compelling in resolving this problem (Burgess and Thompson, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Multivariable MR recognizes the bias caused by horizontal pleiotropy as an omitted-variable bias (OVB), which will disappear automatically if all the omitted exposures are specified in the mul- tivariable MR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The DAG of multivariable MR is exhibited in panel (c) in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' So far, the multivariable versions of the IVW method, MR-Egger, MR-Median, and MR-Lasso/MR-Robust have been developed (Burgess and Thompson, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Rees et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Grant and Burgess, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Sanderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2019) showed that the multivariable MR is able to unbiasedly estimate the causal effects of a target exposure when the other exposure is confounder, collider, or mediator of this exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Weak instrument bias arises when the majority of IVs are weakly associated with the exposures, therefore violating the (IV1) condition and making conventional MR methods unreliable (Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' It is widely recognized that a common trait is often poly- genic affected by hundreds or even thousands of independent variants/genes with small effect sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' With the increasing sample sizes of GWAS, more and more trait-associated variants are being identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Thus, the weak instrument bias is likely to become a con- siderable problem in future MR studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2011) and Sanderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2021) suggested using the F and conditional F statistics to measure the weak instrument bias in MR and multivariable MR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2016) illustrated that the weak instrument bias also depends on the sample overlap in two-sample MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Sadreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2021) examined the impact of sample overlap and winner’s curse when weak instrument bias exists and observed that the weak instrument bias grew dramatically in the presence of winner’s curse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' For the univariate MR model with no sample overlap, Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2020) proposed the robust adjusted profile score to estimate the causal effect unbiasedly, while Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2021) provided the debiased IVW (DIVW) method to remove the weak instrument bias of IVW estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Overall, these aforementioned methods have neither provided a com- prehensive theoretical analysis of weak instrument bias nor a general solution to remove the weak instrument bias in both univariable MR and multivariable MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As the first contribution of this paper, we theoretically characterize the bias in multi- variable IVW causal estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Specifically, we demonstrate that the bias of IVW causal estimate is the product of weak instrument and estimation error biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Meanwhile, we demonstrate that the estimation error bias is a linear combination of measurement error (Yi, 2017) and confounder biases, and the sample overlaps among multiple GWAS cohorts 4 trade off the proportions of these two biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' With moderate conditions on the MR model, we theoretically illustrate how the number of IVs, sample sizes of GWAS studies, and sample overlap among GWAS cohorts influence the asymptotic behavior of multivariable IVW estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' These theoretical findings are summarized in Theorem 2 that to our best knowledge is the first comprehensive investigation of multivariable IVW estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As the second contribution of this paper, we demonstrate our novel multivariable MR approach, MR using Bias-corrected Estimating Equations (MRBEE), can estimate causal effects unbiasedly in the presence of many weak IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Under moderate conditions, we inves- tigate the asymptotic behaviors of IVW and MRBEE, revealing that MRBEE is superior to IVW in terms of strongly asymptotic unbiasedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In particular, only when an estimate is strongly asymptotically unbiased, the inference made based on this estimate is asymp- totically valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Simulations show that only MRBEE can provide unbiased causal effect estimates in the presence of many weak IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Applied to data from the UK Biobank, MR- BEE can successfully remove the weak instrument and estimation error biases and therefore make valid causal inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This paper is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In section 2, we study the asymptotic behavior of multivariate IVW estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In section 3, we introduce MRBEE and examine its asymptotic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In section 4, simulations are conducted to compare MRBEE with the existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In section 5, we apply MRBEE to estimate the causal effects of exposures on cardiovascular disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Discussion is presented in section 6 and proofs of the related theo- rems are shown in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' R package MRBEE (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='com/noahlorinczcomi/ MRBEE) and supplementary materials are available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 2 Mendelian Randomization In this section, we introduce the notations, the model of the multivariable MR, and the bias of the multivariable IVW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Since univariable MR is a special case of multivariable MR, MR refers to the multivariable MR unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 Notation For a vector a = (aj)p×1, ||a||q = (�p j=1 |aj|q)1/q with q ∈ [0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' For a symmetric matrix A = (Aij)p×p, λmax(A) and λmin(A) is its the maximum and minimum eigenvalues, A+ is its Moore–Penrose inverse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' and ||A||q = max{||Aa||q, ||a||q = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Let diag(α) be the diagonalizing operator of vector α and A ⊙ B be the Hadamard product of matrices A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' For a set A, |A| is the number of elements in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Notations O(·) and o(·) are the infinitely large and small quantities, while OP(·) and oP(·) mean that such relationships hold in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 Mendelian randomization model The central aim of MR is to estimate causal effects between exposures and an outcome un- biasedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Let gi = (gi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , gim)⊤ be an (m×1) genotype value vector of m genetic variants, xi = (xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , xip)⊤ be an (p × 1) vector representing p exposures, and yi be an outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 5 Here, m is the number of specified IVs, which is usually the number of independent loci with p-values reaching the genome-wide significant level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Let B = (β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , βm)⊤ be an (m × p) matrix of genetic effects on exposures with βj = (βj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , βjp)⊤ being an (p × 1) vector, and θ = (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , θp)⊤ be an (p × 1) vector of causal effects of the p exposures on the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The MR model is yi = x⊤ i θ + vi, (1) xi = B⊤gi + ui (2) where ui and vi are the noise terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Substituting for xi in (1), we obtain the equation yi = α⊤gi + θ⊤ui + vi, (3) where α = Bθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In the literature, (1) - (3) have been named as the structural form, first-stage, and reduced form, respectively (Stock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In this paper, we assume that the total number of exposures p is fixed and the causal effect θ is fixed and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The genetic variant gij is standardized so that E(gij) = 0 and var(gij) = 1, and all IVs are in linkage equilibrium (LE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', cov(gij, gik) = 0 for j ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The genetic effect βj is random with zero-mean, covariance matrix Σββ, and cumulative covariance matrix Ψββ Σββ = E(βjβ⊤ j ), Ψββ = mΣββ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The covariance matrix Σββ will vanish as m increase, but the cumulative covariance matrix Ψββ is still a constant matrix, representing the total genetic covariance contributed from the m IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Next, the noise terms ui and vj have zero-means and joint covariance matrix Σu×v = cov((u⊤ i , vj)⊤) = �Σuu σuv σ⊤ uv σvv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' � Thus, the exposure vector xi and outcome yi have zero-means and joint covariance matrix Σx×y = cov((x⊤ i , yj)⊤) = �Σxx σxy σ⊤ xy σyy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' � where Σxx = Ψββ + Σuu, σxy = Ψββθ + Σuuθ + σuv, and σyy = θ⊤Ψββθ + θ⊤Σuuθ + 2θ⊤σuv + σvv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Note that σuv ̸= 0 means the confounders simultaneously affect xi and yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In genetics, the genetic effect βjs can be treated as a random variable with mean 0 and variance ψβsβs/m, where ψβsβs is the IV-heritability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', the variance explained by additive effects of specified instrumental variants of the sth exposure (Bulik-Sullivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Since a complex trait is often polygenic with a contribution from thousands of independent variants, the number of causal variants can be regarded as a number approaching infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Subject to this principle, a random effect model can describe the variation of these effects more simply and essentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Although the fixed effect model has also been adopted by some works to study the asymptotic properties of the corresponding MR approaches (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2021), the random effect model is still the most commonly used at 6 genome-wide level studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Moreover, even if all causal variants were identified, the random effect model should still be more efficient than the fixed effect model to characterize the statistical property of the MR model (Diggle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The existing univariable MR methods, such as CAUSE (Morrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2020) and MR- CUE (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2022), can successfully remove the effects of CHP only when a small fraction of IVs have CHP, which is easy to violate because common traits may share a large fraction of pleiotropic variants (Bulik-Sullivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In contrast, multivariable MR resolves the pleiotropic variant problem by specifying all the relevant exposures in the model (1), as the multivariable regression can automatically account for the pleiotropic variants shared by these exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This is one of the greatest advantages of multivariable MR over univariable MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Hence, we assume that all the exposures can be included in the multivariable MR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' therefore, the CHP effect is ignorable in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' On the other hand, some IVs may still present strong UHP effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' To account for potential UHP, we propose using an iterative procedure to remove these invalid IVs, which is similar to detecting outliers in MR analysis (Verbanck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Zhu, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Thus, the bias introduced by UHP and CHP can be greatly alleviated, as demonstrated in our proposed MRBEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 Mendelian randomization with GWAS summary data With the rapid development of GWAS, large GWASs of exposures and disease outcomes have been conducted and their summary statistics including effect sizes, SEs, and variant information are publicly available for download (Sudlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' MacArthur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Thus, many recently developed MR methods are often designed based on GWAS summary statistics, as is in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' With GWAS summary statistics, MR is mainly based on the linear regression model ˆαj = ˆβ⊤ j θ + εj, (4) where ˆαj and ˆβj are estimated from outcome and exposure GWAS for jth IV, and εj repre- sents the residual of this regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Let y[0] = (y[0] 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , y[0] n0)⊤ be the sample vector from outcome GWAS, x[1] = (x[1] 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , x[1] n1)⊤, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , x[p] = (x[p] 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , x[p] np)⊤ be the sample vec- tors of the 1st, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , pth exposure GWAS cohorts, and G[0] = (g[0] ij )n0×m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , G[p] = (g[p] ij )np×m be the sample matrices of m genetic variants of the outcome and 1st, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , pth exposure GWAS cohorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The sample size of the sth cohort is ns, the overlapping sample size between the sth and the kth cohorts is nsk, and the minimum sample size is nmin = min{n0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , np}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The GWAS summary data are generated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Suppose that y[0], {x[s]}, and {G[s]} are centered, and the m genetic variants are in LE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' E(G[s]⊤G[s]/ns) = Im for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This orthogonality enables the following genetic effects to be estimated separately ˆαj = g[0]⊤ j y[0] n0 , ˆβjs = g[s]⊤ j x[s] ns , (5) 7 where the corresponding variance estimates are given by var(ˆαj) = σyy − θ⊤Σββθ n0 ≈ σyy n0 , var(ˆβjs) = σxsxs − σβsβs ns ≈ σxsxs ns .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (6) Then the GWAS summary data are formed by ˆα = (ˆα1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , ˆαm)⊤, ˆβj = (ˆβj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , ˆβjp)⊤, ˆB = ( ˆβ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , ˆβm)⊤, the related SE estimates, the p-values, and sample sizes n0, n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , np, and SNPs information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The IVW method is equivalent to a weighted regression which estimates θ by ˆθIVW = arg min θ � 1 2m m � j=1 (ˆαj − ˆβ⊤ j θ)2 var(ˆαj) � = ( ˆB⊤V ˆB)−1 ˆB⊤V ˆα, (7) where V = diag(1/var(ˆα1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , 1/var(ˆαm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In practice, we often standardize ˆαj and ˆβjs by ˆαj/se(ˆαj) and ˆβjs/se(ˆβjs) to remove the minor allele frequency effect (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Therefore, var(ˆαj) = 1 for all j and (7) reduces to ˆθIVW = arg min θ � 1 2m∥ ˆα − ˆBθ∥2 2 � = ( ˆB⊤ ˆB)−1 ˆB⊤ ˆα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (8) Here, we qualitatively show that the IVW estimate is biased due to the estimation errors of ˆα and ˆB, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', wα = ˆα − α and Wβ = ˆB − B, and meanwhile, the weak IVs can inflate th estimation error bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Specifically, consider the estimating equation and Hessian matrix of ˆθIVW: SIVW(θ) = − ˆB⊤( ˆα − ˆBθ) m = 1 m � − B⊤wα − B⊤Wβθ + W⊤ β wα − W⊤ β Wβθ � , (9) HIVW = ˆB⊤ ˆB m = 1 m � B⊤B + W⊤ β Wβ + B⊤Wβ + W⊤ β B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (10) That is, SIVW(θ) is the score function of (8) and ˆθIVW is estimated by solving SIVW(ˆθIVW) = 0, and HIVW is the second order derivative matrix of (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In particular, since the third derivative of the quadratic loss function (8) is zero, we have ˆθIVW − θ = −H−1 IVWSIVW(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As a result, the expectation of the bias of ˆθIVW is approximately: E(ˆθIVW − θ) ≈ −E(HIVW)−1E(SIVW(θ)) = − � Σββ + ΣWβWβ �−1 � �� � weak instrument bias � ΣWβWβθ − σWβwα � � �� � estimation error bias , (11) where wβj is the jth row of Wβ, wαj is the jth element of wα, and cov((w⊤ βj, wαj)⊤) = ΣWβ×wα = �ΣWβWβ σWβwα σ⊤ Wβwα σwαwα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' � , 8 Intuitively, the bias of ˆθIVW has a product structure “weak instrument bias × estimation error bias”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We call {ΣWβWβθ − σWβwα} the estimation error bias because it comes from the covariance matrix of estimation errors ΣWβ×wα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We term {Σββ + ΣWβWβ} the weak instrument bias because the bias of ˆθIVW is inflated if the covariance matrix of effect sizes Σββ is not considerably larger than the covariance matrix of estimation errors ΣWβWβ, which often happens if the majority of IVs used to infer the causal effect have weak effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='4 Asymptotic behavior of IVW estimate In this subsection, we investigate the asymptotic behavior of the IVW estimate as the num- ber of IVs m and the minimum sample size nmin go to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' To facilitate the theoretical derivation, we specify the following three definitions and four regularity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Definition 1 (Sub-Gaussian variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' A random variable x is sub-Gaussian distributed with sub-Gaussian parameter τx > 0 if for all t > 0, Pr(|x − E(x)| ≥ t) ≤ 2e−t2/τ 2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Definition 2 (Well-conditioned covariance matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' A covariance matrix Σ is well-conditioned if there is a positive constant d0 such that 0 < d−1 0 ≤ λmin(Σ) ≤ λmax(Σ) ≤ d0 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Definition 3 (Strongly asymptotically unbiased estimate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Let ˆθ be a consistent estimate of θ with an asymptotic normal distribution √sn(ˆθ − θ) D −→ N(µθ, Σθ), where µθ is a vector with a bounded ℓ2-norm, Σθ is a well-conditioned covariance matrix, and sn is a sequence of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Then ˆθ is called a strongly asymptotically unbiased estimate of θ if µθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' A sub-Gaussian variable is one of the basic concepts in modern statistics (Vershynin, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' It generalizes the scope of ordinary Gaussian variables to include all bounded dis- crete and common continuous variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The well-conditioned covariance matrix is another important concept (Bickel and Levina, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' A well-conditioned covariance matrix will ensure that the related statistical optimization is nondegenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In addition, we define the strongly asymptotic unbiasedness to distinguish the consistent estimate whose bias square vanishes with an equal and a smaller rate than its variance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' If an estimate is consistent but its bias square and variance vanish at the same rate, the classic confidence interval cannot cover the true parameter with a probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='95, thus leading to invalid statistical inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This problem widely exists in all fields of statistics, especially, in non- parametric statistics and high-dimensional statistics, and many novel methods are derived to reduce the bias such that the bias square vanishes faster than the variance (Hall, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Van de Geer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Jankova and Van De Geer, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Calonico et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Condition 1 (Regularity conditions for multivariable MR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (C1) For gi = (gi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , gim)⊤, each entry gij is a bounded sub-Gaussian variable with E(gij) = 0, var(gij)=1, and sub-Gaussian parameter τg ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' For all (i, j) ̸= (t, s), gij is independent of gts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (C2) For ui = (ui1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , uip)⊤, each entry uij is a sub-Gaussian variable with E(uij) = 0, var(uis) ∈ (0, ∞), and sub-Gaussian parameter τu ∈ (0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' vi is a sub-Gaussian variable with E(vi) = 0, var(vi) ∈ (0, ∞), and sub-Gaussian parameter τv ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 9 Besides, (u⊤ i , vi)⊤ is independent of (u⊤ t , vt)⊤ for all i ̸= t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Furthermore, Σu×v is a well-conditioned covariance matrix of (u⊤ i , vi)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (C3) For βj = (βj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , βjp)⊤, √mβjs is a sub-Gaussian variable with E(√mβjs) = 0, var(√mβjs) ∈ (0, ∞), and sub-Gaussian parameter τβ ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' For all j ̸= t, βj is independent of βt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In addition, Ψββ is a well-conditioned covariance matrix of √mβj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (C4) The genetic variant gij, the genetic effect βj, the noise terms ui and vi, are three mutually independent groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Conditions (C1)-(C4) restrict that all variables involved in this paper are sub-Gaussian distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In practice, gij is standardized from a binomial variable with status 0, 1, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Hence, it is supposedly a bounded sub-Gaussian variable as long as its minor allele frequency is not rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Besides, we assume √mβj to be sub-Gaussian with a well-conditioned covariance matrix Ψββ because the covariance explained by each variant Σββ decreases as the number of instrumental variants m increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Denote wαj = ˆαj − αj and ωjs = ˆβjs − βjs, s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Then for all j, � � � � � √n0wαj √n1wβ1j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' √npwβ1p � � � � � D −→ N � � � � � � � � � � � 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 0 � � � � � , � � � � � � σyy n01 √n0n1σyx1 · · n01 √n0npσyxp n01 √n0n1σyx1 σx1x1 · · n1p √n1npσx1xp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' n0p √n0npσyxp n1p √n1npσx1xp · · σxpxp � � � � � � � � � � � � , if n0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , np and m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Theorem 1 demonstrates the asymptotic normal distribution of the estimation errors, based on which we are able to obtain ΣWβWβ = ∆xx ⊙ Σxx, σWβwα = δxy ⊙ σxy, σwαwα = σyy/n0, (12) where the (j, s)th element of ∆xx is njs/(njns) and the jth element of δxy is nj0/(n0nj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As a result, the expectations of SIVW(θ) and HIVW are given by E(SIVW(θ)) = (∆xx ⊙ Σxx)θ − δxy ⊙ σxy, (13) E(HIVW) = Σββ + ∆xx ⊙ Σxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (14) By expressing σxy = Σxxθ + σuv, we obtain an alternative expectation of SIVW(θ)): E(SIVW(θ)) � �� � estimation bias = {(∆xx − δxy1⊤) ⊙ Σxx}θ � �� � measurement error bias − δxy ⊙ σuv � �� � confounder bias .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (15) From this expectation, it is clear that there are two sources of the estimation error bias: {(∆xx −δxy1⊤)⊙Σxx}θ comes from the measurement error, while {δxy ⊙σuv} is caused by the confounder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Here, we call {(∆xx −δxy1⊤)⊙Σxx}θ the measurement error bias because it has the same statistical impact, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', shrinking the coefficient estimate toward zero, as in 10 measurement error analysis (Yi, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In contrast, we term {δxy ⊙ σuv} the confounder bias because σuv ̸= 0 implies that there are underlying confounders simultaneously affecting both xi and yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In addition, the overlapping fraction vector δxy trades off these two sources of biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Generally, the measurement error bias is dominant when the elements of δxy are small, while the confounder bias dominates when the elements of δxy are large, and there may exist a special sample overlap such that δxy⊙σuv = {(∆xx−δxy1⊤)⊙Σxx}θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In univari- able MR, this special fraction is n01/n0 = σxxθ/σxy, which guarantees that E(SIVW(θ)) = 0 and E(ˆθIVW) = θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This theoretical result explains why in the empirical studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', Figures 1 and 2 in Sadreev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2021)), ˆθIVW has a negative bias when n01/n0 is small, positive bias when n01/n0 is large, and is unbiased at this specific point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Suppose conditions (C1)-(C4) hold and m, nmin → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Then (i) if m/√nmin → 0, √nmin(ˆθIVW − θ) D −→ N(0, ψθΨ−1 ββ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (ii) if m/√nmin → c0, √nmin(ˆθIVW − θ) D −→ N(−c0Ψ−1 ββ(ΨWβWβθ − ψWβwα), ψθΨ−1 ββ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (iii) if m/nmin → c0, ˆθIVW − θ P −→ −c0(Ψββ + c0ΨWβWβ)−1(ΨWβWβθ − ψWβwα);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (iv) if m/nmin → ∞, ˆθIVW P −→ Ψ+ WβWβψWβwα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' where ΨWβ×wα = �ΨWβWβ ψWβwα ψ⊤ Wβwα ψwαwα � = lim nmin→∞ �nminΣWβWβ nminσWβwα nminσ⊤ Wβwα nminσwαwα � , ψθ = ψwαwα + θ⊤ΨWβWβθ − 2θ⊤ψWβwα, and c0 is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Theorem 2 is one of two main theorems in this paper and points out four scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' First, if m goes to infinity with a lower rate than √nmin, ˆθIVW is strongly asymptotically unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In other words, ˆθIVW is able to reliably infer causality only when the sample size of GWAS data is quadratically larger than the number of IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' On the other hand, the asymptotic covariance matrix of ˆθIVW is the inverse of the cumulative covariance matrix Ψββ = �m j=1 cov(βj), therefore, it is optimal to include as many associated variants as possible in order to have Ψββ large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In contrast, using a few top significant variants to perform MR analysis is not recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Second, if m tends to infinity with the same rate as √nmin, √nmin(ˆθIVW − θ) converges to an asymptotic normal distribution with a non-zero asymptotic bias {−c0Ψ−1 ββ(ΨWβWβθ− ψWβwα)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In this asymptotic bias, {−c0(ΨWβWβθ−ψWβwα)} is caused by SIVW(θ) and Ψ−1 ββ is resulted by H−1 IVW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Since the asymptotic bias and asymptotic covariance matrix are of the same order in this scenario, the inference made is invalid although the bias of ˆθIVW is infinitesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Scenario (iii) is more serious than (ii) because the bias of ˆθIVW will not vanish even when √nmin goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In the fourth scenario, ˆθIVW converges to a term irrelevant to θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Scenarios (ii) - (iv) indicate that the IVW method is unlikely to make valid causal inference unless the sample sizes are quadratically larger than the number of IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' It is crucial to understand the asymptotic behaviors of ˆθIVW since the IVW method serves as the foundation for practically all MR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Specifically, IMRP and MR- PRESSO use hypothesis tests to identify invalid IVs and then apply the IVW method to 11 estimate causal effects based on valid IVs only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' MR-Robust and MR-Median replace the quadratic loss function used in IVW by a robust loss function and absolute loss function, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Although there have been literature studying the bias of ˆθIVW empirically (Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2011, 2016), they could not explain what causes the bias and how it behaves asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In contrast, Theorem 2 points out the asymptotic properties of ˆθIVW, representing a significant advance in understanding the IVW method and its extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 3 Bias-corrected Estimating Equation According to (11), it is possible to remove the bias of SIVW(θ) by subtracting the measure- ment error bias {ΣWβWβθ−σWβwα}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Motivated by this principle, we propose MRBEE that estimates the causal effect estimates by solving the new unbiased estimating equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In this section, we introduce the estimation of MRBEE, investigate its asymptotic properties, and discuss three implementation issues including the estimations of the bias-correction terms, the estimation of sandwich formula of causal effect estimate, and the detection of potential pleiotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 Estimation of causal effect There are many methods that can remove the measurement error bias, including max- imum likelihood estimation, unbiased estimating functions, and simulation-extrapolation (SIMEX) methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', Yi (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' MRBEE is a subtraction correction method belong- ing to the class of unbiased estimating function methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Specifically, MRBEE estimates θ by solving the following unbiased estimating equation: SBEE(θ) = SIVW(θ) − (ΣWβWβθ − σWβwα), (16) where SIVW(θ) = − ˆB⊤( ˆα − ˆBθ)/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The solution ˆθBEE such that SBEE(ˆθBEE) = 0 is ˆθBEE = � ˆB⊤ ˆB m − ΣWβWβ �−1� ˆB⊤ ˆα m − σWβwα � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (17) In practice, ˆθBEE is unreliable when the minimum eigenvalue of ˆB⊤ ˆB/m − ΣWβWβ is nega- tive, which is also a common problem for subtraction correction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In this case, we recommend first adjusting the negative eigenvalues to be 0 and then using the generalized inverse of this semi-positive matrix to yield ˆθBEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Suppose conditions (C1)-(C4) hold and m, nmin → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Then (i) if m/nmin → 0, √nmin(ˆθBEE − θ) D −→ N(0, ψθΨ−1 ββ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (ii) if m/nmin → c0, √nmin(ˆθBEE − θ) D −→ N(0, ψθΨ−1 ββ + c0Ψ−1 ββΨBCΨ−1 ββ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (iii) if m/nmin → ∞ and m/n2 min → 0, � n2 min/m(ˆθBEE − θ) D −→ N(0, Ψ−1 ββΨBCΨ−1 ββ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 12 where ψθ is defined in Theorem 2, c0 is a positive constant, and ΨBC is a semi-positive symmetric matrix whose expression is shown in equation (91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Theorem 3 indicates the following three scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' First, if m/n → 0,√nmin(ˆθBEE − θ) converges to a normal distribution with a zero mean and the covariance matrix being exactly the same as ˆθIVW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In other words, ˆθBEE not only enjoys the strongly asymptotic unbiasedness but also loses no efficiency in comparison to ˆθIVW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Second, if m/nmin → c0 ∈ (0, ∞), there is an additional covariance matrix c0Ψ−1 ββΨBCΨ−1 ββ in the asymptotic normal distribution, where ΨBC is introduced by the bias-correction terms: ΨBC = lim nmin→∞ var �nmin √m � (W⊤ β Wβ − mΣWβWβ)θ − (W⊤ β wα − mσWβwα) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In this scenario, ˆθBEE is again strongly asymptotically unbiased with a convergence rate √nmin, while ˆθIVW suffers from a bias not vanishing asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In the third scenario, ˆθBEE is still strongly asymptotically unbiased with a convergence rate � n2 min/m, and the asymptotic distribution is dominated by the bias correction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In contrast, ˆθIVW con- verges to a term irrelevant to θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Note that ˆθIVW is not consistent unless m/n → 0 and the inference made by ˆθIVW is unreliable unless m/√nmin → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Therefore, MRBEE is superior to IVW in terms of both unbiasedness and asymptotic validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Most previous works of MR introduced their methods from the perspective of empirical applications and have not discussed the asymptotic properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', Bowden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2015, 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Verbanck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Morrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Some works (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2021) described the asymptotic behaviors of the causal effect estimates yielded by their univariate MR methods, but the convergence rates and related conditions were not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' For example, Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2020) showed that the convergence rate of their causal effect estimate is O(V1/√V2) where V1 and V2 are two m-concentrations, which may mislead that this estimate has a O(√m) convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' From Theorem 3, it is easy to see that ˆθBEE is strongly asymptotically unbiased, the asymptotic covariance matrix is ψθΨ−1 ββ, ψθΨ−1 ββ + c0Ψ−1 ββΨBCΨ−1 ββ, and Ψ−1 ββΨBCΨ−1 ββ, and the convergence rate is √nmin, √nmin, and � n2 min/m, with respect to scenarios (i), (ii), and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In addition, although our method focuses on the multivariable MR model, the theoretical results can be readily extended to the univariable MR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' To the best of our knowledge, this is the first theoretical work to demonstrate how the convergence rate and asymptotic normal distributions vary with the sample sizes of multiple GWAS cohorts and the number of IVs for univariable and multivariable MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 Estimation of bias-correction terms In this subsection, we discuss how to estimate the bias-correction terms ΣWβWβ and σWβwα in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Specifically, we apply the method provided by Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2015) to estimate the covariance matrix ΣWβ×wα of the vector (w⊤ βj, wαj)⊤ from insignificant GWAS summary statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Let G{0} = (g{0} ij )n1×M, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , G{p} = (g{p} ij )ns×M be the sample matrices of M insignificant and independent genetic variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The insignificance means that the p-value of the genetic variants are larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='05 for all exposures and outcome, and independence 13 means that these variants are in LE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The insignificant GWAS statistics are estimated by ˆα∗ j = g{0}⊤ j y[0] n0 , ˆβ∗ js = g{s}⊤ j x[s] ns , (18) for s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' With these insignificant effect sizes, ΣWβ×wα can be estimated by ˆΣWβ×wα = 1 M M � j=1 (ˆβ∗ j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , ˆβ∗ jp, ˆα∗ j)⊤(ˆβ∗ j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , ˆβ∗ jp, ˆα∗ j), (19) because ˆα∗ j and ˆβ∗ js follow the same distributions of wαj and wβjs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Here, ˆΣWβWβ is the first (p × p) sub-matrix of ˆΣWβ×wα and σWβwα consists of the first p − 1 elements of the last column of ˆΣWβ×wα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Suppose conditions (C1)-(C4) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Let g{s} ij satisfy the condition (C1), E(x[s] i |g{s} ij ) = 0 for all 1 ≤ s ≤ p, and E(y[0] i |g{0} ij ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Then ∥Σ − 1 2 Wβ×wα ˆΣWβ×wαΣ − 1 2 Wβ×wα − Ip+1∥2 = OP � 1 √ M � , if nmin and M → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Theorem 4 shows that ˆΣWβ×wα has a O( √ M) convergence rate after adjusting the scale of ΣWβ×wα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As there may be more than 1 million independent variants in the whole genome, ˆΣWβ×wα has high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In addition, n0, n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', np → ∞ are required such that √n0ˆα∗ j and √ns ˆβ∗ js are asymptotically normally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In addition, many popular GWAS methods such as cross-phenotype association analysis (CPASSOC, Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2015)) and multi-trait analysis of GWAS (MTAG, Turley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2018)) need to estimate the covariance matrix of the estimation errors of GWAS summary statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As far as we are concerned, this theorem is the first one to theoretically guarantee that this covariance matrix can be consistently estimated from the GWAS insignificant statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 Estimation of sandwich formula In this subsection, we illustrate how to estimate the covariance matrix of ˆθBEE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', cov(ˆθBEE)=ΣBEE(θ), through the famous sandwich formula (Liang and Zeger, 1986): ΣBEE(θ) = F−1 BEEVBEE(θ)F−1 BEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (20) Here, the outer matrix FBEE is the Fisher information matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', the expectation of the Hessian matrix of SBEE(θ): FBEE = −E �∂SBEE(θ) ∂θ⊤ � = Σββ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (21) 14 The inner matrix VBEE(θ) is the covariance matrix of SBEE(θ): VBEE(θ) = E � 1 m m � j=1 Sj(θ)Sj(θ)⊤ � , (22) where Sj(θ) = −(ˆαj − θ⊤ ˆβj) ˆβj − ΣWβWβθ + σWβwα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (23) A consistent estimate of ΣBEE(θ) is ˆΣBEE(ˆθBEE) = ˆF −1 BEE ˆVBEE(ˆθBEE)ˆF −1 BEE, (24) where ˆFBEE = ˆB⊤ ˆB m − ˆΣWβWβ, ˆVBEE(ˆθBEE) = 1 m m � j=1 ˆSj(ˆθBEE) ˆSj(ˆθBEE)⊤ ˆSj(ˆθBEE) = −(ˆαj − ˆθ⊤ BEE ˆβj) ˆβj − ˆΣWβWβ ˆθBEE + ˆσWβwα, (25) and ˆΣWβWβ and ˆσWβwα are estimated through (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Under the conditions of Theorem 4, ||Σ − 1 2 BEE(θ) ˆΣBEE(ˆθBEE)Σ − 1 2 BEE(θ) − Ip||2 = OP � max � 1 √nmin , √m nmin , � log m m �� if nmin, m and M → ∞ and m/n2 min → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Theorem 5 shows that ˆΣBEE(θ) has a min(√nmin, � n2 min/m, � m/ log m) convergence rate when m/n2 min → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The first two convergence rates are brought by ||ˆFBEE − FBEE||2, while the third convergence rate is yielded by || ˆVBEE(ˆθBEE) − VBEE(θ)||2, where the non- asymptotic analysis tool of random matrices are used to derive them (Vershynin, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Note that the SE estimation should be of the same importance as the causal effect estima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Although the inference is made based on an unbiased estimate, it could still be invalid if the SE estimate is not reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Our simulations show that the vast majority of current univariable and multivariable MR approaches are unable to provide accurate SE estimates, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', MR-median consistently overestimates the SE and others have a tendency to under- estimate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In contrast, the sandwich formula, whose dependability has been extensively investigated empirically, is a reliable technique to obtain the SE estimate for MRBEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This is yet another advantage of MRBEE over current approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='4 Pleiotropy test Due to the complexity of GWAS data, we cannot completely rule out the possibility of the existence of UHP and CHP even in the case of modeling multiple exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Specifically, 15 if UHP and CHP exist, αj = β⊤ j θ + γuj + γcj, (26) where γuj is a UHP satisfying E(γujβj) = 0 and γcj is a CHP satisfying E(γcjβj) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Conventional pleiotropy detection methods such as MR-Robust, MR-PRESSO, and IMRP do not distinguish between UHP and CHP as long as they resemble outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Recently, some novel methods such as CAUSE and MR-CUE have been developed to separate vertical pleiotropy, UHP and CHP by using a mixture model, allowing slightly larger proportions of UHP and CHP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' However, both the conventional and novel methods only focus on one exposure, failing to realize that most CHP and UHP may disappear automatically after specifying all the relevant exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In this paper, we assume that we have excluded all CHP by including all the relevant exposures and we adopt IMRP (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2021) to detect UHP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' First, we define UHP as γj = αj − β⊤ j θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (27) In particular, we assume that γj has a product structure γj = γ∗ j bj, where γ∗ j is a fixed number and bj is a non-random binary indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Let O = {j : bj ̸= 0} be the set of UHP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The number of elements in O (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', |O|) should be relatively small, otherwise the UHP cannot be regarded as outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We specify the following variant-specific hypothesis test: H0 : γj = 0, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' H1 : γj ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (28) A natural estimate of γj is ˆγj = ˆαj − ˆβ⊤ j ˆθBEE = γj + ϵj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (29) where ϵj = wαj − w⊤ βjθ + w⊤ βj(ˆθBEE − θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' It is easy to see that E(ϵj) = 0 and var(ϵj) = θ⊤ΣWβwαθ + σωγωγ − 2θ⊤σWβwα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As a result, tγj = ˆγ2 j /var(ϵj) can be chosen as a feasible testing statistic for the hypothesis in (28), which follows a central χ2 1-distribution under the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In practice, var(ϵj) can be estimated by � var(ϵj) = ˆϑ⊤ BEESEj ˆRWβ×wαSEj ˆϑBEE, (30) where ˆϑBEE = (ˆθ⊤ BEE, −1)⊤, SEj = diag(se(ˆβj1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , se(ˆβjp), se(ˆαj)), and ˆRWβ×wα is the correlation matrix of ˆΣWβ×wα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Then γj is considered as an outlier if Fχ2 1(ˆtγj) > κ, (31) where Fχ2 1(·) is the CDF of χ2 1-distribution, ˆtγj = ˆγ2 j / � var(ϵj), and κ is a given threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Assume that |O| is fixed and bounded and γ1∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , γ∗ m are a series of non- random numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Then under the conditions of Theorem 5, there exists a threshold κ = Fχ2 1(C0 log m) such that Pr(O = ˆO) → 1 16 where ˆO = {j : Fχ2 1(ˆtγj) > κ} and C0 is a sufficiently large constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Theorem 6 indicates that there is a theoretical threshold κ = Fχ2 1(C0 log m) to consis- tently identify all UHP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This threshold increases with a rate O(log m) to reduce the false discovery rate (FDR) and its concrete value can be chosen by a FDR control method (Ben- jamini and Hochberg, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In practice, MRBEE will iteratively apply the hypothesis test (28) to remove the outliers and use the remaining IVs to estimate θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The stable estimate is regarded as ˆθBEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 4 Simulation In this section, we conduct numerical comparisons between MRBEE and existing MR methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Full details of simulation settings and additional simulation results are shown in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 Univariable MR investigation We briefly introduce the simulation settings for univariable MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' First, we generate a bi- nomial variable from Binom(2, bj) where bj ∼ Unif(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5) and standardize it as gij, the direct effect βj from N(0, 1/m), and ui, vi from a normal distribution with correla- tion coefficient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The variances of ui and vi are chosen such that the IV-heritabilities are σββ/σxx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 and θ2 × (σββ/σyy) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We specify the causal effect θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We compare MRBEE with IVW, DIVW, MR-Egger, MR-Lasso, MR-Median, IMRP, MR-ConMix, and MR-MiX, where most are implemented by using the R package MendelianRandomization (Yavorska and Burgess, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Additionally, the IMRP pro- cedure is incorporated into MRBEE in which the threshold κ is chosen by R package FDRestimation (Murray and Blume, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The so-called overlapping fraction is n01/n0, where the special fraction such that E(SIVW(θ)) = 0 is n01/n0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The number of independent replications is 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' First, we study the influences of overlapping fraction n01/n0 and the number of IVs m, with the results displayed in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Here, we fix n0 = n1 = 20000, specify n01 according to the overlapping fraction, and assume no UHP or CHP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' It is easy to see that in general, only MRBEE is able to yield an unbiased estimate of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' For a special overlapping fraction (placed in the second column of Figure 2), all approaches become unbiased except DIVW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' DIVW performs badly because it will further remove IVs based on their significance levels and consequently introduces an extra IV selection bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In addition, the SE of causal effect estimate for all methods increases as the overlapping fraction decreases but remains unchanged by the increase of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The results are consistent with our theoretical expectation and asymptotic properties of MRBEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As for the standard error, we display the boxplot of ˆse(ˆθ)−se(ˆθ) where se(ˆθ) is approxi- mated by the empirical SE calculated from the independent replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' It is evident that the SE estimates produced by all approaches have reduced variances as m grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' However, only MRBEE and DIVW can provide consistent SE estimates, confirming the accuracy of MRBEE and DIVW’s SE formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Additionally, MR-ConMix is extremely likely to 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='26 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='imrp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='conmix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='mix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='Figure 2: Investigation of MR methods for univarate MR with sample sizes n0 = n1 = 20000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' in terms of overlapping fraction and number of instrumental variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 18 underestimate the standard error, while MR-Egger, MR-Lasso, MR-Median, and MR-Mix constantly overestimate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As for IVW, it underestimates the SE when the fraction is large and overestimates it when the fraction is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The coverage frequency refers to the frequency that the confidence interval covers the true causal effect among simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Here, this confidence interval is constructed by dou- bling ˆse(ˆθ), which means that the coverage frequency corresponding to neither an inflated type-I error nor an inflated type-II error should be around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We observed that only MR- BEE enjoys a coverage frequency around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' When m = 250, MR-Egger, MR-Lasso, and MR-Median suffer from inflated type-II error rates, likely because these methods cannot estimate the SE properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' These approaches also result in inflated-type I error rates caused by weak instrument bias as m increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Additionally, because MR-Mix overestimates the SE, it consistently exhibits a substantially inflated type-II error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Furthermore, IMRP and MR-ConMix consistently have inflated type I error rates because they frequently un- derestimate the SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We next verify if the asymptotic normal distributions in Theorem 2 and Theorem 3 are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' For a general estimate ˆθ, the asymptotic bias and SE are √sn(ˆθ − θ) and √snse(ˆθ), respectively, where √sn is the convergence rate of ˆθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' If this estimate is strongly asymptotically unbiased, the asymptotic bias sn(ˆθ − θ) should also be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Besides, if two estimates have equal asymptotic SEs, they are equally powerful in terms of statistical efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We select MRBEE, IVW, MR-Median, and MR-Lasso to compare, only consider two overlapping fractions: 100% and 0%, set n0 = n1 = nmin, and fix the causal effect θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As for m and nmin, we focus on the following four cases: (1) m = 2500, 5000, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , 50000 and m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='9/n = c0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' we examine the direct bias: ˆθ − θ, asymptotic SE: � n2 min/m se(ˆθ), and coverage frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2) m = 250, 500, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , 5000 and m/n = c0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' we examine the direct bias: ˆθ − θ, asymptotic SE: √nmin se(ˆθ), and coverage frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (3) m = 250, 500, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , 5000 and m2/n = c0 = 5 and 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' we examine the asymptotic bias: √nmin(ˆθ − θ), asymptotic SE: √nmin se(ˆθ), and coverage frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (4) m = 250, 500, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , 5000 and m3/n = c0 = 5 and 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' we examine the asymptotic bias: √nmin(ˆθ − θ), asymptotic SE: √nmin se(ˆθ), and coverage frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Note that we directly generate the estimation errors Wβ and wα according to Theorem 1 because nmin in cases (3) and (4) can be larger than one million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The calculations involving individual-data are extremely time-consuming in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Figure 3 demonstrates the simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In case (1), ˆθBEE is unbiased while the other three estimates suffer from non-removable biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As for the asymptotic SE, � n2 min/m se(ˆθBEE) remains unchanged when nmin and m are sufficiently large (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', the bars colored in blue), verifying conclusion (iii) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' However, the coverage fre- quency of MRBEE is a little larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='95, meaning that the SE of ˆθBEE is overestimated in this extreme case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This phenomenon is reasonable because Theorem 4 points out that the convergence rate of the sandwich formula is min(√nmin, � n2 min/m, � m/ log m), which slows down as m increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In case (2), the direct bias of ˆθIVW is unchanged as nmin tends 19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 −0.' metadata={'source': 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='MR−Median MR−Lasso ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='MR−BEE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='IVW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='MR−Median MR−Lasso ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='MR−BEE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='IVW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='MR−Median MR−Lasso ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='MR−BEE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='IVW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='MR−Median MR−Lasso ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m^(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='9)/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 m^(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='9)/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 m^(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='9)/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 m^(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='9)/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 m^(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='9)/n = 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5K 5K 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5K 10K 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5K 15K 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5K 20K 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5K 25K 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5K 30K 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5K 35K 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5K 40K 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5K 45K 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5K 50K m^2/n = 10 m^2/n = 5 m^2/n = 10 m^2/n = 5 m^2/n = 10 m^2/n = 5 n 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='25K 25K 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='25K 100K 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='25K 225K 306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='25K 400K 506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='25K 625K 756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='25K 900K 1056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='25K 1225K 1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='25K 1600K 1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='25K 2025K 2256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='25K 2500K m^3/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 m^3/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 m^3/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 m^3/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 m^3/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 m^3/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 n 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='4M 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='825M 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='8M 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='475M 200M Figure 3: Investigations of MRBEE and IVW in terms of asymptotic bias and covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 20 to infinity, confirming conclusion (iii) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As for ˆθBEE, its asymptotic SE is a little larger than ˆθIVW, verifying item (ii) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In case (3), the asymptotic bias of ˆθIVW is constant as nmin goes to infinity, illustrating that ˆθIVW is not strongly asymptotically unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As a result, the coverage frequencies of ˆθIVW are significantly smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='95, confirming our claim that any inference made based on ˆθIVW is invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Besides, the asymptotic SEs of ˆθBEE and ˆθIVW are essentially the same, indicating that ˆθBEE and ˆθIVW are equally efficient as long as m/nmin → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In case (4), the asymptotic bias of IVW, MR-Median, and MR-Lasso vanish as nmin increases and their coverage frequencies are around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='95, which is consistent with conclusion (i) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The equal asymptotic SEs also indicate that ˆθBEE and ˆθIVW are equally efficient in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In addition, IVW, MR-Median, and MR-Lasso suffer from the same degree of bias when there is no pleiotropy, while MR-Median not only suffers from a large asymptotic SE but also is likely to overestimate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' To understand why MR-Median is always less efficient than IVW when there is no pleiotropy, its asymptotic behavior is worthy of future investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 Multivariable MR investigation For multivariable MR, we consider p = 6 exposures and set the causal effect vector to be θ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3, 0, 0)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' All of the exposures’ IV-heritabilities are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3, while the outcome’s IV-heritability is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We set an AR(1) structured genetic correlation matrix with coefficient ρ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5 for the genetic effect βj, while considering a more intricate correlation structure for the noise terms ui and vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In order to better mimic real data analysis, we take into account the scenario of completely overlapping GWAS samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', nsk = ns = nk for all s, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Other cases of sample overlaps and details of the simulation settings are present in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Figure 4 presents the comparison between the multivariable versions of IVW, MR- Egger, MR-Lasso, MR-Median, and MRBEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In general, MRBEE is the only method that can produce unbiased causal effect estimates in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As m increases, the SE of ˆθBEE remains the same, while the estimation error of the SE estimate becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' However, a very large m may conversely reduce the accuracy of the SE estimate in multivariable MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' For example, the SE estimates of all approaches in the cases of m = 1000 have larger empirical variances than those in the cases of m = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This phenomenon can be explained by Theorem 5, which indicates that the convergence rate of the sandwich formula is min(√nmin, � n2 min/m, � m/ log m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Hence, a larger m may result in a worse SE estimate if nmin is not increased as m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' All the multivariable MR methods except MRBEE suffer from larger weak instrument biases with the increase of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The SE estimates provided by these methods, in particular MR-Median, are less reliable than that of MRBEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Thus, causal inferences based on the existing multivariable MR methods could be even more unreliable than univariable MR methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In addition, ˆθIVW can have a bias toward any direction in multivariable MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' For example, the bias of ˆθ5,IVW is positive while the bias of ˆθ6,IVW is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The actual directions are jointly determined by the correlations of confounders and genetic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We also examine the impact of omitting some important exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We conduct simu- 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='24 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='32 θ^ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='35 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='35 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='30 −0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='bee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='ivw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='egger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='lasso median ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='Figure 4: Investigation of MR methods for multivariable MR with sample sizes n0 = · · · = n6 = 20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='and overlap-sample sizes n01 = · · · = n65 = 20000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' in terms of number of instrumental variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 22 MR−BEE MR−IVW MR−Egger MR−Lasso MR−Median 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='m = 1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='6 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='6 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='6 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='6 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='6 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='6 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='6 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='6 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='6 exposures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='^θ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='θ^ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='θ^ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='θ^ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='θ^ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='θ^ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='Figure 5: Investigation of MR methods for multivariable MR with sample sizes n0 = · · · = n6 = 20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='and overlap-sample sizes n01 = · · · = n65 = 20000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' in terms of number of specified exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' lations when 1, 3, and all 6 exposures are included in the multivariable MR model, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Figure 5 illustrates the results of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We observed that if associated exposures are omitted, the causal effect estimates can suffer severe biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The degree of the biases is jointly determined by the genetic covariance matrix and covariance matrix of confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In conclusion, even though MRBEE has eliminated the estimation error bias and weak instrument bias, OVB still exists if any relevant exposure is not specified in the multivariable MR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 Other Investigations For univariable MR, we also investigated the effects of sample sizes, type-I error, winner’s curse, and outlier detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Regarding multivariable MR, we investigated the impact of different sample overlaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In addition, the precision of estimating ΣWβwα by insignificant GWAS statistics is also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Only by increasing the sample sizes of the exposure and outcome cohorts simultaneously, the accuracy of MRBEE can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The 23 traditional MR methods suffer from inflated type-I errors when the overlapping fraction is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' After accounting for the weak instrument bias and estimate error bias, MRBEE is almost free of the winner curse’s bias when the overlapping fraction is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Furthermore, by applying the iterative method in IMRP, MRBEE can efficiently eliminate pleiotropic outliers and produce an accurate causal effect estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In addition, the estimation error ΣWβwα decreases with the increase of the number of insignificant variants M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Finally, multivariable MRBEE is accurate regardless of sample overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We summarized the findings with the simulation details in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 5 Real Data Analysis Cardiovascular disease including coronary artery disease (CAD) is one of the leading causes of death for both men and women worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' There are many epidemiological studies and MR analyses based on GWAS summary data dedicated to identifying the causal risk factors for CAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' However, the causal effects of the risk factors on CAD are less clear and the existing evidence can be contradictory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' For example, elevated low-density lipoprotein cholesterol level (LDL-C) is a well-established causal risk factor for CAD (Group et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 1994), whereas Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2022) concluded by multivariable MR analysis that LDL-C is not causally related to CAD in Europeans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Additionally, substantial observational analyses and molecular experiments have suggested that uric acid (UA) and red blood cell counts (RBC) contribute to the development of CAD (Bujak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Yu and Cheng, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Nevertheless, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2022) did not observe significant causal effects of the two risk factors on CAD in Europeans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Furthermore, numerous MR analyses have concluded that body mass index (BMI) has a positive causal effect on CAD (Zhu, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' However, recent literature indicates that BMI is likely to influence CAD through the mediation with diseases such as diabetes and hypertension (Gill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' These contradictions may be due to biases in MR methods, including OVB, weak instrument bias, estimation error bias, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We conducted two data analyses to estimate the causal effects of select risk factors on CAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The first analysis uses the 11 exposures in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2022), including BMI, hemoglobin (HB), hemoglobin a1c (Hba1c), hematocrit (HT), high-density lipoprotein cholesterol level (HDL-C), height, LDL-C, RBC, systolic blood pressure (SBP), triglyc- erides (TG), and UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2022), these 11 exposures were divided into two groups and analyzed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In contrast, we analyzed them in one multivariable MR model to avoid the OVB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In the second analysis, we replace HB, Hba1c, HT, and RBC with alcohol consumption (alcohol), diabetes, lifetime never smoking status (never.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='smoking), and sleeplessness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' All the GWAS summary statistics used in our analyses were downloaded from the Neale lab (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='nealelab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='is/uk-biobank/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Quality controls (QCs) are presented in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The total numbers of instrumental variants for the first and second analyses are 5345 and 5301, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Figure 6 displays the causal effect estimates with 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' MR- BEE confirms the causal effects of LDL-C, RBC, and UA on CAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Here, HB, HT, and RBC have high mutual correlations: � cor(xHB, xHT) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='89, � cor(xHB, xRBC) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='63, and 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 causal effect estimates −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 causal effect estimates Multivariable MR Estimation (with 11 exposures in Genome Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=') Multivariable MR Estimation (with alternative 11 exposures) BMI HB HT HDL Height LDL Hba1c RBC SBP TG UA Alcohol BMI Diabetes HDL Height LDL Never.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='smoking SBP Sleepless TG UA MR−BEE MR−IVW MR−Egger MR−Lasso MR−Median Figure 6: Causal effect estimates of CAD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Confidence intervals are yielded by the double SE estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' � cor(xHT, xRBC) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='72, and thus the inferences obtained by the existing methods are not reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' For example, the existing MR methods suggest that RBC is not significant, HB has a significant positive effect, and HT has a significant negative effect, which contradicts the fact that HT and CAD are positively associated (Sorlie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' MRBEE corrects the estimation error bias and thus leads to a reasonable conclusion – HB and RBC have positive causal effects on CAD while HT has a positive but insignificant causal effect on CAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' For the second analysis, MRBEE reveals that BMI is likely to affect CAD through the mediation of SBP and diabetes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In addition, MRBEE indicates that never.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='smoking is protective against CAD, whereas sleeplessness is associated with increasing CAD risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Furthermore, due to the weak instrument bias and estimation error bias, the existing meth- ods overestimate the effects of HDL-C and height and underestimate the causal effects of diabetes, LDL-C, never.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='smoking, SBP, and sleeplessness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' By using MRBEE, we are able to obtain reliable causal effect estimates and therefore make valid inferences on the causal risk factors of CAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 6 Discussion In this paper, we first investigated the asymptotic behavior of the multivariable IVW es- timate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Since almost all MR methods are based on the IVW method, understanding the asymptotic behavior of the IVW estimate has very far-reaching implications for the theo- retical and empirical studies of MR methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We found that the bias of the multivariable 25 IVW estimate is the product of weak instrument bias and estimation error bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Also, we revealed that estimation error bias is a linear combination of measurement error bias and confounder bias, in which the sample overlaps trade off the proportion of these two compo- nents of estimation error bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In the literature, although the phenomenon that the IVW estimate suffers from bias has been observed, a quantitative explanation for its existence is still absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Our work fills the gap, which is a significant theoretical contribution to MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Subsequently, in this paper, we describe MRBEE that can yield the unbiased causal effect estimate ˆθBEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We point out that ˆθBEE is strongly asymptotically unbiased in all scenarios, indicating that ˆθBEE is asymptotically valid when making causal inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We also discuss how to perform MRBEE in practice, including how to estimate the bias- correction terms, how to estimate the sandwich formula, and how to identify possible UHP when multiple exposures are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We present corresponding theorems to confirm that the estimates involved in the implementation of MRBEE are consistent in theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In simulations, we show that MRBEE simultaneously estimates causal effects and the SE unbiasedly, and identifies UHP consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In section 5 and also in (Lorincz-Comi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2022), the practical advances of MRBEE are further demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' It is worth offering guidance on how to properly perform MR analysis from our perspec- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' First, we suggest applying the multivariable MR approach instead of the univariable MR approach because the causal effect estimates obtained by the univariable MR approach are unreliable due to OVB, regardless of the presence of UHP and CHP in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Sec- ond, rather than selecting the optimal number of instrumental variants such that the F statistics and conditional F statistics are larger than 10 (Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Sanderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 2021), we advise including all the independent instrumental variants that are signif- icantly associated with one or more exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Our theory illustrates that the asymptotic variance of a causal effect estimate is related to the cumulative variance explained by all specified IVs instead of the average variance explained by each IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' In particular, there is no need to worry about the issue of weak IVs because MRBEE has demonstrated ef- ficiency to eliminate weak instrument bias through our simulations and theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Third, when performing multivariable MR analysis, it is not necessary to remove variants that are pleiotropic between the exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' For example, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2022) observed that LDL-C was insignificantly associated with CAD in Europeans, which is unlikely to be true because this risk causality has been well established in randomized clinical trials (Group et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The potential reason for this false negative is that Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2022) excluded the IVs associated with RBC, HB, HT, and UA in their multivariable MR analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We believe that the proper way to perform multivariable MR analysis is to simultaneously in- clude all the relevant exposures, as the multivariable regression can automatically account for the pleiotropic variants shared by the specified exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Fourth, among the existing multivariable MR approaches including IVW, MR-Egger, and MR-Lasso, we recommend MRBEE as the primary analysis approach because it has been proven to be the only one that enjoys strongly asymptotic unbiasedness in the presence of many weak IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 26 A Proof A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 Preliminary lemmas In this subsection, we specify some lemmas that can facilitate the proofs, most of which can be found in the existing papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' We first discuss the equivalent characterizations of sub-Gaussian and sub-exponential variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 (Equivalent characterizations of sub-Guassian variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Given any random variable X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' the following properties are equivalent: (I) there is a constant K1 ≥ 0 such that Pr(|X| ≥ t) ≤ 2 exp(−t2/K2 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' for all t ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (II) the moments of X satisfy ||X||Lp = (E(|X|p)) 1 p ≤ K2 √p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' for all p ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (III) the moment generating function (MGF) of X2 satisfies: E{exp(λ2X2)} ≤ exp(K2 3λ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' for all λ staisfying |λ| ≤ K−1 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (IV) the MGF of X2 is bounded at some point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' namely E{exp(X2/K2 4)} ≤ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (V) if E(X) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' the MGF of X satisfies E{exp(λX)} ≤ exp(K2 5λ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' for all λ ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' where K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , K5 are certain strictly positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This lemma summarizes some well-known properties of sub-Guassian and can be found in Vershynin (2018, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 (Equivalent characterizations of sub-exponential variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Given any ran- dom variable X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' the following properties are equivalent: (I) there is a constant K1 ≥ 0 such that Pr(|X| ≥ t) ≤ 2 exp(−t/K1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' for all t ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (II) the moments of X satisfy ||X||Lp = (E(|X|p)) 1 p ≤ K2p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' for all p ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 27 (III) the moment generating function (MGF) of |X| satisfies: E{exp(λ|X|)} ≤ exp(K3λ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' for all λ staisfying 0 ≤ λ ≤ K−1 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (IV) the MGF of |X| is bounded at some point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' namely E{exp(|X|/K4)} ≤ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (V) if E(X) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' the MGF of X satisfies E{exp(λX)} ≤ exp(K2 5λ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' for all λ ≤ K−1 5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' where K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , K5 are certain strictly positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This lemma summarizes some well-known properties of sub-exponential and can be found in Vershynin (2018, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 (Product of sub-Gaussian variable is sub-exponential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Suppose that X, Z are two sub-Gaussian variable, then Y = XZ is a sub-exponential variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Besides, if X is a bounded sub-Gaussian variable, then then Y = XZ is a sub-Gaussian variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The first claim of this lemma is provided by Vershynin (2018, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The second claim of this lemma is a direct inference of Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (2011, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='4 (ℓ2-norm of matrices with sub-Gaussian entries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Let X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , Xn be n (p×1) independent identically distributed random vector with entries xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , xip are sub-Gaussian with zero-mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Besides, define the covariance matrix of Xi as Σ = E(XiX⊤ i ) and the related sample covariance matrix ˆΣ = 1 n n � i=1 XiX⊤ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Then for every positive integer n, E(|| ˆΣ − Σ||2) ≤ C �p n + �p n � ||Σ||2, where C is certain positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This lemma is provided by Vershynin (2018, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' It shows the convergence rate of sample covariance matrix is √(n/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5 (ℓ2-norm of matrices with sub-exponential entries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Let X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , Xn be n (p × 1) independent identically distributed random vector with entries xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , xip are sub- exponential with zero-mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Besides, define the covariance matrix of Xi as Σ = E(XiX⊤ i ) 28 and the related sample covariance matrix ˆΣ = 1 n n � i=1 XiX⊤ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Then for ever t ≥ 0, the following inequality holds with probability at least 1 − p exp(−ct2): || ˆΣ − Σ||2 ≤ max(||Σ||2δ, δ2), where c is certain positive constant and δ = t � p/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This lemma is the direct inference of Vershynin (2010, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Besides, by letting t = √p log n we further obtain E(|| ˆΣ − Σ||2) = O �� p log n n � ||Σ||2, if ˆΣ is the sample covariance matrix of sub-exponential vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Note that in our method, the dimension p is fixed and hence we cannot chose t = √p log p such that the estimation bound becomes � (p log p)/n||Σ||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='6 (Asymptotic normal distribution of Wishart matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Suppose X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , Xn are n IID relaxation of the p-dimensional variable X ∼ N(0, Σ) with a well-conditioned covariance matrix Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Besides, define the sample covariance matrix of Σ as ˆΣ = 1 n n � i=1 XiX⊤ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' If p is a fixed number, then as n → ∞, √n(vec( ˆΣ) − vec(Σ)) D −→ N � 0, (Ip2 + Kp2)(Σ ⊗ Σ) � , where Kp2 is the so-called commutation matrix, which is able to ensure Kp2vec(A) = vec(A′) for all (p × p) matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' This lemma can be found in Muirhead (2009, equation (5), p90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2 Specific Lemmas In this subsection, we specify the following lemmas that are made based on the preliminary lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='7 (Asymptotic normal distribution of sub-Gaussian and sub-exponential vari- ables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Suppose X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , Xn are n independent sub-Gaussian or sub-exponential variables with mean-zero and variance σ2 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , σ2 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Then lim n→∞ 1 √n n � i=1 Xi D −→ N(0, σ2 x), 29 where σ2 x = lim n→∞ 1 n n � i=1 σ2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' It is easy to verify the Lyapunov’s condition: for all fixed δ > 0, lim n→∞ 1 n1+δ n � i=1 E(|Xi|2+2δ) ≤ √2K2 + 2K2δ 2+2δ nδ → 0 by the (II) of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1, if X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , Xn are sub-Gaussian variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' lim n→∞ 1 n1+δ n � i=1 E(|Xi|2+2δ) ≤ (2K2 + 2K2δ)2+2δ nδ → 0 by the (II) of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='2, if X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , Xn are sub-exponential variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' And hence the asymptotic normal distribution holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='8 (Asymptotic normal distribution of estimation error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Let ξ[s] j = 1 √ns ns � i=1 g[s] ij x[s] i,−j, where x[s] i,−j = x[s] i − βjsg[s] i,j, s = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , p, x[0] i,−j represents y[0] i,−j and βj0 represent αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Then ξ[s] j D−→ N(0, σxsxs − σβsβs), where σx0x0 represents σyy and σβ0β0 represents θ⊤Σββθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Note that both g[s] ij and x[s] i,−j are sub-Gaussian (x[s] i,−j is the product of a sub-Gaussian variable and a bounded sub-Gaussian variable), and it holds E(g[s] ij x[s] i,−j) = 0 and var(g[s] ij x[s] i,−j) = var(g[s] ij ) × var(x[s] i,−j) = σxsxs − σβsβs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (32) As a result, ξ[s] j = 1 √ns ns � i=1 g[s] ij x[s] i,−j D−→ N(0, σxsxs − σβsβs), (33) according Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='9 (Asymptotic normality of bias-correction terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Let ζj = �nmin n1 ξ[1] j , nmin n2 ξ[2] j , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , nmin np ξ[p] j , nmin n0 ξ[0] j �⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 30 Under the conditions (C1)-(C4), lim m→∞ 1 √m m � j=1 (vec(ζjζ⊤ j ) − vec(ΨWβ×wα)) D−→ N � 0, (Ip2 + Kp2)(ΨWβ×wα ⊗ ΨWβ×wα) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' as nmin, m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' By using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='7, ζj follows N(0, ΨWβ×wα) as nmin → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Then by using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='6, this lemma holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='10 (Asymptotic normality of residual term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Under the conditions (C1)-(C4), lim m→∞ 1 √m m � j=1 √mβjξ[s] j D−→ N(0, σxsxsΣββ), and lim m→∞ 1 m m � j=1 √mβj √mβ⊤ j ξ[s] j ξ[k] j P−→ nsk √nsnk σxsxkΣββ, for s = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , p, where σx0xk represents σyxk = �p l=1 θlσxlxk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' By condition (C4), √mβj is independent of ξ[s] j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3, √mβjξ[s] j is sub-exponential with mean 0 and covariance matrix cov(√mβjξ[s] j ) = cov(√mβj) × var(ξ[s] j ) = (σxsxs − σβsβs)Σββ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (34) Hence, by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='6, lim m→∞ 1 √m m � j=1 √mβjξ[s] j D−→ N(0, σxsxsΣββ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' On the other hand, βjξ[s] j is sub-exponential variable according to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3, and cov(√mβjξ[s] j , √mβjξ[k] j ) = cov(ξ[s] j , ξ[k] j ) × Σββ = nsk √nsnk (σxsxk − σβsβk)Σββ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (35) Hence, by using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5 lim m→∞ 1 m m � j=1 √mβj √mβ⊤ j ξ[s] j ξ[k] j P−→ nsk √nsnk σxsxkΣββ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' 31 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='3 Proofs of theorems in section 2 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As for the estimation error ωα, we have wαj = g[0]⊤ j y[0] n0 − αj = g[0]⊤ j y[0] −j n0 , (36) where y[0] −j = y[0] − αjg[0] j = m � s̸=j αtg[0] t + U[0]θ + v[0], (37) and U[0] and v[0] are the corresponding noise terms in the outcome GWAS cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Accord- ing to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='8, ξ[0] j = 1 √n0 n0 � i=1 g[0] ij y[0] i,−j D−→ N(0, σyy − θ⊤Σββθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (38) As for the estimation error wβjs, we have wβjs = g[s]⊤ j x[s] ns − βjs = g[s]⊤ j x[s] −j ns , (39) where x[s] −j = x[s] − g[s] j βjs = � t̸=j βtsg[s] t + u[s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (40) Let ξ[s] j = g[s]⊤ j x[s] −j √ns = 1 √ns ns � i=1 g[s] ij x[s] i,−j, (41) where x[s] i,−j is the ith element in vector x[s] −j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' According to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='8, ξ[s] j = 1 √ns ns � i=1 g[s] ij x[s] i,−j D−→ N(0, σxsxs − σβsβs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (42) Now we show the covariance between ξ[s] j and ξ[k] j : cov(ξ[s] j , ξ[k] j ) = E �x[s]⊤ −j g[s] j g[k]⊤ j x[k] −j √nsnk � , (43) where x[0] −j represents y[0] −j for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Denote Q[sk] = (Q[sk] it ) being a (ns × nk) matrix 32 whose (i, t)th element is Q[sk] it = E(g[s] ij g[k] tj ) = � 1, (i, t) ∈ Q[sk], 0, (i, t) /∈ Q[sk], (44) where Q[sk] = {(i, t) : g[s] ij and g[k] tj come from the same individual}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (45) As a result, cov(ξ[s] j , ξ[k] j ) = E �x[s]⊤ −j Q[sk]x[k] −j √nsnk � = 1 √nsnk � (i,t)∈Q[sk] E(x[s] i,−jx[k] t,−j) = nsk √nsnk � σxsxk − σβsβk � , (46) where σx0xk represents σyxk for simplicity, and σβ0βk represents σβ0βk = cov(√mβ⊤ j θ, √mβjk) = p � l=1 θlσβlβk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (47) Finally, we show ξ[s] j is uncorrelated with ξ[s] t for all t ̸= j and s = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Specifically, cov(ξ[s] j , ξ[s] t ) = E �x[s]⊤ −j g[s] j g[s]⊤ t x[s] −j ns � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (48) According the model setting, g[s] j is independent of g[s] t for all t ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Therefore, cov(ξ[s] j , ξ[s] t ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Note that if m → ∞, Σββ = 1 mΨββ vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' And so Theorem 1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' The score function of IVW is − 1 m ˆB⊤(ˆa − ˆBˆθIVW) = − 1 m ˆB⊤(ˆa − ˆBθ) + 1 m ˆB⊤ ˆB(ˆθIVW − θ) (49) which leads to HIVW(ˆθIVW − θ) = −SIVW(θ), (50) where HIVW = 1 m ˆB⊤ ˆB, SIVW(θ) = − 1 m ˆB⊤(ˆa − ˆBθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (51) 33 We first work with the Hessian matrix HIVW: mHIVW = ˆB⊤ ˆB = B⊤B + B⊤Wβ + W⊤ β B + W⊤ β Wβ = J1 + J2 + J3 + J4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (52) As for J1, J1 = m � j=1 βjβ⊤ j P −→ Ψββ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (53) As for J2, ∥√nminJ2∥2 = ���� 1 √m m � j=1 (√nminwβj)(√mβj)⊤ ���� 2 ≤ � � � � ���� 1 m m � j=1 (√nminwβj)(√nminwβj)⊤ ���� 2 × � � � � ���� 1 m m � j=1 (√mβj)(√mβj)⊤ ���� 2 ≤ λ 1 2max(ΨWβWβ) × λ 1 2max(Ψββ), (54) which means ∥J2∥2 = OP(1/√nmin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (55) As for J3, it has the same order as J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As for J4, nmin m J4 = 1 m m � j=1 (√nminwβj)(√nminwβj)⊤ P −→ ΨWβWβ (56) Hence: (1) If m/nmin → 0, ∥J4∥2 ≤ λmax(ΨWβWβ) × m nmin → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (57) Therefore, mHIVW P −→ Ψββ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (58) (2) If m/nmin → c0 ∈ (0, ∞), then J4 = m nmin × 1 m m � j=1 (√nminwβj)(√nminwβj)⊤ P −→ c0ΨWβWβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (59) 34 Therefore, mHIVW P −→ Ψββ + c0ΨWβWβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (60) (3) If m/nmin → ∞ and m/n1+τ min → c0 ∈ (0, +∞) with certain constant τ > 0, then 1 nτ min J4 = m n1+τ min × 1 m m � j=1 (√nminwβj)(√nminwβj)⊤ P −→ c0ΨWβWβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (61) Therefore, m nτ min HIVW = c0nminHIVW P −→ c0ΨWβWβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (62) We then work with SIVW(θ): mSIVW(θ) = −B⊤wα − W⊤ β wα + B⊤Wβθ + W⊤ β Wβθ = K1 + K2 + K3 + K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (63) As for K1 + K3, √nmin(K1 + K3) = 1 √m m � j=1 (−√nminwαj + √nminw⊤ βjθ)(√mβj) D −→ N(0, ψθΨββ), (64) where ψθ = ψwαwα + θ⊤ΨWβWβθ − 2θ⊤ψWβwα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (65) As for K2, nmin m K2 = − 1 m m � j=1 (√nminwαj)(√nminwβj) P −→ −ψWβwα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (66) As for K4, nmin m K4 = � 1 m m � j=1 (√nminwβj √nminwβj � θ P −→ ΨWβWβθ, (67) Jointing these results, we summary the asymptotic behavior of ˆθIVW: (1) If m/√nmin → 0, then √nmin||K2 + K4|| = OP � m √nmin � = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (68) 35 Therefore, √nmin × mSIVW(θ) = √nmin(K1 + K3) + oP(1) D −→ N(0, ψθΨββ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (69) Note that when m/nmin → 0, mHIVW P −→ Ψββ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Therefore, √nmin(ˆθIVW − θ) = −√nmin(mHIVW)−1(mSIVW(θ)) D −→ N(0, ψθΨ−1 ββ), (70) (2) If m/√nmin → c0, then √nmin(K2 + K4) → −c0ψWβwα + c0ΨWβWβθ, (71) and hence √nmin × mSIVW(θ) D −→ N(−c0(ψWβwα + ΨWβWβθ), ψθΨββ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (72) Note that when m/nmin → 0, mHIVW P −→ Ψββ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Therefore, √nmin(ˆθIVW − θ) = −√nmin(mHIVW)−1(mSIVW(θ)) D −→ N(c0Ψ−1 ββ(ψWβwα − ΨWβWβθ), ψθΨ−1 ββ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (73) (3) If m/√nmin → ∞ and m/nmin → c0, then ||K1 + K3||2 = OP(1/√nmin), K2 + K4 P −→ −c0ψWβwα + c0ΨWβWβθ, (74) and mHIVW P −→ Ψββ + c0ΨWβWβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (75) Hence, ˆθIVW − θ P −→ c0(Ψββ + c0ΨWβWβ)−1(ψWβwα − ΨWβWβθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (76) (4) If m/nmin → ∞ and m/n1+τ min → c0, then 1 nτ min (K2 + K4) P −→ −c0ψWβwα + c0ΨWβWβθ (77) and m nτ min HIVW P −→ c0ΨWβWβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (78) Therefore, ˆθIVW P −→ Ψ−1 WβWβψWβwα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (79) 36 Now Theorem 2 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='4 Proofs of theorems in section 3 Proofs of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Note that 0 = SBEE(ˆθBEE) = SBEE(θ) + HBEE(ˆθBEE − θ), (80) where SBEE(θ) = − 1 m ˆB⊤( ˆα − ˆBθ) − ΣWβWβθ + σWβwα, (81) and HBEE = 1 m ˆB⊤ ˆB − ΣWβWβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (82) As for SBEE(θ), mSBEE(θ) = −(B + Wβ)⊤(α + wα − Bθ − Wβθ) − mΣWβWβ + mσWβwα = − � B⊤(wα − Wβθ) � + �� W⊤ β Wβ − mΣWβWβ � θ � − � W⊤ β wα − mσWβwα � = K1 + K2 + K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (83) Here, we define a new vector ϑ = (θ⊤, 1)⊤, an alternative vector ζj = �nmin n1 ξ[1] j , nmin n2 ξ[2] j , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , nmin np ξ[p] j , nmin n0 ξ[0] j �⊤ , where ξ[s] j = 1 √ns ns � i=1 g[s] ij x[s] is , s = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , p, and a new covariance matrix cov(ζj) = ΨWβ×wα = �ΨWβWβ ψWβwα ψ⊤ Wβwα ψwαwα � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (84) As for K1, it can be rewritten as √nminK1 = − m � j=1 √nmin(wαj − w⊤ βjθ)βj = 1 √m m � j=1 (√nminζ⊤ j ϑ)(√mβj) D −→ N(0, ψθΨββ), (85) where ψθ defined in (65) can be rewritten as ψθ = ϑ⊤ΨWβ×wαϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (86) 37 As for K2 + K3, it can be rewritten as K2 + K3 = I1:p p+1 �W⊤ β Wβ − mΣWβWβ W⊤ β wα − mσWβwα w⊤ α Wβ − mσ⊤ Wβwα w⊤ α wα − mσwαwα � � θ −1 � = √m nmin I1:p p+1 � 1 √m m � j=1 ζjζ⊤ j − ΨWβ×wα � ϑ = √m nmin I1:p p+1K4ϑ, (87) where I1:p p+1 is a (p × (p + 1)) matrix consisting of the first p row of Ip+1 and K4 = 1 √m m � j=1 ζjζ⊤ j − ΨWβ×wα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (88) According to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='6, vec(K4) D −→ N � 0, (I(p+1)2 + K(p+1)2)(ΨWβ×wα ⊗ ΨWβ×wα) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (89) As a result, nmin √m (K2 + K3) D −→ N(0, ΣBC) (90) where ΣBC = � ϑ⊤ ⊗ I1:p p+1 � � �� � p×(p+1)2 � (I(p+1)2 + K(p+1)2)(ΨWβ×wα ⊗ ΨWβ×wα) � � �� � (p+1)2×(p+1)2 � ϑ⊤ ⊗ I1:p p+1 �⊤ � �� � (p+1)2×p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (91) So far, we can obtain: (1) If m/nmin → 0, √nmin × mSBEE(θ) = √nminK1 + oP(1) D −→ N(0, ψθΨββ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (92) (2) If m/nmin → c0, √nmin × mSBEE(θ) = √nminK1 + √nmin(K2 + K3) D −→ N(0, ψθΨββ + c0ΣBC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (93) (3) If m/nmin → ∞ and √m/nmin → 0, nmin √m × mSBEE(θ) = nmin √m (K2 + K3) + nmin √m K1 D −→ N(0, ΣBC), (94) 38 where nmin √m K1 = �nmin m × √nminK1 = OP ��nmin m � = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (95) Now we move to HBEE: mHBEE = B⊤B + � W⊤ β Wβ − mΣWβWβ � + B⊤Wβ + W⊤ β B = J1 + J2 + J3 + J4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (96) As for J1 = B⊤B, we have ||J1 − Ψββ||2 = ���� 1 m m � j=1 √mβj √mβ⊤ j − Ψββ ���� 2 = OP � 1 √m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (97) As for J2 = W⊤ β Wβ − mΣWβWβ, we have J2 = m � j=1 � wβjw⊤ βj − ΣWβWβ � = √m nmin 1 √m m � j=1 � ξjξ⊤ j − ΨWβWβ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (98) As a result, nmin √m vec(J2) D −→ N(0, (Ip2 + Kp2)(ΨWβWβ ⊗ ΨWβWβ)), (99) which means ||J2|| = OP(√m/nmin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' As for J3 = B⊤Wβ, √nmin||J3||2 = ���� 1 √m m � j=1 √mβj √nminω⊤ βj ���� 2 ≤ � � � � ���� 1 m m � j=1 √mβj √mβ⊤ j ���� 2 � � � � ���� 1 m m � j=1 √nminωβj √nminω⊤ βj ���� 2 ≤ λ 1 2max(Ψββ) × λ 1 2max(ΨWβWβ), (100) which means ||J3||2 = OP � 1 √nmin � (101) 39 As for J4, it is easy to see ||J4||2 2 = ||J3||2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Hence, for all three scenarios in Theorem 3, ||mHBEE − Ψββ||2 = OP � max � 1 √m, 1 √nmin , √m nmin �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (102) And hence, according to the Slutsky’s theorem, (1) If m/nmin → 0, √nmin(ˆθBEE − θ) = −√nminΨ−1 ββK1 D −→ N(0, ψθΨ−1 ββ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (103) (2) If m/nmin → c0, √nmin(ˆθBEE − θ) = −√nminΨ−1 ββ(K1 + K2 + K3) D −→ N(0, ψθΨ−1 ββ + c0Ψ−1 ββΨBCΨ−1 ββ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (104) (2) If m/nmin → ∞ and m/n2 min → 0, � n2 min/m(ˆθBEE − θ) = −nmin √m Ψ−1 ββ(K2 + K3) D −→ N(0, Ψ−1 ββΨBCΨ−1 ββ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (105) Thus, Theorem 3 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Similar to ξ[s] j , we define η{s} j as η{s} j = g{s}⊤ j x[s] √ns = 1 √ns ns � i=1 g{s} ij x[s] i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (106) By using similar deduction as which in the proof of Theorem 1, η{s} j D−→ N(0, σxsxs) (107) and cov(η{s} j , η{k} j ) = nsk √nsnk σxsxk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (108) Denote ηj = (η{1} j , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , η{p} j , η{0} j ) where η{0} j represents 1 √n0g{s}⊤ j y[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Then we have cov(ηj) = D−1 η ΣWβ×wαD−1 η , (109) where Dη = diag � 1 √n1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' , 1 √np , 1 √n0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (110) 40 By using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='4, ���� 1 M M � j=1 ηjη⊤ j − cov(ηj) ���� 2 = OP � 1 √ M � , (111) and hence ∥Σ − 1 2 Wβ×wα ˆΣWβ×wαΣ 1 2 Wβ×wα − Ip+1∥2 ≤ λ−1 min(cov(ηj)) ���� 1 M M � j=1 ηjη⊤ j − cov(ηj) ���� 2 = OP � 1 √ M � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (112) Thus, Theorem 4 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Note that Sj(θ) = −(ˆαj − θ⊤ ˆβj) ˆβj − ΣWβWβθ + σWβwα = (wαj − θ⊤wβj)βj + � (wαj − θ⊤wβj)wβj − ΣWβWβθ + σWβwα � = J1j + J2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (113) Note that both J1j and J2j are sub-exponential variables with zero mean and covariance matrix cov(J1j) = 1 mnmin ψθΨββ, cov(J2j) = 1 n2 min ΣBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (114) Therefore, we obtain cov(Sj(θ)) = ΣS = � � � � � � � 1 mnminψθΨββ, if m/nmin → 0, 1 mnminψθΨββ + c0 mnminΣBC, if m/nmin → c0, 1 n2 minΣBC, , if m/nmin → ∞ and √m/nmin → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (115) Then by using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='5, ���� 1 m m � j=1 Sj(θ)Sj(θ)⊤ − ΣS ���� 2 = OP �� log m m � ||ΣS||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (116) By using the Slutsky’s theorem, ���� 1 m m � j=1 ˆSj(ˆθBEE) ˆSj(ˆθBEE)⊤ − ΣS ���� 2 = OP �� log m m � ||ΣS||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (117) 41 where ˆSj(ˆθBEE) = −(ˆθ⊤ BEE ˆβj − ˆαj) ˆβj + ˆΣWβWβ ˆθBEE − ˆσWβwα (118) On the other hand, according to the proof of Theorem 3, ∥mˆFBEE − Ψββ||2 = OP � max � 1 √m, 1 √nmin , √m nmin �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (119) Note that ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', A22(p223) illustrates ∥A1A2A3 − B1B2B3∥2 = OP � max � ||A1 − B1||2, ||A2 − B2||2, ||A3 − B3||2 �� , (120) where A1, A2, A3, B1, B2, B3 are six matrices with non-diverging maximum singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Hence, || ˆΣBEE(ˆθBEE) − ΣBEE(θ)||2 = ����(mˆFBEE)−1 � m � j=1 ˆSj(ˆθBEE) ˆSj(ˆθBEE)⊤ � (mˆFBEE)−1 − mΨ−1 ββΣSΨ−1 ββ ���� 2 = OP � max �� log m m , 1 √nmin , √m nmin �� ||mΣS||2, (121) and consequently ||Σ − 1 2 BEE(θ) ˆΣBEE(θ)Σ − 1 2 BEE(θ) − Ip||2 = OP � max �� log m m , 1 √nmin , √m nmin �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (122) Thus, Theorem 5 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Note that ||ˆθBEE − θ||2 = OP(n − 1 2 min) and hence ˆαj − ˆβ⊤ j ˆθBEE and ˆαj − ˆβ⊤ j θ have the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' For j ∈ Oc, ˆγj = εj = ˆαj − ˆβ⊤ j ˆθBEE = wαj − w⊤ βjθ + w⊤ βj(ˆθBEE − θ) ∼ N(0, σεε), (123) where σεε = θ⊤ΣWβwαθ + σωγωγ − 2θ⊤σWβwα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (124) As a result, ˆγ2 j σεε ∼ χ2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (125) 42 Denote κ∗ = F −1 χ2 1 (κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Then by using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='1 of ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', Pr � max j∈Oc ˆγ2 j σεε ≤ κ∗ � = 1 − Pr � max j∈Oc ˆγ2 j σεε > κ∗ � ≥ 1 − (m − |O|) Pr � ˆγ2 j σεε > κ∗ � ≥ 1 − m Pr � ˆγ2 j σεε > κ∗ � ≥ 1 − m exp � − (√2κ∗ − 1 − 1)2 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (126) By letting κ∗ = C0 log m with C0 being a sufficiently large constant, Pr � max j∈Oc ˆγ2 j σεε ≤ κ∗ � ≥ 1 − exp � log m − 2C0 log m − 2√C0 log m − 1 4 � ≥ 1 − exp � − (2C0 − 4) log m − 2√C0 log m − 1 4 � → 1, (127) if m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' On the other hand, for j ∈ O, ˆγj = γj + εj, and hence ˆγ2 j σεε ∼ χ2 1 � γ2 j σεε � , (128) where χ2 1(λ) refers to the noncentral chi-squared distribution with degree of freedom 1 and noncentrality parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Let Fχ2 1(λ)(·) be the CDF of this noncentral chi-squared distribution, which is indeed equal to Fχ2 1(λ)(x) = 1 − � Q(√x − √ λ) + Q(√x + √ λ) � , (129) where Fχ2 1(λ)(·) be the CDF of χ2 1(λ) and Q(x) is the Gaussian Q-function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=', Q(x) = 1 − Φ(x) and Φ(x) is the CDF of standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Note that there should exist a constant D0 such that γ2 j σεε ≥ D0nmin (130) where D0 is a sufficient large constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' And Pr � min j∈O ˆγ2 j σεε ≥ κ∗ � = 1 − Pr � min j∈Oc ˆγ2 j σεε < κ∗ � ≥ 1 − Pr � ˆγ2 j σεε < κ∗ � , j is arbitrary element in O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' (131) 43 Hence, Pr � min j∈O ˆγ2 j σεε ≥ κ∗ � ≥ Q( √ κ∗ − � D0nmin) + Q( √ κ∗ + � D0nmin) ≥ Q( � C0 log m − � D0nmin) + Q( � C0 log m + � D0nmin) → 1 (132) if m, nmin → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Thus, Theorem 6 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' References Benjamini, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfiA17/content/2301.05130v1.pdf'} +page_content=' Hochberg (1995).' metadata={'source': 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0000000000000000000000000000000000000000..94098274025b749b0d2ba8ad0c2fb8cdd78b9df5 --- /dev/null +++ b/LtAyT4oBgHgl3EQfgPig/content/tmp_files/2301.00356v1.pdf.txt @@ -0,0 +1,1175 @@ +Multiorbital effects in high-order harmonic emission from CO2 +Andres Mora, Lauren Bauerle, Yuqing Xia, Agnieszka Jaron1 +1JILA and Department of Physics, University of Colorado, Boulder, CO 80309-0440, USA +We study the ellipticity of high-order harmonics emitted from CO2 molecule driven by linearly po- +larized laser fields using numerical simulations within the time-dependent density functional theory. +We find that the overall ellipticity of the harmonics is small, which is in agreement with experimen- +tal data. On the other hand, our analysis of the numerical results indicates that several valence +orbitals contribute significantly to the harmonic emission and some of these contributions show a +strong ellipticity of the harmonics. The small ellipticity in the total harmonics signal arises from a +combination of factors, namely, the fact that harmonic emission from different orbitals is strongest +at different alignment angles of the molecular axis with respect to the laser polarization direction, +as well as interference effects and a strong laser coupling between several inner valence orbitals. +PACS numbers: 32.80.Fb,32.80.Wr +I. +INTRODUCTION +High-order harmonic generation (HHG) is one of the +highly nonlinear, nonperturbative processes that occur +when an atom or molecule is irradiated by an intense +laser field [1, 2]. It results from the distortion of the elec- +tron density in the presence of the strong electromagnetic +field of the laser and the power spectrum of the emitted +harmonic radiation corresponds to the Fourier transform +of the electron dipole acceleration. The intensity spectra +of the emitted high harmonics shows some general char- +acteristic features, such as a fast decrease of the signal +over the first few harmonics followed by a region with +fairly constant plateau harmonic intensities ending by a +sharp cutoff, beyond which the harmonic intensity drops +quickly. +Over the past few decades HHG has been an active area +of research since it provides a source for coherent short- +wavelength light, extending into the soft-X-ray regime +[3], and for ultrashort laser pulses and waveforms in the +attosecond [4, 5]. Furthermore, it has been shown that +HHG spectra contain information about the atomic and +electronic structure of the target (e.g., [6–11]), ultrafast +molecular and intra-molecular electron dynamics (e.g., +[12–15]) as well as time resolution of chemical processes +(e.g., [16]). +Basic intuitive picture of HHG is provided by the semi- +classical three-step model [17–19], according to which an +electron tunnels through the barrier created by the laser +field and the Coulomb field into the continuum, is accel- +erated by the electric field of the laser first away from +and then back to the parent ion. +Upon return it re- +combines, emitting excess energy in the form of high- +order harmonic radiation. Since the process occurs every +half cycle of the driving laser field, an attosecond pulse +train is produced. +In recent years, it has been shown +that, in particular for high-order harmonic generation +from molecules, the generated harmonic spectra incorpo- +rate more features than predicted by the basic three-step +single-active-electron model. One example are polarime- +try measurements of high-order harmonic emission from +aligned diatomic and linear triatomic molecules driven by +linearly polarized laser fields. Surprisingly, strong ellipti- +cally polarized harmonics were observed for N2 [20, 21], +while in contrast CO2 exhibited a much lower ellipticity +in the harmonic emission [20]. Structural effects [22–26], +such as the symmetry of the Highest Occupied Molecular +Orbital (HOMO) as well as interference effects, and ultra- +fast multielectron dynamics involving lower-lying orbitals +in the molecule [27] or in the molecular ion [21] have been +put forward as potential origins for the observed elliptic- +ity. +In this article we focus on the role of multielectron +and multiorbital effects in the neutral CO2 molecule on +the polarization state of high-order harmonics. We have +shown previously [27], that results based on the time- +dependent density functional theory (TDDFT) are in +excellent agreement with the experimental data for N2 +[20, 21], if contributions from at least three Kohn-Sham +orbitals are taken into account. Similar strong influence +of inner shell contributions has been observed and pre- +dicted for other strong-field processes as well [28–37]. +Our results of numerical TDDFT simulations show +that indeed the contributions from several valence or- +bitals contribute to the higher-order harmonic emission +from CO2. Moreover, we find that the emission from each +of the orbitals is elliptically polarized. However, our re- +sults for the total high-order harmonic spectrum, which +includes the contributions of up to six orbitals, surpris- +ingly shows, in agreement with the experimental data +[20], almost no ellipticity. +Thus, despite the fact that +high-order harmonic generation from CO2 appears to be +a multielectron process with several orbitals actively in- +volved, signatures in the ellipticity of the harmonic emis- +sion from the different orbitals fade away in the total +signal. +The article is organized as follows: In the next sec- +tion we briefly outline the basics of the time-dependent +density functional approach used for our numerical sim- +ulations. We then discuss the application to calculations +of the ellipticity of high-order harmonic generation of +molecules, including the proper account of the distribu- +tion of alignment in the molecular ensemble. Next, we +arXiv:2301.00356v1 [physics.atm-clus] 1 Jan 2023 + +2 +compare the results of our calculations with the experi- +mental data and analyze the contributions from the dif- +ferent valence orbitals to the total harmonic spectra. We +end with a brief summary of our results. +II. +THEORY +In the nonperturbative intensity regime the theoret- +ical study of the interaction of multielectron targets, +e.g. molecules, with ultrashort laser pulses is challenging. +An approximative approach to analyze multielectron and +multiorbital effects in strong-field processes utilizes the +framework of the time-dependent density functional the- +ory (TDDFT). In this section we outline the application +of TDDFT to the calculation of high-harmonic genera- +tion in molecules, focusing in particular on the evaluation +of the ellipticity of the radiation in an ensemble of aligned +molecules. +A. +TDDFT for strong-field induced molecular +processes +The TDDFT approach is based on the one-to-one cor- +respondence between the time-dependent electron den- +sity ρ(r, t) and the time-dependent potential in multi- +electron Schr¨odinger equation [38]. The density is calcu- +lated from the time-dependent multielectron Schr¨odinger +equation expressed as system of auxiliary time-dependent +noninteracting single-electron Kohn-Sham equations: +i ∂ +∂tφk(r, t) = +� +−∇2 +2 + VKS(r, t) +� +φk(r, t) +(1) +with +ρ(r, t) = +n +� +k=1 +fk|φk(r, t)|2 +(2) +where r is the electronic coordinate, fk is the electron +population in the k-th Kohn-Sham orbital φk(r, t) and +n is the number of orbitals. For a molecule interacting +with a time-dependent intense laser field the Kohn-Sham +potential +VKS(r, t) = Vext(r, t) + +� +ρ(r′, t) +|r − r′|dr′ + Vxc(r) +(3) +includes the external potential due to the interaction of +the electron with the N nuclei in the molecule and with +the time-dependent electric field: +Vext(r, t) = +N +� +i=1 +− +Zi +|Ri − r| + E0(t) sin(ωt) +n +� +k=1 +rk · ˆϵ (4) +where Zi is the charge of the ith nucleus, ˆϵ is the polar- +ization direction, ω and E0(t) are the angular frequency +and the time-dependent amplitude of the laser field. In +the present calculations we considered a sin2-shaped en- +velope. +The exact form of the exchange-correlation potential +Vxc, which takes account of the multielectron effects, is +unknown. To use TDDFT for practical calculations, dif- +ferent approaches have been proposed to design density +functionals for the exchange-correlation energy (for an +overview, see e.g., [39]). For the present calculations, we +have performed systematic studies with various function- +als and found that functionals based on the local density +approximation (LDA), +ELDA +xc +[ρ] = +� +ρ(r)Vxc(r)dr , +(5) +provide, in general, good results. +An improvement +is to take into account the correct asymptotic behav- +ior (1/r), which can be done, for example, via the +exchange-correlation potential proposed by van Leeuwen +and Baerends [40], +V LB +xc (α, β; r) = αV LDA +x +(r) + βV LDA +c +(r) +(6) +− +βx2(r)ρ1/3(r) +1 + 3βx(r) ln[x2(r) + (x2(r) + 1)1/2], +where V LDA +x +and V LDA +c +are the LDA exchange and cor- +relation potentials and x(r) = |∇ρ(r)|/[ρ(r)]4/3. α and +β are parameters obtained by fit to the exact exchange- +correlation function of a certain atomic or molecular sys- +tem. A similar TDDFT approach for the interaction of +molecules with strong fields has been used recently by +Chu and co-workers [41, 42]. +In order to solve the Kohn-Sham equations, Eq. (1), +we have discretized the wavefunction in space and time +with uniform step ∆x = 0.03 a.u. and ∆t = 0.03 a.u., +which converts the ansatz into a matrix equation using +the Octopus code [43, 44]. The initial wavefunctions for +the molecules considered in our study have been obtained +by solving the eigenvalue problem self-consistently using +an initial guess and geometry optimized using Octopus +code as well (this ensures consistency and minimizes risk +for errors). The wavefunction for each orbital is prop- +agated forward in time using the enforced time-reversal +symmetry method. We used grids that extend over 120 +a.u. in polarization direction and 36 a.u. in the trans- +verse directions. To suppress reflection of the wavefunc- +tions at the boundary of the grid an imaginary absorbing +potential has been applied. +B. +High-order harmonic generation from an +ensemble of aligned molecules +High-order harmonic generation is determined through +the Fourier transform of the laser induced dipole moment +in the target. Within the TDDFT formalism, the laser + +3 +FIG. 1: Configuration of pump (aligning) pulse in the y − z +plane, probe (driver) pulse along the ˆz-direction and molecu- +lar axis. +induced dipole moment is given by: +dtot = +n +� +k=1 +dk, +(7) +where dk is the contribution to the dipole moment from +the kth Kohn-Sham orbital, +dk = ⟨φk(r, t)|r|φk(r, t)⟩ . +(8) +The HHG spectrum is then found using the Fourier trans- +form of the dipole moment, d(ω): +P(ω) = +ω4 +12πϵ0c3 d(ω) · d∗(ω) . +(9) +For the molecules studied below, the laser induced dipole +moment has two components, parallel (d||) and perpen- +dicular (d⊥) with respect to the direction of the electric +field of the driving laser. The ellipticity of a given har- +monic is then determined by: +ϵ = +� +1 + r2 − +� +1 + 2r2 cos(2δ) + r4 +1 + r2 + +� +1 + 2r2 cos(2δ) + r4 +(10) +where +r = |d⊥(ω)| +|d||(ω)| +(11) +is the amplitude ratio and +δ = arg[d⊥(ω)] − arg[d||(ω)] +(12) +is the relative phase between the two components. Maxi- +mum ellipticity, i.e. circular polarization, occurs for r = 1 +and δ = π. +In the experimental observations of the ellipticity in +high-order harmonic generation of linear molecules, the +molecules are often aligned by a pump laser pulse. The +distribution of the alignment, achieved in the experi- +ments, is typically measured via ⟨cos2(θ)⟩, where θ is +the angle between the polarization direction of the pump +laser and the internuclear axis of the molecule (see Fig. +1). +In our simulations we have accounted for the ex- +perimental alignment of molecular ensemble by solving +the Kohn-Sham equations for different alignment angles. +For each angle, we obtained the parallel and perpendicu- +lar components of the dipole moment and then averaged +them using the reported alignment distributions. +III. +RESULTS +In this section we present our results for the po- +larization and ellipticity of high-order harmonics from +molecules H+ +2 , H2, and CO2. +The data for the differ- +ent molecules provide us with the opportunity to com- +pare our results with those from other theoretical anal- +ysis (for the one-electron system H+ +2 ) and demonstrate +how multielectron effects and inner valence shell contri- +butions influence the harmonics’ ellipticity for the larger +molecules. +A. +Harmonic generation from H+ +2 and H2 +In order to test our numerical calculations, we first +present results for the one-electron system H+ +2 . In Fig. +2 we show results for the amplitudes (upper panel) and +the phase difference (lower panel) for the 57th harmonics +emitted from H+ +2 as a function of the alignment angle be- +tween the molecular axis and the polarization direction of +a driving laser pulse at 800 nm and 3×1014 W/cm2 with +a pulse duration of 30 fs. The laser parameters are cho- +sen to be the same as in a recent work by Son et al. [26], +who studied the ellipticity of high-order harmonic gen- +eration from H+ +2 using the time-dependent generalized +pseudospectral method. Our results are in good agree- +ment with those previously reported for the overall shape +of the components with a minimum at about 50o for the +parallel component and a phase jump at the same align- +ment angle. It has been shown before [24–26], that these +characteristic features are related to the two-center in- +terference effect occurring in the parallel component. +In order to get an impression of the influence of multi- +electron effects on the ellipticity of high-order harmonics, +we compare results for H+ +2 (Fig. 3, (a-c)) and H2 (Fig. 3, +(d-f)) obtained at the same set of laser parameters (800 +nm, 3×1014 W/cm2). In each case we present theoretical +predictions for four consecutive odd harmonics. For the +single-electron molecule we observe, in agreement with +our results in Fig. 2, a maximum close to 1 in the ratio +of the amplitude in parallel and perpendicular direction +(a), a rapid change in the phase difference (b) and corre- + +Probe +Pump +aser +a +molecule +1 +m +x4 +FIG. 2: Amplitudes of parallel and perpendicular components +(a) and phase difference (b) of 57th harmonic order of H+ +2 as +a function of the alignment angle. Laser parameters: 800 nm, +3 × 1014 W/cm2 and 30 fs. +spondingly a maximum in the ellipticity (c) around the +alignment angle, at which the interference minimum in +the specific harmonic occurs. For H2, one would expect +a similar pattern for the amplitude and the phase dif- +ference, since both electrons are in the same molecular +orbital as in the case of H+ +2 . Indeed, some features in +the overall trend of the results in Fig. 3 are similar, in +particular we still note a maximum amplitude ratio (d) +and a quick phase change (e) at about the same angles +as for H+ +2 . +However, for the ratio we observe a much +narrower structure and for the lowest harmonic a second +maximum. On the other hand, the data for the phase dif- +ference are not as smooth as those for the single-electron +molecule. +As a result, we observe a much more com- +plex pattern for the ellipticity of the harmonics generated +from H2 (f), although some maxima in the structures still +occur near the alignment angle for the interference min- +imum. Thus, the comparison for the simplest molecules +indicates that the ellipticity of high-order harmonics can +be strongly influenced by multielectron effects. For larger +molecules we may therefore expect even more complex +features in the overall ellipticity patterns, since interfer- +ences from orbitals with different symmetry as well as +coupling between different orbitals [27, 36] may play ad- +ditional role. +FIG. 3: Comparison of amplitude ratio r (a, d), phase differ- +ence δ (b, e), and ellipticity (c, f) of high order harmonics from +H+ +2 (a-c) and H2 (d-f) as a function of the alignment angle: +27th (solid lines), 29th (dashed lines), 31st (dashed-dotted +lines), and 33th harmonic (dotted lines). Laser parameters as +in Fig. 2. +FIG. 4: +TDDFT results for the intensity ratio of perpendic- +ular to parallel component of four consecutive harmonics in +CO2 as a function of the angle between the pump and the 30 +fs probe laser pulse at 800 nm and 1.5 × 1014 W/cm2: 17th +(red line), 19th (blue line), 21st (green line) and 23rd har- +monic (black line). For each angle, the experimental reported +alignment distribution [20] was considered in the calculations. +B. +Harmonic generation from CO2 +Next, we analyze the results of our calculations for the +ellipticity in the harmonic generation from the more com- +plex but linear triatomic molecule CO2, which has been +also studied experimentally [20]. +In order to compare +with the experimental data, we have obtained the inten- + +0.008 +Amplitude [arb. units] +0.007 +(a) +0.006 +0.005 +0.004 +0.003 +0.002 +0.001 +0.000 +10 +30 +50 +70 +90 +Alignment angle [degree] +1.0 +(b) +Phase difference [π] +0.5 +0.0 +0.5 +-1.0 +10 +30 +50 +70 +90 +Alignment angle [degree]2.5 +16 +(a) +(d) +amplitude ratio +2.0 +amplitude ratio +12 +1.5 +8 +1.0 +4 +0.5 +0.0 +0 +10 +30 +50 +70 +90 +10 +30 +50 +70 +90 +1.0 +1.0 +(b) +(e) +0.5 +0.5 +0.0 +0.0 +-0.5 +-0.5 +-1.0 +-1.0 +10 +30 +50 +70 +90 +10 +30 +50 +70 +90 +1.0 +1.0 +(c) +(f) +0.8 +0.8 +ellipticity +ellipticity +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +10 +30 +50 +70 +90 +10 +30 +50 +70 +90 +alignment angle (degree) +alignment angle (degree)0.075 +H17 +-H19 +H21 +Intensity Ratio +0.05 +一H23 +0.025 +-100 +-50 +0 +50 +100 +Pump-probe +angle (degree)5 +FIG. 5: +Comparison of results for the ellipticity of high-order +harmonics as a function of alignment angle for CO2: without +(a) and with averaging (b). Laser parameters as in Fig. 4. +sity ratio of the perpendicular to parallel component of +the harmonic emission as a function of the angle between +the pump and probe laser pulse. +For each orientation +angle considered, we have taken into account the exper- +imentally reported alignment distribution by performing +an average over the simulation results for the respective +alignment angles in the distribution. Our results in Fig. 4 +show a rather small intensity ratio and, hence, relatively +small ellipticity with a maximum at about a relative an- +gle of about 60o between polarization direction of pump +and probe pulse for each of the harmonics studied exper- +imentally. The absolute values as well as the position of +the maxima are in very good agreement with the obser- +vations by Zhao et al. [20]. The observed and calculated +rather weak perpendicular component of the harmonics +in CO2 is in contrast to results for N2, for which both +experiment [20, 21] and TDDFT [27] as well as other +calculations [21, 24, 26] show a strong ellipticity for the +emitted harmonics at certain alignment angles. +Part of the explanation for the weak ellipticity is due +to the ensemble angle average effect, which reduces the +overall ellipticity, as observed before in N2 [27]. The ef- +fect can be seen from the comparison of the harmonics +ellipticity as a function of the alignment angle without (a) +and with (b) average in Fig. 5. It is clearly seen that, +in particular for the lower-order harmonics (below 15th +harmonics), without averaging there is a strong elliptic- +ity for certain alignment angles which disappears after +alignment average is taken into account. In contrast, for +the experimentally reported data in the range of 17th to +23rd harmonics the averaging process does have a smaller +effect only. +In this latter range of harmonics from CO2 the main +origin for the weak ellipticity is actually the role of mul- +tielectron effects involving contributions from several or- +bitals. In order to analyze these contributions, we com- +pare in Fig. ?? the ellipticity of the harmonic response +from the HOMO only (a) with those when adding subse- +quently the contributions from the inner valence orbitals +up to HOMO-5 (f). The comparison shows that the ellip- +ticity of high-order harmonics from CO2 is influenced by +the six valence orbitals considered. While the ellipticity +of 17th to 23rd harmonics generated from the HOMO is +rather large for certain alignments angles, the ellipticity +FIG. 6: +Ellipticity of high-order harmonics as a func- +tion +of +alignment +angle +for +CO2. +Starting +with +the +results +from +HOMO +only +(a): +(1πg)4, +contribu- +tions +from +inner +valence +orbitals +are +added +subse- +quently +in +the +other +panels: +(b) +(3σu)2(1πg)4, +(c) +(1πu)4(3σu)2(1πg)4, +(d) +(4σg)2(1πu)4(3σu)2(1πg)4, +(e) +(2σu)2(4σg)2(1πu)4(3σu)2(1πg)4, +(f) +(3σg)2(2σu)2(4σg)2 +(1πu)4(3σu)2(1πg)4. Laser parameters as in Fig. 4. +gradually gets weaker as more contributions are added. +In contrast, for harmonics around the cutoff there re- +mains a strong ellipticity at some alignment angles. +The ellipticity of higher-order harmonics at certain +alignment angles from the HOMO (3πg) can be under- +stood based on the two-center interference effect, similar +as in the case of H+ +2 and H2 above. The importance of +such orbital structure effect for the harmonic generation +from the HOMO of CO2 has been pointed out before +[24]. The strong contributions from the inner valence or- +bitals originate on a variety of effects. Both, HOMO-1 +(2σu) and HOMO-2 (1πu) have a different orbital symme- +try than the HOMO of CO2. Therefore, ionization and, +hence, harmonic generation, from HOMO is suppressed +due to destructive interference at alignment angles of 0◦ +and 90◦ while it is at maximum around 45◦ [45]. In con- +trast, the ionization rate is largest at 0◦ for HOMO-1 and +90◦ for HOMO-2. +Consequently, high-order harmonic +generation from these two orbitals contributes strongly +close to alignment angles at which the signal from the +HOMO is weakest, despite the fact that the ionization + +30 +0.9 +25 +0.8 +0.7 +Harmonic order +20 +0.6 +0.5 +15 +0.4 +10 +0.3 +0.2 +5 +0.1 +0 +20 +40 +60 +80 +Angle(Degrees25 +0.9 +20 +0.8 +0.7 +Harmonic order +15 +0.6 +0.5 +0.4 +10 +0.3 +0.2 +5 +0.1 +0 +20 +40 +60 +80 +Angle (Degrees)25 +25 +(a) +(d) +Harmonic order +20 +0.8 +Harmonicorder +20 +0.8 +15 +0.6 +15 +0.6 +10 +0.4 +10 +0.4 +0.2 +0.2 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +Angle[degrees] +Angle[degrees] +25 +25 +(e) +(b) +0.8 +20 +0.8 +Harmonic +15 +0.6 +15 +0.6 +10 +0.4 +10 +0.4 +0.2 +0.2 +0 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +Angledegrees +Angle [degrees] +25 +25 +(c) +(f) +Harmonic order +20 +0.8 +order +20 +0.8 +15 +0.6 +Harmonic +15 +0.6 +10 +0.4 +10 +0.4 +0.2 +0.2 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +Angle[degrees +Angle[degrees6 +FIG. 7: Rotational averaging assuming distribution 1.The rest +of the notation and parameters as in fig.6. +potential for the inner valence orbitals is smaller than +that of the HOMO. +As for the other inner valence orbitals, that have an +even higher ionization potential, we have found that these +are either strongly coupled to one of the higher lying +states or among each other by the driving field. In the +case of the HOMO-3 state (2σg), the projection onto the +HOMO-1 state is shown in Fig. 9(a). We observe a strong +coupling driven by the field although the frequency is +non-resonant. +This explains the significant change in +the ellipticity pattern upon inclusion of the HOMO-3 +state (Fig. ??(d)). Finally, HOMO-4 and HOMO-5 states +slightly contribute to the 17th to 23rd harmonic genera- +tion at the given parameters and, hence, to the ellipticity +pattern, since these two orbitals are coupled with each +other, leading to a population transfer of about 40% (see +Fig. 9(b)). +To summarize, our results obtained within the time- +dependent density functional theory indicate that high- +order harmonic generation from CO2 is influenced by +multielectron effects with contributions from a significant +number of inner-valence orbitals, besides the contribution +from the HOMO. The harmonic emission from these or- +bitals is strongest at different alignment angles due to +interference effects arising from the specific orbital struc- +tures and there is a strong laser driven coupling between +certain orbitals. As a result, the overall ellipticity of the +FIG. 8: Rotational averaging assuming disributions 2. The +rest of the notation and parameters as in fig.6. +higher-order harmonics is rather small, except for the +cutoff harmonics. The partial alignment and the related +averaging of the results for different orientation angles +further diminishes the ellipticity. +Acknowledgments +This work was supported by the U.S. National Science +Foundation (Grants Nos. Grant No. PHY-1734006 and +Grant No. PHY-2110628). This work utilized the Sum- +mit supercomputer, which was supported by the U.S. Na- +tional Science Foundation and the University of Colorado +Boulder. +[1] A. McPherson, G. Gibson, H. Jara, T.S. Luk, I.A. McIn- +tyre, K. Boyer and C.K. Rhodes, J. Opt. 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Main- + +25 +25 +a +(d) +Harmonic order +20 +0.8 +Harmonicorder +20 +0.8 +5 +0.6 +15 +0.6 +10 +0.4 +10 +0.4 +0.2 +5 +0.2 +0 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +Angle[degrees] +Angle[degrees +25 +25 +(b) +e +0.8 +Harmonic order +20 +0.8 +15 +0.6 +15 +0.6 +10 +0.4 +10 +0.4 +0.2 +0.2 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +Angle [degrees] +Angle[degrees +25 +25 +(c) +(f) +0.8 +Harmonicorder +20 +0.8 +Harmonic +15 +0.6 +15 +0.6 +10 +0.4 +10 +0.4 +5 +0.2 +5 +0.2 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +Angle[degrees +Angle[degrees25 +25 +a +(d) +Harmonic order +20 +0.8 +20 +0.8 +Harmonic orde +15 +0.6 +15 +0.6 +10 +0.4 +10 +0.4 +0.2 +5 +0.2 +0 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +Angle[degrees] +Angle[degrees] +25 +25 +(b) +le +0.8 +20 +0.8 +15 +0.6 +15 +0.6 +10 +0.4 +10 +0.4 +0.2 +0.2 +0 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +Angle [degrees] +Angle[degrees +25 +25 +(c) +(f) +0.8 +e 20 +0.8 +0.6 +Harmonic +15 +0.6 +10 +0.4 +10 +0.4 +5 +0.2 +0.2 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +Angle[degrees +Angle[degrees]7 +FIG. 9: +Projection of coupled inner valence orbitals (a) +HOMO-3 (4σg) to HOMO-1 (3πu) (a) and (b) HOMO-5 (3σg) +to HOMO-4 (2πu) (b) for an alignment angle of 20o. 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Lee and C.D. Lin, Phys. +Rev. Lett. 102, 203001 (2009). + diff --git a/LtAyT4oBgHgl3EQfgPig/content/tmp_files/load_file.txt b/LtAyT4oBgHgl3EQfgPig/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f5247e9dd20165b96ed6dd13ca6f6c86296ffd76 --- /dev/null +++ b/LtAyT4oBgHgl3EQfgPig/content/tmp_files/load_file.txt @@ -0,0 +1,828 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf,len=827 +page_content='Multiorbital effects in high-order harmonic emission from CO2 Andres Mora, Lauren Bauerle, Yuqing Xia, Agnieszka Jaron1 1JILA and Department of Physics, University of Colorado, Boulder, CO 80309-0440, USA We study the ellipticity of high-order harmonics emitted from CO2 molecule driven by linearly po- larized laser fields using numerical simulations within the time-dependent density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' We find that the overall ellipticity of the harmonics is small, which is in agreement with experimen- tal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' On the other hand, our analysis of the numerical results indicates that several valence orbitals contribute significantly to the harmonic emission and some of these contributions show a strong ellipticity of the harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The small ellipticity in the total harmonics signal arises from a combination of factors, namely, the fact that harmonic emission from different orbitals is strongest at different alignment angles of the molecular axis with respect to the laser polarization direction, as well as interference effects and a strong laser coupling between several inner valence orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' PACS numbers: 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='Fb,32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='Wr I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' INTRODUCTION High-order harmonic generation (HHG) is one of the highly nonlinear, nonperturbative processes that occur when an atom or molecule is irradiated by an intense laser field [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' It results from the distortion of the elec- tron density in the presence of the strong electromagnetic field of the laser and the power spectrum of the emitted harmonic radiation corresponds to the Fourier transform of the electron dipole acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The intensity spectra of the emitted high harmonics shows some general char- acteristic features, such as a fast decrease of the signal over the first few harmonics followed by a region with fairly constant plateau harmonic intensities ending by a sharp cutoff, beyond which the harmonic intensity drops quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Over the past few decades HHG has been an active area of research since it provides a source for coherent short- wavelength light, extending into the soft-X-ray regime [3], and for ultrashort laser pulses and waveforms in the attosecond [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Furthermore, it has been shown that HHG spectra contain information about the atomic and electronic structure of the target (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=', [6–11]), ultrafast molecular and intra-molecular electron dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=', [12–15]) as well as time resolution of chemical processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=', [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Basic intuitive picture of HHG is provided by the semi- classical three-step model [17–19], according to which an electron tunnels through the barrier created by the laser field and the Coulomb field into the continuum, is accel- erated by the electric field of the laser first away from and then back to the parent ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Upon return it re- combines, emitting excess energy in the form of high- order harmonic radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Since the process occurs every half cycle of the driving laser field, an attosecond pulse train is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In recent years, it has been shown that, in particular for high-order harmonic generation from molecules, the generated harmonic spectra incorpo- rate more features than predicted by the basic three-step single-active-electron model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' One example are polarime- try measurements of high-order harmonic emission from aligned diatomic and linear triatomic molecules driven by linearly polarized laser fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Surprisingly, strong ellipti- cally polarized harmonics were observed for N2 [20, 21], while in contrast CO2 exhibited a much lower ellipticity in the harmonic emission [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Structural effects [22–26], such as the symmetry of the Highest Occupied Molecular Orbital (HOMO) as well as interference effects, and ultra- fast multielectron dynamics involving lower-lying orbitals in the molecule [27] or in the molecular ion [21] have been put forward as potential origins for the observed elliptic- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In this article we focus on the role of multielectron and multiorbital effects in the neutral CO2 molecule on the polarization state of high-order harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' We have shown previously [27], that results based on the time- dependent density functional theory (TDDFT) are in excellent agreement with the experimental data for N2 [20, 21], if contributions from at least three Kohn-Sham orbitals are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Similar strong influence of inner shell contributions has been observed and pre- dicted for other strong-field processes as well [28–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Our results of numerical TDDFT simulations show that indeed the contributions from several valence or- bitals contribute to the higher-order harmonic emission from CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Moreover, we find that the emission from each of the orbitals is elliptically polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' However, our re- sults for the total high-order harmonic spectrum, which includes the contributions of up to six orbitals, surpris- ingly shows, in agreement with the experimental data [20], almost no ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Thus, despite the fact that high-order harmonic generation from CO2 appears to be a multielectron process with several orbitals actively in- volved, signatures in the ellipticity of the harmonic emis- sion from the different orbitals fade away in the total signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The article is organized as follows: In the next sec- tion we briefly outline the basics of the time-dependent density functional approach used for our numerical sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' We then discuss the application to calculations of the ellipticity of high-order harmonic generation of molecules, including the proper account of the distribu- tion of alignment in the molecular ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Next, we arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='00356v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='atm-clus] 1 Jan 2023 2 compare the results of our calculations with the experi- mental data and analyze the contributions from the dif- ferent valence orbitals to the total harmonic spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' We end with a brief summary of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' THEORY In the nonperturbative intensity regime the theoret- ical study of the interaction of multielectron targets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' molecules, with ultrashort laser pulses is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' An approximative approach to analyze multielectron and multiorbital effects in strong-field processes utilizes the framework of the time-dependent density functional the- ory (TDDFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In this section we outline the application of TDDFT to the calculation of high-harmonic genera- tion in molecules, focusing in particular on the evaluation of the ellipticity of the radiation in an ensemble of aligned molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' TDDFT for strong-field induced molecular processes The TDDFT approach is based on the one-to-one cor- respondence between the time-dependent electron den- sity ρ(r, t) and the time-dependent potential in multi- electron Schr¨odinger equation [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The density is calcu- lated from the time-dependent multielectron Schr¨odinger equation expressed as system of auxiliary time-dependent noninteracting single-electron Kohn-Sham equations: i ∂ ∂tφk(r, t) = � −∇2 2 + VKS(r, t) � φk(r, t) (1) with ρ(r, t) = n � k=1 fk|φk(r, t)|2 (2) where r is the electronic coordinate, fk is the electron population in the k-th Kohn-Sham orbital φk(r, t) and n is the number of orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' For a molecule interacting with a time-dependent intense laser field the Kohn-Sham potential VKS(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' t) = Vext(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' t) + � ρ(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' t) |r − r′|dr′ + Vxc(r) (3) includes the external potential due to the interaction of the electron with the N nuclei in the molecule and with the time-dependent electric field: Vext(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' t) = N � i=1 − Zi |Ri − r| + E0(t) sin(ωt) n � k=1 rk · ˆϵ (4) where Zi is the charge of the ith nucleus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' ˆϵ is the polar- ization direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' ω and E0(t) are the angular frequency and the time-dependent amplitude of the laser field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In the present calculations we considered a sin2-shaped en- velope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The exact form of the exchange-correlation potential Vxc, which takes account of the multielectron effects, is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' To use TDDFT for practical calculations, dif- ferent approaches have been proposed to design density functionals for the exchange-correlation energy (for an overview, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=', [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' For the present calculations, we have performed systematic studies with various function- als and found that functionals based on the local density approximation (LDA), ELDA xc [ρ] = � ρ(r)Vxc(r)dr , (5) provide, in general, good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' An improvement is to take into account the correct asymptotic behav- ior (1/r), which can be done, for example, via the exchange-correlation potential proposed by van Leeuwen and Baerends [40], V LB xc (α, β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' r) = αV LDA x (r) + βV LDA c (r) (6) − βx2(r)ρ1/3(r) 1 + 3βx(r) ln[x2(r) + (x2(r) + 1)1/2], where V LDA x and V LDA c are the LDA exchange and cor- relation potentials and x(r) = |∇ρ(r)|/[ρ(r)]4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' α and β are parameters obtained by fit to the exact exchange- correlation function of a certain atomic or molecular sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' A similar TDDFT approach for the interaction of molecules with strong fields has been used recently by Chu and co-workers [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In order to solve the Kohn-Sham equations, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' (1), we have discretized the wavefunction in space and time with uniform step ∆x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='03 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' and ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='03 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=', which converts the ansatz into a matrix equation using the Octopus code [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The initial wavefunctions for the molecules considered in our study have been obtained by solving the eigenvalue problem self-consistently using an initial guess and geometry optimized using Octopus code as well (this ensures consistency and minimizes risk for errors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The wavefunction for each orbital is prop- agated forward in time using the enforced time-reversal symmetry method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' We used grids that extend over 120 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' in polarization direction and 36 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' in the trans- verse directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' To suppress reflection of the wavefunc- tions at the boundary of the grid an imaginary absorbing potential has been applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' High-order harmonic generation from an ensemble of aligned molecules High-order harmonic generation is determined through the Fourier transform of the laser induced dipole moment in the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Within the TDDFT formalism, the laser 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 1: Configuration of pump (aligning) pulse in the y − z plane, probe (driver) pulse along the ˆz-direction and molecu- lar axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' induced dipole moment is given by: dtot = n � k=1 dk, (7) where dk is the contribution to the dipole moment from the kth Kohn-Sham orbital, dk = ⟨φk(r, t)|r|φk(r, t)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' (8) The HHG spectrum is then found using the Fourier trans- form of the dipole moment, d(ω): P(ω) = ω4 12πϵ0c3 d(ω) · d∗(ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' (9) For the molecules studied below, the laser induced dipole moment has two components, parallel (d||) and perpen- dicular (d⊥) with respect to the direction of the electric field of the driving laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The ellipticity of a given har- monic is then determined by: ϵ = � 1 + r2 − � 1 + 2r2 cos(2δ) + r4 1 + r2 + � 1 + 2r2 cos(2δ) + r4 (10) where r = |d⊥(ω)| |d||(ω)| (11) is the amplitude ratio and δ = arg[d⊥(ω)] − arg[d||(ω)] (12) is the relative phase between the two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Maxi- mum ellipticity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' circular polarization, occurs for r = 1 and δ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In the experimental observations of the ellipticity in high-order harmonic generation of linear molecules, the molecules are often aligned by a pump laser pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The distribution of the alignment, achieved in the experi- ments, is typically measured via ⟨cos2(θ)⟩, where θ is the angle between the polarization direction of the pump laser and the internuclear axis of the molecule (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In our simulations we have accounted for the ex- perimental alignment of molecular ensemble by solving the Kohn-Sham equations for different alignment angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' For each angle, we obtained the parallel and perpendicu- lar components of the dipole moment and then averaged them using the reported alignment distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' RESULTS In this section we present our results for the po- larization and ellipticity of high-order harmonics from molecules H+ 2 , H2, and CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The data for the differ- ent molecules provide us with the opportunity to com- pare our results with those from other theoretical anal- ysis (for the one-electron system H+ 2 ) and demonstrate how multielectron effects and inner valence shell contri- butions influence the harmonics’ ellipticity for the larger molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Harmonic generation from H+ 2 and H2 In order to test our numerical calculations, we first present results for the one-electron system H+ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 2 we show results for the amplitudes (upper panel) and the phase difference (lower panel) for the 57th harmonics emitted from H+ 2 as a function of the alignment angle be- tween the molecular axis and the polarization direction of a driving laser pulse at 800 nm and 3×1014 W/cm2 with a pulse duration of 30 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The laser parameters are cho- sen to be the same as in a recent work by Son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' [26], who studied the ellipticity of high-order harmonic gen- eration from H+ 2 using the time-dependent generalized pseudospectral method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Our results are in good agree- ment with those previously reported for the overall shape of the components with a minimum at about 50o for the parallel component and a phase jump at the same align- ment angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' It has been shown before [24–26], that these characteristic features are related to the two-center in- terference effect occurring in the parallel component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In order to get an impression of the influence of multi- electron effects on the ellipticity of high-order harmonics, we compare results for H+ 2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 3, (a-c)) and H2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 3, (d-f)) obtained at the same set of laser parameters (800 nm, 3×1014 W/cm2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In each case we present theoretical predictions for four consecutive odd harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' For the single-electron molecule we observe, in agreement with our results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 2, a maximum close to 1 in the ratio of the amplitude in parallel and perpendicular direction (a), a rapid change in the phase difference (b) and corre- Probe Pump aser a molecule 1 m x4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 2: Amplitudes of parallel and perpendicular components (a) and phase difference (b) of 57th harmonic order of H+ 2 as a function of the alignment angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Laser parameters: 800 nm, 3 × 1014 W/cm2 and 30 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' spondingly a maximum in the ellipticity (c) around the alignment angle, at which the interference minimum in the specific harmonic occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' For H2, one would expect a similar pattern for the amplitude and the phase dif- ference, since both electrons are in the same molecular orbital as in the case of H+ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Indeed, some features in the overall trend of the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 3 are similar, in particular we still note a maximum amplitude ratio (d) and a quick phase change (e) at about the same angles as for H+ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' However, for the ratio we observe a much narrower structure and for the lowest harmonic a second maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' On the other hand, the data for the phase dif- ference are not as smooth as those for the single-electron molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' As a result, we observe a much more com- plex pattern for the ellipticity of the harmonics generated from H2 (f), although some maxima in the structures still occur near the alignment angle for the interference min- imum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Thus, the comparison for the simplest molecules indicates that the ellipticity of high-order harmonics can be strongly influenced by multielectron effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' For larger molecules we may therefore expect even more complex features in the overall ellipticity patterns, since interfer- ences from orbitals with different symmetry as well as coupling between different orbitals [27, 36] may play ad- ditional role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 3: Comparison of amplitude ratio r (a, d), phase differ- ence δ (b, e), and ellipticity (c, f) of high order harmonics from H+ 2 (a-c) and H2 (d-f) as a function of the alignment angle: 27th (solid lines), 29th (dashed lines), 31st (dashed-dotted lines), and 33th harmonic (dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Laser parameters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 4: TDDFT results for the intensity ratio of perpendic- ular to parallel component of four consecutive harmonics in CO2 as a function of the angle between the pump and the 30 fs probe laser pulse at 800 nm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='5 × 1014 W/cm2: 17th (red line), 19th (blue line), 21st (green line) and 23rd har- monic (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' For each angle, the experimental reported alignment distribution [20] was considered in the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Harmonic generation from CO2 Next, we analyze the results of our calculations for the ellipticity in the harmonic generation from the more com- plex but linear triatomic molecule CO2, which has been also studied experimentally [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In order to compare with the experimental data, we have obtained the inten- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='008 Amplitude [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' units] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='007 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='000 10 30 50 70 90 Alignment angle [degree] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='0 (b) Phase difference [π] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='0 10 30 50 70 90 Alignment angle [degree]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='5 16 (a) (d) amplitude ratio 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='0 amplitude ratio 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='5 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='0 4 0.' metadata={'source': 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='8 ellipticity ellipticity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='0 10 30 50 70 90 10 30 50 70 90 alignment angle (degree) alignment angle (degree)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='075 H17 H19 H21 Intensity Ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='05 一H23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='025 100 50 0 50 100 Pump-probe angle (degree)5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 5: Comparison of results for the ellipticity of high-order harmonics as a function of alignment angle for CO2: without (a) and with averaging (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Laser parameters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' sity ratio of the perpendicular to parallel component of the harmonic emission as a function of the angle between the pump and probe laser pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' For each orientation angle considered, we have taken into account the exper- imentally reported alignment distribution by performing an average over the simulation results for the respective alignment angles in the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Our results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 4 show a rather small intensity ratio and, hence, relatively small ellipticity with a maximum at about a relative an- gle of about 60o between polarization direction of pump and probe pulse for each of the harmonics studied exper- imentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The absolute values as well as the position of the maxima are in very good agreement with the obser- vations by Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The observed and calculated rather weak perpendicular component of the harmonics in CO2 is in contrast to results for N2, for which both experiment [20, 21] and TDDFT [27] as well as other calculations [21, 24, 26] show a strong ellipticity for the emitted harmonics at certain alignment angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Part of the explanation for the weak ellipticity is due to the ensemble angle average effect, which reduces the overall ellipticity, as observed before in N2 [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The ef- fect can be seen from the comparison of the harmonics ellipticity as a function of the alignment angle without (a) and with (b) average in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' It is clearly seen that, in particular for the lower-order harmonics (below 15th harmonics), without averaging there is a strong elliptic- ity for certain alignment angles which disappears after alignment average is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In contrast, for the experimentally reported data in the range of 17th to 23rd harmonics the averaging process does have a smaller effect only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In this latter range of harmonics from CO2 the main origin for the weak ellipticity is actually the role of mul- tielectron effects involving contributions from several or- bitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In order to analyze these contributions, we com- pare in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' the ellipticity of the harmonic response from the HOMO only (a) with those when adding subse- quently the contributions from the inner valence orbitals up to HOMO-5 (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The comparison shows that the ellip- ticity of high-order harmonics from CO2 is influenced by the six valence orbitals considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' While the ellipticity of 17th to 23rd harmonics generated from the HOMO is rather large for certain alignments angles, the ellipticity FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 6: Ellipticity of high-order harmonics as a func- tion of alignment angle for CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Starting with the results from HOMO only (a): (1πg)4, contribu- tions from inner valence orbitals are added subse- quently in the other panels: (b) (3σu)2(1πg)4, (c) (1πu)4(3σu)2(1πg)4, (d) (4σg)2(1πu)4(3σu)2(1πg)4, (e) (2σu)2(4σg)2(1πu)4(3σu)2(1πg)4, (f) (3σg)2(2σu)2(4σg)2 (1πu)4(3σu)2(1πg)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Laser parameters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' gradually gets weaker as more contributions are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In contrast, for harmonics around the cutoff there re- mains a strong ellipticity at some alignment angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The ellipticity of higher-order harmonics at certain alignment angles from the HOMO (3πg) can be under- stood based on the two-center interference effect, similar as in the case of H+ 2 and H2 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The importance of such orbital structure effect for the harmonic generation from the HOMO of CO2 has been pointed out before [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The strong contributions from the inner valence or- bitals originate on a variety of effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Both, HOMO-1 (2σu) and HOMO-2 (1πu) have a different orbital symme- try than the HOMO of CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Therefore, ionization and, hence, harmonic generation, from HOMO is suppressed due to destructive interference at alignment angles of 0◦ and 90◦ while it is at maximum around 45◦ [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In con- trast, the ionization rate is largest at 0◦ for HOMO-1 and 90◦ for HOMO-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Consequently, high-order harmonic generation from these two orbitals contributes strongly close to alignment angles at which the signal from the HOMO is weakest, despite the fact that the ionization 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='9 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='7 Harmonic order 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='5 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='4 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='1 0 20 40 60 80 Angle(Degrees25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='9 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='7 Harmonic order 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='4 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='1 0 20 40 60 80 Angle (Degrees)25 25 (a) (d) Harmonic order 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='8 Harmonicorder 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='8 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='6 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='6 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='4 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='2 0 20 40 60 80 0 20 40 60 80 Angle[degrees] Angle[degrees] 25 25 (e) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='8 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='8 Harmonic 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='6 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='6 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='4 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='2 0 0 20 40 60 80 0 20 40 60 80 Angledegrees Angle [degrees] 25 25 (c) (f) Harmonic order 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='8 order 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='8 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='6 Harmonic 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='6 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='4 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='2 0 20 40 60 80 0 20 40 60 80 Angle[degrees Angle[degrees6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 7: Rotational averaging assuming distribution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='The rest of the notation and parameters as in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' potential for the inner valence orbitals is smaller than that of the HOMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' As for the other inner valence orbitals, that have an even higher ionization potential, we have found that these are either strongly coupled to one of the higher lying states or among each other by the driving field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' In the case of the HOMO-3 state (2σg), the projection onto the HOMO-1 state is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 9(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' We observe a strong coupling driven by the field although the frequency is non-resonant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' This explains the significant change in the ellipticity pattern upon inclusion of the HOMO-3 state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Finally, HOMO-4 and HOMO-5 states slightly contribute to the 17th to 23rd harmonic genera- tion at the given parameters and, hence, to the ellipticity pattern, since these two orbitals are coupled with each other, leading to a population transfer of about 40% (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 9(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' To summarize, our results obtained within the time- dependent density functional theory indicate that high- order harmonic generation from CO2 is influenced by multielectron effects with contributions from a significant number of inner-valence orbitals, besides the contribution from the HOMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The harmonic emission from these or- bitals is strongest at different alignment angles due to interference effects arising from the specific orbital struc- tures and there is a strong laser driven coupling between certain orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' As a result, the overall ellipticity of the FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 8: Rotational averaging assuming disributions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The rest of the notation and parameters as in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' higher-order harmonics is rather small, except for the cutoff harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' The partial alignment and the related averaging of the results for different orientation angles further diminishes the ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Acknowledgments This work was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' National Science Foundation (Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' PHY-1734006 and Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' PHY-2110628).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' This work utilized the Sum- mit supercomputer, which was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Na- tional Science Foundation and the University of Colorado Boulder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' McPherson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Gibson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Jara, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Luk, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' McIn- tyre, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Boyer and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Rhodes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' B 4, 595 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Ferray, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' L’Huillier, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Lompre, G.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 9: Projection of coupled inner valence orbitals (a) HOMO-3 (4σg) to HOMO-1 (3πu) (a) and (b) HOMO-5 (3σg) to HOMO-4 (2πu) (b) for an alignment angle of 20o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Laser parameters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' fray and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Manus, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' B 21, L31 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Popmintchev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Popmintchev, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Arpin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Brown, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Alisauskas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Andriukaitis, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Balciu- nas, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' M¨ucke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Pugzlys, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Baltuska, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Shim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Schrauth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Gaeta, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Hernandez-Garcia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Plaja, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Becker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Jaron-Becker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Murnane and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Kapteyn H C, Science 91, 1287 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Hentschel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Kienberger, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Spielmann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} +page_content=' Reider, N.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfgPig/content/2301.00356v1.pdf'} diff --git a/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf b/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..551f3decea3ca81943a5d4d1d3708a6231e20ef1 --- /dev/null +++ b/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ec0d46acbb62950030c1782c3c9961aed5775a8411032bd4c974ee682aef601d +size 2121406 diff --git a/TdE5T4oBgHgl3EQfAQ7r/content/tmp_files/2301.05378v1.pdf.txt b/TdE5T4oBgHgl3EQfAQ7r/content/tmp_files/2301.05378v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..36b401977e9afc155ee83cdb8c9c230da69e74db --- /dev/null +++ b/TdE5T4oBgHgl3EQfAQ7r/content/tmp_files/2301.05378v1.pdf.txt @@ -0,0 +1,2484 @@ +Dynamical Signatures of Liouvillian Flat Band +Yu-Guo Liu1 and Shu Chen1, 2, 3, ∗ +1Beijing National Laboratory for Condensed Matter Physics, +Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China +2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China +3Yangtze River Delta Physics Research Center, Liyang, Jiangsu 213300, China +(Dated: January 16, 2023) +Although flat-band structures have attracted intensive studies in condensed matter and optical physics due to +their eigenstates exhibiting huge degeneracy and allowing for the localization of wave packet, it is not clear how +the flat band of Liouvillian influences the relaxation dynamics of open quantum systems. To this end, we study +the dynamical signatures of Liouvillian flat band in the scheme of Lindblad master equation. Considering a chain +model with gain and loss, we demonstrate three kinds of band dispersion of Liouvillian: flat bland, dispersionless +only in the real part and imaginary part, and capture their dynamical signatures: when the rapidity spectrum of +Liouvillian is flat, the particle numbers in different sites relax to its steady state value with the same decay rate; +when the real or imaginary part of rapidity spectrum is dispersionless, the relaxation behaviors have oscillating +or forked characteristics. We also unveil that the Liouvillian flat band can lead to dynamical localization, which +is characterized by the halt of propagation of a local perturbation on the steady state. +Introduction.— Band structure of a Hamiltonian plays an +important role in understanding the motion of particles in pe- +riodic crystals. Usually, special band structures may give rise +to exotic quantum phenomena, for example, low-energy ex- +citations of electrons on a linear dispersive band in graphene +behave like massless Dirac fermions [1, 2]. Another instance +is the flat band (FB) in which all electrons carry the same en- +ergy regardless of their momentum. Due to the dispersionless +band structure, particles in FB have arbitrarily large effective +mass, so they will be localized in real space. Especially, in +strongly correlated systems, heavy degeneracy and zero ki- +netic energy in FB can increase density of electronic states +and highlight Coulomb interaction, leading to rich many-body +phenomena [3–5]. +In open quantum systems, dynamics of density matrix ρ is +described by Lindblad master equation (LME) under Born- +Markov approximation [6–8]: +dρ +dt = L(ρ) := −i[H, ρ] + +� +µ +� +LµρL† +µ − 1 +2{L† +µLµ, ρ} +� +, +(1) +where L is called the Liouvillian superoperator, H is the +Hamiltonian of system, and Lµ are Lindblad operators which +reflect the coupling between system and environment. The +Planck constant ℏ is set to unity throughout this Letter. There +have been several methods developed to obtain the spectrum +of L, especially for quadratic systems [9–15]. In Ref.[15], +a route for realizing dispersionless bands is proposed based +on the underlying mechanism with the emergence of a dis- +sipationless dark space. Generally speaking, the short-time +dynamics is related to the Liouvillian eigenvalues with large +modulus of the real part, whereas the long-time relaxation to +the smallest modulus beyond zero (the so called Liouvillian +gap) [16–22]. However, how the structure of Liouvillian, es- +pecially the Liouvillian flat band (LFB), influences dynamics +is still a subtle and unexplored question. +In this Letter, we focus on the dynamics of open quan- +tum systems with LFB. In comparison with the real spectrum +of Hamiltonian system, the Liouvillian spectrum is complex, +and thus the corresponding rapidity spectrum can exhibit more +rich structures with dispersionless band in both imaginary and +real part or either of them. To make our study concrete, we +shall first apply a geometrically intuitive method to construct +lattice with correlated gain and loss, which supports LFB, and +explore the generality of dynamical signatures associated with +the structure of Liouvillian spectrum. We show that the ra- +pidity spectra from Liouvillian and damping-matrix spectra +of correlation functions have the same dispersion characteris- +tics, which lead to different signatures of damping dynamics +of local particle number distribution: oscillating, forked, syn- +chronous damping are related to the band dispersionless only +in imaginary part, real part and in both parts, respectively. Fur- +thermore, we exactly solve the model and show that the LFB +can induce dynamical localization, which is characterized by +the halt of the propagation of a local perturbation on the non- +equilibrium steady state (NESS). +Formalism.— The density matrix ρ and Liouvillian super- +operator L in Eq. (1) can be formally expressed as +ρ = +� +IJ +ρIJ|I⟩a⟨J|a, L(ρ) = +� +i j +Fi(a, a†) ρ Fj(a, a†), (2) +where a is the set of fermionic annihilation operators i.e. a = +(a1, a2, · · · ), Fi(a, a†) is a function with variables among a +and a†, I = (I1, I2, · · · ), J = (J1, J2, · · · ) and +|I⟩a⟨J|a = (a† +1)I1(a† +2)I2 · · · (a† +L)IL|0⟩a⟨0|a(aL)JL · · · (a1)J1, (3) +where |0⟩a is the vacuum state for all a−fermions. For the +convenience of analysis and calculation, we map fermionic +LME into a new representation referred to as C by following +the method in Ref. [10]: +ρ → | ρ⟩C = +� +IJ +ρIJ(a† +1)I1 · · · (a† +L)IL(c† +1 ˆP)J1 · · · (c† +L ˆP)JL |0⟩, +(4a) +L → ˆLC = +� +i j +Fi(a, a†) F T +j ( ˆPc, c† ˆP), +(4b) +arXiv:2301.05378v1 [cond-mat.other] 13 Jan 2023 + +2 +where c = (c1, c2, · · · ) is the set of annihilation operators +of c−fermions, which is a one-to-one mapping from a, T +means matrix transpose, and |0⟩ is the vacuum state of both +a− and c−fermions. +ˆP is the parity operator defined by +ˆP = exp +� +iπ � +j(a† +jaj + c† +jcj) +� +, which is introduced to ensure +fermionic anticommutation relations between a−fermions and +c−fermions. Full mapping process is shown in the Supple- +mental Material [25]. +Model.— We consider a Liouvillian in a periodic chain: +L(ρ) = −i[H, ρ] + (1 − w)DL(ρ) + (1 + w)DR(ρ), +(5) +where H = � +l J(a† +l+1al + h.c.), w ∈ [−1, 1], and +DL(ρ) = +� +l +� +2AlρA† +l − A† +l Alρ − ρA† +l Al +� +, +DR(ρ) = +� +l +� +2A† +l ρAl − AlA† +l ρ − ρAlA† +l +� +, +(6) +where Al = √γ1a† +l + √γ2al+1. The operators Al and A† +l tie the +gain and loss of neighboring sites together, which could be re- +alized by optical superlattice with Bose-Einstein condensate +reservoir [23]. The role of w ∈ [−1, 1] is analogous to the sta- +tistical distribution from temperature [24]. Mapping Eq. (5) +into the representation C, we get a ladder model consisting of +a−fermion chain and c−fermion chain (see the Supplemental +Material [25]). The L is mapped to ˆL = ˆH + (1 − w) ˆDL + (1 + +w) ˆDR, where ˆH = � +l +� +−iJ(a† +l+1al + h.c.) + iJ(c† +l+1cl + h.c.) +� +. +ˆDL and ˆDR are illustrated in Fig. 1 (a) and (b), which have +leftward and rightward hoppings, respectively, along two di- +agonals of every plaquette in the ladder. The cross-stitch-type +hopping is crucial for generating FB because it can form a +destructive-interference structure, which consists with our ex- +perience in the FB ladder models [34–37]. +In momentum space, +ˆL can be expressed in BdG +form +as +ˆL += +0.5 ˆLk=0 + �π− +k=0+ ˆLk, +where +ˆLk += +(a† +k c† +k a−k c−k) Lk (ak ck a† +−k c† +−k)T − 4γ and γ = γ1 + γ2. +Due to parity conservation in ˆL, the operator ˆP can be substi- +tuted by a constant P which equals 1(−1) when ˆL acts on the +state with even (odd) fermions. Then we have +Lk = −i2J cos kσz ⊗ σz − 4 √γ1γ2 cos kPσz ⊗ σx − +2γPσy ⊗ σy + 2w +� +(γ2 − γ1)σz ⊗ I + 2 √γ1γ2 sin kσy ⊗ σz ++i(γ2 − γ1)Pσx ⊗ σy + i2 √γ1γ2 sin kPI ⊗ σx +� +, +(7) +where I and σi are identity and Pauli matrices. +ˆLk can +be diagonalized as ˆLk += λ−(k) +� +ζ +′ +1(k)ζ1(k) + ζ +′ +4(k)ζ4(k) +� ++ +λ+(k) +� +ζ +′ +2(k)ζ2(k) + ζ +′ +3(k)ζ3(k) +� +, where ζ +′ +i(k) and ζj(k +′) ful- +fill anticommutation relations: {ζ +′ +i(k), ζj(k +′)} = δi jδkk′ and +{ζ +′ +i(k), ζ +′ +j(k +′)} = {ζi(k), ζj(k +′)} = 0 [9]. The λ±(k) is called +rapidity spectrum given by λ±(k) = −2γ ± 2mk for both odd +and even parity [38], where +mk = +����� +� +(4γ1γ2 − J2) cos2 k, +J2 ≤ 4γ1γ2, +i +� +(J2 − 4γ1γ2) cos2 k, +J2 > 4γ1γ2. +(8) +FIG. 1. ˆDL, ˆDR and ˆL = ˆH + ˆDL + ˆDR are sketched by (a), (b) and +(c), where the color ovals, straight lines (with or without arrow) and +wavy lines represent onsite loss, particle hopping and pair production +and annihilation. The bule, red and orange ovals are corresponding +to terms (γ1 − γ2)ˆna/c, l − γ1, (γ2 − γ1)ˆna/c, l − γ2 and constant loss +−γ. +ˆna/c, l is the particle number operator of a− or c−fermion on +the site l. Horizontal black wavy lines represent ± √γ1γ2 ˆP(alal+1 + +h.c.) or ± √γ1γ2 ˆP(clcl+1 +h.c.). The black arrows indicate directional +hoppings with strength −2 √γ1γ2 ˆP. The bule, red and orange vertical +wavy lines are corresponding to 2γ1 ˆPa† +l c† +l + 2γ2 ˆPclal, 2γ2 ˆPa† +l c† +l + +2γ1 ˆPclal and 2γ ˆP(a† +l c† +l + clal). (d) shows the (c) in even parity and +under flat band condition, where J = 2 √γ1γ2 = 1. The dashed wavy +lines indicate the pairing terms have no effect on single particle- or +hole- excitation on its steady state. +The λ±(k) is independent with w and we show it in Fig. 2. +When J2 = 4 √γ1γ2, λ is a FB of k. When J2 < 4 √γ1γ2 +(J2 > 4 √γ1γ2), λ is dispersionless in its imaginary (real) part. +Especially, in Fig. 2 (c) and (f) the spectrum is pure real, which +indicates Lk possessing a pseudo-Hermiticity [39–41], while +in Fig. 2 (a) and (d) the complex spectrum shows the break- +ing of pseudo-Hermiticity. Since the Liouvillian spectrum is +obtained by sum of different number of λ±(k), it inherits the +characteristics of rapidity spectrum, as shown in Fig. 2 (g)∼(i). +When J2 = 4 √γ1γ2, Liouvillian spectrum consists of some +highly degenerate discrete points (Fig. 2 (h)), corresponding +to different occupations of the FB of rapidity spectrum, so we +call this kind of Liouvillian spectrum as the LFB. +Two-operator correlation functions.— By making Fourier +transform, Eq. (5) becomes +L(ρ) = +π +� +k=−π +� +−i2J cos k[ˆnk, ρ]+(1−w)DL +k(ρ)+(1+w)DR +k (ρ) +� +, +(9) +where DL +k(ρ) = 2BkρB† +k − {B† +kBk, ρ}, DR +k (ρ) = 2B† +kρBk − +{BkB† +k, ρ} and Bk = √γ1eika† +k + √γ2a−k. +We define two- +operator correlation functions: Gk1, k2 = Tr(a† +k1ak2ρ), Dk1, k2 = +Tr(ak1ak2ρ), and D∗ +k1, k2 = Tr(a† +k2a† +k1ρ). In terms of the correla- +tion function vector Ψk1k2 = (Gk1,k2,G−k2,−k1, Dk2,−k1, D∗ +k1,−k2)T, +the dynamical evolution is governed by the following closed + +(a) +(b) +Y102. +Y12P +a +a +a +C +C +C +h121 +(c) +(d) +iJ +2J +2 +2 +a +a +a +a +a +a +c +c +C +c +C +iJ +iJ +2 +23 +(b) +(a) +(d) +(e) +(c) +(f) +(g) +(h) +(i) +FIG. 2. (a)∼(c) the real part of rapidity spectra λ±(k). (d)∼(f) the +imaginary part of λ±(k). (g)∼(i) the Liouvillian spectra obtained by +exactly diagonalizing 6-site lattice with w = 0, J = 1 and γ1 = 0.25 +for all subfigures. γ2 = 0.5 in (a), (d) and (g). γ2 = 1 in (b), (e) and +(h). γ2 = 1.5 in (c), (f) and (i). +equation: +d +dtΨk1k2 = Xk1k2Ψk1k2 + Vk1k2, +(10) +where +Xk1k2 = −4γI ⊗ I + i2J cos k1σz ⊗ σz − i2J cos k2I ⊗ σz ++ 4 √γ1γ2 cos k1σx ⊗ σz − 4 √γ1γ2 cos k2σy ⊗ σy (11) +and Vk1k2 += +δk1,k2 +� +2γ + 2w(γ2 − γ1), 2γ + 2w(γ2 − +γ1), i4w √γ1γ2 sin k1, −i4w √γ1γ2 sin k1 +�T. +The +damping +matrix +Xk1k2 +has +four +eigenstates +which +fulfill +the +equation +Xk1k2|Γ±± +k1k2⟩ += +Γ±± +k1k2|Γ±± +k1k2⟩ +with +the +eigenvalues +given +by +Γ±± +k1k2 += +−4γ ± +2 +� +4γ1γ2 − J2 � +(| cos k1| ± | cos k2|sgn(4γ1γ2 − J2))2, +where +sgn(x) is a sign function. Γ also has a transition from the +complex to the real by decreasing J due to the PT −symmetry +of Xk1k2. +In the Supplemental Material [25] we show that +Xk1k2 has higher symmetry than ˆLk, which makes Xk1k2 have a +similar band structure as ˆLk. In Fig. 3, we see that Γ±± +k1k2 fully +inherits the dispersion characteristics of real and imaginary +part from the rapidity spectra in Fig. 2. +(b) +(a) +(d) +(e) +(c) +(f) +FIG. 3. (a)∼(c) the real part of Γ±± +k1k2. (d)∼(f) the imaginary part of +Γ±± +k1k2. J = 1 and γ1 = 0.25 are for all subfigures. γ2 = 0.5 in (a) and +(d). γ2 = 1 in (b) and (e). γ2 = 1.5 in (c) and (f). +Flat-band damping dynamics.— Damping dynamics dis- +plays the converging processes from initial state to NESS [45]. +Here, we show that the “flat band” in real or imaginary +or both parts will effectively influence the damping behav- +iors in real space. +We concentrate on the vector Ψl1l2 = +(Gl1,l2,Gl2,l1, Dl2,l1, D∗ +l1,l2)T consisting of real-space correlation +functions: +Gl1, l2 = Tr(a† +l1al2ρ), Dl1, l2 = Tr(al1al2ρ), D∗ +l1, l2 = Tr(a† +l2a† +l1ρ). +Introduce the deviating expectation of operator ˆO as � +O(t) = +⟨ ˆO⟩(t) − ⟨ ˆO⟩S to describe the deviation from steady state ex- +pectation value ⟨ ˆO⟩S += ⟨ ˆO⟩(∞). +From Eq. (10), we get +d +dt �Ψk1k2 = Xk1k2�Ψk1k2. +Making Fourier transformation, we +have �Ψl1l2(t) = � +k1k2 ei(−k1l1+k2l2)�Ψk1k2(t). +Decomposing ar- +bitrary initial state �Ψk1k2(0) by the eigenstates of Xk1k2 i.e. +�Ψk1k2(0) = � +αβ Cαβ +k1k2|Γαβ +k1k2⟩, where α and β take ±, then we +have +�Ψl1l2(t) = +� +k,µ +eik·˜rCµ +ket Γµ +k |Γµ +k⟩, +(12) +where k = (k1, k2), ˜r = (−l1, l2) and µ = (α, β). For non-zero +Liouvillian gap, the system exponentially decays to NESS +with time, so we can define instantaneous decay rate K(t) of +the j component of �Ψl1l2(t) as +K j +l1l2 = d +dt log +� +|�Ψj +l1l2(t)| +� +. +(13) +Below we unveil how K(t) is affected by the dispersion of +Γµ +k through Fig. 4, in which the damping behaviors of local +deviating particle number �nl = � +Gll from the initial state with a +single excitation on site 1 are shown: +(i) When FB appears, Γµ +k becomes a constant, denoted +by Γ0. Then we have �Ψl1l2(t) = eΓ0t � +k,µ eik·˜rCµ +k|Γµ +k⟩ and +K j +l1l2(t) = Re(Γ0), which means for arbitrary initial state dif- +ferent two-operator correlation functions will synchronously +relax to their steady state expectation values with the same +decay rate, as demonstrated in Fig. 4 (b) and (e), where dif- +ferent curves of log(˜nl) as a function with γt have the same +constant slope, i.e. K1 +ll = 4γ for all l. +(ii) When Γµ +k is only dispersionless in its real part, we set +Γµ +k = −x0 − iyµ(k), where x0 and yµ(k) are real. Then we +have �Ψl1l2(t) = e−x0t � +k,µ Cµ +keik·˜re−iyµ(k)t|Γµ +k⟩ and +K j +l1l2 = −x0 + d +dt log +�������� +��� +� +k,µ +Cµ +k|Γµ +k⟩ jei� +k·˜r−yµ(k)t���� +�������� . +(14) +The right side of Eq. (14) contains sum of a series of plane +waves, which leads to K j +l1l2(t) oscillating around x0, as shown +in Fig. 4 (d). The oscillating slopes lead to continuously inter- +secting curves in Fig. 4 (a). +(iii) When Γµ +k is only dispersionless in its imaginary part, +we set Γµ +k = −(xc + δxµ(k)) − iy0, where xc and δxµ(k) +are the central value and the offset function of Re(Γµ +k), + +-1.5 +-2-2 +-2.5 +-3k(元)k(元)J2 +> +412J2 + = 412J2 +412Im(入±ReReReIm2 +-4Re(X±5 +0 +5 +-20 +-10 +05 +5 +-30 +-15 +05 +5 +-40 +-20 +02 +0 +2 +0 +0.5 +10.5 +0 +-0.5 +0 +0.5 +10.5 +0 +-0.5 +0 +0.5k(元)2 +0 +-2 +1 +0 +0 +1 +-11 +0 +1 +1 +0 +0 +-1 -1J2 +412k1(元)k2(元)k1(元)k1(元)k1(元)k2(元)k1(元)k1(元)k2(元)1 +0 +1 +1 +0 +0k2(元)k2(元)k2(元)-2 +-3 +-4 +1 +1 +0 +0 +.1.4 +-5 +6 +1 +1 +0 +0 +1-5 +.10 +1 +1 +0 +0 +-1 +-1HH +Re(T) +k1k2HH +k1k2J2 +> +412J2 + = 4124 +and y0 is the imaginary part. +Then we have �Ψl1l2(t) = +e−(xc+iy0)t � +k,µ Cµ +keik·˜re−δxµ(k)t|Γµ +k⟩ and +K j +l1l2 = −xc + d +dt log +�������� +��� +� +k,µ +Cµ +k|Γµ +k⟩ jeik·˜re−δxµ(k)t��� +�������� . +(15) +Since δxµ(k) is real, the relaxation process does not display +oscillating decay rates (see Fig. 4 (f)). This induces the forked +damping curves typically as shown in Fig. 4 (c). +(b) +(a) +(c) +(d) +(e) +(f) +FIG. 4. The damping of particle number at different sites. The lattice +has 15 sites under the periodic boundary condition. Initial state is a +single excitation on the first site from vacuum. The time evolutions of +log(|˜nl|) are shown in (a), (b), (c), and their derivatives K1 +ll are shown +in (d), (e) and (f). The blue, red and orange lines are corresponding +to l = 1, l = 2 and l = 3, respectively. In (a) and (d), γ2 is set as 0.5. +In (b) and (e), γ2 = 1. In (c) and (f), γ2 = 1.5. Others parameters are +the same in all subfigures with J = 1, γ1 = 0.25 and w = 0.25. The +black dashed line represents a constant decay rate as ˜nl ∝ e−4γt. +The above damping dynamics is directly related to disper- +sion of damping-matrix spectra. The damping-matrix spec- +tra reflect the decay of correlation functions, however, the Li- +ovillian spectra reflect the decay of the whole system. We +prove that the damping-matrix spectra are included in Liouvil- +lian spectra in the Supplemental Material [25]. Therefore, for +more general models with closed evolution equations of two- +operator correlation functions, the dispersionless Liouvillian +bands will lead to dispersionless damping-matrix spectra, and +then give rise to the same dynamical signatures as shown in +our model. +Localized +normal +master +modes +and +dynamic +localization.— In isolated system, FBs lead to localized +eigenstates by destructive interference. +Now, we exactly +solve our model (see the Supplemental Material [25]) to show +that the LFB can induce dynamic localization by localized +normal master modes (LNMMs), which suppress propagation +of local perturbation on NESS. +Usually, the odd parity part of ˆL has no effect on the ex- +pectation value of observation in pure fermionic system [25]. +Therefore, we focus on the balanced model (w = 0) with even +parity (P = 1), whose Liouvillian is illustrated in Fig. 1 (c). +By solving the equation ζi(k)|Ω⟩ = 0 for i = 1 ∼ 4, we get the +steady state |Ω⟩ as +|Ω⟩ = 1 +N +π +� +k=−π +(1 + a† +kc† +−k)|0⟩ = 1 +N +L +� +l=1 +(1 + a† +l c† +l )|0⟩, +(16) +where N = 2L and this state is independent with γ1 and γ2. +At the FB point with J = 2 √γ1γ2, the exceptional degeneracy +occurs in the non-Hermitian matrix Lk of Eq. (7) with four +eigenstates coalescing into two. Then ˆLk is reduced to ˆLk = +−2γ +� +ζ +′ +A(k)ζA(k) + ζ +′ +B(k)ζB(k) +� +, where +ζ +′ +A(k) = −a† +k + c−k, +ζA(k) = 1 +2(−ak + ick + ia† +−k + c† +−k), +ζ +′ +B(k) = ak + c† +−k, +ζB(k) = 1 +2(a† +k − ic† +k + ia−k + c−k). +(17) +Making Fourier transformation, we get +ζ +′ +A(l) = +� +k +e−iklζ +′ +A(k) = −a† +l +cl, ζ +′ +B(l) = +� +k +eiklζ +′ +B(k) = al+c† +l , +(18) +which create local eigenstate ζ +′ +A,B(l)|Ω⟩ of ˆL with eigenvalue +−2γ and are coined as LNMMs. +We can also understand LNMMs intuitively from the per- +spective of destructive interference. Writing the real-space Li- +ouvillian with w = 0, J = 2 √γ1γ2 = 1 as ˆL = � +l(ˆhl + ˆfl −2γ), +where the hopping term hl is defined as ˆhl = −i(a† +l+1al + +h.c.) + i(c† +l+1cl + h.c.) − (a† +l+1cl + c† +l+1al + h.c) and the pair- +ing term ˆfl is defined as fl = 2γ(a† +l c† +l + clal), we can check +that ˆflal|Ω⟩ = +ˆflcl|Ω⟩ = +ˆfla† +l |Ω⟩ = +ˆflc† +l |Ω⟩ = 0. This im- +plies that the pairing terms do not affect a single particle or +hole excited on the NESS. Therefore, for these states only +hopping terms make sense. We schematically plot this re- +duced ladder in Fig. 1 (d). It is easy to find another LNMM +as ζ +′ +C(l) = a† +l − ic† +l from the view of destructive interference, +which forbids the state ζ +′ +C(l)|Ω transferring to other sites. We +can also check that ˆL ζ +′ +C(l)|Ω⟩ = −2γ ζ +′ +C(l)|Ω⟩. +The LNMMs contain decay information of quantum +jumps. To see it clearly, we map the C−representation state +ζ +′ +A(l)ζ +′ +B(l)|Ω⟩, for example, back to density-matrix representa- +tion: +ζ +′ +A(l)ζ +′ +B(l)|Ω⟩ → −a† +l alρs + ρsala† +l + alρsa† +l − a† +l ρsal, +(19) +where ρs is the density matrix of NESS. The terms alρsa† +l +and a† +l ρsal are exactly corresponding to local quantum jumps +on NESS. Eq. (19) implies that the local perturbation on +NESS from quantum jumps will relax to NESS without ex- +panding its territory. To see it clearly, we simulate the evo- +lution from an initial state described by the density matrix +ρ0 = a† +1ρsa1/Tr(a† +1ρsa1), which is created by a quantum jump +on the first site of NESS. In Fig 5, we demonstrate the time +evolution of particle numbers of the first three sites in a lattice +with 15 sites. In the initial time, a jump occurs on the first +site of NESS, increasing only the particle number on the first +site n1 to 1 with others sites keeping their steady state value + +2 +-6 +0 +2-2 +-4 +-6 +0 +1 +2K= +-4ttK= +-4K= +-4tttK +1 +1log +(1i +nt +1)-2 +-6 +0 +1 +20 +n1 +n2 +n3 +-5 +-10 +0 +1 +20 +n1 +n2 +n3 +-5 +-10 +0 +1 +20 +n1 +n2 +n3 +~ +-5 +-10 +0 +1 +2J2 +> +412J2 + = 412J2 +412t5 +(a) +(b) +(c) +FIG. 5. The time evolution of particle number on the first site n1 +shown in (a), second site n2 in (b) and third site n3 in (c). Initial +state is a† +1ρsa1/Tr(a† +1ρsa1) corresponding to a quantum jump on the +first site of steady state. The periodic lattice has 15 sites with w = 0, +J = 1, γ1 = 0.25 in all subfigures. The black dotted, red solid, and +blue dashed lines are corresponding to γ2 = 0.5, γ2 = 1 and γ2 = 1.5, +respectively. +0.5. The red solid line, black dotted line and blue dashed line +are corresponding to the situation with J2 = 4γ1γ2 (LFB), +J2 > 4γ1γ2, and J2 < 4γ1γ2, respectively. We can see that +when J2 � 4γ1γ2, the perturbation can spread from n1 to n3. +However, for the case with LFB, the perturbation excitation +decays locally without going through to n2 and n3, indicating +the occurrence of dynamical localization. +Final remarks.— (i) We use a geometrically intuitive +method to construct flat band models in open system and +demonstrate that the dispersion of Liouvillian band can ef- +fectively affect the damping dynamics of local particle num- +ber, intermediated by damping matrix of correlation function +vector. When the Liouvillian flat band appears, the particle +number in different sites will relax to their stable values syn- +chronously. When only the real or imaginary part of rapidity +spectrum is dispersionless, the damping behaviors show the +oscillating or forked characteristic. +(ii) We show flat-band Liouvillian can induce dynamical +localization on NESS by the localized normal master modes, +which halt the propagation of perturbation from other sites to +the target sites. +(iii) Our model does not exhibit non-Hermitian skin ef- +fect [42, 43], which was uncovered to cause many abnormal +phenomena such as boundary sensitivity [44], chiral and he- +lical damping [45, 46] and slowing down of relaxation pro- +cesses [20]. The interplay between Liouvillian flat band and +non-Hermitian skin effect is an interesting topic for future +studies. +Acknowledgments.— We thank X. L. Wang, Z. Y. Zheng +and C. X. 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Schaller, Non-equilibrium +boundary driven quantum systems: models, methods and prop- +erties, arXiv:2104.14350 [quant-ph] (2022). +[25] See Supplemental Material for (i) Mapping of Lindblad master +equation, (ii) Diagonalization, exceptional point and symmetry +of the Liouvillian, (iii) Exactly solution and discussion on parity, +(iv) Evolution equations of correlation functions and the symme- +try of damping matrix, (v) Particle number distribution of steady +state, (vi) The relationship between the damping-matrix spectra +and the Liouvillian spectra. The supplemental materias include +also references [26–33]. +[26] M.-D. Choi, Completely positive linear maps on complex ma- +trices, Linear Algebra Applications 10, 285 (1975). +[27] A. Jamiołkowski, Linear transformations which preserve trace +and positive semidefiniteness of operators, Rep. Math. Phys. 3, +275 (1972). +[28] J. E. 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Wang, Non-Hermitian Skin Effect and +Chiral Damping in Open Quantum Systems, Phys. Rev. Lett. +123, 170401 (2019). +[46] C.-H. Liu, K. Zhang, Z. Yang, and S. Chen, Helical damp- +ing and dynamical critical skin effect in open quantum systems, +Physical Review Research 2, 043167 (2020). + +7 +SUPPLEMENTAL MATERIAL: Dynamics Signatures of Liouvillian Flat Band +S1. Mapping of Lindblad master equation +FIG. S1. Mapping of Lindblad master equation. +The Lindblad master equation, formalized density matrix ρ and Liouvillian superoperator L is shown in Eq. (1) and Eq. (2) in +the main text. First we carry out the Choi-Jamiolkwski isomorphism [1–4] to map the fermionic LME into representation B as +d +dt| ρ⟩B = ˆLB| ρ⟩B, +(S1) +where | ρ⟩B is vectorized from ρ and ˆLB is mapped from L. Specifically, the mapping is +ρ → | ρ⟩B = +� +IJ +ρIJ|I⟩a ⊗ |J⟩b, +(S2a) +L → ˆLB = +� +i j +Fi(a, a†) ⊗ F T +j (b, b†), +(S2b) +where b = (b1, b2, · · · ) is the set of annihilation operators of b−fermions, which is one-to-one mapping from a, and T means +matrix transpose. |I⟩a and |J⟩b are defined as +|I⟩a = (a† +1)I1(a† +2)I2 · · · (a† +L)IL|0⟩a, +(S3a) +|J⟩b = (b† +1)J1(b† +2)J2 · · · (b† +L)JL|0⟩b, +(S3b) +where |0⟩a and |0⟩b are vacuum state of all a−fermions and b−fermions, respectively. In this representation, the expectation +value of observable becomes +⟨ ˆOa⟩ =B ⟨S0| ˆOa ⊗ Ib| ρ⟩B, +(S4) +where B⟨S0| is a special state defined as: +B⟨S0| = +� +S +⟨S|a ⊗ ⟨S|b = +� +S +� +⟨0|a(aL)S L · ·(a1)S 1 ⊗ ⟨0|b(bL)S L · ·(b1)S 1� +, +(S5) +and Ib is a unit operator of all b−fermions. The element S i of S = (S 1, S 2, · · · ) can take 0 or 1, and � +S requires a sum over all +possible configurations of S. Let us prove Eq. (S4): +⟨ ˆOa⟩ = +� +IJS +ρIJ ⟨S|a ˆOa|I⟩a⟨S|bIb|J⟩b += +� +IJS +ρIJ a⟨0|aS L +L · · · aS 1 +1 ˆOa(a† +1)I1 · · · (a† +L)IL|0⟩a δSJ += +� +IJ +ρIJ a⟨0|aJL +L · · · aJ1 +1 ˆOa(a† +1)I1 · · · (a† +L)IL|0⟩a += +� +IJ +ρIJ a⟨0| ˆOa|0⟩a = Tr( ˆOaρ). +(S6) + +p=p [I)ac +278 +In representation B, operators satisfy the following relations: +{ai, a† +j} = {bi, b† +j} = δi j, +{a† +i , a† +j} = {ai, a j} = {b† +i , b† +j} = {bi, bj} = 0, +(S7a) +[a† +i , b j] = [a† +i , b† +j] = [ai, bj] = [ai, b† +j] = 0. +(S7b) +The commutation relations in Eq. (S7b) are from the direct product between a−fermions and b−fermions, which are unfavorable +for further analysis. To enforce fermionic anticommutation relations over all operators, we define operators of c−fermions as +c† = b† ˆP and c = ˆPb, where ˆP is a parity operator defined as +ˆP := exp +� +iπ +� +l +(a† +l al + b† +l bl) +� += exp +� +iπ +� +l +(a† +l al + c† +l cl) +� +. +(S8) +It is easy to check the fermionic anticommutation relations in a−fermions and c−fermions: +{ci, c† +j} = δi j, +{c† +i , c† +j} = {ci, c j} = 0 +(S9a) +{a† +i , cj} = {a† +i , c† +j} = {ai, cj} = {ai, c† +j} = 0 +(S9b) +By c we can fully fermionize system from representation B to representation C. The mapping is +| ρ⟩B → | ρ⟩C = +� +IJ +ρIJ(a† +1)I1 · · · (a† +L)IL(c† +1 ˆP)J1 · · · (c† +L ˆP)JL |0⟩, +(S10a) +ˆLB → ˆLC = +� +ij +Fi(a, a†) F T +j ( ˆPc, c† ˆP). +(S10b) +The LME and the expectation value of observable in representation C are +d +dt| ρ⟩C = ˆLC| ρ⟩C, +(S11a) +⟨ ˆOa⟩ =C ⟨S0| ˆOa| ρ⟩C, +(S11b) +where C⟨S0| is defined as: +C⟨S0| = +� +S +⟨0| ( ˆPcL)S L · · · ( ˆPc1)S 1aS L +L · · · aS 1 +1 . +(S12) +Combining the mappings in Eq. (S2) and Eq. (S10), we get the final mapping, i.e., Eq. (4) in the main text. The mapping +process is schematically shown in Fig. S1. +S2. Model in representation C: diagonalization, exceptional point and symmetry +In this section we map our Liouvillian in Eq. (5) in the main text into representation C and get its BdG form in momentum +space. Based on the BdG form, we show the exceptional point and symmetry of our Liouvillian. +A. The derivation of ˆL +Our Liouvillian L in Eq. (5) is mapped into ˆL by the mapping (4b) in the main text: +L(·) = −i[H, ·] + (1 − w)DL(·) + (1 + w)DR(·) → ˆL = ˆH + (1 − w) ˆDL + (1 + w) ˆDR. +(S13) +Note that our matrix representation of creation and annihilation operator is real, thus we have aT = a†, cT = c†, ˆPT = ˆP. Then +we get +− i[H, ·] → ˆH = −iH(a, a†) + iHT( ˆPc, c† ˆP) = −iJ +� +l +(a† +l+1al + a† +l al+1) + iJ +� +l +(c† +l cl+1 + c† +l+1cl). +(S14) + +9 +Note that our matrix representation of Al = √γ1a† +l + √γ2al+1 is real, thus we have AT +l = A† +l . Then we get +DL(·) → ˆDL = +� +l +� +2Al(a, a†)Al( ˆPc, c† ˆP) − A† +l (a, a†)Al(a, a†) − A† +l ( ˆPc, c† ˆP)Al( ˆPc, c† ˆP) +� += +� +l +� +2( √γ1a† +l + √γ2al+1)( √γ1c† +l ˆP + √γ2 ˆPcl+1) − ( √γ1al + √γ2a† +l+1)( √γ1a† +l + √γ2al+1) +− ( √γ1 ˆPcl + √γ2c† +l+1 ˆP)( √γ1c† +l ˆP + √γ2 ˆPcl+1) +� += +� +l +� +− 2 √γ1γ2 ˆP(a† +l cl+1 + c† +l al+1) + 2γ1 ˆPa† +l c† +l + 2γ2 ˆPcl+1al+1 − √γ1γ2(alal+1 + a† +l+1a† +l ) ++ √γ1γ2(clcl+1 + c† +l+1c† +l ) − γ2(a† +l+1al+1 + c† +l+1cl+1) − γ1(ala† +l + clc† +l ) +� +, +(S15) +DR(·) → ˆDR = +� +l +� +2A† +l (a, a†)A† +l ( ˆPc, c† ˆP) − Al(a, a†)A† +l (a, a†) − Al( ˆPc, c† ˆP)A† +l ( ˆPc, c† ˆP) +� += +� +l +� +2( √γ1al + √γ2a† +l+1)( √γ1 ˆPcl + √γ2c† +l+1 ˆP) − ( √γ1a† +l + √γ2al+1)( √γ1al + √γ2a† +l+1) +− ( √γ1c† +l ˆP + √γ2 ˆPcl+1)( √γ1 ˆPcl + √γ2c† +l+1 ˆP) +� += +� +l +� +− 2 √γ1γ2 ˆP(a† +l+1cl + c† +l+1al) + 2γ1 ˆPclal + 2γ2 ˆPa† +l+1c† +l+1 + √γ1γ2(alal+1 + a† +l+1a† +l ) +− √γ1γ2(clcl+1 + c† +l+1c† +l ) − γ2(al+1a† +l+1 + cl+1c† +l+1) − γ1(a† +l al + c† +l cl) +� +. +(S16) +Due to [ ˆP, ˆL] = 0, the state will keep its parity in the evolution governed by the Lindblad master equation. Therefore, ˆP can +reduce to a constant P, which equals 1 in even parity channel and −1 in odd parity channel. +By Fourier transformation +a† +l = +π +� +k=−π +e−ikla† +k, +al = +π +� +k=−π +eiklak, +c† +l = +π +� +k=−π +e−iklc† +k, +cl = +π +� +k=−π +eiklck, +(S17) +we get ˆL in BdG form as +ˆL = 1 +2 +ˆLk=0 + +π− +� +k=0+ +ˆLk, +(S18) +where +ˆLk = (a† +k c† +k a−k c−k) Lk (ak ck a† +−k c† +−k)T − 4γ, +(S19) +and +Lk = −i2J cos kσz ⊗ σz − 4 √γ1γ2 cos kPσz ⊗ σx − 2γPσy ⊗ σy + 2w +� ++(γ2 − γ1)σz ⊗ I + 2 √γ1γ2 sin kσy ⊗ σz ++i(γ2 − γ1)Pσx ⊗ σy + i2 √γ1γ2 sin kPI ⊗ σx +� +. +(S20) +B. Diagonalization of ˆLk +We make a similarity transformation for ˆLk by matrix W: +ˆLk = (a† +k c† +k a−k c−k) W W−1 Lk W W−1 (ak ck a† +−k c† +−k)T − 4γ += (ζ +′ +1(k) ζ +′ +2(k) ζ3(k) ζ4(k)) Λ (ζ1(k) ζ2(k) ζ +′ +3(k) ζ +′ +4(k))T − 4γ += λ1(k)ζ +′ +1(k)ζ1(k) + λ2(k)ζ +′ +2(k)ζ2(k) + λ3(k)ζ3(k)ζ +′ +3(k) + λ4(k)ζ4(k)ζ +′ +4(k) − 4γ, +(S21) +where +(a† +k c† +k a−k c−k) W = (ζ +′ +1(k) ζ +′ +2(k) ζ3(k) ζ4(k)), +W−1 (ak ck a† +−k c† +−k)T = (ζ1(k) ζ2(k) ζ +′ +3(k) ζ +′ +4(k))T +(S22) + +10 +and Λ is a diagonal matrix given by +Λ = W−1 Lk W = diag(λ1(k), λ2(k), λ3(k), λ4(k)). +(S23) +We write W and W−1 as +W = (⃗v1 ⃗v2 ⃗v3 ⃗v4), +W−1 = +�������������� +⃗u t +1 +⃗u t +2 +⃗u t +3 +⃗u t +4 +�������������� +, +(S24) +where the column vector ⃗vi and row vector ⃗u t +j satisfy ⃗u t +j · ⃗vi = δi j. Then we have +ζ +′ +1(k) = (a† +k c† +k a−k c−k) · ⃗v1, +ζ +′ +2(k) = (a† +k c† +k a−k c−k) · ⃗v2, +ζ +′ +3(k) = (ak ck a† +−k c† +−k) · ⃗u3, +ζ +′ +4(k) = (ak ck a† +−k c† +−k) · ⃗u4, +ζ1(k) = (ak ck a† +−k c† +−k) · ⃗u1, +ζ2(k) = (ak ck a† +−k c† +−k) · ⃗u2, +ζ3(k) = (a† +k c† +k a−k c−k) · ⃗v3, +ζ4(k) = (a† +k c† +k a−k c−k) · ⃗v4. +(S25) +ζ +′ +i(k) and ζj(k) hold anticommutation relations: +{ζ +′ +i(k), ζj(k)} = δi j, +{ζ +′ +i(k), ζ +′ +j(k)} = {ζi(k), ζj(k)} = 0 +(S26) +Calculating the eigenvalues of Eq. (S20), we get the same values for both even and odd parity: λ1(k) = −2γ − 2mk, λ2(k) = +−2γ + 2mk, λ3(k) = 2γ − 2mk and λ4(k) = 2γ + 2mk , where +mk = +����� +� +(4γ1γ2 − J2) cos2 k, +4γ1γ2 ≥ J2 +i +� +(J2 − 4γ1γ2) cos2 k, +4γ1γ2 < J2 +(S27) +Then Lk can be diagonalized as +ˆLk = λ−(k) +� +ζ +′ +1(k)ζl(k) + ζ +′ +4(k)ζ4(k) +� ++ λ+(k) +� +ζ +′ +2(k)ζ2(k) + ζ +′ +3(k)ζ3(k) +� +, +(S28) +where +λ±(k) = −2γ ± mk. +(S29) +C. Exceptional point +When J2 = 4γ1γ2, the exceptional point of Lk emerges. To see it clearly, we show real and imaginary part of the rapidity +λ±(k) in Fig. S2. When the flat band condition is satisfied (γ2 = 1), it occurs exceptional degeneracy between λ+ and λ−. +(a) +(b) +FIG. S2. The real (a) and imaginary (b) part of λ±(k) as a function with k and γ2. Other parameters are taken as J = 1 and γ1 = 0.25 + +2 +0 +-2 +0 +0 +0.5 +1 +2 +10 +-5 +0 +0 +0.5 +1 +2 +1Im(入±Re(入±)k(元)k(元)11 +D. The symmetry of Liouvillian +Due to ˆLk = ˆL−k, we can write ˆL in Eq. (S18) as ˆL = 1 +2 +�π +k=−π ˆLk. Therefore, we can study the symmetry of the Liouvillian +from Lk with k ∈ (−π, π). It is easy to check that Lk in Eq. (S20) has time-reversal symmetry (TRS), particle-hole symmetry +(PHS) and chiral symmetry (CS)[5–8]: +TRS : T+ L∗ +k T −1 ++ += L−k +=⇒ T+ = σz ⊗ σx; T+T ∗ ++ = 1 +PHS : C− LT +k C−1 +− = −L−k +=⇒ C− = σx ⊗ I; C−C∗ +− = 1 +CS : Γ L† +k Γ−1 = −Lk +=⇒ Γ = σy ⊗ σx; Γ2 = 1. +(S30) +Due to that Lk has full real spectrum in the region J2 < 4γ1γ2, the mathematical theorem ensures the Liouvillian having pseudo- +Hermiticity i.e. there exists a Hermitian matrix η that η L† +kη−1 = Lk. Especially, when w = 0, the system will additionally have +inversion symmetry (IS) and the pseudo-Hermiticity will be enhanced to the parity-time symmetry (PTS): +IS : P Lk P−1 = L−k +=⇒ P = σy ⊗ σy +PTS : PT L∗ +k PT −1 = Lk +=⇒ PT = σx ⊗ σz. +(S31) +S3. Exactly solution of the model when w = 0 +In this section, we exactly solve our model both in even and odd channels. We show steady state and all the excited states +of the open system. In addition, we prove that the odd parity states have no contribution on observations with even fermionic +operators. Last, we calculate the correlation functions of steady state and local-quantum-jump states beyond the steady state. +A. All the eigenstates of ˆL +First, we diagonalize ˆLk in even channel (P = 1). Then normal master modes are show in Eq. (S25). The vectors ⃗v and ⃗u can +be solved as +⃗v1 = 1 +2 +� +− 1 − iJ cos k/mk, −2 √γ1γ2 cos k/mk, −2 √γ1γ2 cos k/mk, 1 + iJ cos k/mk +�T +⃗v2 = 1 +2 +� +− 1 + iJ cos k/mk, 2 √γ1γ2 cos k/mk, 2 √γ1γ2 cos k/mk, 1 − iJ cos k/mk +�T +⃗v3 = 1 +2 +� +1, −iJ + mk/ cos k +2 √γ1γ2 +, iJ − mk/ cos k +2 √γ1γ2 +, 1 +�T +⃗v4 = 1 +2 +� +1, −iJ − mk/ cos k +2 √γ1γ2 +, iJ + mk/ cos k +2 √γ1γ2 +, 1 +�T +⃗u1 = 1 +2 +� +− 1, iJ − mk/ cos k +2 √γ1γ2 +, iJ − mk/ cos k +2 √γ1γ2 +, 1 +�T +⃗u2 = 1 +2 +� +− 1, iJ + mk/ cos k +2 √γ1γ2 +, iJ + mk/ cos k +2 √γ1γ2 +, 1 +�T +⃗u3 = 1 +2 +� +1 + iJ cos k/mk, 2 √γ1γ2 cos k/mk, −2 √γ1γ2 cos k/mk, 1 + iJ cos k/mk +�T +⃗u4 = 1 +2 +� +1 − iJ cos k/mk, −2 √γ1γ2 cos k/mk, 2 √γ1γ2 cos k/mk, 1 − iJ cos k/mk +�T. +(S32) +We make an ansatz for steady state |Ω⟩ as +|Ω⟩ = +π +� +k=0 +(z1 + z2a† +kc† +−k)(z3 + z4a† +−kc† +k)|0⟩. +(S33) +Solving the steady state equations: ζi|Ω⟩ = 0 for i = 1 ∼ 4, we get z1 = z2 and z3 = z4. Therefore, the solution of steady state +(the Eq. (16) in the main text) is given by +|Ω⟩ = 1 +N +π +� +k=−π +(1 + a† +kc† +−k)|0⟩. +(S34) + +12 +By using Tr(ρs) = 1, we get the normalization factor N as +N =C ⟨S0| +π +� +k=−π +(1 + a† +kc† +−k)|0⟩ = 2L, +(S35) +where L is the length of the chain. The details of N = 2L is given in subsection D. In addition, we get steady state in real space +given by +|Ω⟩ = 1 +N exp +� +π +� +k=−π +a† +kc† +−k +� +|0⟩ = 1 +N exp +� +L +� +l=1 +a† +l c† +l +� +|0⟩ = 1 +N +L +� +l=1 +(1 + a† +l c† +l )|0⟩. +(S36) +Under the parity constraint, valid eigenstates in even parity channel are |Ω⟩, ζ +′ +α1(ki)ζ +′ +α2(kj)|Ω⟩, ζ +′ +α1(ki)ζ +′ +α2(kj)ζ +′ +α3(km)ζ +′ +α4(kn)|Ω⟩, · · · +Secondly, we diagonalize ˆLk in the odd channel (P = −1). The process of diagonalization is the same as it in the even channel, +however, the eigenvectors ⃗v and ⃗u of odd channel are different from them in even channel. We mark the eigenvectors and normal +master modes of the odd channel with ’∗’: +ζ +′ +1∗(k) = (a† +k c† +k a−k c−k) · ⃗v1∗, +ζ +′ +2∗(k) = (a† +k c† +k a−k c−k) · ⃗v2∗, +ζ +′ +3∗(k) = (ak ck a† +−k c† +−k) · ⃗u3∗, +ζ +′ +4∗(k) = (ak ck a† +−k c† +−k) · ⃗u4∗, +ζ1∗(k) = (ak ck a† +−k c† +−k) · ⃗u1∗, +ζ2∗(k) = (ak ck a† +−k c† +−k) · ⃗u2∗, +ζ3∗(k) = (a† +k c† +k a−k c−k) · ⃗v3∗, +ζ4∗(k) = (a† +k c† +k a−k c−k) · ⃗v4∗, +(S37) +where +⃗v1∗ = 1 +2 +� +1 + iJ cos k/mk, −2 √γ1γ2 cos k/mk, 2 √γ1γ2 cos k/mk, 1 + iJ cos k/mk +�T +⃗v2∗ = 1 +2 +� +1 − iJ cos k/mk, 2 √γ1γ2 cos k/mk, −2 √γ1γ2 cos k/mk, 1 − iJ cos k/mk +�T +⃗v3∗ = 1 +2 +� +− 1, −iJ + mk/ cos k +2 √γ1γ2 +, −iJ + mk/ cos k +2 √γ1γ2 +, 1 +�T +⃗v4∗ = 1 +2 +� +− 1, −iJ − mk/ cos k +2 √γ1γ2 +, −iJ − mk/ cos k +2 √γ1γ2 +, 1 +�T +⃗u1∗ = 1 +2 +� +1, iJ − mk/ cos k +2 √γ1γ2 +, −iJ + mk/ cos k +2 √γ1γ2 +, 1 +�T +⃗u2∗ = 1 +2 +� +1, iJ + mk/ cos k +2 √γ1γ2 +, −iJ − mk/ cos k +2 √γ1γ2 +, 1 +�T +⃗u3∗ = 1 +2 +� +− 1 − iJ cos k/mk, 2 √γ1γ2 cos k/mk, 2 √γ1γ2 cos k/mk, 1 + iJ cos k/mk +�T +⃗u4∗ = 1 +2 +� +− 1 + iJ cos k/mk, −2 √γ1γ2 cos k/mk, −2 √γ1γ2 cos k/mk, 1 − iJ cos k/mk +�T. +(S38) +Solving the equation, ζi∗|Ω∗⟩ = 0 for i = 1 ∼ 4, we get +|Ω∗⟩ = 1 +N +π +� +k=−π +(1 − a† +kc† +−k)|0⟩ = 1 +N +L +� +l=1 +(1 − a† +l c† +l )|0⟩. +(S39) +Note that |Ω∗⟩ is even parity ( ˆP|Ω∗⟩ = +1|Ω∗⟩). Therefore, the valid eigenstates in odd parity channel are the states with odd +numbers of excitations on the |Ω∗⟩, i.e. ζ +′ +α1∗(ki)|Ω∗⟩, ζ +′ +α1∗(ki)ζ +′ +α2∗(k j)ζ +′ +α3∗(km)|Ω∗⟩, · · · +In summary, the full eigenstates of ˆL are +Steady state: +|Ω⟩ +Single excitation: +ζ +′ +α1∗(ki) |Ω∗⟩ +Double excitation: +ζ +′ +α1(ki)ζ +′ +α2(k j) |Ω⟩ +Triple excitation: +ζ +′ +α1∗(ki)ζ +′ +α2∗(kj)ζ +′ +α3∗(km) |Ω∗⟩ +Quadruple excitation: +ζ +′ +α1(ki)ζ +′ +α2(kj)ζ +′ +α3(km)ζ +′ +α4(kn) |Ω⟩ +· · · +(S40) + +13 +B. Flat band condition +When the condition J2 = 4γ1γ2 is satisfied, Liouvillian flat band occurs. We have λ1 = λ2 = −2γ, λ3 = λ4 = 2γ and mk = 0, +which leads to divergence of eigenvectors ⃗v1, ⃗v2, ⃗u3, ⃗u4, ⃗v1∗, ⃗v2∗, ⃗u3∗ and ⃗u4∗. This indicates the exceptional point of ˆL. However, +we can eliminate divergence by summing of these eigenvectors. Setting J = 2 √γ1γ2, we can get the normal master modes in +even parity +ζ +′ +A(k) = (a† +k c† +k a−k c−k) · (⃗v1 + ⃗v2) = −a† +k + c−k +ζA(k) = (ak ck a† +−k c† +−k) · (⃗u1 + ⃗u2)/2 = 1 +2(−ak + ick + ia† +−k + c† +−k) +ζ +′ +B(k) = ak ck a† +−k c† +−k) · (⃗u3 + ⃗u4) = ak + c† +−k +ζB(k) = (a† +k c† +k a−k c−k) · (⃗v3 + ⃗v4)/2 = 1 +2(a† +k − ic† +k + ia−k + c−k), +(S41) +and in odd parity +ζ +′ +A∗(k) = (a† +k c† +k a−k c−k) · (⃗v1∗ + ⃗v2∗) = a† +k + c−k +ζA∗(k) = (ak ck a† +−k c† +−k) · (⃗u1∗ + ⃗u2∗)/2 = 1 +2(ak + ick − ia† +−k + c† +−k) +ζ +′ +B∗(k) = ak ck a† +−k c† +−k) · (⃗u3∗ + ⃗u4∗) = −ak + c† +−k +ζB∗(k) = (a† +k c† +k a−k c−k) · (⃗v3∗ + ⃗v4∗)/2 = 1 +2(−a† +k − ic† +k − ia−k + c−k). +(S42) +C. Ineffectiveness of odd parity +Given an arbitrary state | ρ⟩, it can be decomposed into even and odd eigenstate of ˆL: +| ρ⟩ = +� � +i +Ce +i |i⟩e +� ++ +� � +j +Co +j | j⟩o +� +, +(S43) +where |i⟩e and | j⟩o represents even and odd parity state in Eq. (S40). The expectation value of observation ˆO is +C⟨S0| ˆO| ρ⟩ = +� � +i +Ce +i C⟨S0| ˆO|i⟩e +� ++ +� � +j +Co +j C⟨S0| ˆO| j⟩o +� +. +(S44) +When ˆO has even fermionic operators, we have C⟨S0| ˆO| j⟩o = 0. When ˆO has odd fermionic operators, we have C⟨S0| ˆO|i⟩e = 0. +Usually, in pure fermionic system, fermionic operators appear in pairs, so the odd parity part of ˆL does not influence the +expectation value of observation. +D. Correlation functions of steady state and quantum jump states +Firstly, we show the details for the calculation of normalization factor N: +N =C ⟨S0| +L +� +l=1 +(1 + a† +l c† +l )|0⟩ += +� +S +⟨0|( ˆPcL)S L · · · ( ˆPc1)S 1aS L +L · · · aS 1 +1 (1 + a† +1c† +1) · · · (1 + a† +Lc† +L)|0⟩ += +� +S +⟨0|( ˆPcLaL)S L · · · ( ˆPc1a1)S 1 (1 + a† +1c† +1) · · · (1 + a† +Lc† +L)|0⟩ += ⟨0|(1 + ˆPcLaL) · · · (1 + ˆPc1a1) (1 + a† +1c† +1) · · · (1 + a† +Lc† +L)|0⟩ += ⟨0| +L +� +l=1 +� +(1 + ˆPclal)(1 + a† +l c† +l ) +� +|0⟩ += 2L. +(S45) + +14 +Secondly, we show the particle number distribution of the steady state ns +j +ns +j =C ⟨S0|a† +jaj|Ω⟩ += 1 +N +� +S +⟨0|( ˆPcL)S L · · · ( ˆPc1)S 1aS L +L · · · aS 1 +1 a† +ja j (1 + a† +1c† +1) · · · (1 + a† +Lc† +L)|0⟩ += 2L−1 +N ⟨0|(1 + ˆPc jaj)a† +jaj(1 + a† +jc† +j)|0⟩ += 1 +2 +(S46) +The other correlation functions of steady state can be calculated by the same method. The results are +Gs +j1,j2 = 0 (j1 � j2), Ds +j1,j2 = 0, Ds∗ +j1, j2 = 0. +(S47) +Thirdly, we focus on a state from a quantum jump on the site l of the steady state. We denote this state as |φl⟩: +|φl⟩ := +a† +l ρsal +Tr(a† +l ρsal) += +a† +l c† +l |Ω⟩ +C⟨S0|a† +l c† +l |Ω⟩ +. +(S48) +The particle number on site j of |φl⟩, denoted as nl +j: +nl +j=l =C ⟨S0|a† +l al|φl⟩ += +⟨0|(1 + ˆPclal) a† +l al a† +l c† +l (1 + a† +l c† +l )|0⟩ +⟨0|(1 + ˆPclal)a† +l c† +l (1 + a† +l c† +l )|0⟩ += 1. +(S49) +nl +j�l =C ⟨S0|a† +jaj|φl⟩ += +⟨0|(1 + ˆPclal)a† +l c† +l (1 + a† +l c† +l ) (1 + ˆPcjaj)a† +jc† +j(1 + a† +jc† +j)|0⟩ +⟨0|(1 + ˆPclal)a† +l c† +l (1 + a† +l c† +l ) (1 + ˆPcja j)(1 + a† +jc† +j)|0⟩ += 1 +2. +(S50) +By the same way, we get the other correlation functions of |φl⟩. The results are +Gl +j1,j2 = 0 (j1 � j2), Dl +j1,j2 = 0, Dl∗ +j1, j2 = 0. +(S51) +S4. Evolution equations of correlation functions +In this section, we derive the evolution equations of two-operator correlation functions both in real space and momentum +space and show the symmetry of damping matrix in momentum space. +A. Evolution equations of correlation functions in real space +The evolution equation of the expectation value of operator ˆO in the open system is +d +dtTr( ˆOρ(t)) = Tr( ˆO d +dtρ) = Tr( ˆOL(ρ)). +(S52) +By considering the Liouvillian L in Eq. (5) in the main text, the equation becomes +d +dtTr( ˆOρ(t)) = −iTr( ˆO[H, ρ]) + (1 − w)Tr( ˆODL(ρ)) + (1 + w)Tr( ˆODR(ρ)). +(S53) + +15 +Using the relation Tr(ABC) = Tr(CAB), we have +Tr( ˆO[H, ρ]) = Tr([ ˆO, H] ρ) = J +� +l +Tr([ ˆO, a† +l+1al + a† +l al+1] ρ), +(S54) +Tr( ˆODL(ρ)) = +� +l +� +Tr( ˆO2Al ρ A† +l ) − Tr( ˆOA† +l Al ρ) − Tr( ˆO ρ A† +l Al) +� += +� +l +� +Tr([A† +l , ˆO]Al ρ) + Tr(A† +l [ ˆO, Al] ρ) +� +, +(S55) +Tr( ˆODR(ρ)) = +� +l +� +Tr( ˆO2A† +l ρ Al) − Tr( ˆOAlA† +l ρ) − Tr( ˆO ρ AlA† +l ) +� += +� +l +� +Tr([Al, ˆO]A† +l ρ) + Tr(Al[ ˆO, A† +l ] ρ) +� +. +(S56) +Substituting ˆO = a† +l1al2, ˆO = al1al2 and ˆO = a† +l2a† +l1 into Eq.(S52) ∼ Eq.(S55), we get the evolution equations of Gl1,l2, Dl1,l2 and +D∗ +l1,l2, respectively. Namely, the evolution equations of correlation functions in real space are +d +dtGl1,l2 = − 4γGl1,l2 + iJ(Gl1−1,l2 + Gl1+1,l2 − Gl1,l2−1 − Gl1,l2+1) + 2�γ + w(γ2 − γ1)� δl1,l2 ++ √γ1γ2 (−Dl1−1,l2 − Dl1+1,l2 + Dl2,l1−1 + Dl2,l1+1) + √γ1γ2 (D∗ +l1,l2−1 + D∗ +l1,l2+1 − D∗ +l2−1,l1 − D∗ +l2+1,l1), +(S57) +d +dt Dl1,l2 = + 2 √γ1γ2(−Gl1−1,l2 − Gl1+1,l2 + Gl2−1,l1 + Gl2+1,l1) + 2w √γ1γ2 +�δl1,l2−1 − δl2,l1−1 +� +− 4γDl1,l2 − iJ/2 (Dl1−1,l2 + Dl1+1,l2 + Dl1,l2−1 + Dl1,l2+1) + iJ/2 (Dl2,l1−1 + Dl2,l1+1 + Dl2−1,l1 + Dl2+1,l1), +(S58) +d +dt D∗ +l1,l2 = + 2 √γ1γ2(−Gl2,l1−1 − Gl2,l1+1 + Gl1,l2−1 + Gl1,l2+1) + 2w √γ1γ2 +�δl1,l2−1 − δl2,l1−1 +� +− 4γD∗ +l1,l2 + iJ/2 (D∗ +l1−1,l2 + D∗ +l1+1,l2 + D∗ +l1,l2−1 + D∗ +l1,l2+1) − iJ/2 (D∗ +l2,l1−1 + D∗ +l2,l1+1 + D∗ +l2−1,l1 + D∗ +l2+1,l1). +(S59) +B. Evolution equations of correlation functions in momentum space +The Liouvillian of our model in momentum space is shown in Eq. (9) in the main text. Substituting this equation into Eq. (S52), +we have +d +dtTr( ˆOρ(t)) = +π +� +k=−π +� +− i2J cos kTr( ˆO[ˆnk, ρ]) + (1 − w)Tr( ˆODL +k(ρ)) + (1 + w)DR +k (ρ) +� += +π +� +k=−π +� +− i2J cos kTr([ ˆO, ˆnk]ρ) + (1 − w)� Tr([B† +k, ˆO]Bkρ) + Tr(B† +k[ ˆO, Bk]ρ) � ++ (1 + w)� Tr([Bk, ˆO]B† +kρ) + Tr(Bk[ ˆO, B† +k]ρ) �� +. +(S60) +Substituting ˆO = a† +k1ak2, ˆO = a† +−k2a−k1, ˆO = ak2a−k1 and ˆO = a† +−k2a† +k1 into Eq.(S60), we get the evolution equations of correlation +functions Gk1,k2, G−k2,−k1, Dk2,−k1 and D∗ +k1,−k2: +d +dt +��������������� +Gk1,k2 +G−k2,−k1 +Dk2,−k1 +D∗ +k1,−k2 +��������������� += Xk1k2 +��������������� +Gk1,k2 +G−k2,−k1 +Dk2,−k1 +D∗ +k1,−k2 +��������������� ++ Vk1k2, +(S61) +where +Xk1k2 = +�������������� +−4γ + i2J(cos k1 − cos k2) +0 +4 √γ1γ2 cos k1 +4 √γ1γ2 cos k2 +0 +−4γ + i2J(cos k2 − cos k1) +−4 √γ1γ2 cos k2 +−4 √γ1γ2 cos k1 +4 √γ1γ2 cos k1 +−4 √γ1γ2 cos k2 +−4γ − i2J(cos k1 + cos k2) +0 +4 √γ1γ2 cos k2 +−4 √γ1γ2 cos k1 +0 +−4γ + i2J(cos k1 + cos k2) +�������������� +(S62) + +16 +and +Vk1k2 = δk1,k2 +�������������� +2γ + 2w(γ2 − γ1) +2γ + 2w(γ2 − γ1) +i4w √γ1γ2 sin k1 +−i4w √γ1γ2 sin k1 +�������������� +. +(S63) +Eq. (S61) ∼ Eq. (S63) are just the same equations as Eq. (10) and Eq. (11) in the main text. +C. Symmetry of damping matrix +Denoting k = (k1, k2), then we can check the damping matrix Xk has TRS, PHS, CS, IS and PTS: +TRS : UT X∗ +k U−1 +T = X−k +=⇒ +UT = σz ⊗ σx; UT U∗ +T = 1 +PHS : UC XT +k U−1 +C = −X−k +=⇒ UC = I ⊗ σx; UCU∗ +C = 1 +CS : UΓ X† +k U−1 +Γ = −Xk +=⇒ +UΓ = σz ⊗ I; U2 +Γ = 1 +IS : UP Xk U−1 +P = X−k +=⇒ +UP = I ⊗ I +PTS : UPT X∗ +k U−1 +PT = Xk +=⇒ UPT = σz ⊗ σx. +(S64) +Compared with the symmetry of the Liouvillian in Eq.(S30), Xk has higher symmetry. +S5. Particle number distribution of steady state +As for steady state, the Eq. (S61) equals to 0, so we can get the correlation functions of steady state by +� +Gs +k1,k2 Gs +−k2,−k1 Ds +k2,−k1 Ds∗ +k1,−k2 +�T = −X−1 +k1k2Vk1k2 +(S65) +When k1 = k2 = k, we get particle number distribution of steady state in momentum space ns +k: +ns +k = Gs +kk = (1 − w)γ1 + (1 + w)γ2 +2γ +− +2Jwγ1γ2 cos2 k sin k +γ3 + γ(J2 − 4γ1γ2) cos2 k. +(S66) +Due to the translation invariance of our system, the particle number distributes uniformly on each site. Therefore, particle +number on site l in the thermodynamic limit can be calculated by +ns +l = 1 +L +L +� +j=1 +ns +j = 1 +L +� +k +ns +k = 1 +2π +� π +k=−π +dk ns +k = 1 +2 + w(γ2 − γ1) +2γ +. +(S67) +S6. The relationship between the damping-matrix spectra and the Liouvillian spectra +In this section, we demonstrate that for a real physical process with closed evolution equations of correlation functions the +damping-matrix spectra are the subset of the Liouvillian spectra. +The general form of closed evolution equations of correlation functions is +d +dtΨ = X Ψ + V, +(S68) +where +X +is +the +damping +matrix, +Ψ +is +the +vector +of +correlation +functions, +for +example, +Ψ +is +taken +as +(Gk1,k2,G−k2,−k1, Dk2,−k1, D∗ +k1,−k2)T in our model. The vector V induces the correlation function vector of steady state ΨS as +ΨS = −X−1V. By deducting ΨS , we have +d +dt(Ψ(t) − ΨS ) = X (Ψ(t) − ΨS ). +(S69) + +17 +If the correlation function vector ΨΓ is governed by the eigen equation of damping matrix, we have +X (ΨΓ(t) − ΨS ) = Γ (ΨΓ(t) − ΨS ), +(S70) +where Γ is the eigenvalue of X. The equation in the initial time is +X (ΨΓ(0) − ΨS ) = Γ (ΨΓ(0) − ΨS ), +(S71) +Then from Eq. (S69), we obtain +ΨΓ(t) − ΨS = eΓt (ΨΓ(0) − ΨS ). +(S72) +If ΨΓ is in a real physical process, we will have +ΨΓ(t) = C⟨S0| ˆΨ e ˆLC t| ρ(0)⟩C, +(S73a) +ΨS = C⟨S0| ˆΨ e ˆLC t|Ω⟩C = C⟨S0| ˆΨ |Ω⟩C, +(S73b) +where ˆLC is the Liouvillian of system in representation C, | ρ(0)⟩C is the initial state of system and |Ω⟩C is the steady state +of system. +ˆΨ is the vector of operators in terms of correlation function vector Ψ, for example, in our model ˆΨ equals to +(a† +k1ak2, a† +−k2a−k1, a−k1ak2, a† +−k2a† +k1)T. Substituting Eq.(S73) into Eq.(S72), we obtain +C⟨S0| ˆΨ e ˆLC t� | ρ(0)⟩C − |Ω⟩C +� = C⟨S0| ˆΨ eΓ t� | ρ(0)⟩C − |Ω⟩C +�. +(S74) +Comparing the two sides of the above equation, we have +e ˆLC t� | ρ(0)⟩C − |Ω⟩C +� = eΓ t� | ρ(0)⟩C − |Ω⟩C +�, +(S75) +and thus the eigenvalue Γ of damping matrix X is also the eigenvalue of Liouvillian ˆLC. +∗ schen@iphy.ac.cn +[1] M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Applications 10, 285 (1975). +[2] A. 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Chen, Topological classification of defects in non-Hermitian systems, Phys. Rev. B 100, 144106 (2019). + diff --git a/TdE5T4oBgHgl3EQfAQ7r/content/tmp_files/load_file.txt b/TdE5T4oBgHgl3EQfAQ7r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e1dcd7dfe1abbba565d41137132c630075e35878 --- /dev/null +++ b/TdE5T4oBgHgl3EQfAQ7r/content/tmp_files/load_file.txt @@ -0,0 +1,889 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf,len=888 +page_content='Dynamical Signatures of Liouvillian Flat Band Yu-Guo Liu1 and Shu Chen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' ∗ 1Beijing National Laboratory for Condensed Matter Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' China 2School of Physical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' China 3Yangtze River Delta Physics Research Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Liyang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Jiangsu 213300,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' China (Dated: January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 2023) Although flat-band structures have attracted intensive studies in condensed matter and optical physics due to their eigenstates exhibiting huge degeneracy and allowing for the localization of wave packet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' it is not clear how the flat band of Liouvillian influences the relaxation dynamics of open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' To this end, we study the dynamical signatures of Liouvillian flat band in the scheme of Lindblad master equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Considering a chain model with gain and loss, we demonstrate three kinds of band dispersion of Liouvillian: flat bland, dispersionless only in the real part and imaginary part, and capture their dynamical signatures: when the rapidity spectrum of Liouvillian is flat, the particle numbers in different sites relax to its steady state value with the same decay rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' when the real or imaginary part of rapidity spectrum is dispersionless, the relaxation behaviors have oscillating or forked characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' We also unveil that the Liouvillian flat band can lead to dynamical localization, which is characterized by the halt of propagation of a local perturbation on the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='— Band structure of a Hamiltonian plays an important role in understanding the motion of particles in pe- riodic crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Usually, special band structures may give rise to exotic quantum phenomena, for example, low-energy ex- citations of electrons on a linear dispersive band in graphene behave like massless Dirac fermions [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Another instance is the flat band (FB) in which all electrons carry the same en- ergy regardless of their momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Due to the dispersionless band structure, particles in FB have arbitrarily large effective mass, so they will be localized in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Especially, in strongly correlated systems, heavy degeneracy and zero ki- netic energy in FB can increase density of electronic states and highlight Coulomb interaction, leading to rich many-body phenomena [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' In open quantum systems, dynamics of density matrix ρ is described by Lindblad master equation (LME) under Born- Markov approximation [6–8]: dρ dt = L(ρ) := −i[H, ρ] + � µ � LµρL† µ − 1 2{L† µLµ, ρ} � , (1) where L is called the Liouvillian superoperator, H is the Hamiltonian of system, and Lµ are Lindblad operators which reflect the coupling between system and environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The Planck constant ℏ is set to unity throughout this Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' There have been several methods developed to obtain the spectrum of L, especially for quadratic systems [9–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' [15], a route for realizing dispersionless bands is proposed based on the underlying mechanism with the emergence of a dis- sipationless dark space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Generally speaking, the short-time dynamics is related to the Liouvillian eigenvalues with large modulus of the real part, whereas the long-time relaxation to the smallest modulus beyond zero (the so called Liouvillian gap) [16–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' However, how the structure of Liouvillian, es- pecially the Liouvillian flat band (LFB), influences dynamics is still a subtle and unexplored question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' In this Letter, we focus on the dynamics of open quan- tum systems with LFB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' In comparison with the real spectrum of Hamiltonian system, the Liouvillian spectrum is complex, and thus the corresponding rapidity spectrum can exhibit more rich structures with dispersionless band in both imaginary and real part or either of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' To make our study concrete, we shall first apply a geometrically intuitive method to construct lattice with correlated gain and loss, which supports LFB, and explore the generality of dynamical signatures associated with the structure of Liouvillian spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' We show that the ra- pidity spectra from Liouvillian and damping-matrix spectra of correlation functions have the same dispersion characteris- tics, which lead to different signatures of damping dynamics of local particle number distribution: oscillating, forked, syn- chronous damping are related to the band dispersionless only in imaginary part, real part and in both parts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Fur- thermore, we exactly solve the model and show that the LFB can induce dynamical localization, which is characterized by the halt of the propagation of a local perturbation on the non- equilibrium steady state (NESS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='— The density matrix ρ and Liouvillian super- operator L in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (1) can be formally expressed as ρ = � IJ ρIJ|I⟩a⟨J|a, L(ρ) = � i j Fi(a, a†) ρ Fj(a, a†), (2) where a is the set of fermionic annihilation operators i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' a = (a1, a2, · · · ), Fi(a, a†) is a function with variables among a and a†, I = (I1, I2, · · · ), J = (J1, J2, · · · ) and |I⟩a⟨J|a = (a† 1)I1(a† 2)I2 · · · (a† L)IL|0⟩a⟨0|a(aL)JL · · · (a1)J1, (3) where |0⟩a is the vacuum state for all a−fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' For the convenience of analysis and calculation, we map fermionic LME into a new representation referred to as C by following the method in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' [10]: ρ → | ρ⟩C = � IJ ρIJ(a† 1)I1 · · · (a† L)IL(c† 1 ˆP)J1 · · · (c† L ˆP)JL |0⟩, (4a) L → ˆLC = � i j Fi(a, a†) F T j ( ˆPc, c† ˆP), (4b) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='05378v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='other] 13 Jan 2023 2 where c = (c1, c2, · · · ) is the set of annihilation operators of c−fermions, which is a one-to-one mapping from a, T means matrix transpose, and |0⟩ is the vacuum state of both a− and c−fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' ˆP is the parity operator defined by ˆP = exp � iπ � j(a† jaj + c† jcj) � , which is introduced to ensure fermionic anticommutation relations between a−fermions and c−fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Full mapping process is shown in the Supple- mental Material [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='— We consider a Liouvillian in a periodic chain: L(ρ) = −i[H, ρ] + (1 − w)DL(ρ) + (1 + w)DR(ρ), (5) where H = � l J(a† l+1al + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='), w ∈ [−1, 1], and DL(ρ) = � l � 2AlρA† l − A† l Alρ − ρA† l Al � , DR(ρ) = � l � 2A† l ρAl − AlA† l ρ − ρAlA† l � , (6) where Al = √γ1a† l + √γ2al+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The operators Al and A† l tie the gain and loss of neighboring sites together, which could be re- alized by optical superlattice with Bose-Einstein condensate reservoir [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The role of w ∈ [−1, 1] is analogous to the sta- tistical distribution from temperature [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Mapping Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (5) into the representation C, we get a ladder model consisting of a−fermion chain and c−fermion chain (see the Supplemental Material [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The L is mapped to ˆL = ˆH + (1 − w) ˆDL + (1 + w) ˆDR, where ˆH = � l � −iJ(a† l+1al + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=') + iJ(c† l+1cl + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=') � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' ˆDL and ˆDR are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 1 (a) and (b), which have leftward and rightward hoppings, respectively, along two di- agonals of every plaquette in the ladder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The cross-stitch-type hopping is crucial for generating FB because it can form a destructive-interference structure, which consists with our ex- perience in the FB ladder models [34–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' In momentum space, ˆL can be expressed in BdG form as ˆL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5 ˆLk=0 + �π− k=0+ ˆLk, where ˆLk = (a† k c† k a−k c−k) Lk (ak ck a† −k c† −k)T − 4γ and γ = γ1 + γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Due to parity conservation in ˆL, the operator ˆP can be substi- tuted by a constant P which equals 1(−1) when ˆL acts on the state with even (odd) fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Then we have Lk = −i2J cos kσz ⊗ σz − 4 √γ1γ2 cos kPσz ⊗ σx − 2γPσy ⊗ σy + 2w � (γ2 − γ1)σz ⊗ I + 2 √γ1γ2 sin kσy ⊗ σz +i(γ2 − γ1)Pσx ⊗ σy + i2 √γ1γ2 sin kPI ⊗ σx � , (7) where I and σi are identity and Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' ˆLk can be diagonalized as ˆLk = λ−(k) � ζ ′ 1(k)ζ1(k) + ζ ′ 4(k)ζ4(k) � + λ+(k) � ζ ′ 2(k)ζ2(k) + ζ ′ 3(k)ζ3(k) � , where ζ ′ i(k) and ζj(k ′) ful- fill anticommutation relations: {ζ ′ i(k), ζj(k ′)} = δi jδkk′ and {ζ ′ i(k), ζ ′ j(k ′)} = {ζi(k), ζj(k ′)} = 0 [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The λ±(k) is called rapidity spectrum given by λ±(k) = −2γ ± 2mk for both odd and even parity [38], where mk = ����� � (4γ1γ2 − J2) cos2 k, J2 ≤ 4γ1γ2, i � (J2 − 4γ1γ2) cos2 k, J2 > 4γ1γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (8) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' ˆDL, ˆDR and ˆL = ˆH + ˆDL + ˆDR are sketched by (a), (b) and (c), where the color ovals, straight lines (with or without arrow) and wavy lines represent onsite loss, particle hopping and pair production and annihilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The bule, red and orange ovals are corresponding to terms (γ1 − γ2)ˆna/c, l − γ1, (γ2 − γ1)ˆna/c, l − γ2 and constant loss −γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' ˆna/c, l is the particle number operator of a− or c−fermion on the site l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Horizontal black wavy lines represent ± √γ1γ2 ˆP(alal+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=') or ± √γ1γ2 ˆP(clcl+1 +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The black arrows indicate directional hoppings with strength −2 √γ1γ2 ˆP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The bule, red and orange vertical wavy lines are corresponding to 2γ1 ˆPa† l c† l + 2γ2 ˆPclal, 2γ2 ˆPa† l c† l + 2γ1 ˆPclal and 2γ ˆP(a† l c† l + clal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (d) shows the (c) in even parity and under flat band condition, where J = 2 √γ1γ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The dashed wavy lines indicate the pairing terms have no effect on single particle- or hole- excitation on its steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The λ±(k) is independent with w and we show it in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' When J2 = 4 √γ1γ2, λ is a FB of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' When J2 < 4 √γ1γ2 (J2 > 4 √γ1γ2), λ is dispersionless in its imaginary (real) part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Especially, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 2 (c) and (f) the spectrum is pure real, which indicates Lk possessing a pseudo-Hermiticity [39–41], while in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 2 (a) and (d) the complex spectrum shows the break- ing of pseudo-Hermiticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Since the Liouvillian spectrum is obtained by sum of different number of λ±(k), it inherits the characteristics of rapidity spectrum, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 2 (g)∼(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' When J2 = 4 √γ1γ2, Liouvillian spectrum consists of some highly degenerate discrete points (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 2 (h)), corresponding to different occupations of the FB of rapidity spectrum, so we call this kind of Liouvillian spectrum as the LFB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Two-operator correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='— By making Fourier transform, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (5) becomes L(ρ) = π � k=−π � −i2J cos k[ˆnk, ρ]+(1−w)DL k(ρ)+(1+w)DR k (ρ) � , (9) where DL k(ρ) = 2BkρB† k − {B† kBk, ρ}, DR k (ρ) = 2B† kρBk − {BkB† k, ρ} and Bk = √γ1eika† k + √γ2a−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' We define two- operator correlation functions: Gk1, k2 = Tr(a† k1ak2ρ), Dk1, k2 = Tr(ak1ak2ρ), and D∗ k1, k2 = Tr(a† k2a† k1ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' In terms of the correla- tion function vector Ψk1k2 = (Gk1,k2,G−k2,−k1, Dk2,−k1, D∗ k1,−k2)T, the dynamical evolution is governed by the following closed (a) (b) Y102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Y12P a a a C C C h121 (c) (d) iJ 2J 2 2 a a a a a a c c C c C iJ iJ 2 23 (b) (a) (d) (e) (c) (f) (g) (h) (i) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (a)∼(c) the real part of rapidity spectra λ±(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (d)∼(f) the imaginary part of λ±(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (g)∼(i) the Liouvillian spectra obtained by exactly diagonalizing 6-site lattice with w = 0, J = 1 and γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='25 for all subfigures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5 in (a), (d) and (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' γ2 = 1 in (b), (e) and (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' γ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5 in (c), (f) and (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' equation: d dtΨk1k2 = Xk1k2Ψk1k2 + Vk1k2, (10) where Xk1k2 = −4γI ⊗ I + i2J cos k1σz ⊗ σz − i2J cos k2I ⊗ σz + 4 √γ1γ2 cos k1σx ⊗ σz − 4 √γ1γ2 cos k2σy ⊗ σy (11) and Vk1k2 = δk1,k2 � 2γ + 2w(γ2 − γ1), 2γ + 2w(γ2 − γ1), i4w √γ1γ2 sin k1, −i4w √γ1γ2 sin k1 �T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The damping matrix Xk1k2 has four eigenstates which fulfill the equation Xk1k2|Γ±± k1k2⟩ = Γ±± k1k2|Γ±± k1k2⟩ with the eigenvalues given by Γ±± k1k2 = −4γ ± 2 � 4γ1γ2 − J2 � (| cos k1| ± | cos k2|sgn(4γ1γ2 − J2))2, where sgn(x) is a sign function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Γ also has a transition from the complex to the real by decreasing J due to the PT −symmetry of Xk1k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' In the Supplemental Material [25] we show that Xk1k2 has higher symmetry than ˆLk, which makes Xk1k2 have a similar band structure as ˆLk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 3, we see that Γ±± k1k2 fully inherits the dispersion characteristics of real and imaginary part from the rapidity spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (b) (a) (d) (e) (c) (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (a)∼(c) the real part of Γ±± k1k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (d)∼(f) the imaginary part of Γ±± k1k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' J = 1 and γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='25 are for all subfigures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5 in (a) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' γ2 = 1 in (b) and (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' γ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5 in (c) and (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Flat-band damping dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='— Damping dynamics dis- plays the converging processes from initial state to NESS [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Here, we show that the “flat band” in real or imaginary or both parts will effectively influence the damping behav- iors in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' We concentrate on the vector Ψl1l2 = (Gl1,l2,Gl2,l1, Dl2,l1, D∗ l1,l2)T consisting of real-space correlation functions: Gl1, l2 = Tr(a† l1al2ρ), Dl1, l2 = Tr(al1al2ρ), D∗ l1, l2 = Tr(a† l2a† l1ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Introduce the deviating expectation of operator ˆO as � O(t) = ⟨ ˆO⟩(t) − ⟨ ˆO⟩S to describe the deviation from steady state ex- pectation value ⟨ ˆO⟩S = ⟨ ˆO⟩(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (10), we get d dt �Ψk1k2 = Xk1k2�Ψk1k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Making Fourier transformation, we have �Ψl1l2(t) = � k1k2 ei(−k1l1+k2l2)�Ψk1k2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Decomposing ar- bitrary initial state �Ψk1k2(0) by the eigenstates of Xk1k2 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' �Ψk1k2(0) = � αβ Cαβ k1k2|Γαβ k1k2⟩, where α and β take ±, then we have �Ψl1l2(t) = � k,µ eik·˜rCµ ket Γµ k |Γµ k⟩, (12) where k = (k1, k2), ˜r = (−l1, l2) and µ = (α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' For non-zero Liouvillian gap, the system exponentially decays to NESS with time, so we can define instantaneous decay rate K(t) of the j component of �Ψl1l2(t) as K j l1l2 = d dt log � |�Ψj l1l2(t)| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (13) Below we unveil how K(t) is affected by the dispersion of Γµ k through Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 4, in which the damping behaviors of local deviating particle number �nl = � Gll from the initial state with a single excitation on site 1 are shown: (i) When FB appears, Γµ k becomes a constant, denoted by Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Then we have �Ψl1l2(t) = eΓ0t � k,µ eik·˜rCµ k|Γµ k⟩ and K j l1l2(t) = Re(Γ0), which means for arbitrary initial state dif- ferent two-operator correlation functions will synchronously relax to their steady state expectation values with the same decay rate, as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 4 (b) and (e), where dif- ferent curves of log(˜nl) as a function with γt have the same constant slope, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' K1 ll = 4γ for all l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (ii) When Γµ k is only dispersionless in its real part, we set Γµ k = −x0 − iyµ(k), where x0 and yµ(k) are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Then we have �Ψl1l2(t) = e−x0t � k,µ Cµ keik·˜re−iyµ(k)t|Γµ k⟩ and K j l1l2 = −x0 + d dt log �������� ��� � k,µ Cµ k|Γµ k⟩ jei� k·˜r−yµ(k)t���� �������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (14) The right side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (14) contains sum of a series of plane waves, which leads to K j l1l2(t) oscillating around x0, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 4 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The oscillating slopes lead to continuously inter- secting curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (iii) When Γµ k is only dispersionless in its imaginary part, we set Γµ k = −(xc + δxµ(k)) − iy0, where xc and δxµ(k) are the central value and the offset function of Re(Γµ k), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5 2-2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5 3k(元)k(元)J2 > 412J2 = 412J2 412Im(入±ReReReIm2 4Re(X±5 0 5 20 10 05 5 30 15 05 5 40 20 02 0 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5k(元)2 0 2 1 0 0 1 11 0 1 1 0 0 1 -1J2 412k1(元)k2(元)k1(元)k1(元)k1(元)k2(元)k1(元)k1(元)k2(元)1 0 1 1 0 0k2(元)k2(元)k2(元)-2 3 4 1 1 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='4 5 6 1 1 0 0 1-5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='10 1 1 0 0 1 1HH Re(T) k1k2HH k1k2J2 > 412J2 = 4124 and y0 is the imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Then we have �Ψl1l2(t) = e−(xc+iy0)t � k,µ Cµ keik·˜re−δxµ(k)t|Γµ k⟩ and K j l1l2 = −xc + d dt log �������� ��� � k,µ Cµ k|Γµ k⟩ jeik·˜re−δxµ(k)t��� �������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (15) Since δxµ(k) is real, the relaxation process does not display oscillating decay rates (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 4 (f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' This induces the forked damping curves typically as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 4 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (b) (a) (c) (d) (e) (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The damping of particle number at different sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The lattice has 15 sites under the periodic boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Initial state is a single excitation on the first site from vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The time evolutions of log(|˜nl|) are shown in (a), (b), (c), and their derivatives K1 ll are shown in (d), (e) and (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The blue, red and orange lines are corresponding to l = 1, l = 2 and l = 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' In (a) and (d), γ2 is set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' In (b) and (e), γ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' In (c) and (f), γ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Others parameters are the same in all subfigures with J = 1, γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='25 and w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The black dashed line represents a constant decay rate as ˜nl ∝ e−4γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The above damping dynamics is directly related to disper- sion of damping-matrix spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The damping-matrix spec- tra reflect the decay of correlation functions, however, the Li- ovillian spectra reflect the decay of the whole system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' We prove that the damping-matrix spectra are included in Liouvil- lian spectra in the Supplemental Material [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Therefore, for more general models with closed evolution equations of two- operator correlation functions, the dispersionless Liouvillian bands will lead to dispersionless damping-matrix spectra, and then give rise to the same dynamical signatures as shown in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Localized normal master modes and dynamic localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='— In isolated system, FBs lead to localized eigenstates by destructive interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Now, we exactly solve our model (see the Supplemental Material [25]) to show that the LFB can induce dynamic localization by localized normal master modes (LNMMs), which suppress propagation of local perturbation on NESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Usually, the odd parity part of ˆL has no effect on the ex- pectation value of observation in pure fermionic system [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Therefore, we focus on the balanced model (w = 0) with even parity (P = 1), whose Liouvillian is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' By solving the equation ζi(k)|Ω⟩ = 0 for i = 1 ∼ 4, we get the steady state |Ω⟩ as |Ω⟩ = 1 N π � k=−π (1 + a† kc† −k)|0⟩ = 1 N L � l=1 (1 + a† l c† l )|0⟩, (16) where N = 2L and this state is independent with γ1 and γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' At the FB point with J = 2 √γ1γ2, the exceptional degeneracy occurs in the non-Hermitian matrix Lk of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (7) with four eigenstates coalescing into two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Then ˆLk is reduced to ˆLk = −2γ � ζ ′ A(k)ζA(k) + ζ ′ B(k)ζB(k) � , where ζ ′ A(k) = −a† k + c−k, ζA(k) = 1 2(−ak + ick + ia† −k + c† −k), ζ ′ B(k) = ak + c† −k, ζB(k) = 1 2(a† k − ic† k + ia−k + c−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (17) Making Fourier transformation, we get ζ ′ A(l) = � k e−iklζ ′ A(k) = −a† l +cl, ζ ′ B(l) = � k eiklζ ′ B(k) = al+c† l , (18) which create local eigenstate ζ ′ A,B(l)|Ω⟩ of ˆL with eigenvalue −2γ and are coined as LNMMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' We can also understand LNMMs intuitively from the per- spective of destructive interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Writing the real-space Li- ouvillian with w = 0, J = 2 √γ1γ2 = 1 as ˆL = � l(ˆhl + ˆfl −2γ), where the hopping term hl is defined as ˆhl = −i(a† l+1al + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=') + i(c† l+1cl + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=') − (a† l+1cl + c† l+1al + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='c) and the pair- ing term ˆfl is defined as fl = 2γ(a† l c† l + clal), we can check that ˆflal|Ω⟩ = ˆflcl|Ω⟩ = ˆfla† l |Ω⟩ = ˆflc† l |Ω⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' This im- plies that the pairing terms do not affect a single particle or hole excited on the NESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Therefore, for these states only hopping terms make sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' We schematically plot this re- duced ladder in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 1 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' It is easy to find another LNMM as ζ ′ C(l) = a† l − ic† l from the view of destructive interference, which forbids the state ζ ′ C(l)|Ω transferring to other sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' We can also check that ˆL ζ ′ C(l)|Ω⟩ = −2γ ζ ′ C(l)|Ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The LNMMs contain decay information of quantum jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' To see it clearly, we map the C−representation state ζ ′ A(l)ζ ′ B(l)|Ω⟩, for example, back to density-matrix representa- tion: ζ ′ A(l)ζ ′ B(l)|Ω⟩ → −a† l alρs + ρsala† l + alρsa† l − a† l ρsal, (19) where ρs is the density matrix of NESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The terms alρsa† l and a† l ρsal are exactly corresponding to local quantum jumps on NESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (19) implies that the local perturbation on NESS from quantum jumps will relax to NESS without ex- panding its territory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' To see it clearly, we simulate the evo- lution from an initial state described by the density matrix ρ0 = a† 1ρsa1/Tr(a† 1ρsa1), which is created by a quantum jump on the first site of NESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' In Fig 5, we demonstrate the time evolution of particle numbers of the first three sites in a lattice with 15 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' In the initial time, a jump occurs on the first site of NESS, increasing only the particle number on the first site n1 to 1 with others sites keeping their steady state value 2 6 0 2-2 4 6 0 1 2K= 4ttK= 4K= 4tttK 1 1log (1i nt 1)-2 6 0 1 20 n1 n2 n3 5 10 0 1 20 n1 n2 n3 5 10 0 1 20 n1 n2 n3 ~ 5 10 0 1 2J2 > 412J2 = 412J2 412t5 (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The time evolution of particle number on the first site n1 shown in (a), second site n2 in (b) and third site n3 in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Initial state is a† 1ρsa1/Tr(a† 1ρsa1) corresponding to a quantum jump on the first site of steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The periodic lattice has 15 sites with w = 0, J = 1, γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='25 in all subfigures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The black dotted, red solid, and blue dashed lines are corresponding to γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5, γ2 = 1 and γ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The red solid line, black dotted line and blue dashed line are corresponding to the situation with J2 = 4γ1γ2 (LFB), J2 > 4γ1γ2, and J2 < 4γ1γ2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' We can see that when J2 � 4γ1γ2, the perturbation can spread from n1 to n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' However, for the case with LFB, the perturbation excitation decays locally without going through to n2 and n3, indicating the occurrence of dynamical localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Final remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='— (i) We use a geometrically intuitive method to construct flat band models in open system and demonstrate that the dispersion of Liouvillian band can ef- fectively affect the damping dynamics of local particle num- ber, intermediated by damping matrix of correlation function vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' When the Liouvillian flat band appears, the particle number in different sites will relax to their stable values syn- chronously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' When only the real or imaginary part of rapidity spectrum is dispersionless, the damping behaviors show the oscillating or forked characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (ii) We show flat-band Liouvillian can induce dynamical localization on NESS by the localized normal master modes, which halt the propagation of perturbation from other sites to the target sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (iii) Our model does not exhibit non-Hermitian skin ef- fect [42, 43], which was uncovered to cause many abnormal phenomena such as boundary sensitivity [44], chiral and he- lical damping [45, 46] and slowing down of relaxation pro- cesses [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The interplay between Liouvillian flat band and non-Hermitian skin effect is an interesting topic for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content='— We thank X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Y.' metadata={'source': 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ing and dynamical critical skin effect in open quantum systems, Physical Review Research 2, 043167 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' 7 SUPPLEMENTAL MATERIAL: Dynamics Signatures of Liouvillian Flat Band S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Mapping of Lindblad master equation FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Mapping of Lindblad master equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The Lindblad master equation, formalized density matrix ρ and Liouvillian superoperator L is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (1) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (2) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' First we carry out the Choi-Jamiolkwski isomorphism [1–4] to map the fermionic LME into representation B as d dt| ρ⟩B = ˆLB| ρ⟩B, (S1) where | ρ⟩B is vectorized from ρ and ˆLB is mapped from L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Specifically, the mapping is ρ → | ρ⟩B = � IJ ρIJ|I⟩a ⊗ |J⟩b, (S2a) L → ˆLB = � i j Fi(a, a†) ⊗ F T j (b, b†), (S2b) where b = (b1, b2, · · · ) is the set of annihilation operators of b−fermions, which is one-to-one mapping from a, and T means matrix transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' |I⟩a and |J⟩b are defined as |I⟩a = (a† 1)I1(a† 2)I2 · · · (a† L)IL|0⟩a, (S3a) |J⟩b = (b† 1)J1(b† 2)J2 · · · (b† L)JL|0⟩b, (S3b) where |0⟩a and |0⟩b are vacuum state of all a−fermions and b−fermions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' In this representation, the expectation value of observable becomes ⟨ ˆOa⟩ =B ⟨S0| ˆOa ⊗ Ib| ρ⟩B, (S4) where B⟨S0| is a special state defined as: B⟨S0| = � S ⟨S|a ⊗ ⟨S|b = � S � ⟨0|a(aL)S L · ·(a1)S 1 ⊗ ⟨0|b(bL)S L · ·(b1)S 1� , (S5) and Ib is a unit operator of all b−fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' The element S i of S = (S 1, S 2, · · · ) can take 0 or 1, and � S requires a sum over all possible configurations of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' Let us prove Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (S4): ⟨ ˆOa⟩ = � IJS ρIJ ⟨S|a ˆOa|I⟩a⟨S|bIb|J⟩b = � IJS ρIJ a⟨0|aS L L · · · aS 1 1 ˆOa(a† 1)I1 · · · (a† L)IL|0⟩a δSJ = � IJ ρIJ a⟨0|aJL L · · · aJ1 1 ˆOa(a† 1)I1 · · · (a† L)IL|0⟩a = � IJ ρIJ a⟨0| ˆOa|0⟩a = Tr( ˆOaρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfAQ7r/content/2301.05378v1.pdf'} +page_content=' (S6) p=p [I)a T), the ben- +zene molecule is in a superposition of the A1g ground +state and the E1u excited state and also, to an extent, +ionized. This causes oscillations in the charge and current +densities ρ(r, t) and J(r, t), respectively, with period 612 +as (corresponding to the energy difference between the +ground state and excited states), which are an example +of attosecond charge migration [26]. +In order to visualize the current we isolate the sta- +tionary component of the current density, by computing +an angle averaged cross section defined by the following +integral (in cylindrical coordinates ρ, z, φ), +J(x, z, t) = 1 +2π +� 2π +0 +ˆφ · J(|x|, z, φ)dφ. +(2) +The geometric interpretation of this integral is given in +Fig. +1. +The angle averaging procedure for the few +level model causes that the fast-oscillating component +effectively vanishes. Within the few-level model, the fast- +oscillating component of the current density is zeroed out +by this averaging procedure because of its parity. +It has +similar effect on TDDFT results, and therefore J(x, z, y) +has only a very gradual time dependence for t > T. The +same is true for TDDFT. These integrated current densi- +ties are plotted in Fig. 1b. At low intensities the current +is a combination of a strong co-rotating current (red) and +a weak counter-rotating current (blue), while at high in- +tensities the counter-rotating current dominates. As we +explain below (see Fig. 4), the reversal is a signature of +the transition from electron to hole current regime. +The oscillatory component of the charge motion is best +visualized by plotting the charge displacement, +∆ρ(r, t) = ρ(r, t) − ρ(r, 0), +(3) +shown in Fig. 2. The cloud of displaced charge circulates +around the molecule with the expected period of 612 as, +and this continues even after the pulse ends. +Overall, +both the magnitude and shape of the charge displace- +ment are remarkably similar between the two models, +however there are some subtle differences. +First, long +after the laser pulse the two models gradually become +desynchronized. Second, in TDDFT there appears to be +a rearrangement of charge in the plane of the molecule, +whereas the few level model only predicts the dynamics +above and below the plane. + +(a) +(b) +3.8 × 1011 W/cm2 +Few-level Model +4.0 +2.0 +('n +-0.0 +N +-2.0 +-4.0 +5 × 1012 W/cm² +1013 W/cm² +4.0 +2.0 +-0.0 +-2.0 +-4.0 +-6.0 +-3.0 +0.0 +3.0 +-6.0 +-3.0 +0.0 +3.0 +6.0 +x (a.u.) +x (a.u.)3 +FIG. 2. +Snapshots of the charge displacement induced by a circularly-polarized laser pulse with peak intensity 5×1012 W/cm2 +taken around the peak of the laser pulse (first three columns t ≈ 100 a.u.) and after the laser pulse (last three columns t ≈ 400 +a.u.). Light areas indicate excess electrons while dark areas indicate fewer electrons, as compared to the ground state charge +density before the laser pulse. We compare the results between the two theoretical models, TDDFT (top row) and the few +level model (bottom row). +FIG. 3. +Comparison of full TDDFT simulations (solid blue +line) to the few level model (orange dashed line). For peak +intensities, when ionization (dotted green line) becomes non- +negligible, the two models begin to disagree. +The smooth +lines have been interpolated between the calculated intensities +using the method described in [25]. +Another important observation about the density dif- +ference is that the dark areas are generally larger than +the light areas. In the TDDFT results one reason for this +is ionization, with the ionization probability given by +P ionize = − +� +∆ρ(r, 2T)d3r, +(4) +where the integral ranges over the simulation box. Unex- +pectedly, the few level model also appears to have dark +areas larger than light areas even though it does not +include ionization, and in fact the charge displacement +must integrate to zero in that model. The reason for this +is that the E1u is of mixed character, part of which in- +volves excitation to LUMO + 3 [25]. +Note: The excess +of darker areas in the TDDFT model is a combination of +both ionization and excitation to LUMO +3 orbital. +The intensity dependence of the dynamics is illustrated +in Fig. 3. using the current. Note: this current is directly +proportional to z-component of the magnetic moment as +well as z-component of electronic angular momentum), +Since the domain of integration is the simulation box, +ionized electrons are not included. +For this reason we +plot Lz(2T) so that the ionizing wavepacket has enough +time to leave the box. Whereas in the few level model +the magnetic moment increases monotonically with the +laser intensity (up to about 1013 W/cm2, after which +the system Rabi oscillates back to the ground state), in +TDDFT the current starts to decrease already around +1012 W/cm2, and reverses sign for even higher intensities. +We also plot the ionization probability (defined in Eq. 4), +and conclude that the reversal occurs precisely when the +ionization probability becomes non-negligible. +The implications of the transition from electron to hole +current on the charge dynamics, and the underlying phys- +ical mechanism responsible for that transition, can be +understood in more detail describe in more detail using +complex molecular orbitals, as illustrated schematically +in Fig. 4. These orbitals represent a change of basis from +the usual real-valued Kohn-Sham orbitals ψn(r) (defined +in [25]), +ψHOMO +± +(r) = [ψ14(r) ± iψ15(r)] / +√ +2, +(5) +ψLUMO +± +(r) = [ψ16(r) ± iψ17(r)] / +√ +2. +(6) +The advantage of using complex orbitals is that they are +eigenfunctions of the 6-fold symmetry operator (rotation + +t = 105 a.u. +t = 110 a.u. +t = 390 a.u. +t = 395 a.u. +t = 100 a.u. +t = 400 a.u. +TDDFT +Few +Level +Model0.8 +Lz (TDDFT) +Lz (Few level) +0.6 +lonizationprobability(TDDFT) +or probability +0.4 +0.2 +(a.u.) +0.0 +0.2 +1011 +1012 +1013 +Peak intensity (W/cm2)4 +FIG. 4. +Schematic illustrating the complex molecular or- +bitals and the physical mechanism for the transition from +electron to hole current. Color indicates the complex phase. +about the molecular axis by 60◦), +exp +� +− iπ +3¯h +ˆLz +¯h +� +ψHOMO +± +(r) = exp +� +∓iπ +3 +� +ψHOMO +± +(r),(7) +exp +� +−iπ +3 +ˆLz +¯h +� +ψLUMO +± +(r) = exp +� +∓2iπ +3 +� +ψLUMO +± +(r).(8) +The complex orbitals have magnetic quantum numbers +m defined modulo 6: ψHOMO +± +have m = ±1 and ψLUMO +± +have m = ±2. Just as for atomic orbitals, the sign of +m indicates the direction the electron circulates around +the molecule, and the magnitude indicates more-or-less +the angular speed. We have chosen our conventions such +that m > 0 electrons are co-rotating with the laser field, +and m < 0 electrons are counter-rotating. +Using the notation of complex orbitals, Fig. 4 illus- +trates how in the ground state, both ψHOMO +± +are dou- +bly occupied, and consequently there is zero net cur- +rent. When the benzene molecule is exposed to a cir- +cularly polarized laser pulse, the usual selection rule +∆m = 1 applies (here we assume the laser is polar- +ized in the molecular plane, see [25] for the more gen- +eral case), so that the only dipole-allowed transition is +ψHOMO ++ +to ψLUMO ++ +, which is the dominant component of +the E1u excited state. The electron excited to LUMO +contributes a strong co-rotating current (m = +2), but +the imbalance of electrons in the HOMO contributes a +weaker counter-rotating current (m = −1). This can al- +ternatively be interpreted as a positively charged hole +occupying ψHOMO ++ +producing a co-rotating hole current +(rather than a counter-rotating electron current). This +is precisely what we see in the top row of Fig. 1b, two +components to the current with opposite sign (red and +blue). +In order to explain the reversal of the current at higher +intensity (bottom row of Fig. 1b), we simply recognize +that the electron previously excited to ψLUMO ++ +can ab- +sorb a second photon from the same laser pulse, ionizing, +and leaving behind only the hole current. The balance +between the one-photon excitation and the two-photon +ionization processes can be controlled by varying the laser +intensity, because the first process scales with I while the +second process scales with I2 (with I ∝ E2 the laser in- +tensity). Furthermore, it is now apparent that the sign +reversal can be interpreted as a change in the dominant +charge carrier from electrons to holes. +In conclusion, we have shown that both electron and +hole currents are present during resonance-enhanced two- +photon ionization of benzene, and the balance between +the two current regimes can be controlled by varying the +peak laser intensity. We have proposed a simple expla- +nation for the effect in terms of molecular orbitals, which +is consistent with the results of full TDDFT simulations. +Variants of complex orbital model should apply to a wide +variety of molecules other than benzene, meaning that +the structure of the complex molecular orbitals can be +used to predict the interplay between electron and hole +currents during REMPI. In order to measure this effect +in experiment, several pump-probe schemes have been +proposed that are sensitive to the magnitude and direc- +tion of the ring current [7, 11]. In [25], we demonstrate +that the reversal is independent of the orientation of the +molecule, which greatly simplifies any potential exper- +iment. +Finally, our results suggest that the few level +model typically used to study photoinduced ring currents +may be insufficient even for moderate laser intensities +around 1012 W/cm2. A more ab initio nonperturbative +theory such as TDDFT, as used in present paper, is more +appropriate for this regime. +This work was supported by the NSF Grant No. PHY- +1734006 and Grant No. PHY-2110628. This work uti- +lized resources from the University of Colorado Boulder +Research Computing Group, which is supported by the +National Science Foundation. +[1] J. A. N. F. Gomes and R. B. Mallion, Chemical Reviews +101, 1349 (2001). +[2] T. M. Krygowski, H. Szatylowicz, O. A. Stasyuk, J. Do- +minikowska, +and M. Palusiak, Chemical Reviews 114, +6383 (2014). +[3] T. Heine, C. Corminboeuf, and G. Seifert, Chemical Re- +views 105, 3889 (2005). +[4] I. 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Rubio, Physical Chemistry Chemical +Physics 17, 31371 (2015). +[25] See Supplemental Material for details on the TDDFT +and few-level models, orientation dependence, and the +method for interpolating over intensity. +[26] H. J. W¨orner, C. A. Arrell, N. Banerji, A. Cannizzo, +M. Chergui, A. K. Das, P. Hamm, U. Keller, P. M. Kraus, +E. Liberatore, P. Lopez-Tarifa, M. Lucchini, M. Meuwly, +C. Milne, J.-E. Moser, U. Rothlisberger, G. Smolentsev, +J. Teuscher, J. A. van Bokhoven, and O. Wenger, Struc- +tural Dynamics 4, 061508 (2017). + diff --git a/VdAyT4oBgHgl3EQfhfjt/content/tmp_files/load_file.txt b/VdAyT4oBgHgl3EQfhfjt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..166dc0d6952b2f07548d8163bab29923fe46c88d --- /dev/null +++ b/VdAyT4oBgHgl3EQfhfjt/content/tmp_files/load_file.txt @@ -0,0 +1,447 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf,len=446 +page_content='Ultrafast switching of persistent electron and hole currents in ring molecules Tennesse Joyce and Agnieszka Jaron JILA and Department of Physics, University of Colorado, Boulder, CO-80309, USA (Dated: January 3, 2023) A circularly polarized laser pulse can induce persistent intra-molecular currents by either exciting or ionizing molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' These two cases are identified as electron currents and hole currents, respec- tively, and up to now they have been studied only separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' We report ab initio time-dependent density-functional theory (TDDFT) simulations of currents during resonance-enhanced two-photon ionization of benzene, which reveal for the first time that both electron and hole currents can be present simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' By adjusting the intensity of the laser pulse, the balance between the two types of current can be controlled, and the overall sign of the current can be switched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' We provide a physical explanation for the effect in terms of complex molecular orbitals which is consistent with the TDDFT simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' It has long been understood that, in response to an ap- plied magnetic field, the delocalized electrons of an aro- matic molecule circulate in so-called aromatic ring cur- rent [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' This effect is important in nuclear magnetic resonance spectroscopy, where the internal magnetic field generated by the ring current is responsible for diamag- netic shielding [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' In 2006, it was proposed that ring currents in molecules could also be induced by ultra- short laser pulses with circular or elliptical polarization [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' The basic mechanism is that angular momentum carried by light is transfered to electrons in a molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Due to conservation of angular momentum, the current persists after the pulse has ended—even without an ex- ternal magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Various experiments on atomic targets have confirmed the existence of the effect [6, 7], although no direct observational data is available in the case of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Recent interest in photoinduced ring currents is motivated by the rapid technological advances in polarization control of high-harmonic radiation made in the last few years [8–10], which may enable experimen- tal study of these phenomena in the near future [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' There are several major advantages of photoinduced ring currents compared to those induced by static mag- netic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' First, the current is expected to be orders of magnitude stronger, and so is the induced magnetic field [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Second, they enable femtosecond (or even attosec- ond) time-resolved studies of aromaticity and magnetism [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Lastly, they establish the possibility for coherent control of ring currents [15], which may have applications for controlling chemical reactions or the operation of ad- vanced opto-electronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' In this Letter we predict a novel effect which causes the dominant charge carrier of the ring current to transi- tion from electrons to holes as the peak laser intensity in- creases past around 1012 W/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' We illustrate the effect with a series of ab initio time-dependent density func- tional theory (TDDFT) simulations of benzene (C6H6), which is the prototypical aromatic molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Lastly, we demonstrate that the effect is not accounted for in the commonly used few level model of ring currents, due to the fact that it neglects ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' This calls into ques- tion the results of several previous studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' [4, 5, 15]) where it was assumed that the few level model is accurate for laser intensities on the order of 1012 W/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' We begin by introducing the distinction between elec- tron and hole current: when an electron is promoted to an orbital with nonzero angular momentum, this creates an electron current;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' when an electron is removed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=', ionized) from an orbital with nonzero angular momen- tum, this creates a hole current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' So far, hole currents have mostly been studied in the context of strong field ionization of atoms by circularly polarized laser pulses, and it was recently confirmed experimentally that a hole can be created with a specific angular momentum relative to the laser polarization [16–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Electron currents on the other hand do not involve ionization, only excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' However, in the interaction of atoms and molecules with strong laser fields, excitation and ionization are of- ten closely related and occur together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' A typical example is resonance-enhanced multiphoton ionization (REMPI) [20, 21], a two-step ionization process wherein an atom or molecule is first excited to an intermediate state (that must be resonant with some multiple of the laser fre- quency) and then subsequently ionized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Now consider REMPI in a system where the intermediate excited state corresponds to an electron current, and the final ionized state corresponds to a hole current (we will show that benzene is such a system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' The balance between excita- tion and ionization (and therefore electron and hole cur- rent) will depend on the laser intensity because the pro- cesses involve different numbers of photons (and therefore scale with different powers of intensity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' In particular at low intensities we expect electron current to dominate (excitation), and at high intensities we expect hole cur- rent to dominate (ionization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Our main theoretical method is TDDFT, as imple- mented by Octopus [22–24], which provides a fully non- perturbative description of the light-matter interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' As a reference point to compare against the full TDDFT simulations, we also consider the few level model of ring currents (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' We discuss the implementations of both models in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Because the few level model does not include ionization, we expect the two models to di- verge at high enough laser intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' The laser pulse in our simulations is described in the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='00380v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='chem-ph] 1 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' (a) Visualization of the current density based on the component passing through a plane bisecting the molecule as shown (averagea over all possible orientations of that plane [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' (2)]) (b) Cross sections of the current density taken at the end of the laser pulse (t = 200 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=') for several different simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' At low laser intensity the co-rotating current (red) dominates, while at high intensity the counter-rotating current (blue) dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Note: Each plot is scaled individually relative to the maximum absolute value within that plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' The nuclei lie in the plane z = 0 with the carbon ring at x = ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='63 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' and the hydrogen ring at x = ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='69 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='. dipole approximation by the following electric field, E(t) = � E sin2 (πt/T) Re � ˆϵeiω(t−T/2)� , 0 < t < T, 0, otherwise, (1) with central frequency ω = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='76 eV (183 nm), dura- tion T = 16π/ω = 202 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='9 fs, circular polar- ization ˆϵ = (ˆx + iˆy)/ √ 2 (with the molecule in the xy- plane), and a variable peak amplitude E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' The central frequency was chosen to be resonant with the doubly degenerate E1u state (as computed with linear response TDDFT [25]), which is predominantly associated with the HOMO-LUMO transition (HOMO = Highest Occu- pied Molecular Orbital;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' LUMO = Lowest Unoccupied Molecular Orbital).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Because the computed ionization threshold is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 eV < 2ω, this laser pulse is designed to drive 1+1 REMPI where one photon is enough to promote electron to the excited state and one additional photon to ionize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' After interacting with the laser pulse (t > T), the ben- zene molecule is in a superposition of the A1g ground state and the E1u excited state and also, to an extent, ionized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' This causes oscillations in the charge and current densities ρ(r, t) and J(r, t), respectively, with period 612 as (corresponding to the energy difference between the ground state and excited states), which are an example of attosecond charge migration [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' In order to visualize the current we isolate the sta- tionary component of the current density, by computing an angle averaged cross section defined by the following integral (in cylindrical coordinates ρ, z, φ), J(x, z, t) = 1 2π � 2π 0 ˆφ · J(|x|, z, φ)dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' (2) The geometric interpretation of this integral is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' The angle averaging procedure for the few level model causes that the fast-oscillating component effectively vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Within the few-level model, the fast- oscillating component of the current density is zeroed out by this averaging procedure because of its parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' It has similar effect on TDDFT results, and therefore J(x, z, y) has only a very gradual time dependence for t > T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' The same is true for TDDFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' These integrated current densi- ties are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' At low intensities the current is a combination of a strong co-rotating current (red) and a weak counter-rotating current (blue), while at high in- tensities the counter-rotating current dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' As we explain below (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' 4), the reversal is a signature of the transition from electron to hole current regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' The oscillatory component of the charge motion is best visualized by plotting the charge displacement, ∆ρ(r, t) = ρ(r, t) − ρ(r, 0), (3) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' The cloud of displaced charge circulates around the molecule with the expected period of 612 as, and this continues even after the pulse ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Overall, both the magnitude and shape of the charge displace- ment are remarkably similar between the two models, however there are some subtle differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' First, long after the laser pulse the two models gradually become desynchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Second, in TDDFT there appears to be a rearrangement of charge in the plane of the molecule, whereas the few level model only predicts the dynamics above and below the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' (a) (b) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='8 × 1011 W/cm2 Few-level Model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content="0 ('n 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 5 × 1012 W/cm² 1013 W/cm² 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 x (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=') x (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' )3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Snapshots of the charge displacement induced by a circularly-polarized laser pulse with peak intensity 5×1012 W/cm2 taken around the peak of the laser pulse (first three columns t ≈ 100 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=') and after the laser pulse (last three columns t ≈ 400 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Light areas indicate excess electrons while dark areas indicate fewer electrons, as compared to the ground state charge density before the laser pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' We compare the results between the two theoretical models, TDDFT (top row) and the few level model (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Comparison of full TDDFT simulations (solid blue line) to the few level model (orange dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' For peak intensities, when ionization (dotted green line) becomes non- negligible, the two models begin to disagree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' The smooth lines have been interpolated between the calculated intensities using the method described in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Another important observation about the density dif- ference is that the dark areas are generally larger than the light areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' In the TDDFT results one reason for this is ionization, with the ionization probability given by P ionize = − � ∆ρ(r, 2T)d3r, (4) where the integral ranges over the simulation box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Unex- pectedly, the few level model also appears to have dark areas larger than light areas even though it does not include ionization, and in fact the charge displacement must integrate to zero in that model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' The reason for this is that the E1u is of mixed character, part of which in- volves excitation to LUMO + 3 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Note: The excess of darker areas in the TDDFT model is a combination of both ionization and excitation to LUMO +3 orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' The intensity dependence of the dynamics is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' using the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Note: this current is directly proportional to z-component of the magnetic moment as well as z-component of electronic angular momentum), Since the domain of integration is the simulation box, ionized electrons are not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' For this reason we plot Lz(2T) so that the ionizing wavepacket has enough time to leave the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Whereas in the few level model the magnetic moment increases monotonically with the laser intensity (up to about 1013 W/cm2, after which the system Rabi oscillates back to the ground state), in TDDFT the current starts to decrease already around 1012 W/cm2, and reverses sign for even higher intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' We also plot the ionization probability (defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' 4), and conclude that the reversal occurs precisely when the ionization probability becomes non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' The implications of the transition from electron to hole current on the charge dynamics, and the underlying phys- ical mechanism responsible for that transition, can be understood in more detail describe in more detail using complex molecular orbitals, as illustrated schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' These orbitals represent a change of basis from the usual real-valued Kohn-Sham orbitals ψn(r) (defined in [25]), ψHOMO ± (r) = [ψ14(r) ± iψ15(r)] / √ 2, (5) ψLUMO ± (r) = [ψ16(r) ± iψ17(r)] / √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' (6) The advantage of using complex orbitals is that they are eigenfunctions of the 6-fold symmetry operator (rotation t = 105 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' t = 110 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' t = 390 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' t = 395 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' t = 100 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' t = 400 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' TDDFT Few Level Model0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='8 Lz (TDDFT) Lz (Few level) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='6 lonizationprobability(TDDFT) or probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='2 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content='2 1011 1012 1013 Peak intensity (W/cm2)4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Schematic illustrating the complex molecular or- bitals and the physical mechanism for the transition from electron to hole current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Color indicates the complex phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' about the molecular axis by 60◦), exp � − iπ 3¯h ˆLz ¯h � ψHOMO ± (r) = exp � ∓iπ 3 � ψHOMO ± (r),(7) exp � −iπ 3 ˆLz ¯h � ψLUMO ± (r) = exp � ∓2iπ 3 � ψLUMO ± (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' (8) The complex orbitals have magnetic quantum numbers m defined modulo 6: ψHOMO ± have m = ±1 and ψLUMO ± have m = ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Just as for atomic orbitals, the sign of m indicates the direction the electron circulates around the molecule, and the magnitude indicates more-or-less the angular speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' We have chosen our conventions such that m > 0 electrons are co-rotating with the laser field, and m < 0 electrons are counter-rotating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Using the notation of complex orbitals, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' 4 illus- trates how in the ground state, both ψHOMO ± are dou- bly occupied, and consequently there is zero net cur- rent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' When the benzene molecule is exposed to a cir- cularly polarized laser pulse, the usual selection rule ∆m = 1 applies (here we assume the laser is polar- ized in the molecular plane, see [25] for the more gen- eral case), so that the only dipole-allowed transition is ψHOMO + to ψLUMO + , which is the dominant component of the E1u excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' The electron excited to LUMO contributes a strong co-rotating current (m = +2), but the imbalance of electrons in the HOMO contributes a weaker counter-rotating current (m = −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' This can al- ternatively be interpreted as a positively charged hole occupying ψHOMO + producing a co-rotating hole current (rather than a counter-rotating electron current).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' This is precisely what we see in the top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' 1b, two components to the current with opposite sign (red and blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' In order to explain the reversal of the current at higher intensity (bottom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' 1b), we simply recognize that the electron previously excited to ψLUMO + can ab- sorb a second photon from the same laser pulse, ionizing, and leaving behind only the hole current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' The balance between the one-photon excitation and the two-photon ionization processes can be controlled by varying the laser intensity, because the first process scales with I while the second process scales with I2 (with I ∝ E2 the laser in- tensity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Furthermore, it is now apparent that the sign reversal can be interpreted as a change in the dominant charge carrier from electrons to holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' In conclusion, we have shown that both electron and hole currents are present during resonance-enhanced two- photon ionization of benzene, and the balance between the two current regimes can be controlled by varying the peak laser intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' We have proposed a simple expla- nation for the effect in terms of molecular orbitals, which is consistent with the results of full TDDFT simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Variants of complex orbital model should apply to a wide variety of molecules other than benzene, meaning that the structure of the complex molecular orbitals can be used to predict the interplay between electron and hole currents during REMPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' In order to measure this effect in experiment, several pump-probe schemes have been proposed that are sensitive to the magnitude and direc- tion of the ring current [7, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' In [25], we demonstrate that the reversal is independent of the orientation of the molecule, which greatly simplifies any potential exper- iment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' Finally, our results suggest that the few level model typically used to study photoinduced ring currents may be insufficient even for moderate laser intensities around 1012 W/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' A more ab initio nonperturbative theory such as TDDFT, as used in present paper, is more appropriate for this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' This work was supported by the NSF Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' PHY- 1734006 and Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' PHY-2110628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' This work uti- lized resources from the University of Colorado Boulder Research Computing Group, which is supported by the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} +page_content=' A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'} diff --git a/WNAyT4oBgHgl3EQfu_m7/vector_store/index.pkl b/WNAyT4oBgHgl3EQfu_m7/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..afc399b6605efdd180ef8ab383ba3ece64959a93 --- /dev/null +++ b/WNAyT4oBgHgl3EQfu_m7/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1bb1ce92669537ffd3c2fb588842ec0edd29960fa52396311baf26e1727f63b4 +size 299618 diff --git a/WtAyT4oBgHgl3EQfWPdu/content/tmp_files/2301.00159v1.pdf.txt b/WtAyT4oBgHgl3EQfWPdu/content/tmp_files/2301.00159v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..25324efce91071dc855936f28c5bf09b629e8a31 --- /dev/null +++ b/WtAyT4oBgHgl3EQfWPdu/content/tmp_files/2301.00159v1.pdf.txt @@ -0,0 +1,2049 @@ +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT +EPIDEMIC MODELS +I: THE REPLACEMENT NUMBER DYNAMICS +FLORIAN NILL +31-DEC-2022 +Abstract. As shown recently by the author, constant population SI(R)S models map to +Hethcote’s classic endemic model originally proposed in 1973. This unifies a whole class +of models with up to 10 parameters being all isomorphic to a simple 2-parameter master +model for endemic bifurcation. In this work this procedure is extended to a 14-parameter +SSISS Model, including social behavior parameters, a (diminished) susceptibility of the +R-compartment and unbalanced constant per capita birth and death rates, thus covering +many prominent models in the literature. Under mild conditions, in the dynamics for +fractional variables in this model all vital parameters become redundant at the cost of +possibly negative incidence rates. There is a symmetry group GS acting on parameter +space A, such that systems with GS-equivalent parameters are isomorphic and map to the +same normalized system. Using (Xrep, I) as canonical coordinates, Xrep the replacement +number, normalization reduces to parameter space A/GS with 5 parameters only. This +approach reveals unexpected relations between various models in the literature. Part two +of this work will analyze equilibria, stability and backward bifurcation and part three +will further reduce the number of essential parameters from 5 to 3. +Contents +1. +Introduction +2 +2. +The SSISS model +6 +2.1. +Constant population +8 +2.2. +Time varying population +9 +2.3. +Classifying parameter space +10 +2.4. +Examples from the literature +12 +2.5. +Absence of periodic solutions +14 +3. +Normalization +15 +3.1. +Phase space +15 +3.2. +Canonical coordinates +16 +3.3. +Main results +17 +3.4. +Examples revisited +22 +4. +Summary and outlook +23 +Appendix A. +Normalizing linear vital dynamics +24 +Appendix B. +Scaling the SI(R)S model +24 +Appendix C. +The case α1 = α2 = 0 +27 +References +27 +E-mail address: nill.florian@gmail.com. +2020 Mathematics Subject Classification. 34C23, 34C26, 37C25, 92D30. +Key words and phrases. SIRS model, SSISS model, normalization, symmetry, stability, endemic bifur- +cation, backward bifurcation. +The author is retired physicist, Dr.rer.nat.habil., formerly senior research fellow at Inst. theor. Physik, +Freie Universität Berlin. +1 +arXiv:2301.00159v1 [q-bio.PE] 31 Dec 2022 + +2 +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +1. Introduction +Building mathematical models to describe phenomena in natural sciences one typically +encounters dynamical variables and external parameters. Within the model values for +external parameters are considered to be given from outside, like fundamental natural +constants (speed of light c, Planck’s constant ℏ), parameters describing material or bi- +ological properties (spring constant κ, birth rate δ, recovery rate γ) or social behavior +(contact rate β). Naturally, reducing the number of essential parameters is always a goal +to detect redundancies within parameter space and to simplify computations by unload- +ing formulas. In the simplest case a pure dimensional scale parameter may without loss +be put equal to one by choosing dimensional units appropriately. For example, putting +c = 1 amounts to measuring spatial distances by light running times and masses in units +of energies, putting ℏ = 1 amounts to measuring energies by angular frequencies and +putting γ = 1 amounts to measuring time in units of the recovery time in an epidemic +model. +More generally a normalization program consists of finding appropriate coordinate +transformations in variable+parameter space such that the transformed system only de- +pends on a maximally reduced subset of transformed parameters. Examples are1 +Harmonic oscillator +Predator-prey model +˙u += +v +˙u += +−uv + c1u +˙v += +−u +˙v += +uv − v +Classic SIR model +Classic endemic model +˙u += +−uv +˙u += +−uv − c1u + c2 +˙v += +uv − v +˙v += +uv − v +(1.1) +Following this strategy the 6-parameter SI(R)S model (≡ combined SIRS/SIS model) +with standard incidence, constant vaccination and immunity waning rates and a balanced +birth and death rate has recently been shown by the author (Nill 2022) to admit a nor- +malized version looking like the classic endemic model above2. +In this work (including two follow ups to be denoted as parts II and III (Nill n.d.[b],[c])) +this method is extended to the case where immunity after recovery (or vaccination) is +incomplete right from the onset and where also compartment dependent constant per +capita birth and death rates lead to a time varying population size N. +In this way +one is naturally lead to replacing the SI(R)S model by a SSISS model, where in place +of the usual S, I and R compartments we have two susceptible compartments S1 and +S2 and one infectious compartment I. Infection transmission from I to S2 is diminished +as compared to transmission to S1. There is a vaccination flow from S1 to S2 and an +immunity waning flow from S2 to S1. The model could also be interpreted by considering +1The variables in these examples are: +- Harmonic oscillator: u = q, v = p/ +√ +mk, where q, p, κ, m are coordinate, momentum, spring constant +and particle mass and where the oscillation period is normalized to T = 2π by putting m/k = 1. +- Predator-prey model: (u, v) denote appropriately rescaled prey and predator populations, respectively, +and the predator mortality rate is normalized to one. +- SIR model: u = r0S, v = r0I, where r0 is the basic reproduction number, (S, I) are susceptible and +infectious fractions of the population and where the recovery rate is normalized to γ = 1. +- Endemic model: +(u, v, r0, γ) as above, c1 = δ/(γ + δ) and c2 = r0c1, where δ is the balanced +birth/mortality rate and where now time scale is normalized to γ + δ = 1. +2Aapart from allowing also values u ∈ R and an enlarged parameter range (c1, c2) ∈ R+ × R ∪ {0, 0}. + +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +3 +S2 as the “lock-down” fraction and S1 as the “freedom fraction”. In this picture flows from +S1 to S2 and vice-versa are described by an I-linear (respectively (N −I)-linear) flow with +rate parameters θi, i = 1, 2, modeling social behavior in reaction to published prevalence +data. +Combining both interpretations it turns out to be convenient to start with an +abstract version of a SSISS model staying completely symmetric under interchanging S1 +and S2, see Fig. 1. +The present part I provides a normalization prescription reducing the number of inde- +pendent parameters in this model from initially fourteen to essentially five (four in the +SI(R)S model sub-case). Based on this approach, part II will give a complete review on +equilibria and stability in the master SSISS model, thereby also recovering an exceptional +scenario which had been overlooked in the literature so far. In part III the scaling sym- +metry for SI(R)S models mentioned above will be generalized to the full SSISS model, +thereby reducing the number of parameters again by two. So, the total reduction from +fourteen to three reveals a great hidden redundancy in parameter space. It also provides +a unifying view on results in the literature concerning equilibrium states, endemic bifur- +cation and stability properties for all kinds of sub-classes of this model. Put differently, +in the presence of a common normalized version presenting basically repeated arguments +for various subsets of non-vanishing parameters becomes obsolete. +Relating this work to the literature, let me focus on deterministic SIR-type 3-compartment +dynamical systems, which conveniently may be classified according to +A) constant vs. time-varying total population size N, +B) infection transmission only from I to S vs. also from I to R (in which case it makes +sense to rename S ≡ S1 and R ≡ S2). +Also, I will restrict this survey to models with standard bi-linear incidence flows βiSiI/N, +such that the vector field ˙Y = V(Y), Y = (S1, S2, I), is homogeneous of first order. This +applies to diseases where the number of effective contacts per capita is independent of N. +ad A) Endemic models with constant population have first been constructed by adding +a non-zero balanced birth and death rate to the classic SIR model of (Kermack and +McKendrick 1927). As shown by (Hethcote 1974) (see also (Hethcote 1976, 1989)), in +this way already the simplest model without vaccination and loss of immunity shows +a bifurcation from a stable disease-free equilibrium point (DFE) to a stable endemic +scenario when raising the basic reproduction number R0 above one. Nowadays this is +considered as Hethcote’s classic endemic model. Including linear vaccination and/or loss +of immunity terms and optionally also considering recovery without immunity one ends up +with various types of constant population SI(R)S models without changing this picture, +see for example (Batistela et al. 2021; Chauhan, Misra, and Dhar 2014; Korobeinikov and +Wake 2002; O’Regan et al. 2010). As remarked above (and reviewed in more detail in +Appendix B), the true reason lies in the fact that constant population SI(R)S models with +up to 10 parameters all map to the same normalized 2-parameter version of the classic +endemic model as given in Eq. (1.1). +Models with variable population are mostly studied under the assumption of a constant +(i.e. N-independent) birth flow. Heuristically this may be justified by assuming that +N varies slowly on characteristic epidemic time scales. But truly speaking, as already +pointed out by (Mena-Lorca and Hethcote 1992), this Ansatz rather models a constant +immigration scenario. So in this work I will follow the more natural proposal of modeling +vital dynamics by possibly department dependent constant per capita birth and death + +4 +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +rates. Note that, unless fine tuning parameters, this implies that either N(t) → ∞ or +N(t) → 0 as t → ∞. So in this type of models one always analyzes the dynamics of +fractional variables Si := Si/N, I := I/N, which is well known to be independent of N(t). +Apparently, this stream of models has been initiated by (Busenberg and Driessche 1990, +1991; Derrick and Driessche 1993). (Razvan 2001) has studied a SIRS model in this sense +with infection transmission also from outside and a SIS-version with varying population +size has been analyzed by (J. Li and Ma 2002). For generalizations to SEIR models see +e.g. (Greenhalgh 1997; M. Y. Li et al. 1999; G. Lu and Z. Lu 2018; Sun and Hsieh 2010). +ad B) A different approach to modeling partial and/or waning immunity consists of +introducing a diminished incidence flow with rate βR ≡ β2 > 0 directly from R ≡ S2 +to I. This has presumably first been proposed in the so-called SIRI model of (Derrick +and Driessche 1993), see above. In addition, the authors also introduced a time varying +population size N(t) and an excess mortality ∆µI in compartment I to this model. In turn, +they didn’t use linear vaccination nor immunity waning terms. In this way they identified +a range of parameters in the domain R0 < 1, for which besides the locally asymptotically +stable disease free equilibrium there also coexist two endemic equilibria, one being a +saddle and the other one also being locally asymptotically stable. Later (Hadeler and +Castillo-Chavez 1995) found the same phenomenon in their combined SIS/SIRS core group +model with linear vaccination, constant population and also two incidence rates βi for +S → I and R → I. Meanwhile it is well known that models with infection incidents +from several compartments may show a so-called backward bifurcation from the disease- +free to an endemic scenario (Hadeler and Driessche 1997). This means that two locally +asymptotically stable equilibrium states may coexist for some range below threshold, +causing also hysteresis effects upon varying parameters. Apparently, a varying population +size is not needed for this. In (Kribs-Zaleta and Velasco-Hernandez 2000) the authors have +improved and extended these results by adding also a linear immunity waning rate to the +model of (Hadeler and Driessche 1997). +One may also distinguish vaccinated and recovered people into separate compartments. +This leads to 4-compartment models, where similar results have been obtained by, e.g. +(J. Arino, Mccluskey, and Driessche 2003; Yang, Sun, and Julien Arino 2010). +Backward bifurcation has lately also been observed in SEIRS-type models for Covid- +19 by considering two distinguished susceptible compartments. +In (Nadim and Chat- +topadhyay 2020) the less susceptible compartment had been interpreted as an incomplete +lockdown and in (Diagne et al. 2021) as an incomplete vaccination efficacy. +More recently, in (Avram, Adenane, Basnarkov, et al. 2021; Avram, Adenane, Bianchin, +et al. 2022) the authors have given a thorough stability analysis of an eight parameter +SIRS-type model by adding a varying population size to the model of (Kribs-Zaleta and +Velasco-Hernandez 2000) (apparently without being aware of that paper). +Closing this overview I should also remark that backward bifurcation is also observed +when considering I-dependent contact or recovery rates to model reactive behavior or +infection treatment. However the list of papers on this topic over the last 20 years becomes +too huge to be quoted at this place. +This paper extends the normalization algorithm for constant population SI(R)S models to +models as above, i.e. with time varying population size and/or a non-zero incidence rate +βR ≡ β2 from R ≡ S2 to I. As a starting observation, there is an ambiguity in deriving the +dynamics ˙y = F(y) for fractional variables y = (S1, S2, I), see Appendix A. This allows +choosing the vector field F such that all vital dynamics parameters become redundant, + +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +5 +provided the birth-minus-death rates νi = δi −µi in S1 and S2 coincide, ν1 = ν2 = ν. This +redundancy already reduces the number of parameters in the master SSISS model from +fourteen to eight. More than that, F depends on the incidence rates βi only as a function +of ˜βi = βi + νI − ν, where νI = δI − µI is the birth-minus-death rate in I. Assuming for +simplicity compartment independent birth rates gives ˜βi = βi − ∆µI, where ∆µI denotes +the excess mortality in I. In this way models with variable population, ∆µI > 0, and +absence of a incidence rate from R, β2 = 0, look like models with constant population, +∆µI = 0, and a negative incidence rate β2 = ˜β2 < 0. Conversely, models with positive +incidence rates βi > 0 and excess mortality ∆µI < min{β1, β2} behave like models with +constant population size and incidence rates βi = ˜βi > 0. So, the above classification +schemes A) and B) become blurred and, instead, it is more expedient to view all models +as if they had constant population size and two distinguished and possibly also negative +incidence rates ˜βi ∈ R. +In this way most of the above bench marking 3-compartment models (if necessary after +imposing the constraint ν1 = ν2) become comparable as sub-cases of the master SISS +model, with tilde parameters swallowing all birth and death rates and possibly with +negative incidence rates ˜βi ∈ R. As an example, the models of (Hadeler and Castillo- +Chavez 1995) and (Kribs-Zaleta and Velasco-Hernandez 2000) become isomorphic and +they completely cover the sub-case µ1 = µ2 and 0 < min{˜β1, ˜β2} in (Avram, Adenane, +Bianchin, et al. 2022). Also, apart from an irrelevant boundary case, the complementary +sub-case µ1 = µ2 and 0 > min{˜β1, ˜β2} in (Avram, Adenane, Bianchin, et al. 2022) is +covered by the model of (J. Li and Ma 2002). So, applying the normalization procedure +of this paper, all results in Section 5 and 6 of (Avram, Adenane, Bianchin, et al. 2022) +already follow from the previous literature. A more detailed list of unexpected relations +between the above models is given in Section 2.4. +The plan of this paper is as follows. In Sections 2.1 and 2.2 we pass to fractional com- +partment variables, Si = Si/N and I = I/N, and prove redundancy of all vital dynamics +parameters at the cost of possibly negative incidence rates ˜βi. For convenience, time scale +is also normalized by putting the total expected waiting time in compartment I equal to +one. In this way the number of essential parameters is already reduced from fourteen to +seven. Thus, denoting A the space of essential parameters, we have dim A = 7. +Section 2.3 classifies various useful subsets in parameter space like Aphys ⊂ A, guaran- +teeing forward invariance of the physical triangle +Tphys := {(S1, S2, I) ∈ R3 +≥0 | S1 + S2 + I = 1}, +and Abio ⊂ Aphys, guaranteeing an epidemiological interpretation of parameters by re- +quiring in particular θ1 ≥ 0 ≥ θ2. +Section 2.4 identifies eight examples from the above list of models as sub-cases of the +master SSISS model. In this way we obtain various relations between these models as +indicated above, which apparently have not been recognized before. +In Section 2.5 we adapt methods from (Busenberg and Driessche 1990) to prove ab- +sence of periodic solutions for all parameters non-negative, except βi. The extension to +parameters a ∈ Abio (requiring θ2 ≤ 0) heavily relies on the symmetry results in Section +3 and will be proven in Section 3.3. +Section 3 starts from the observation, that the time-normalized equation of motion for +I takes the generic form ˙I = (Xrep − 1)I, where Xrep = β1S1 + β2S2 is the replacement + +6 +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +number (Hethcote 2000), i.e. +the expected number of secondary cases produced by a +typical infectious individual during its time of infectiousness (nowadays mostly called +effective reproduction number). +A coordinate free formulation of the model naturally +leads to taking (Xrep, I) as independent canonical coordinates3 in the physical triangle +Tphys. In this way, we arrive at formulating the SSISS model as a dynamical system in +(Xrep, I)-space, called the replacement number (RN) dynamics (Section 3.2). +˙Xrep = f(Xrep, I), +˙I = (Xrep − 1)I. +(1.2) +Since f(Xrep, I) turns out to be a 5-parameter quadratic polynomial with no term ∼ X2 +rep, +the number of free parameters is now reduced from seven to five. +The main results of this paper are derived in Section 3.3. Denoting D the new parameter +set, dim D = 5, the above approach yields a surjective submersion A ∋ a �→ x(a) ∈ D. +Moreover, A becomes a principal fibre bundle with respect to a group right action ◁ : +A × GS → A such that x(a ◁ g) = x(a) and D ∼= A/GS. Here GS ⊂ GL+(R2) is the +group acting on (S1, S2) ∈ R2 and leaving S1 + S2 invariant. Eq. (1.2) implies that SSISS +dynamical systems at parameter values a, a′ ∈ A are isomorphic whenever a and a′ are +GS-equivalent, i.e. x(a) = x(a′) or equivalently a′ = a ◁ g for some g ∈ GS. In this way +we also get +- +Absence of periodic solutions also for parameters a ∈ Abio, +- +Conditions under which the social behavior parameters θi can be “gauged to zero”, i.e. +there exists g ∈ GS such that a ◁ g ∈ Aθ=0. +Section 3.4 revisits the examples from the literature within the new formalism and Sec- +tion 4 gives a summary and outlook to parts II and III of this work. Finally, Appen- +dix A provides a normalization prescription for the dynamics of fractional variables in +n-compartment models with linear (i.e. constant per capita) birth and death rates, Ap- +pendix B reviews the scaling symmetry in SI(R)S models introduced in (Nill 2022) and +Appendix C discusses a boundary case in parameter space. +Acknowledgement I would like to thank Florin Avram for encouraging interest and +useful discussions. +2. The SSISS model +This Section starts with proposing an abstract completely symmetrized SSISS model +consisting of three compartments, S1, S2 and I, with total population N = S1 + S2 + I. +Members of I are infectious, members of S1 are highly susceptible (socially active or not +immune) and members of S2 are less susceptible (partly immune or reducing contacts). +The flow diagram between compartments is depicted in Fig. 1. +The parameters in this model may be given the following interpretations +3Here “canonical” is not meant in the sense of Hamiltonian systems. + +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +7 +Figure 1. Completely symmetric flow diagram of the SSISS model. +All pa- +rameters are nonnegative except θ2 ∈ [−α2, 0]. Also q1 + q2 = 1, γ1 + γ2 > 0 +and β1 > β2. Generalizing to compartment dependent birth rates amounts to +replacing δN by δ1S1 + δ2S2 + δII. +α1 +: +Vaccination rate of susceptibles moving from S1 → S2 (assuming +θ1 = θ2 = 0, see below). +α2 +: +Immunity waning rate inducing a flow from S2 → S1 (assuming +θ2 = 0, see below). +βi +: +Number of effective contacts per unit time of a susceptible from Si. +γi +: +Recovery rate from I → Si. +θ1 +: +Willingness to get vaccinated (alternatively to reduce contacts) +given the actual prevalence I/N. +In reality only one of the two +parameters α1 and θ1 should be chosen non-zero. +θ2 +: +Epidemiologically one should restrict to θ2 = 0 or (θ2 = −α2 < 0 +and α1 = 0). In this latter case the meaning of the S2-compartment +is “contact reducing” and α2 = −θ2 parametrizes the readiness to +increase contacts proportional to 1 − I/N. +µi +: +Mortality rate in Si. +µI +: +Mortality rate in I. One could also consider vertical transmission, +in which case µI would be the mortality rate diminished by the rate +of infected newborns. +∆µI +: +Mortality excess ∆µI = µI − µ in case µ1 = µ2 = µ, which will be +assumed most of the time. +δ +: +Rate of not infected newborns. Generalizing to compartment de- +pendent birth rates amounts to replacing δN = δ1S1 + δ2S2 + δII. +qi +: +Split ratio of newborns between S1 and S2, q1 + q2 = 1. In the +reduced-immunity interpretation q2 would be the portion of vacci- +nated newborns. + +8 +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +So in total this model counts 15 independent parameters (12 if we require constant total +population, δi = µi, δI = µI). Epidemiologically all parameters except 0 ≥ θ2 ≥ −α2 +are assumed non-negative and also β2 < β1. A more technical classification of admissible +parameter ranges will be given below. Here is a list of prominent examples in the literature +- +Hethcotes classic 3-parameter endemic model (Hethcote 1974, 1976, 1989) by putting +δ = µi = µI > 0, q1 = 1, β1 > 0, γ2 > 0 and all other parameters vanishing. +- +The 7-parameter SIRS model with time varying population size in (Busenberg and +Driessche 1990), adding to Hethcote’s model an immunity waning rate α2 and allowing +different (constant per capita) mortality and birth rates. +- +The 6-parameter SIRI model of (Derrick and Driessche 1993), replacing the immunity +waning rate α2 in (Busenberg and Driessche 1990) by the incidence rate β2 > 0 and +also requiring µ1 = µ2. +- +An extended 10-parameter constant population SI(R)S (i.e. mixed SIRS/SIS) model +with constant and I-linear vaccination rates α1, θ1, an immunity waning rate α2 and +two recovery flows I ← Si. Hence δi = µi, δI = µI and θ2 = β2 = 04. +- +The 6-parameter isolated core system in (Hadeler and Castillo-Chavez 1995), with +two incidence and recovery rates, βi, γi > 0, a vaccination term α1 > 0 and a constant +population with balanced birth and death rates, δ = µi = µI > 0 and q1 = 1. +- +The 7-parameter vaccination models of (Kribs-Zaleta and Velasco-Hernandez 2000) +adding an immunity waning rate α2 > 0 to the model of (Hadeler and Castillo-Chavez +1995). As we will see in Eq. (2.24) below, due to a redundancy of parameters the two +models actually stay isomorphic. +- +The 8-parameter SIS-model with vaccination and varying population size of (J. Li and +Ma 2002) keeping only θi = γ2 = β2 = 0 and assuming µ1 = µ2 = µ.5 As we will see +in (2.25), after a parameter transformation this model becomes isomorphic to the case +where only θi = 0 and β2 ≤ 0. +- +The 8-parameter SIRS-type model analyzed recently by (Avram, Adenane, Bianchin, +et al. 2022), keeping only γ1 = θ1 = θ2 = q2 = 0 and all other parameters positive. +The authors allow a varying population size by first discussing the general case of all +mortality rates being different and then concentrate on µ1 = µ2 ̸= δ and ∆µI > 0. +Their paper is closest to the present work and in fact initiated it. +In a “zeroth normalization” step I will now show that passing to fractional variables and +requiring δ1 − µ1 = δ2 − µ2 all vital dynamic parameters in the SSISS model become +redundant6. In this way the number of essential parameters reduces from 14 to 8. The +price to pay in the non-constant population case is possibly getting negative incidence +rates βi. +2.1. Constant population. To get a constant population N the birth rates have to obey +δi = µi and δI = µI, or more generally +δ = (µ1S1 + µ2S2 + µII)/N . +(2.1) +In case µ1 = µ2 = µ this would read δ = µ+I∆µI. Heuristically this should be understood +as an approximation for ∆µI/µ ≪ 1. Under this assumption, denoting fractions of the +4Here I have chosen enlarge the conventional setting for SI(R)S models by also allowing θ1 > 0. +5Actually the authors let µ be a function of N, which however disappears when passing to fractional +variables. +6Redundancy of constant per capita birth and death rates may in fact be shown under quite general +assumptions in n-compartment models, see Appendix A. + +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +9 +total population by Si = Si/N and I = I/N and introducing the notations +˜α1 +:= +α1 + q2µ1 , +˜γ1 +:= +γ1 + q1µI , +˜α2 +:= +α2 + q1µ2 , +˜γ2 +:= +γ2 + q2µI , +(2.2) +S = +� +S1 +S2 +� +, +D(β) = +� +β1 +0 +0 +β2 +� +, +E(α) = +� +α1 +−α2 +−α1 +α2 +� +, +˜γ = +� +˜γ1 +˜γ2 +� +(2.3) +the dynamical system described by the flow diagram Fig. 1 becomes +˙S += +− [E( ˜α) + IE(θ) + ID(β)] S + I ˜γ , +(2.4) +˙I += +˜γ(Xrep − 1)I , +˜γ = ˜γ1 + ˜γ2 +(2.5) +Xrep +:= +(β1S1 + β2S2)/˜γ . +(2.6) +Note that ˜γ−1 ≡ (γ1 + γ2 + µI)−1 is the expected waiting time in I and hence Xrep is the +replacement number (Hethcote 2000), i.e. the expected number of secondary cases pro- +duced by a typical infectious individual during its time of infectiousness. In conventional +SI(R)S models, i.e. for β2 = θ2 = 0, the replacement number in the limit S1 = 1 would +become the basic reproduction number r0 = β1/γ. This is why nowadays the replacement +number is mostly called effective reproduction number. Later we will also have the notion +of a reduced reproduction number R0 as the value of Xrep at the disease-free equilibrium. +To avoid misunderstandings, I prefer to keep the various notions of “reproduction num- +bers” for parameters, whereas the replacement number Xrep is considered as a dynamical +variable. +Now obviously, by (2.2), all vital dynamics parameters become redundant and may be +absorbed by redefining αi and γi. Note that this observation is independent of the choice +of βi and θi, i.e. it already holds in a combined SI(R)S model. +2.2. Time varying population. To derive the equations of motion in case of a time vary- +ing population keep compartment dependent per capita birth and death rates δi, δI, µi, µI +constant and put Y = (S1, S2, I), y = N−1Y and +ν ≡ (ν1, ν2, νI) := (δ1 − µ1, δ2 − µ2, δI − µI). +Then ˙y = ˙Y/N − y ˙N/N and ˙N/N = ⟨ν | y⟩. Using S1 + S2 + I = 1 we may rewrite +S1 ˙N/N = S1[ν1 + (ν2 − ν1)S2 + (νI − ν1)I] +S2 ˙N/N = S2[ν2 + (ν1 − ν2)S1 + (νI − ν2)I] +I ˙N/N = I[νI + (ν1 − νI)S1 + (ν2 − νI)S2]. +So now introduce +˜α1 +:= +α1 + q2δ1 , +˜α2 +:= +α2 + q1δ2 , +˜γ1 +:= +γ1 + q1δI , +˜γ2 +:= +γ2 + q2δI , +˜β1 +:= +β1 + νI − ν1 , +˜β2 +:= +β2 + νI − ν2 . +(2.7) +With the same notation as in Eq. (2.3) and e(ν) := +� +ν1 − ν2 +ν2 − ν1 +� +we then get +˙S += +− +� +E( ˜α) + IE(θ) + ID( ˜β) +� +S + I ˜γ + S1S2e(ν) , +(2.8) +˙I += +˜γ(Xrep − 1)I , +(2.9) +Xrep +:= +(˜β1S1 + ˜β2S2)/˜γ , +˜γ := ˜γ1 + ˜γ2. +(2.10) + +10 +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +So, imposing the condition ν1 = ν2 =: ν and putting ∆νI := ν−νI we get e(ν) = 0 and the +equations of motion look exactly as in the case of constant population (2.4)-(2.6). Again +all vital dynamics parameters become redundant and may be absorbed by redefining βi, +αi and γi. The difference this time is that ˜βi = βi − ∆νI may become negative! Thus we +arrive at +Proposition 2.1. Assume ν1 = ν2. +i) +If ∆νI ≤ min{β1, β2} the SSISS model with variable population maps to the model with +constant population. +ii) If ∆νI > min{β1, β2} it maps to the model with min{β1, β2} = 0 and variable popula- +tion with � +∆νI = ∆νI − min{β1, β2}. +iii) If ∆νI = β2 < β1 and θ2 = 0 it becomes the extended SI(R)S model with θ1 ≥ 0 and +two recovery flows I → S1 and I → S2. +Remark 2.2. Note that under the usual assumptions δi = δI = δ and µ1 = µ2 = µ, ∆νI +coincides with the excess mortality in the infectious compartment, ∆νI = µI −µ = ∆µI. +Remark 2.3. The observation that on the level of fractional variables in both scenarios +(constant vs. variable population, the latter provided ν1 = ν2) all vital dynamics param- +eters are redundant seems to be new7. Essential for this is allowing all four parameters +(αi, γi) being positive and βi possibly being negative. The introduction of parameters θi +is not needed to assure this. Redundancy of constant per capita birth and death rates +may in fact be shown under quite general assumptions in n-compartment models, see +Appendix A. +2.3. Classifying parameter space. In this subsection assume ν1 = ν2. Then the re- +formulation in terms of possibly negative incidence rates ˜βi leads to a new classification +scheme identifying seven sectors in this model. For θi = 0 these are labeled by the signa- +tures of ˜β1 + ˜β2 and ˜β1 ˜β2 (in case of a compartment independent birth rate δ equivalently +by the size of the excess mortality ∆µI), see Table 1. For θi ̸= 0 this classification will be +refined in Section 3, Table 3. +To simplify notation, in what follows let me drop the tilde above parameters. The case +β1 = β2 will be ignored, since in this case putting S = S1 + S2 one easily checks that +(S, I) obeys the dynamics of a SIS model, which can immediately be solved by separation +of variables. Also, due to the permutation symmetry 1 ↔ 2, there is no loss assuming +β1 > β2. Next, choosing time scale to be measured in units of γ−1, we may without +loss also put γ = 1. Thus, assume γi ∈ [0, 1] and γ1 + γ2 = 1. So, having started from +fourteen, essentially we are now left with seven free parameters (think of all greek symbols +of dimension [time]−1 being divided by γ). +To further classify the space of admissible parameters some formalism will be needed. Put +C := {(αi, γi, θi) ∈ R6 | α1 + α2 > 0 ∧ γ1 + γ2 = 1} +(2.11) +C+ := C ∩ {(αi, γi) ∈ R4 +≥0} +(2.12) +Csplit := C ∩ {θ1 ≥ 0 ≥ θ2} +(2.13) +Cphys := C+ ∩ {θi + αi ≥ 0 , i = 1, 2} +(2.14) +Cbio := Csplit ∩ Cphys +(2.15) +7As communicated privately this had also been realized recently in a talk by Florin Avram. + +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +11 +Table 1. Seven sectors in the SSISS-model at θi = 0 and for compartment +independent birth rate δ. By Corollary 2.9 Sector I is isomorphic to the models +of (Hadeler and Castillo-Chavez 1995; Kribs-Zaleta and Velasco-Hernandez 2000) +and Sectors III-VII are largely covered by (J. Li and Ma 2002). Sector II is a +mixed SI(R)S model with two recovery flows I → R and I → S. +Sector +sign(˜β1 + ˜β2) +sign(˜β1 ˜β2) +Interval [˜β1, ˜β2] +Excess mortality ∆µI +I ++ ++ +0 < ˜β2 < ˜β1 +∆µI < β2 +II (SIRS) ++ +0 +0 = ˜β2 < ˜β1 +∆µI = β2 +III ++ +− +0 < −˜β2 < ˜β1 +β2 < ∆µI < (β1 + β2)/2 +IV +0 +− +0 < −˜β2 = ˜β1 +∆µI = (β1 + β2)/2 +V +− +− +˜β2 < −˜β1 < 0 +(β1 + β2)/2 < ∆µI < β1 +VI +− +0 +˜β2 < ˜β1 = 0 +β1 = ∆µI +VII +− ++ +˜β2 < ˜β1 < 0 +β1 < ∆µI +Note that for θi = 0 we have C+ = Cphys = Cbio. Denoting +B := {β = (β1, β2) ∈ R2 | β2 < β1}. +(2.16) +the full parameter sets are then given by A := C × B or Ax := Cx × B, respectively. I will +also use obvious notations like Aθ=0 := A ∩ {θi = 0} and Aα≥0 := A ∩ {αi ≥ 0}. +Remark 2.4. In the definition of C in (2.11) the border case α1 = α2 = 0 (i.e. absence of +constant vaccination and waning immunity rates) has been excluded, see Appendix C for +a short discussion. For the body of this paper I will stick with the assumption α1+α2 > 0. +Next, it is easy to check, that for a ∈ Aphys the physical triangle +Tphys := {(S1, S2, I) ∈ R3 +≥0 | S1 + S2 + I = 1} +(2.17) +stays forward invariant under the dynamics (2.8)-(2.9), i.e. on Tphys we have I = 0 ⇒ ˙I = +0 and Si = 0 ⇒ ˙Si ≥ 0. Note that θi + αi ≥ 0 in (2.14) is sufficient but not necessary to +assure this. +Lemma 2.5. In the SSISS model (2.8)-(2.9) the physical triangle stays forward invariant +for all parameters (αi, βi, γi, θi) ∈ Aphys, also including the border case α1 = α2 = 0. +□ +We are now ready to state a main result of this paper. Assuming ν1 = ν2 the normaliza- +tion procedure to be introduced in Section 3 will further reduce the number of essential +parameters from seven to five. This means, SSISS models fall into isomorphy classes map- +ping to the same normalized system. It turns out, that these isomorphy classes coincide +with orbits under a parameter symmetry group GS acting simultaneously on phase P +and parameter space A, such that parameters for the normalized system are naturally +identified as elements of A/GS. +Theorem 2.6. For y = (S1, S2, I)T ∈ R3 and parameter values a = (α, β, γ, θ) ∈ A +denote ˙y = Fa(y) the dynamical system (2.8)-(2.9) with vector field Fa : R3 → R3. Let +GS ⊂ GL+(R2) be the subgroup acting on S ∈ R2 from the left and leaving S1 + S2 +invariant. + +12 +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +i) +Then there exists a free right action ◁ : A × GS → A such that A becomes a principal +GS-bundle and +Fa ◦ Tg = Tg ◦ Fa◁ g , +Tg := +� +� g +0 +0 +0 +0 +1 +� +� , +∀(a, g) ∈ A × GS. +(2.18) +ii) Put j := ( 0 1 +1 0 ) and for g ∈ GS denote ¯g := jgj ∈ GS. Viewing α, γ, θ ∈ R2 as column +vectors and β ∈ B as a row vector and writing a ◁ g = a′ = (α′, β′, γ′, θ′) we have +α′ = ¯g−1α, +θ′ = ¯g−1θ + ϑ +γ′ = g−1γ, +ϑ = +1 +β′ +1 − β′ +2 +� +−(β1 − β′ +1)(β2 − β′ +1) +(β1 − β′ +2)(β2 − β′ +2) +� +β′ = βg +iii) The GS-right action B × GS ∋ (β, g) �→ βg ∈ B is free and transitive and A ∼= +A/GS × B as trivial principal fiber bundles. +iv) Put S′ = g−1S. Then ⟨β|S⟩ = ⟨β′|S′⟩ ≡ Xrep and therefore ˙Xrep = fa(Xrep, I) where +fa = fa◁ g is GS-invariant, i.e. it only depends on A/GS. +v) If θ1 ≥ θ2 or θ1θ2 > 08, then there exists g ∈ GS such that a′ := a ◁ g ∈ Aθ=0, i.e. the +parameters θi may be “gauged to zero”. If in this case a ∈ Abio then also a′ ∈ Abio. +Remark 2.7. As we will see, although the linear transformation Tg preserves the condition +S1 + S2 + I = 1, it does not necessarily leave R3 +≥0 (and hence Tphys) invariant. +Remark 2.8. Since dim GS = 2 we have dim A/GS = dim A − 2. +So, using (Xrep, I) +as independent coordinates in Tphys, the number of essential parameters of the SSISS +dynamical system reduces from seven to five. +Parts i)-iv) of Theorem 2.6 will be proven in Corollary 3.7 and Lemma 3.8 and part v) in +Lemma 3.18. Before coming to this let me close this Section +- +in Subsection 2.4 with shortly revisiting some bench-marking models in the literature +within the present framework, +- +in Subsection 2.5 with proving absence of periodic solutions by optimizing the methods +of (Busenberg and Driessche 1990). +2.4. Examples from the literature. For simplicity, in this subsection let me assume a +compartment independent birth rate δ. Formulating the dynamics for fractional variables +y = (S1, S2, I) there always remains an ambiguity by adding a vectorfield vanishing on +Tphys. In Eqs. (2.8)-(2.9) the vector field F ≡ Fa has the special form +F(y) = My + Γ(y ⊗ y), +⟨1|M = ⟨1|Γ = 0, +(2.19) +where M ∈ R3×3, 1 = (1, 1, 1) and Γ ∈ Hom (R3 ⊗ R3, R3). As is shown in Appendix A, +n-compartment models with at most quadratic terms and population size varying only +due to constant per capita birth and death rates may always be normalized in this way. +Using different conventions bears the risk of overlooking redundancies in parameter space. +Moreover, it also makes it tedious to pin down the differences between (or equivalence of) +various models in the literature. Table 2 shows how the examples quoted at the beginning +of this Section9 compare with each other when mapped to the present set of parameters. +8Actually these conditions are sufficient but not necessary. For a weaker condition see Section 3.3. +9Heth = (Hethcote 1974, 1976, 1989); SIRI = (Derrick and Driessche 1993); BuDr = (Busenberg +and Driessche 1990); SI(R)S = 10-parameter mixed SIRS/SIS model with constant population size and + +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +13 +Table 2. Mapping models in the literature9 expressed in non-normalized vari- +ables (S1, S2, I) to the present choice of parameters. The column # counts the +number of free parameters in the original models. +After passing to fractional +variables (S1, S2, I) and tilde parameters, Eq. (2.2) or Eq. (2.7), and resetting +time scale to ˜γ = 1, the column #eff counts the number of effectively independent +parameters as determined in Eqs. (2.20)-(2.26). +α1 +α2 +β1 +β2 +γ1 +γ2 +δ +µ1 +µ2 +µI +q1 +q2 +# +#eff +Heth +0 +0 +✓ +0 +0 +✓ +δ = µ1 = µ2 = µI +1 +0 +3 +2 +SIRI1 +0 +0 +✓ +✓ +0 +✓ +✓ +µ1 = µ2 +✓ +1 +0 +6 +3 +SIRI2 +0 +0 +✓ +✓ +✓ +0 +✓ +µ1 = µ2 +✓ +0 +1 +6 +3 +BuDr +0 +✓ +✓ +0 +0 +✓ +✓ +✓ +✓ +✓ +1 +0 +7 +5 +SI(R)S +✓ +✓ +✓ +0 +✓ +✓ +δ = µ1 = µ2 = µI +✓ +✓ +7 +4 +HaCa +✓ +0 +✓ +✓ +✓ +✓ +δ = µ1 = µ2 = µI +1 +0 +6 +5 +KZVH +✓ +✓ +✓ +✓ +✓ +✓ +δ = µ1 = µ2 = µI +1 +0 +7 +5 +LM +✓ +✓ +✓ +0 +✓ +0 +✓ +µi = f(N) +✓ +✓ +✓ +8 +5 +AABH1 +✓ +✓ +✓ +✓ +0 +✓ +✓ +µ1 = µ2 +10 +✓ +1 +0 +8 +5 +AABH2 +✓ +✓ +✓ +✓ +✓ +0 +✓ +µ1 = µ2 +10 +✓ +0 +1 +8 +5 +Applying the transformations (2.2) or (2.7), respectively, maps the above 11-parameter +set to the redundancy-free 6-parameter set (˜αi, ˜βi, ˜γi). After resetting time scale to ˜γ ≡ +˜γ1 + ˜γ2 = 1 the classification of the above models looks as follows: +AHeth = Abio ∩ Aθ=0 ∩ {˜α1 = 0 ∧ ˜γ2 > 0 ∧ ˜γ1 = ˜α2 ∧ ˜β2 = 0} +(2.20) +ASIRIi = Abio ∩ Aθ=0 ∩ {˜αi = 0 ∧ ˜γj > 0 ∧ ˜γi = ˜αj, j ̸= i} +(2.21) +ABuDr = Abio ∩ Aθ=0 ∩ {˜α1 = 0 ∧ ˜γ2 > 0 ∧ ˜β2 < 0}11 +(2.22) +ASIRS = Abio ∩ Aθ2=0 ∩ {˜β2 = 0} +(2.23) +AKZVH = Abio ∩ Aθ=0 ∩ {˜β2 > 0} = AHaCa +(2.24) +ALM = Abio ∩ Aθ=0 ∩ {˜β2 < 0 ∧ ˜γ1 > 0} +(2.25) +AAABHi = Abio ∩ Aθ=0 ∩ {˜γj > 0, j ̸= i} +(2.26) +The dimensions of these parameter spaces are displayed in the last column of Table 211. +To verify Eqs. (2.20)-(2.26) the following explanations should suffice. +• +The SIRI model of (Derrick and Driessche 1993) with varying population requires +αi = γ1 = 0. Since for βR > βS the mapping to the SISS model permutes 1 ↔ 2 (i.e. +maps R → S1 and S → S2), if βR < βS we get ˜α1 = 0, ˜α2 = ˜γ1 = δ and ˜γ2 = γ2 > 0, +and if βR > βS we get ˜α2 = 0, ˜α1 = ˜γ2 = δ and ˜γ1 = γ1 > 0. +• +The SIRS model of (Busenberg and Driessche 1990) differs from SIRI by allowing +α2 > 0 and µ1 < µ2, but in turn it requires βS > βR = 0. Thus, we have ˜α1 = 0 +θ2 = β2 = 0; HaCa = core system in (Hadeler and Castillo-Chavez 1995); KZVH = (Kribs-Zaleta and +Velasco-Hernandez 2000); LM = (J. Li and Ma 2002); AABH = (Avram, Adenane, Bianchin, et al. 2022). +BuDr and AABH come in two versions, the subscript 1 refers to βS > βR and 2 to βS < βR. +10 The bulk of results in Section 5 and 6 of (Avram, Adenane, Bianchin, et al. 2022) assumes µ1 = µ2. +11To be comparable Eq. (2.22) refers to the sub-case µ1 = µ2 in (Busenberg and Driessche 1990), so +dim ABuDr = 4. Allowing also an excess mortality µ2 − µ1 > 0 gives #eff = 5 in Table 2. + +14 +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +and ˜γ1 = δ as in SIRI1, but ˜α2 = α2 + δ becomes independent. If, for comparison, we +restrict to µ1 = µ2 = µ then β2 = 0 implies ˜β2 = −∆µI ≤ 0. +• +If q1 > 0 then one of the three parameters (γ1, α2, δ) always becomes redundant. +So the models of (Hadeler and Castillo-Chavez 1995) and (Kribs-Zaleta and Velasco- +Hernandez 2000) are isomorphic, in spite of the latter containing the additional im- +munity waning rate α2. Also, they both satisfy ˜β2 = β2 > 0. +• +Putting q2 = 1 in the SIS-type model of (J. Li and Ma 2002) the mapping (α1, α2, γ1, δ) �→ +(˜αi, ˜γi) is bijective. Also, the authors have defined µi = f(N) and µI = f(N) + ∆µI. +Hence, the only restrictions in this model are ˜β2 = −∆µI < 0 and ˜γ1 > 0. +In summary we get the following conclusions, which apparently have not yet been realized +in the literature. +Corollary 2.9. Assume µ1 = µ2 =: µ and put ∆µI := µI − µ. +i) +For β1 > β2 = ∆µI the SIRI model of (Derrick and Driessche 1993) is isomorphic to +Hethcote’s classic endemic model. +Moreover, restricting to ˜γ1 > 0 and β2 ̸= ∆µI we have +ii) The SIRS-type model of (Busenberg and Driessche 1990) reduces to a sub-case of the +SIS-type model of (J. Li and Ma 2002), which in turn covers Sectors III-VII of the +SSISS model at θi = 0. +iii) The models of (Hadeler and Castillo-Chavez 1995) and (Kribs-Zaleta and Velasco- +Hernandez 2000) are isomorphic and cover Sector I of the SSISS model at θi = 0. +iv) The models of (J. Li and Ma 2002) and (Hadeler and Castillo-Chavez 1995; Kribs- +Zaleta and Velasco-Hernandez 2000) only differ by the sign of ˜β2. +v) Their disjoint union covers the SIRI model of (Derrick and Driessche 1993) and co- +incides with the model of (Avram, Adenane, Bianchin, et al. 2022). +An equivalent formulation of Corollary 2.9 based on normalized parameters and vari- +ables is given in Corollary 3.19 in Section 3.4. +2.5. Absence of periodic solutions. In this subsection I will specify parameter ranges +guaranteeing absence of periodic solutions by optimizing methods from (Busenberg and +Driessche 1990) (see also (Busenberg and Driessche 1991; Derrick and Driessche 1993)) for +the present situation, including θi ̸= 0. To start with, the Busenberg-Driessche version of +the classical Bendixson–Dulac Theorem may be given the following alternative formulation +Lemma 2.10. (Busenberg and Driessche 1990) Let F : R3 → R3 be smooth in a neigh- +borhood of Tphys and assume Tphys forward invariant under the flow of ˙y = F(y). Assume +there exists a smooth scalar function u(y) defined in a neighborhood of Tphys such that +Ψ(y) := ∇ · (uF)(y) − (y · ∇)(u +� +i +Fi)(y) ≤ 0 , +∀y ∈ Tphys +(2.27) +and Ψ(y) < 0 for some y ∈ Tphys. Then in Tphys \ ∂Tphys there exist no periodic solutions, +homoclinic loops or oriented phase polygons of the dynamical system ˙y = F(y). +Proof. Put 1 := (1, 1, 1) and g := y × uF. Then g · F = 0 and ⟨1 | ∇ × g⟩|Tphys = Ψ|Tphys, +where the second identity easily follows from ⟨1 | F⟩|Tphys = 0. Now the claim follows by +Stoke’s Theorem as in the proof of Theorem 4.1 of (Busenberg and Driessche 1990). +□ +Remark 2.11. In Lemma A.1 in Appendix A it is shown that for models with constant +per capita birth and death rates one may always replace F by ˜F obeying F|Tphys = ˜F|Tphys + +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +15 +and ⟨1 | ˜F⟩ = 0 also outside Tphys. So in this case the second term in (2.27) vanishes and +the condition ∇(u˜F) ≤ 0 looks like in the classical Bendixson-Dulac theorem. +As in (Busenberg and Driessche 1990) putting y = (S1, S2, I) and u = 1/(S1S2I) we now +apply this to the dynamical system Eqs. (2.8)-(2.10). We have uF(y) = uMy + uf(y) +where +M = +� +� +−˜α1 +˜α2 +˜γ1 +˜α1 +−˜α2 +˜γ2 +0 +0 +−1 +� +� , +(uf)(y) = +� +� +−( ˜β1 + θ1)/S2 + θ2/S1 + (ν1 − ν2)/I +−( ˜β2 + θ2)/S1 + θ1/S2 + (ν2 − ν1)/I +˜β1/S2 + ˜β2/S1 +� +� . +(2.28) +Here the time scale normalization ˜γ1 + ˜γ2 = 1 is understood. +Theorem 2.12. Under the following conditions there exist no periodic solutions, homo- +clinic loops or oriented phase polygons of the SSISS system (2.8)-(2.10) in Tphys. +i) (˜αi, ˜γi, θi) ∈ R6 +≥0. +ii) (˜αi, ˜γi, θi) ∈ Cbio and ν1 = ν2. +Proof. First note that ˜γ1 + ˜γ2 = 1 implies that the boundary lines {S1 = 0} and {S2 = 0} +cannot both be forward invariant. Hence, ∂Tphys cannot be a phase polygon. Next, the +second term in (2.27) vanishes, because we have ⟨1 | F⟩ = 0 also outside of Tphys. We are +left to compute ∇·(u(y)My) = − � +i̸=j Mi,jyj/yi < 0 and ∇·f = −θ2/S2 +1 −θ1/S2 +2. Part i) +follows by Lemma 2.5 and Lemma 2.10. The proof of part ii) relies on the normalization +formalism of Section 3 and follows from Corollary 3.17. +□ +Remark 2.13. Note that Theorem 2.12ii) doesn’t follow directly from Theorem 2.6, because +there the equivalence transformation Tg need not preserve Tphys, see also Remark 2.7. +Remark 2.14. Usually in the literature on models with constant per capita birth and death +rates the vector field F appears in the form F = FM + f, where FM = My − ⟨1 | My⟩y, +the second term being nonzero. This makes computations more involved but still yields +ΨM|Tphys ≡ ∇ · (uFM)|Tphys − (y · ∇)⟨1 | uFM⟩|Tphys = − � +i̸=j Mi,jyj/yi, see Eq. (3.8) +in (Derrick and Driessche 1993). The fact that M may be chosen to satisfy ⟨1|M = 0 +(Lemma A.1 in Appendix A, see also remark 2.11) is rarely noticed in the literature. +3. Normalization +3.1. Phase space. From now on we drop again the tilde above parameters and also +require ν1 = ν2. To proceed one has to choose suitable coordinates (X, Y ) on a phase space +P ⊃ Tphys. Let’s first do some linear algebra. Put V = R2 and consider S ≡ |S⟩ = +� S1 +S2 +� +, +α ≡ |α⟩ = ( α1 +α2 ), γ ≡ |γ⟩ = ( γ1 +γ2 ), θ ≡ |θ⟩ = +� θ1 +θ2 +� +as elements of V (“ket-” or “column-” +vectors). Denote +e ≡ ⟨e| := (1, 1) , +β ≡ ⟨β| := (β1, β2) +(3.1) +as a basis in the dual space V ∗ (“bra-” or “row-” vectors). Putting L(β, θ) := D(β)+E(θ) +we then have +⟨e|E(α) = 0, +⟨e|L(β, θ) = ⟨β|, +⟨e | γ⟩ = 1 +(3.2) +where ⟨· | ·⟩ denotes the dual pairing V ∗ ⊗ V → R. Generalizing this setting, pick (e, β) +any oriented12 basis in V ∗ and γ ∈ V satisfying ⟨e | γ⟩ = 1. Denote E ⊂ End V the right +12The requirement of being oriented (with respect to a given orientation in V ) is a coordinate free +version of the condition β2 < β1. + +16 +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +ideal anihilated by ⟨e| and L := {L ∈ End V | ⟨e|L = ⟨β|}. On V × R = R3 consider the +dynamical system +˙S = − [E + IL] S + Iγ, +S ∈ V, E ∈ E, L ∈ L, +(3.3) +˙I = (X − 1)I, +I ∈ R, X := ⟨β | S⟩. +(3.4) +Fixing e and varying (β, E, L, γ) under the above constraints defines a 7-parameter dy- +namical system which in fact provides a coordinate free reformulation of the SSISS model +(2.4). Note that the conditions imply ⟨e | ˙S⟩ + ˙I ≡ ˙S1 + ˙S2 + ˙I = 0, so the dynamics +(3.3)-(3.4) leaves the cosets {⟨e | S⟩ + I = const.} ⊂ R3 invariant. Since I = 0 implies +˙I = 0 also the half spaces {I ∈ R±} as well as the plane {I = 0} stay invariant. +Definition 3.1. The dynamical system (3.3)-(3.4) on phase space P = {(S, I) ∈ V ×R≥0 | +⟨e | S⟩ + I = 1} with parameter space A = C × B is called the extended SSISS model. +Remark 3.2. The extension to negative values of variables Si and parameters a is needed +to construct the symmetry operation of GS in Theorem 2.6. +3.2. Canonical coordinates. Putting I := 1 − ⟨e | S⟩ and using S as independent +coordinates on P Eq. (3.4) becomes redundant and we end up with a two-dimensional +system. However, based on the coordinate free formulation (3.3)-(3.4), there is another +natural set of canonical coordinates for this system. Put +X := ⟨β | S⟩, +Y := ⟨e | S⟩, +(3.5) +or equivalently choose the basis dual to (3.1) in V +e⊥ ≡ |e⊥⟩ := +1 +β1 − β2 +� +1 +−1 +� +, +β⊥ ≡ |β⊥⟩ := +1 +β1 − β2 +� +−β2 +β1 +� +(3.6) +Hence we have X ≡ Xrep, Y ≡ S1 + S2 and +S = Xe⊥ + Y β⊥. +(3.7) +Lemma 3.3. In canonical coordinates the extended SSISS model becomes +˙X += +(−aX + b) + (−cX + d)I − ϵI2 , +(3.8) +˙Y += +(1 − X)I = − ˙I , +(3.9) +where I = 1 − Y and where the new parameters are given by +a := α1 + α2 +(3.10) +b := α2β1 + α1β2 +(3.11) +c := β1 + β2 + θ1 + θ2 +(3.12) +d := γ1β1 + γ2β2 − b + ϵ +(3.13) +ϵ := β1β2 + β1θ2 + β2θ1 . +(3.14) +Proof. In canonical coordinates the matrices E(α) and L(β, θ) := D(β) + E(θ) take the +normal form +E(α) = +� +a +−b +0 +0 +� +, +L(β, θ) = +� +c +−ϵ +1 +0 +� +. +(3.15) +Using |γ⟩ = (β1γ1 +β2γ2)|e⊥⟩+|β⊥⟩ the claim follows by straightforward calculation. +□ + +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +17 +The canonical form of the SSISS dynamical system (3.8)-(3.9) will also be called the RN- +dynamical system (RN = replacement number). Beware that unless β2 ≥ 0 the “would-be” +replacement number X may take negative values even for Si ≥ 0. In fact, in canonical +coordinates the physical triangle takes the form +Tphys(β) = {(X, Y ) ∈ R × [0, 1] | β2Y ≤ X ≤ β1Y } += {(X, I) ∈ R × [0, 1] | β2(1 − I) ≤ X ≤ β1(1 − I)}. +(3.16) +So in (X, I)-space Tphys is given by the corners T< = (β2, 0), T> = (β1, 0) and T∧ = (0, 1). +To stay with epidemiological conventions, from now on I will use X ≡ Xrep and I ≡ 1−Y +as independent variables, in terms of which phase space is now given by +P = {(X, I) ∈ R × R≥0}. +Also note that in canonical coordinates the dynamics is reduced from seven to five pa- +rameters, i.e. the system no longer depends on β. So, the role of β is reduced to fixing +the image of physical triangles Tphys in canonical coordinates. Equivalently this means +that fixing x = (a, b, c, d, ϵ) and varying β ∈ B we get an equivalence class of isomorphic +dynamical systems, albeit physical triangles are not mapped onto each other under these +isomorphisms. +Proposition 3.4. For a, a′ ∈ A, a = (α, β, γ, θ) and a′ = (α′, β′, γ′, θ′), assume x(a) = +x(a′). Following Eq. (3.7) put +S := Xe⊥(β) + (1 − I)β⊥ , +S′ := Xe⊥(β′) + (1 − I)β′⊥ . +(3.17) +Then S1 + S2 = S′ +1 + S′ +2 = 1 − I and S = gS′ where g ∈ GL+(R2) is uniquely defined by +g = |β⊥⟩⟨e| + |e⊥(β)⟩⟨β′| = +1 +β1 − β2 +� +β′ +1 − β2 +β′ +2 − β2 +β1 − β′ +1 +β1 − β′ +2 , +� +(3.18) +implying det g = (β′ +1 − β′ +2)/(β1 − β2) > 0. Moreover, (S, I) satisfies the SSISS dynamics +(3.3)-(3.4) at parameter values a iff (S′, I) satisfies it at parameter values a′. +Proof. Eq. (3.17) implies ⟨e|S⟩ = ⟨e|S′⟩ = 1−I and ⟨β|S⟩ = ⟨β′|S′⟩ = X. Hence, g must +satisfy ⟨e|g = ⟨e| and ⟨β|g = ⟨β′| with unique solution (3.18). +□ +Remark 3.5. Apparently we have g ∈ GS := {g ∈ GL+(R2) | ⟨e|g = ⟨e|} and by Eq. +(3.18) β �→ βg defines a transitive and free right action of GS on B13. In Corollary 3.7 +below this action will be transported to a free GS-action on A, thus proving parts i)-iv) +of Theorem 2.6. +3.3. Main results. In this subsection we study the constraints on the new parameters +x := (a, b, c, d, ϵ) and admissible ranges of β - or equivalently Tphys(β) - for given values +of x, which will finally lead to a proof of Theorems 2.6 and 2.12. Recalling A ≡ C × B +denote +φ : A ∋ a �→ (x(a), β) ∈ D × B, +D := R+ × R4 +(3.19) +where x(a) is given by (3.10)-(3.14). The proof of the following Lemma is by straight +forward calculation and hence omitted. +13Note that dim GS = 2. +The parametrization of g in (3.18) is redundant by invariance under +(β1, β2) �→ (β1 + λ, β2 + λ) and (β1, β2) �→ (χβ1, χβ2), (λ, χ) ∈ R × R+. + +18 +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +Lemma 3.6. The map φ : A → D × B provides a diffeomorphism with φ−1 given by +αi = b − aβi +βj − βi +, +γi = d + b − ϵ − βj +βi − βj +, +θi = β2 +i − cβi + ϵ +βj − βi +, +j ̸= i +(3.20) +□ +Corollary 3.7. Consider D×B as a trivial principal GS-bundle with fiber B and GS right +action (x, β) ◁ g := (x, βg), see Remark 3.5. Defining a ◁ g := φ−1(x(a), βg) we get an +isomorphic GS-bundle structure on A. Putting y := (S1, S2, I) and writing the dynamical +system (3.3)-(3.4) with parameters a ∈ A as ˙y = Fa(y), Proposition 3.4 becomes +Fa ◦ Tg = Tg ◦ Fa◁ g , +Tg := g ⊕ id , +g ∈ GS. +This proves parts i), iii) and iv) of Theorem 2.6. +□ +The remaining transformation rules in part ii) of Theorem 2.6 now boil down to an exercise +in linear algebra. +Lemma 3.8. Let D(β) and E(α) be given as in Eq. (2.3) and ϑ(β, β′) as in part ii) of +Theorem 2.6. Then for all g ∈ GS, α ∈ R2 and β′ = βg ∈ B +E(¯gα)g = gE(α), +D(β)g = g [D(β′) + E(ϑ(β, β′))] +Applying these identities to the dynamical system (3.3)-(3.4) proves Theorem 2.6ii). +□ +Remark 3.9. Beware that the transformation matrix g preserves S1+S2 but not necessarily +R2 +≥0. +Also, if a ∈ Aphys (or Abio) and x(a) = x(a′) then it depends on β′ whether +a′ ∈ Aphys (or Abio), see Proposition 3.15 below. Hence, the above equivalencies may +produce scenarios where a ∈ Aphys and a′ = a ◁ g ̸∈ Aphys and T−1 +g Tphys ̸∈ R3 +≥0 but still +T−1 +g Tphys is forward invariant under the flow of Fa′. +Next, on D define the functions +R0(x) := b/a +≡ α2β1 + α1β2 +α1 + α2 +, +(3.21) +R1(x) := d + b − ϵ ≡ γ1β1 + γ2β2 . +(3.22) +Obviously we may also use x ≡ (a, R0, R1, c, ϵ) ∈ R+ × R4 as independent parameters in +D. Moreover we clearly have +φ(A+) = {(x, β) ∈ D × B | β2 ≤ Ri ≤ β1 , i = 1, 2} , +(3.23) +i.e. on A+ the functions Ri may be interpreted as two kinds of mean values of β1 and β2. +Again beware that for β2 < 0 we may have Ri < 0 even on A+. To explain the meaning +of R0 note that for a > 0 the value of the replacement number X at the disease-free +equilibrium (DFE) of the RN-dynamical system (3.8)-(3.9) is precisely given by X∗ +0 = R0. +Following results of (Driessche and Watmough 2002) this leads to +Definition 3.10. R0 is called the reduced reproduction number. +Remark 3.11. As has been shown by (Driessche and Watmough 2002, 2008), in models +with just one infectious compartment the more general notion of R0 as the spectral +radius of the next generation matrix ((Diekmann, Heesterbeek, and Metz 1990), see also +(Diekmann and Heesterbeek 2000)) reduces to the above definition. Denoting the values + +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +19 +of Si at the DFE by S∗ +i we have R0 = β1S∗ +1 + β2S∗ +2, which is the usual formula, see e.g. +(Kribs-Zaleta and Velasco-Hernandez 2000) or (Avram, Adenane, Bianchin, et al. 2022). +Remark 3.12. Mostly in the literature R0 is called the basic reproduction number. But in +case β2 = 0 this terminology is already occupied by r0 := β1/γ as the expected number +of secondary cases produced by a typical infectious individual in a totally susceptible +population. So to avoid confusion I prefer to call R0 the reduced reproduction number. +Next put Dx := πD(φ(Ax)), x = phys or x = bio, where πD : D × B → D denotes +the canonical projection. We look for suitable coordinates describing Dx and then derive +additional bounds on β to describe φ(Ax). Consider the following functions on D. +A±(x) := 1 +2 +� +a + c ± +� +(a + c)2 − 4(b + ϵ) +� +, +(3.24) +B±(x) := 1 +2 +� +c ± +√ +c2 − 4ϵ +� +. +(3.25) +Then by (3.15) and the trace-det formula A± and B± provide the eigenvalues of E + L +and L, respectively. The meaning of these eigenvalues becomes clear by looking at (3.20) +β1 = A+ ⇔ α1 + θ1 = 0 +β1 = B+ ⇔ θ1 = 0 +(3.26) +β2 = A− ⇔ α2 + θ2 = 0 +β2 = B− ⇔ θ2 = 0 +(3.27) +βi = R0 ⇔ αi = 0 +βi = R1 ⇔ γj = 0, j ̸= i +(3.28) +More generally from (3.20) we get +θi = (βi − B−)(βi − B+) +βj − βi +, +j ̸= i , +(3.29) +αi + θi = (βi − A−)(βi − A+) +βj − βi +, +j ̸= i . +(3.30) +Hence A± will serve to fix the constraints on (x, β) ∈ φ(Aphys) and B± (B ≡ “bio”) to fix +constraints on (x, β) ∈ φ(Abio). First we gather some trivial identities. +c = B+ + B− = A+ + A− − a , +ϵ = B+B− = A+A− − b , +(3.31) +a = A+ + A− − B+ − B− , +aR0 ≡ b = A+A− − B+B− +(3.32) +From these one immediately computes +a(A± − R0) = (A± − B+)(A± − B−) = A2 +± − cA± + ϵ +(3.33) +a(R0 − B±) = (B± − A+)(B± − A−) = B2 +± − (a + c)B± + (b + ϵ) +(3.34) +Now let’s introduce the notation +DA := D ∩ {A± ∈ R} +(3.35) +DB := D ∩ {B± ∈ R ∧ B− < B+} +(3.36) +DAB := DA ∩ DB ∩ {B− ≤ A− ≤ B+ ≤ A+}. +(3.37) +Lemma 3.13. The following identities hold +DAB = {x ∈ DA | A− ≤ R0 ≤ A+ ∧ c2 ̸= 4ϵ} = {x ∈ DB | B− ≤ R0 ≤ B+} +(3.38) +Hence in DAB we always have the additional bound +B− ≤ A− ≤ R0 ≤ B+ ≤ A+ . +(3.39) + +20 +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +Proof. By Eqs. (3.33) and (3.34) on DAB we always have A− ≤ R0 ≤ A+ and B− ≤ +R0 ≤ B+. +Conversely, if A± ∈ R, c2 ̸= ϵ and A− ≤ R0 ≤ A+ then (3.33) implies +A2 +− − cA− + ϵ ≤ 0 ≤ A2 ++ − cA+ + ϵ and therefore c2 > 4ϵ. Hence B− < B+ ∈ R and again +by (3.33) B− ≤ A− ≤ B+ ≤ A+. The second identity follows analogously. +□ +Lemma 3.14. Denoting iB : DB ∋ x �→ (x, B+, B−) ∈ DB × B the following identities +hold +φ(Asplit) = {(x, β) ∈ DB × B | B− ≤ β2 < β1 ≤ B+} , +(3.40) +φ(Aθ+α≥0) = {(x, β) ∈ DA × B | β2 ≤ A− ≤ β1 ≤ A+} . +(3.41) +φ(Aθ=0) = iB(DB) +(3.42) +φ(Aθ=0 ∩ Aα≥0) = iB(DAB) +(3.43) +Proof. We have B± ∈ R iff there exists β ∈ R such that β2 − cβ + ϵ ≤ 0. Hence, by +(3.20), if θ1 ≥ 0 and θ2 ≤ 0 then B± ∈ R and B− ≤ β2 < β1 ≤ B+, proving the “⊂”-part +in (3.40). The opposite direction follows from (3.29). Similarly, A± ∈ R iff there exists +β ∈ R such that β2 − (a + c)β + b + ϵ ≤ 0. Hence, by (3.20), if α1 + θ1 ≥ 0 then A± ∈ R +and A− ≤ β1 ≤ A+. If in addition α2 + θ2 ≥ 0 then also β2 ≤ A−, proving the “⊂”-part +in (3.41). The opposite direction follows from (3.30). Eq. (3.42) follows since in Aθ=0 we +have β1 = B+ and β2 = B−. If in addition αi ≥ 0 then (3.30) implies Eq. (3.43). +□ +We are now in the position to summarize the constraints describing φ(Aphys) and φ(Abio). +Proposition 3.15. For Ax = Cx × B as defined in (2.14) - (2.15) we have +φ(Aphys) ≡ φ(A+) ∩ φ(Aθ+α≥0) += (DA × B) ∩ {β2 ≤ {A−, R0, R1} ≤ β1 ≤ A+} , +(3.44) +Dphys = DA ∩ {R0,1 ≤ A+} , +(3.45) +φ(Abio) ≡ φ(Asplit) ∩ φ(Aphys) += (DAB × B) ∩ {B− ≤ β2 ≤ A− ≤ R0 ≤ β1 ≤ B+} ∩ {R1 ∈ [β2, β1]} , +(3.46) +Dbio = DAB ∩ {R1 ∈ [B−, B+]} ⊂ Dphys . +(3.47) +Proof. This is a summary of Eq. (3.23) and Lemmas 3.13 - 3.14. +□ +Proposition 3.15 motivates the following notation and definition +Definition 3.16. For x ∈ Dbio put +βmax +2 +(x) := min{A−, R1}, +βmin +1 +(x) := max{R0, R1}. +(3.48) +Then β ∈ B is called bio-compatible with x if B− ≤ β2 ≤ βmax +2 +and βmin +1 +≤ β1 ≤ B+, +equivalently if φ−1(x, β) ∈ Abio. +Similarly, β is called compatible if β2 ≤ βmax +2 +and +βmin +1 +≤ β1 ≤ A+, equivalently if φ−1(x, β) ∈ Aphys. A physical triangle Tphys(β) is called +(bio)-compatible, if β is (bio)-compatible. +Hence, (bio-)compatible physical triangles are always forward invariant under the RN- +dynamics (3.8)-(3.9) and the smallest one is just Tphys(βmin +1 +, βmax +2 +). The following Corollary +also proves part ii) of Theorem 2.12. + +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +21 +Corollary 3.17. Let x ∈ Dbio and let β ∈ B be compatible with x. Then there exist +no periodic solutions, homoclinic loops or oriented phase polygons of the RN-dynamical +system (3.8)-(3.9) in Tphys(β). +Proof. Let Z ⊂ Tphys(β) be a solution cycle (image of a periodic solution, a homoclinic +loop or an oriented phase polygon). As argued in the proof of Theorem 2.12, we must +have Z ̸= ∂Tphys(β′) for all β′ ∈ B. Hence, by forward invariance, Z must lie inside the +smallest compatible triangle, Z ⊂ Tphys(βmin +1 +, βmax +2 +) ⊂ Tphys(B+, B−). But, by Proposition +3.15 and Eq. +(3.43), φ−1(x, B+, B−) ∈ Abio ∩ Aθ=0 and we get a contradiction with +Theorem 2.12i). +□ +Finally, to prove Theorem 2.6v), note that Lemma 3.14 and Proposition 3.15 in particular +imply (use that GS acts transitively on B) +Aθ=0 ◁ GS = Asplit ◁ GS = φ−1(DB × B) +(3.49) +Aθ+α≥0 ◁ GS = φ−1(DA × B) +(3.50) +(Aθ=0 ∩ Aα≥0) ◁ GS = φ−1(DAB × B) +(3.51) +Aphys ◁ GS = φ−1(Dphys × B) +(3.52) +Abio ◁ GS = φ−1(Dbio × B) +(3.53) +Aθ=0 ∩ Abio ◁ GS ⊂ Abio +(3.54) +where the last equation follows from (Dbio × B) ∩ iB(DB) = iB(Dbio) ⊂ φ(Abio). Part v) +of Theorem 2.6 now follows from Eqs. (3.49), (3.54) and Lemma 3.18 below. +Lemma 3.18. Put AB := φ−1(DB × B) = Aθ=0 ◁ GS, then +AB ⊃ A ∩ {θ1 ≥ θ2 ∨ θ1θ2 > 0} ⊃ Asplit ⊃ Abio. +Proof. The second and third inclusions are obvious from the definitions (2.13) and (2.15) +and the first inclusion follows from DB = D ∩ {c2 > 4ϵ} and +c2 − 4ϵ = (β1 − β2)2 + (θ1 + θ2)2 + 2(β1 − β2)(θ1 − θ2) += (β1 − β2 + θ1 − θ2)2 + 4θ1θ2. +□ +Table 3. Sector classification in Abio generalizing Table 1. +Sector +c = B− + B+ +ϵ = B−B+ +Interval [B−, B+] +I ++ ++ +0 < B− < B+ +II (SIRS) ++ +0 +0 = B− < B+ +III ++ +− +0 < −B− < B+ +IV +0 +− +0 < −B− = B+ +V +− +− +B− < −B+ < 0 +VI +− +0 +B− < B+ = 0 +VII +− ++ +B− < B+ < 0 + +22 +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +Let me close by mentioning that the parametrizations (3.31) can now be used to generalize +the Sector classification of Table 1 from the special case Aθ=0 to all of Abio (more generally +to AB := φ−1(DB × B) ⊃ Abio) as shown in Table 3. +3.4. Examples revisited. For completeness let us revisit the examples in Section 2.4 +within the present setting. Eqs. (2.20)-(2.26) translate into14 +DHeth = Dbio ∩ {R0 = B+ ∧ a < 1 ∧ d = B− = 0} +(3.55) +DSIRI1,2 = Dbio ∩ {R0 = B± ∧ a < 1 ∧ d = B∓(B± + 1 − a)} +(3.56) +DBuDr = Dbio ∩ {R1 < R0 = B+ ∧ B− < 0}15 +(3.57) +DSIRS = Dbio ∩ {B− = 0} +(3.58) +DLM = Dbio ∩ {B− < min{0, R1}} +(3.59) +DKZVH = Dbio ∩ {B− > 0} = DHaCa +(3.60) +DAABH1 = Dbio ∩ {B− ≤ R1 < B+}15 +(3.61) +DAABH2 = Dbio ∩ {B− < R1 ≤ B+}15 +(3.62) +Note that all models except SI(R)S already satisfy θi = 0 whence ˜β1 = B+, ˜β2 = B− +by Eqs. (3.26)-(3.28). In the SI(R)S model we have instead 0 = ˜β2 = B− < ˜β1 ≤ B+. +Corollary 2.9 may now be reformulated as follows +Corollary 3.19. Referring to the sub-cases µ1 = µ2 in (Avram, Adenane, Bianchin, et al. +2022; Busenberg and Driessche 1990) and putting DAABH := DAABH1 ∪ DAABH2 we have +DHeth = DSIRI1 ∩ {B− = 0} +(3.63) += DSIRS ∩ {a < 1 ∧ R0 = c ∧ d = 0} +(3.64) +DLM ⊃ DBuDr ∩ {B− ̸= R1} +(3.65) +DLM = DAABH2 ∩ {B− < 0} +(3.66) +DKZVH = DAABH ∩ {B− > 0} +(3.67) +Finally, we are now in the position to generalize the scaling symmetry for SI(R)S models +of (Nill 2022) to the present setting. First note that having started from the 10-parameter +extended SI(R)S model we now have arrived at dim DSIRS = 4. Also, dim DHeth = 2 with +independent parameters a ∈ (0, 1) and c = R0 = B+ > 0. In particular, if x ∈ DHeth +then putting (u, v) := (X, cI) the RN-dynamical system (3.8)-(3.9) reduces to the classic +endemic model in Eq. (1.1). In a second normalization step the number of parameters in +the SI(R)S case may now be reduced again by two. In this way, for c > d,16 the normalized +SI(R)S model also looks like the classic endemic model +˙u = −uv − c1u + c2 , +˙v = uv − v , +(3.68) +the difference being that coming from DHeth we have c1 = a ∈ (0, 1) and c2 = aR0 ≥ 0, +whereas coming from DSIRS gives (c1, c2) ∈ R+ × R17. However, since endemic bifurcation +14Heth = (Hethcote 1974, 1976, 1989); SIRI = (Derrick and Driessche 1993); BuDr = (Busenberg +and Driessche 1990); SIRS = 10-parameter mixed SIRS/SIS model with constant population size and +θ2 = β2 = 0; HaCa = core system in (Hadeler and Castillo-Chavez 1995); KZVH = (Kribs-Zaleta and +Velasco-Hernandez 2000); LM = (J. Li and Ma 2002); AABH = (Avram, Adenane, Bianchin, et al. 2022). +BuDr and AABH come in two versions, the subscript 1 refers to βS > βR and 2 to βS < βR. +15Referring to the sub-case µ1 = µ2 in these models, see Footnotes 10 and 11. +16Note that in DSIRS we have c = B+ ≥ R1 − aR0 = d where equality implies R0 = 0 and R1 = B+. +17In case a = 0 we would get c1 = c2 = 0. + +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +23 +in the model (3.68) occurs at R0 = c2/c1 = 1, extending this model to the SI(R)S case +by including also values c2 < 0 and c1 ≥ 1 doesn’t change its characteristic behavior. +In particular, various proofs in the literature on variants of constant population SI(R)S +models with standard incidence become obsolete, it’s all contained in Hethcote’s work. +Eq. (3.68) is proven in Appendix B. In principle, the proof relies on the same structure +as in Theorem 2.6, with the symmetry group GS acting on A replaced by a dilatation +group Gdil = R2 ++ acting on D. Since these dilatations may blow up physical triangles to +arbitrary size, we also get the following +Lemma 3.20. For x ∈ DSIRS the forward flow of the RN-dynamical system (3.8)-(3.9) +stays bounded for all initial conditions (X0, I0) ∈ R × R≥0. +This result may be used to prove, that SI(R)S models as above are always Hamiltonian +(Nill n.d.[a]). Lemma 3.20 is also proven in Appendix B. +4. Summary and outlook +In summary we have seen, that in canonical coordinates the 14-parameter SSISS model, +constraint by ν1 = ν2, effectively depends on at most five parameters x = (a, b, c, d, ϵ). De- +pending on natural model restrictions like “phys” or “bio” these parameters obey various +relations which can be encoded by further reparametrizations like x = (a, R0, R1, B+, B−), +see Eqs. (3.21), (3.22), (3.31) and Proposition 3.15. The incidence rates βi have disap- +peared from the equations of motion. Their role is reduced to fixing physical triangles +Tphys(β) in (X, I)-space, see Eq. +(3.16). +If x ∈ Dbio, then for all compatible values +β = (β1, β2) the triangles Tphys(β) stay forward invariant under the RN-dynamics (3.8)- +(3.9). Independence of β also means that SSISS models at parameter values φ−1(x, β) +for fixed x ∈ D and varying β ∈ B are all isomorphic to each other18 (Proposition 3.4). +The isomorphisms are provided by a parameter symmetry group GS ⊂ GL+(R2) acting +simultaneously on phase space P and parameter space A (Theorem 2.6i-iv). If x ∈ DB +then a representative in A of the equivalence class x may always be chosen by putting +β1 = B+ and β2 = B− and hence θi = 0 (Theorem 2.6v). In combination with methods +from (Busenberg and Driessche 1990) this also leads to a proof of absence of periodic +solutions for all a ∈ Abio (Theorem 2.12). +In part III of this work it will be shown, that the model also admits an additional scaling +symmetry leading to a second normalization step, similar as described for the SI(R)S +model in Appendix B, see also (Nill 2022). In this way the number of essential parameters +will further reduce from five to three (respectively two in Sectors II and VI). +Part II of this work will reanalyze equilibrium points and their stability properties in +all Sectors of Abio, thereby recovering and extending the results of (Avram, Adenane, +Bianchin, et al. 2022; Hadeler and Castillo-Chavez 1995; Kribs-Zaleta and Velasco-Hernandez +2000; J. Li and Ma 2002), which had been obtained for θi = 0 and some more parameter +restrictions, see Table 2 and Corollary 2.9/3.19. This approach will differ from previ- +ous papers by relying on the normalization formalism and sector classification of the +present work. In this way the search for endemic equilibria (X∗, I∗) simplifies consider- +ably, since always X∗ = 1. So one is left with analyzing roots of the quadratic equation +h(I∗) := ˙X(X∗ = 1, I∗) = 0. This will also uncover an exceptional scenario in Sectors +III-V, which apparently has been overlooked in the literature so far. +18By Remark 3.9, physical triangles are not mapped onto each other under these isomorphisms. + +24 +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +Appendix A. Normalizing linear vital dynamics +This Appendix gives a normalization prescription for the dynamics of fractional vari- +ables in an n-compartment model with linear vital dynamics. Let the vectorfield V : +Rn → Rn be homogeneous of degree one and assume there exists ν = (ν1, · · · , νn) such +that ⟨1|V(Y)⟩ ≡ � +i Vi(Y) = ⟨ν|Y⟩ for all Y ∈ Rn, where 1 := (1, · · · , 1). +Call +N(Y) := ⟨1|Y⟩ the total population and y := N−1Y the fractional compartment vari- +ables, then the dynamical system ˙Y = V(Y) implies +˙y = V(y) − ⟨ν | y⟩y =: F(y). +Denote S := {y ∈ Rn | ⟨1|y⟩ = 1}, then clearly ⟨1|F⟩|S = 0. The aim is to substitute F +by ˜F such that F|S = ˜F|S and ⟨1|˜F⟩ = 0 holds as an identity on all of Rn. The following +Lemma holds by straight forward calculation. +Lemma A.1. Put Λijk := (δij − δik)(νk − νj) and Λi(y) := � +j,k Λijkyjyk. +i) +For all y ∈ Rn and i = 1, · · · , n we have +1 +2Λi(y) = +� +k +(νk − νi)yiyk ≡ yi⟨ν|y⟩ − νiyi⟨1|y⟩. +(A.1) +ii) Put +˜F := V − diag(ν) − 1 +2Λ. +(A.2) +Then F|S = ˜F|S and ⟨1|˜F⟩ = 0 as an identity on Rn. +By this method we also get conditions guaranteeing that constant per capita birth and +death rates become redundant as in Eq. (2.7). +Lemma A.2. Let V(Y) be of the form +Vi(Y) = +� +j +MijYj + 1 +2 +� +j,k +ΓijkYjYk/N + +� +j +LijYj +where without loss Γijk = Γikj and where � +i Mij = � +i Γijk = 0. Hence, all vital dynamics +parameters are encoded in (Lij) and νj := � +i Lij satisfies ⟨1|V = ⟨ν|. If in this case +Lij ̸= νiδij ⇒ Mij ̸= 0 and νj ̸= νk ⇒ (Γjjk ̸= 0 ∧ Γkkj ̸= 0), then for the dynamics of +fractional variables all parameters Lij are redundant. +Proof. Applying (A.2) we have ˜Fi(y) = � +j ˜ +Mijyj+ 1 +2 ˜Γijkyjyk, where ˜ +Mij = Mij+Lij−νiδij +and ˜Γijk = Γijk − Λijk. The claim follows since Λijk = Λikj, Λjjk = −Λkkj and Λijk = 0 if +νj = νk or if j ̸= i ̸= k, which also yields � +i Λijk = 0 . +□ +Appendix B. Scaling the SI(R)S model +In this appendix we extend the dilatation symmetry as proposed for a 6-parameter +SI(R)S model in (Nill 2022) to the 10-parameter extended SI(R)S model as classified in +this paper. Denote Sector II in DB by DII := DB ∩ {B− = 0} and DSIRS := DII ∩ Dbio. +Recall that in DII we have c = B+ > 0 and in DSIRS we have 0 ≤ Ri ≤ B+ and hence +d − c = R1 − aR0 − B+ ≤ 0, where equality implies R0 = 0 and R1 = B+. Hence the +following Lemma in particular includes Lemma 3.20. + +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +25 +Lemma B.1. Consider the RN-dynamical system (3.8) - (3.9) on phase space P ≡ R×R≥0 +for parameter values x = (a, b, c, d, ϵ = 0) ∈ DII ∩ {d ≤ c ∧ d = c ⇒ R0 < 1} ⊃ DSIRS. +Let T ⊂ P be a rectangular triangle with corners T◁ = (X◁, 0), T▷ = (X▷, 0) and T△ = +(X◁, I△), where X◁ < X▷. Call T compatible with x if +I△ = (X▷ − X◁)/c +X◁ ≤ min{R0, d/c} +R0 − X◁ ≤ I△ min{c, (c − d)/a} +i) +Then every x-compatible triangle T is forward invariant. +ii) The forward flow for arbitrary initial conditions (X0, I0) ∈ P stays bounded. +Proof. To prove part i), the upper bounds on X◁ imply ˙X > 0 on the line {X = X◁}. We +are left to show ˙X + c ˙I ≤ 0 on the hypotenuse X(I) = X◁ + c(I△ − I), 0 ≤ I ≤ I△. +˙X + c ˙I = a(R0 − X(I)) + (d − c)I += a(R0 − X◁ − c(I△ − I)) + (d − c)I +≤ I△ min{ac, c − d} − ac(I△ − I) + (d − c)I +≤ 0 +Part ii) follows since for d < c we may always choose X◁ < X0 and X▷ large enough, +such T is x-compatible and (X0, I0) ∈ T . For d = c and R0 < 1 x-compatibility requires +X◁ = R0. If in this case X0 < R0 glue the rectangle R = [X0, R0]×[0, I△] to the left of T . +Then (X0, I0) ∈ R ∪ T for X▷ large enough and R ∪ T is forward invariant, since ˙I < 0 +and ˙X > 0 for (X, I) ∈ R. +□ +Given x ∈ DII ∩ {d ≤ c ∧ d = c ⇒ R0 < 1} as above and T compatible with x +we now show that the RN-dynamical system (3.8) - (3.9) may always be rescaled to an +isomorphic system with parameters x′ ∈ DSIRS such that T maps to the physical triangle +Tphys(B′ ++, 0) of the SI(R)S system. Following (Nill 2022) the dilatation symmetry group +Gdil ≡ GX × GI ≡ R2 ++ is defined by rescaling (X, I) variables according to +X(ξ,λ)(t) − 1 := ξ(X(ξt) − 1), +I(ξ,λ)(t) := λI(ξt), +(ξ, λ) ∈ R2 ++ +The following Lemma is easily verified by straightforward calculation. +Lemma B.2. Let the group action ▷ : Gdil × D ∋ (ξ, λ, x) �→ (ξ, λ) ▷ x ∈ D be given by +(ξ, λ) ▷ (a, R0 − 1, c, d − c, ϵ) := (ξa, ξ(R0 − 1), ξc/λ, ξ2(d − c)/λ, ξ2ϵ/λ2) +(B.1) +and for x ∈ D let fx(X, I) denote the vector field of the system (3.8) - (3.9). Then +( ˙X, ˙I) = fx(X, I) ⇐⇒ ( ˙X(ξ,λ), ˙I(ξ,λ)) = fx′(X(ξ,λ), I(ξ,λ)), +x′ = (ξ, λ) ▷ x. +□ +Note that this action leaves all Sectors in DB invariant, but in general not Dbio ⊂ DB. We +now determine Gdil ▷ DSIRS, thereby also providing an alternative proof of Lemma B.1i). +Proposition B.3. +i) +Let T be compatible with x ∈ DII ∩ {d ≤ c ∧ d = c ⇒ R0 < 1} in the sense of Lemma +B.1. +Then there exists a unique dilatation transformation (ξ, λ) ∈ Gdil such that +x′ := (ξ, λ) ▷ x ∈ Dbio and such that the rescaled triangle satisfies T(ξ,λ) = Tphys(B′ ++, 0). +ii) Gdil ▷ DSIRS = DII ∩ {d ≤ c ∧ d = c ⇒ R0 < 1}. + +26 +SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I +Proof. To prove part i) denote transformed quantities by a prime. +The requirements +T′ +◁ = (0, 0) and T′ +△ = (0, 1) fix ξ = (1 − X◁)−1 and λ = I−1 +△ . +Hence X▷ maps to +ξcI△ = c′ = B′ ++ and therefore T(ξ,λ) = Tphys(B′ ++, 0). To show 0 ≤ R′ +i ≤ B′ ++ use R′ +0 = +ξ(R0 − 1) + 1 = ξ(R0 − X◁) and therefore +0 ≤ R′ +0 ≤ ξ +λ min{c, (c − d)/a} = min{c′, (c′ − d′)/a′} ≤ B′ ++ +By the above we also have R′ +1 = a′R′ +0 + d′ ≤ c′ = B′ ++ and we are left to show R′ +1 ≥ 0. +Sufficient is d′ ≥ 0 which follows from 1−d′/c′ = ξ(1−d/c) ≤ ξ(1−X◁) = 1. This proves +part i) and therefore also the “⊃”-direction of part ii). To prove the “⊂”-direction use that +the action of Gdil on D preserves the sign of d − c and in case d = c we have R0 = 0 and +therefore R′ +0 = ξ(R0 − 1) + 1 = 1 − ξ < 1. +□ +As in (Nill 2022), the above dilatation symmetry leads to a second normalization step +for the SIRS-Sector, thus further reducing its number of essential parameters from four +to two. Equivalently this means, that equivalence classes of Gdil-isomorphic systems with +parameters in Gdil ▷ DSIRS are naturally parametrized by KSIRS := (Gdil ▷ DSIRS)/Gdil. A +convenient realization of the normalized system on phase space P = {(q, p) ∈ R × R≥0} is +given by putting +q(t) := 1 +a(X(t/a) − 1) , +p(t) := c +aI(t/a) +(B.2) +In terms of these variables the RN-dynamical system (3.8) - (3.9) becomes +˙q = −q(p + 1) + κ0 − κ1p , +˙p = qp , +(B.3) +where the new Gdil-invariant parameters are given by +κ0 := R0 − 1 +a +, +κ1 := c − d +ac +. +(B.4) +The only remaining constraint on the reduced parameter space says +KSIRS = {(κ0, κ1) ∈ R × R≥0 | κ1 = 0 ⇒ κ0 < 0} . +(B.5) +Thus, after normalization the whole SIRS Sector just looks like Hethcote’s classic endemic +model except for a somewhat less restricted parameter space. In fact, by Eq. (3.55), +DHeth ⊂ DSIRS is already two-dimensional with independent parameters a ∈ (0, 1) and +c = R0 = B+ > 0. These map injectively to KSIRS via κ0 = (c − 1)/a and κ1 = 1/a, +whence +DHeth ∼= KHeth = KSIRS ∩ {κ1 > 1 ∧ κ0 + κ1 > 0} +(B.6) +The normalization convention in Eq. (3.68) is obtained under the restriction c > d or +equivalently κ1 > 0. In this case one may alternatively use +u(t) − 1 := +c +c − d(X(ct/(c − d)) − 1) = 1 +κ1 +q(t/κ1) , +(B.7) +v(t) := +c2 +c − dI(ct/(c − d)) = 1 +κ1 +p(t/κ1) . +(B.8) +In terms of these variables we recover the normalization convention (1.1), (3.68) +˙u = −uv − c1u + c2 , +˙v = uv − v , +(B.9) +where c1 = 1/κ1 and c2 = 1/κ1 + κ0/κ2 +1, which is also the version given in (Nill 2022). +In part III of this work the above normalization step will be generalized to all Sectors of +Dbio. In this way the equation for ˙q in (B.3) gets an additional term −κ2p2, and so our + +REFERENCES +27 +initial 14-parameter19 SSISS model boils down to a much simpler 3-parameter dynamical +system. +Appendix C. The case α1 = α2 = 0 +This Appendix shortly discusses the border case α1 = α2 = 020. In this case define +parameter spaces C0 +x as in Eqs. (2.11)-(2.15) with αi = 0 and A0 +x := C0 +x ×B. In particular, +in A0 +bio we have θ1 ≥ 0, θ2 = 0, γi ≥ 0 and γ1 + γ2 = 1. Lemma 3.3 still holds with +a = b = 0 and d = R1 + ϵ, i.e. the replacement number dynamics becomes +˙X = (d − cX)I − ϵI2 , +˙I = (X − 1)I . +(C.1) +In this case R0 is undefined and there is a continuum of disease free equilibria at I = 0, +which are locally stable for X < 1 and unstable for X > 1. Proposition 3.4 remains +unchanged provided a = a′ ∈ A0. Putting D0 = {(c, d, ϵ) ∈ R3} Lemma 3.6 still holds +with A replaced by A0 and D replaced by D0. Moreover, in A0 +bio we get A+ = B+ = β1+θ1, +A− = B− = β2, c = β1+β2+θ1, ϵ = β2(β1+θ1) and putting D0 +A = D0 +B = D0 +AB := D0∩{c2 > +4ϵ} Proposition 3.15 becomes +φ(A0 +phys) = (D0 +B × B) ∩ {β2 ≤ {B−, R1} ≤ β1 ≤ B+} , +(C.2) +D0 +phys = D0 +B ∩ {R1 ≤ B+} , +(C.3) +φ(A0 +bio) = (D0 +B × B) ∩ {B− = β2 ≤ R1 ≤ β1 ≤ B+} , +(C.4) +D0 +bio = D0 +B ∩ {B− ≤ R1 ≤ B+} ⊂ D0 +phys . +(C.5) +So, for x ∈ D0 +phys physical triangles Tphys(β1, β2) are forward invariant provided (β1, β2) +satisfy the bounds C.2. Finally, Eq. (3.42) becomes φ−1(iB(D0 +B)) = φ(Aθ=0 ∩ Aα=0) and +Theorem 2.12, Theorem 2.6 and Corollary 3.17 stay valid also for α = 0. +References +Arino, J., C.C. Mccluskey, and P. van den Driessche (2003). “Global results for an epidemic +model with vaccination that exhibits backwad bifurcation.” In: SIAM J. Appl. Math. +64, pp. 260–276. doi: 10.1137/S0036139902413829. +Avram, F., R. Adenane, L. Basnarkov, et al. (Dec. 2021). “On matrix-SIR Arino models +with linear birth rate, loss of immunity, disease and vaccination fatalities, and their +approximations.” In: arXiv preprint. url: http://arxiv.org/abs/2112.03436. +Avram, F., R. Adenane, G. Bianchin, et al. (2022). “Stability analysis of an eight parameter +SIR- type model including loss of immunity, and disease and vaccination fatalities.” In: +Mathematics 10.3, p. 402. doi: 10.3390/math10030402. +Batistela, C.M. et al. (2021). “Vaccination and social distance to prevent Covid-19.” In: +IFAC PapersOnLine 54-15, pp. 151–156. +Busenberg, S. N. and P. van den Driessche (1990). “Analysis of a disease transmission +model in a population with varying size.” In: J. Math. Biol. 28, pp. 257–270. +Busenberg, S. N. and P. van den Driessche (1991). “Nonexistence of periodic solutions +for a class of epidemiological models.” In: Biology, Epidemiology, and Ecology. Ed. by +S.N. Busenberg and M. Martelli. Vol. 92. Lecture Notes in Biomath. 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(2000). “The Mathematics of Infectious Diseases.” In: SIAM Rev. 42, +p. 599. +Kermack, W. O. and A. G. McKendrick (1927). “Contribution to the mathematical theory +of epidemics, part I.” In: Proc. Roy. Soc. Lond A 115, pp. 700–721. +Korobeinikov, A. and G.C. Wake (2002). “Lyapunov Functions and Global Stability for +SIR, SIRS, and SIS Epidemiological Models.” In: Appl. Math. Lett. 15, pp. 955–960. +Kribs-Zaleta, C.M. and J.X. Velasco-Hernandez (2000). “A simple vaccination model with +multiple endemic states.” In: Mathematical Biosciences 164, pp. 183–201. doi: 10. +1007/s00285-021-01629-8. + +REFERENCES +29 +Li, Jianquan and Zhien Ma (2002). “Qualitative analyses of SIS epidemic model with vac- +cination and varying total population size.” In: Mathematical and Computer Modelling +35, pp. 1235–1243. +Li, Michael Y et al. 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(2001). “Multiple Equilibria for an SIRS Epidemiological System.” In: arXiv +preprint arXiv:math/0101051v1. doi: 10.48550/arXiv.math/0101051. +Sun, Chengjun and Ying-Hen Hsieh (2010). “Global analysis of an SEIR model with vary- +ing population size and vaccination.” In: Applied Mathematical Modelling 34, pp. 2685– +2697. +Yang, Wei, Chengjun Sun, and Julien Arino (2010). “Global analysis for a general epidemi- +ological model with vaccination and varying population.” In: Journal of Mathematical +Analysis and Applications 372.1, pp. 208–223. + diff --git a/WtAyT4oBgHgl3EQfWPdu/content/tmp_files/load_file.txt b/WtAyT4oBgHgl3EQfWPdu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bc884e99de7b458c9532e81ad5da06a21eaa87cf --- /dev/null +++ b/WtAyT4oBgHgl3EQfWPdu/content/tmp_files/load_file.txt @@ -0,0 +1,1603 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf,len=1602 +page_content='SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I: THE REPLACEMENT NUMBER DYNAMICS FLORIAN NILL 31-DEC-2022 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' As shown recently by the author, constant population SI(R)S models map to Hethcote’s classic endemic model originally proposed in 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This unifies a whole class of models with up to 10 parameters being all isomorphic to a simple 2-parameter master model for endemic bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this work this procedure is extended to a 14-parameter SSISS Model, including social behavior parameters, a (diminished) susceptibility of the R-compartment and unbalanced constant per capita birth and death rates, thus covering many prominent models in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Under mild conditions, in the dynamics for fractional variables in this model all vital parameters become redundant at the cost of possibly negative incidence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' There is a symmetry group GS acting on parameter space A, such that systems with GS-equivalent parameters are isomorphic and map to the same normalized system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Using (Xrep, I) as canonical coordinates, Xrep the replacement number, normalization reduces to parameter space A/GS with 5 parameters only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This approach reveals unexpected relations between various models in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Part two of this work will analyze equilibria, stability and backward bifurcation and part three will further reduce the number of essential parameters from 5 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Introduction 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The SSISS model 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Constant population 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Time varying population 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Classifying parameter space 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Examples from the literature 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Absence of periodic solutions 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Normalization 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Phase space 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Canonical coordinates 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Main results 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Examples revisited 22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Summary and outlook 23 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Normalizing linear vital dynamics 24 Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Scaling the SI(R)S model 24 Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The case α1 = α2 = 0 27 References 27 E-mail address: nill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='florian@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 34C23, 34C26, 37C25, 92D30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' SIRS model, SSISS model, normalization, symmetry, stability, endemic bifur- cation, backward bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The author is retired physicist, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='rer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='habil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=', formerly senior research fellow at Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Physik, Freie Universität Berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='00159v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='PE] 31 Dec 2022 2 SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Introduction Building mathematical models to describe phenomena in natural sciences one typically encounters dynamical variables and external parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Within the model values for external parameters are considered to be given from outside, like fundamental natural constants (speed of light c, Planck’s constant ℏ), parameters describing material or bi- ological properties (spring constant κ, birth rate δ, recovery rate γ) or social behavior (contact rate β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Naturally, reducing the number of essential parameters is always a goal to detect redundancies within parameter space and to simplify computations by unload- ing formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In the simplest case a pure dimensional scale parameter may without loss be put equal to one by choosing dimensional units appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' For example, putting c = 1 amounts to measuring spatial distances by light running times and masses in units of energies, putting ℏ = 1 amounts to measuring energies by angular frequencies and putting γ = 1 amounts to measuring time in units of the recovery time in an epidemic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' More generally a normalization program consists of finding appropriate coordinate transformations in variable+parameter space such that the transformed system only de- pends on a maximally reduced subset of transformed parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Examples are1 Harmonic oscillator Predator-prey model ˙u = v ˙u = −uv + c1u ˙v = −u ˙v = uv − v Classic SIR model Classic endemic model ˙u = −uv ˙u = −uv − c1u + c2 ˙v = uv − v ˙v = uv − v (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1) Following this strategy the 6-parameter SI(R)S model (≡ combined SIRS/SIS model) with standard incidence, constant vaccination and immunity waning rates and a balanced birth and death rate has recently been shown by the author (Nill 2022) to admit a nor- malized version looking like the classic endemic model above2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this work (including two follow ups to be denoted as parts II and III (Nill n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='[b],[c])) this method is extended to the case where immunity after recovery (or vaccination) is incomplete right from the onset and where also compartment dependent constant per capita birth and death rates lead to a time varying population size N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this way one is naturally lead to replacing the SI(R)S model by a SSISS model, where in place of the usual S, I and R compartments we have two susceptible compartments S1 and S2 and one infectious compartment I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Infection transmission from I to S2 is diminished as compared to transmission to S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' There is a vaccination flow from S1 to S2 and an immunity waning flow from S2 to S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The model could also be interpreted by considering 1The variables in these examples are: Harmonic oscillator: u = q, v = p/ √ mk, where q, p, κ, m are coordinate, momentum, spring constant and particle mass and where the oscillation period is normalized to T = 2π by putting m/k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Predator-prey model: (u, v) denote appropriately rescaled prey and predator populations, respectively, and the predator mortality rate is normalized to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' SIR model: u = r0S, v = r0I, where r0 is the basic reproduction number, (S, I) are susceptible and infectious fractions of the population and where the recovery rate is normalized to γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Endemic model: (u, v, r0, γ) as above, c1 = δ/(γ + δ) and c2 = r0c1, where δ is the balanced birth/mortality rate and where now time scale is normalized to γ + δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2Aapart from allowing also values u ∈ R and an enlarged parameter range (c1, c2) ∈ R+ × R ∪ {0, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I 3 S2 as the “lock-down” fraction and S1 as the “freedom fraction”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this picture flows from S1 to S2 and vice-versa are described by an I-linear (respectively (N −I)-linear) flow with rate parameters θi, i = 1, 2, modeling social behavior in reaction to published prevalence data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Combining both interpretations it turns out to be convenient to start with an abstract version of a SSISS model staying completely symmetric under interchanging S1 and S2, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The present part I provides a normalization prescription reducing the number of inde- pendent parameters in this model from initially fourteen to essentially five (four in the SI(R)S model sub-case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Based on this approach, part II will give a complete review on equilibria and stability in the master SSISS model, thereby also recovering an exceptional scenario which had been overlooked in the literature so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In part III the scaling sym- metry for SI(R)S models mentioned above will be generalized to the full SSISS model, thereby reducing the number of parameters again by two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' So, the total reduction from fourteen to three reveals a great hidden redundancy in parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' It also provides a unifying view on results in the literature concerning equilibrium states, endemic bifur- cation and stability properties for all kinds of sub-classes of this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Put differently, in the presence of a common normalized version presenting basically repeated arguments for various subsets of non-vanishing parameters becomes obsolete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Relating this work to the literature, let me focus on deterministic SIR-type 3-compartment dynamical systems, which conveniently may be classified according to A) constant vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' time-varying total population size N, B) infection transmission only from I to S vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' also from I to R (in which case it makes sense to rename S ≡ S1 and R ≡ S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Also, I will restrict this survey to models with standard bi-linear incidence flows βiSiI/N, such that the vector field ˙Y = V(Y), Y = (S1, S2, I), is homogeneous of first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This applies to diseases where the number of effective contacts per capita is independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' ad A) Endemic models with constant population have first been constructed by adding a non-zero balanced birth and death rate to the classic SIR model of (Kermack and McKendrick 1927).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' As shown by (Hethcote 1974) (see also (Hethcote 1976, 1989)), in this way already the simplest model without vaccination and loss of immunity shows a bifurcation from a stable disease-free equilibrium point (DFE) to a stable endemic scenario when raising the basic reproduction number R0 above one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Nowadays this is considered as Hethcote’s classic endemic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Including linear vaccination and/or loss of immunity terms and optionally also considering recovery without immunity one ends up with various types of constant population SI(R)S models without changing this picture, see for example (Batistela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Chauhan, Misra, and Dhar 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Korobeinikov and Wake 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' O’Regan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' As remarked above (and reviewed in more detail in Appendix B), the true reason lies in the fact that constant population SI(R)S models with up to 10 parameters all map to the same normalized 2-parameter version of the classic endemic model as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Models with variable population are mostly studied under the assumption of a constant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' N-independent) birth flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Heuristically this may be justified by assuming that N varies slowly on characteristic epidemic time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' But truly speaking, as already pointed out by (Mena-Lorca and Hethcote 1992), this Ansatz rather models a constant immigration scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' So in this work I will follow the more natural proposal of modeling vital dynamics by possibly department dependent constant per capita birth and death 4 SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Note that, unless fine tuning parameters, this implies that either N(t) → ∞ or N(t) → 0 as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' So in this type of models one always analyzes the dynamics of fractional variables Si := Si/N, I := I/N, which is well known to be independent of N(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Apparently, this stream of models has been initiated by (Busenberg and Driessche 1990, 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Derrick and Driessche 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (Razvan 2001) has studied a SIRS model in this sense with infection transmission also from outside and a SIS-version with varying population size has been analyzed by (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Li and Ma 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' For generalizations to SEIR models see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (Greenhalgh 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Lu and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Lu 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Sun and Hsieh 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' ad B) A different approach to modeling partial and/or waning immunity consists of introducing a diminished incidence flow with rate βR ≡ β2 > 0 directly from R ≡ S2 to I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This has presumably first been proposed in the so-called SIRI model of (Derrick and Driessche 1993), see above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In addition, the authors also introduced a time varying population size N(t) and an excess mortality ∆µI in compartment I to this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In turn, they didn’t use linear vaccination nor immunity waning terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this way they identified a range of parameters in the domain R0 < 1, for which besides the locally asymptotically stable disease free equilibrium there also coexist two endemic equilibria, one being a saddle and the other one also being locally asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Later (Hadeler and Castillo-Chavez 1995) found the same phenomenon in their combined SIS/SIRS core group model with linear vaccination, constant population and also two incidence rates βi for S → I and R → I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Meanwhile it is well known that models with infection incidents from several compartments may show a so-called backward bifurcation from the disease- free to an endemic scenario (Hadeler and Driessche 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This means that two locally asymptotically stable equilibrium states may coexist for some range below threshold, causing also hysteresis effects upon varying parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Apparently, a varying population size is not needed for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In (Kribs-Zaleta and Velasco-Hernandez 2000) the authors have improved and extended these results by adding also a linear immunity waning rate to the model of (Hadeler and Driessche 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' One may also distinguish vaccinated and recovered people into separate compartments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This leads to 4-compartment models, where similar results have been obtained by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Arino, Mccluskey, and Driessche 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Yang, Sun, and Julien Arino 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Backward bifurcation has lately also been observed in SEIRS-type models for Covid- 19 by considering two distinguished susceptible compartments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In (Nadim and Chat- topadhyay 2020) the less susceptible compartment had been interpreted as an incomplete lockdown and in (Diagne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2021) as an incomplete vaccination efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' More recently, in (Avram, Adenane, Basnarkov, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Avram, Adenane, Bianchin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2022) the authors have given a thorough stability analysis of an eight parameter SIRS-type model by adding a varying population size to the model of (Kribs-Zaleta and Velasco-Hernandez 2000) (apparently without being aware of that paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Closing this overview I should also remark that backward bifurcation is also observed when considering I-dependent contact or recovery rates to model reactive behavior or infection treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' However the list of papers on this topic over the last 20 years becomes too huge to be quoted at this place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This paper extends the normalization algorithm for constant population SI(R)S models to models as above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' with time varying population size and/or a non-zero incidence rate βR ≡ β2 from R ≡ S2 to I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' As a starting observation, there is an ambiguity in deriving the dynamics ˙y = F(y) for fractional variables y = (S1, S2, I), see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This allows choosing the vector field F such that all vital dynamics parameters become redundant, SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I 5 provided the birth-minus-death rates νi = δi −µi in S1 and S2 coincide, ν1 = ν2 = ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This redundancy already reduces the number of parameters in the master SSISS model from fourteen to eight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' More than that, F depends on the incidence rates βi only as a function of ˜βi = βi + νI − ν, where νI = δI − µI is the birth-minus-death rate in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Assuming for simplicity compartment independent birth rates gives ˜βi = βi − ∆µI, where ∆µI denotes the excess mortality in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this way models with variable population, ∆µI > 0, and absence of a incidence rate from R, β2 = 0, look like models with constant population, ∆µI = 0, and a negative incidence rate β2 = ˜β2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Conversely, models with positive incidence rates βi > 0 and excess mortality ∆µI < min{β1, β2} behave like models with constant population size and incidence rates βi = ˜βi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' So, the above classification schemes A) and B) become blurred and, instead, it is more expedient to view all models as if they had constant population size and two distinguished and possibly also negative incidence rates ˜βi ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this way most of the above bench marking 3-compartment models (if necessary after imposing the constraint ν1 = ν2) become comparable as sub-cases of the master SISS model, with tilde parameters swallowing all birth and death rates and possibly with negative incidence rates ˜βi ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' As an example, the models of (Hadeler and Castillo- Chavez 1995) and (Kribs-Zaleta and Velasco-Hernandez 2000) become isomorphic and they completely cover the sub-case µ1 = µ2 and 0 < min{˜β1, ˜β2} in (Avram, Adenane, Bianchin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Also, apart from an irrelevant boundary case, the complementary sub-case µ1 = µ2 and 0 > min{˜β1, ˜β2} in (Avram, Adenane, Bianchin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2022) is covered by the model of (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Li and Ma 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' So, applying the normalization procedure of this paper, all results in Section 5 and 6 of (Avram, Adenane, Bianchin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2022) already follow from the previous literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' A more detailed list of unexpected relations between the above models is given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The plan of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2 we pass to fractional com- partment variables, Si = Si/N and I = I/N, and prove redundancy of all vital dynamics parameters at the cost of possibly negative incidence rates ˜βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' For convenience, time scale is also normalized by putting the total expected waiting time in compartment I equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this way the number of essential parameters is already reduced from fourteen to seven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Thus, denoting A the space of essential parameters, we have dim A = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3 classifies various useful subsets in parameter space like Aphys ⊂ A, guaran- teeing forward invariance of the physical triangle Tphys := {(S1, S2, I) ∈ R3 ≥0 | S1 + S2 + I = 1}, and Abio ⊂ Aphys, guaranteeing an epidemiological interpretation of parameters by re- quiring in particular θ1 ≥ 0 ≥ θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4 identifies eight examples from the above list of models as sub-cases of the master SSISS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this way we obtain various relations between these models as indicated above, which apparently have not been recognized before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='5 we adapt methods from (Busenberg and Driessche 1990) to prove ab- sence of periodic solutions for all parameters non-negative, except βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The extension to parameters a ∈ Abio (requiring θ2 ≤ 0) heavily relies on the symmetry results in Section 3 and will be proven in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Section 3 starts from the observation, that the time-normalized equation of motion for I takes the generic form ˙I = (Xrep − 1)I, where Xrep = β1S1 + β2S2 is the replacement 6 SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I number (Hethcote 2000), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' the expected number of secondary cases produced by a typical infectious individual during its time of infectiousness (nowadays mostly called effective reproduction number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' A coordinate free formulation of the model naturally leads to taking (Xrep, I) as independent canonical coordinates3 in the physical triangle Tphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this way, we arrive at formulating the SSISS model as a dynamical system in (Xrep, I)-space, called the replacement number (RN) dynamics (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' ˙Xrep = f(Xrep, I), ˙I = (Xrep − 1)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2) Since f(Xrep, I) turns out to be a 5-parameter quadratic polynomial with no term ∼ X2 rep, the number of free parameters is now reduced from seven to five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The main results of this paper are derived in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Denoting D the new parameter set, dim D = 5, the above approach yields a surjective submersion A ∋ a �→ x(a) ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Moreover, A becomes a principal fibre bundle with respect to a group right action ◁ : A × GS → A such that x(a ◁ g) = x(a) and D ∼= A/GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Here GS ⊂ GL+(R2) is the group acting on (S1, S2) ∈ R2 and leaving S1 + S2 invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2) implies that SSISS dynamical systems at parameter values a, a′ ∈ A are isomorphic whenever a and a′ are GS-equivalent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' x(a) = x(a′) or equivalently a′ = a ◁ g for some g ∈ GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this way we also get Absence of periodic solutions also for parameters a ∈ Abio, Conditions under which the social behavior parameters θi can be “gauged to zero”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' there exists g ∈ GS such that a ◁ g ∈ Aθ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4 revisits the examples from the literature within the new formalism and Sec- tion 4 gives a summary and outlook to parts II and III of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Finally, Appen- dix A provides a normalization prescription for the dynamics of fractional variables in n-compartment models with linear (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' constant per capita) birth and death rates, Ap- pendix B reviews the scaling symmetry in SI(R)S models introduced in (Nill 2022) and Appendix C discusses a boundary case in parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Acknowledgement I would like to thank Florin Avram for encouraging interest and useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The SSISS model This Section starts with proposing an abstract completely symmetrized SSISS model consisting of three compartments, S1, S2 and I, with total population N = S1 + S2 + I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Members of I are infectious, members of S1 are highly susceptible (socially active or not immune) and members of S2 are less susceptible (partly immune or reducing contacts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The flow diagram between compartments is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The parameters in this model may be given the following interpretations 3Here “canonical” is not meant in the sense of Hamiltonian systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I 7 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Completely symmetric flow diagram of the SSISS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' All pa- rameters are nonnegative except θ2 ∈ [−α2, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Also q1 + q2 = 1, γ1 + γ2 > 0 and β1 > β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Generalizing to compartment dependent birth rates amounts to replacing δN by δ1S1 + δ2S2 + δII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' α1 : Vaccination rate of susceptibles moving from S1 → S2 (assuming θ1 = θ2 = 0, see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' α2 : Immunity waning rate inducing a flow from S2 → S1 (assuming θ2 = 0, see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' βi : Number of effective contacts per unit time of a susceptible from Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' γi : Recovery rate from I → Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' θ1 : Willingness to get vaccinated (alternatively to reduce contacts) given the actual prevalence I/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In reality only one of the two parameters α1 and θ1 should be chosen non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' θ2 : Epidemiologically one should restrict to θ2 = 0 or (θ2 = −α2 < 0 and α1 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this latter case the meaning of the S2-compartment is “contact reducing” and α2 = −θ2 parametrizes the readiness to increase contacts proportional to 1 − I/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' µi : Mortality rate in Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' µI : Mortality rate in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' One could also consider vertical transmission, in which case µI would be the mortality rate diminished by the rate of infected newborns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' ∆µI : Mortality excess ∆µI = µI − µ in case µ1 = µ2 = µ, which will be assumed most of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' δ : Rate of not infected newborns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Generalizing to compartment de- pendent birth rates amounts to replacing δN = δ1S1 + δ2S2 + δII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' qi : Split ratio of newborns between S1 and S2, q1 + q2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In the reduced-immunity interpretation q2 would be the portion of vacci- nated newborns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 8 SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I So in total this model counts 15 independent parameters (12 if we require constant total population, δi = µi, δI = µI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Epidemiologically all parameters except 0 ≥ θ2 ≥ −α2 are assumed non-negative and also β2 < β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' A more technical classification of admissible parameter ranges will be given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Here is a list of prominent examples in the literature Hethcotes classic 3-parameter endemic model (Hethcote 1974, 1976, 1989) by putting δ = µi = µI > 0, q1 = 1, β1 > 0, γ2 > 0 and all other parameters vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The 7-parameter SIRS model with time varying population size in (Busenberg and Driessche 1990), adding to Hethcote’s model an immunity waning rate α2 and allowing different (constant per capita) mortality and birth rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The 6-parameter SIRI model of (Derrick and Driessche 1993), replacing the immunity waning rate α2 in (Busenberg and Driessche 1990) by the incidence rate β2 > 0 and also requiring µ1 = µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' An extended 10-parameter constant population SI(R)S (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' mixed SIRS/SIS) model with constant and I-linear vaccination rates α1, θ1, an immunity waning rate α2 and two recovery flows I ← Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Hence δi = µi, δI = µI and θ2 = β2 = 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The 6-parameter isolated core system in (Hadeler and Castillo-Chavez 1995), with two incidence and recovery rates, βi, γi > 0, a vaccination term α1 > 0 and a constant population with balanced birth and death rates, δ = µi = µI > 0 and q1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The 7-parameter vaccination models of (Kribs-Zaleta and Velasco-Hernandez 2000) adding an immunity waning rate α2 > 0 to the model of (Hadeler and Castillo-Chavez 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' As we will see in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='24) below, due to a redundancy of parameters the two models actually stay isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The 8-parameter SIS-model with vaccination and varying population size of (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Li and Ma 2002) keeping only θi = γ2 = β2 = 0 and assuming µ1 = µ2 = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='5 As we will see in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='25), after a parameter transformation this model becomes isomorphic to the case where only θi = 0 and β2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The 8-parameter SIRS-type model analyzed recently by (Avram, Adenane, Bianchin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2022), keeping only γ1 = θ1 = θ2 = q2 = 0 and all other parameters positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The authors allow a varying population size by first discussing the general case of all mortality rates being different and then concentrate on µ1 = µ2 ̸= δ and ∆µI > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Their paper is closest to the present work and in fact initiated it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In a “zeroth normalization” step I will now show that passing to fractional variables and requiring δ1 − µ1 = δ2 − µ2 all vital dynamic parameters in the SSISS model become redundant6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this way the number of essential parameters reduces from 14 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The price to pay in the non-constant population case is possibly getting negative incidence rates βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Constant population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' To get a constant population N the birth rates have to obey δi = µi and δI = µI, or more generally δ = (µ1S1 + µ2S2 + µII)/N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1) In case µ1 = µ2 = µ this would read δ = µ+I∆µI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Heuristically this should be understood as an approximation for ∆µI/µ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Under this assumption, denoting fractions of the 4Here I have chosen enlarge the conventional setting for SI(R)S models by also allowing θ1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 5Actually the authors let µ be a function of N, which however disappears when passing to fractional variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 6Redundancy of constant per capita birth and death rates may in fact be shown under quite general assumptions in n-compartment models, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I 9 total population by Si = Si/N and I = I/N and introducing the notations ˜α1 := α1 + q2µ1 , ˜γ1 := γ1 + q1µI , ˜α2 := α2 + q1µ2 , ˜γ2 := γ2 + q2µI , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2) S = � S1 S2 � , D(β) = � β1 0 0 β2 � , E(α) = � α1 −α2 −α1 α2 � , ˜γ = � ˜γ1 ˜γ2 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3) the dynamical system described by the flow diagram Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 1 becomes ˙S = − [E( ˜α) + IE(θ) + ID(β)] S + I ˜γ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4) ˙I = ˜γ(Xrep − 1)I , ˜γ = ˜γ1 + ˜γ2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='5) Xrep := (β1S1 + β2S2)/˜γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6) Note that ˜γ−1 ≡ (γ1 + γ2 + µI)−1 is the expected waiting time in I and hence Xrep is the replacement number (Hethcote 2000), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' the expected number of secondary cases pro- duced by a typical infectious individual during its time of infectiousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In conventional SI(R)S models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' for β2 = θ2 = 0, the replacement number in the limit S1 = 1 would become the basic reproduction number r0 = β1/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This is why nowadays the replacement number is mostly called effective reproduction number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Later we will also have the notion of a reduced reproduction number R0 as the value of Xrep at the disease-free equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' To avoid misunderstandings, I prefer to keep the various notions of “reproduction num- bers” for parameters, whereas the replacement number Xrep is considered as a dynamical variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Now obviously, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2), all vital dynamics parameters become redundant and may be absorbed by redefining αi and γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Note that this observation is independent of the choice of βi and θi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' it already holds in a combined SI(R)S model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Time varying population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' To derive the equations of motion in case of a time vary- ing population keep compartment dependent per capita birth and death rates δi, δI, µi, µI constant and put Y = (S1, S2, I), y = N−1Y and ν ≡ (ν1, ν2, νI) := (δ1 − µ1, δ2 − µ2, δI − µI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Then ˙y = ˙Y/N − y ˙N/N and ˙N/N = ⟨ν | y⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Using S1 + S2 + I = 1 we may rewrite S1 ˙N/N = S1[ν1 + (ν2 − ν1)S2 + (νI − ν1)I] S2 ˙N/N = S2[ν2 + (ν1 − ν2)S1 + (νI − ν2)I] I ˙N/N = I[νI + (ν1 − νI)S1 + (ν2 − νI)S2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' So now introduce ˜α1 := α1 + q2δ1 , ˜α2 := α2 + q1δ2 , ˜γ1 := γ1 + q1δI , ˜γ2 := γ2 + q2δI , ˜β1 := β1 + νI − ν1 , ˜β2 := β2 + νI − ν2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='7) With the same notation as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3) and e(ν) := � ν1 − ν2 ν2 − ν1 � we then get ˙S = − � E( ˜α) + IE(θ) + ID( ˜β) � S + I ˜γ + S1S2e(ν) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8) ˙I = ˜γ(Xrep − 1)I , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9) Xrep := (˜β1S1 + ˜β2S2)/˜γ , ˜γ := ˜γ1 + ˜γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='10) 10 SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I So, imposing the condition ν1 = ν2 =: ν and putting ∆νI := ν−νI we get e(ν) = 0 and the equations of motion look exactly as in the case of constant population (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Again all vital dynamics parameters become redundant and may be absorbed by redefining βi, αi and γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The difference this time is that ˜βi = βi − ∆νI may become negative!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Thus we arrive at Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Assume ν1 = ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' i) If ∆νI ≤ min{β1, β2} the SSISS model with variable population maps to the model with constant population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' ii) If ∆νI > min{β1, β2} it maps to the model with min{β1, β2} = 0 and variable popula- tion with � ∆νI = ∆νI − min{β1, β2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' iii) If ∆νI = β2 < β1 and θ2 = 0 it becomes the extended SI(R)S model with θ1 ≥ 0 and two recovery flows I → S1 and I → S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Note that under the usual assumptions δi = δI = δ and µ1 = µ2 = µ, ∆νI coincides with the excess mortality in the infectious compartment, ∆νI = µI −µ = ∆µI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The observation that on the level of fractional variables in both scenarios (constant vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' variable population, the latter provided ν1 = ν2) all vital dynamics param- eters are redundant seems to be new7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Essential for this is allowing all four parameters (αi, γi) being positive and βi possibly being negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The introduction of parameters θi is not needed to assure this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Redundancy of constant per capita birth and death rates may in fact be shown under quite general assumptions in n-compartment models, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Classifying parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this subsection assume ν1 = ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Then the re- formulation in terms of possibly negative incidence rates ˜βi leads to a new classification scheme identifying seven sectors in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' For θi = 0 these are labeled by the signa- tures of ˜β1 + ˜β2 and ˜β1 ˜β2 (in case of a compartment independent birth rate δ equivalently by the size of the excess mortality ∆µI), see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' For θi ̸= 0 this classification will be refined in Section 3, Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' To simplify notation, in what follows let me drop the tilde above parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The case β1 = β2 will be ignored, since in this case putting S = S1 + S2 one easily checks that (S, I) obeys the dynamics of a SIS model, which can immediately be solved by separation of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Also, due to the permutation symmetry 1 ↔ 2, there is no loss assuming β1 > β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Next, choosing time scale to be measured in units of γ−1, we may without loss also put γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Thus, assume γi ∈ [0, 1] and γ1 + γ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' So, having started from fourteen, essentially we are now left with seven free parameters (think of all greek symbols of dimension [time]−1 being divided by γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' To further classify the space of admissible parameters some formalism will be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Put C := {(αi, γi, θi) ∈ R6 | α1 + α2 > 0 ∧ γ1 + γ2 = 1} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='11) C+ := C ∩ {(αi, γi) ∈ R4 ≥0} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='12) Csplit := C ∩ {θ1 ≥ 0 ≥ θ2} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='13) Cphys := C+ ∩ {θi + αi ≥ 0 , i = 1, 2} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='14) Cbio := Csplit ∩ Cphys (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='15) 7As communicated privately this had also been realized recently in a talk by Florin Avram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I 11 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Seven sectors in the SSISS-model at θi = 0 and for compartment independent birth rate δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9 Sector I is isomorphic to the models of (Hadeler and Castillo-Chavez 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Kribs-Zaleta and Velasco-Hernandez 2000) and Sectors III-VII are largely covered by (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Li and Ma 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Sector II is a mixed SI(R)S model with two recovery flows I → R and I → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Sector sign(˜β1 + ˜β2) sign(˜β1 ˜β2) Interval [˜β1, ˜β2] Excess mortality ∆µI I + + 0 < ˜β2 < ˜β1 ∆µI < β2 II (SIRS) + 0 0 = ˜β2 < ˜β1 ∆µI = β2 III + − 0 < −˜β2 < ˜β1 β2 < ∆µI < (β1 + β2)/2 IV 0 − 0 < −˜β2 = ˜β1 ∆µI = (β1 + β2)/2 V − − ˜β2 < −˜β1 < 0 (β1 + β2)/2 < ∆µI < β1 VI − 0 ˜β2 < ˜β1 = 0 β1 = ∆µI VII − + ˜β2 < ˜β1 < 0 β1 < ∆µI Note that for θi = 0 we have C+ = Cphys = Cbio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Denoting B := {β = (β1, β2) ∈ R2 | β2 < β1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='16) the full parameter sets are then given by A := C × B or Ax := Cx × B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' I will also use obvious notations like Aθ=0 := A ∩ {θi = 0} and Aα≥0 := A ∩ {αi ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In the definition of C in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='11) the border case α1 = α2 = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' absence of constant vaccination and waning immunity rates) has been excluded, see Appendix C for a short discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' For the body of this paper I will stick with the assumption α1+α2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Next, it is easy to check, that for a ∈ Aphys the physical triangle Tphys := {(S1, S2, I) ∈ R3 ≥0 | S1 + S2 + I = 1} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='17) stays forward invariant under the dynamics (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' on Tphys we have I = 0 ⇒ ˙I = 0 and Si = 0 ⇒ ˙Si ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Note that θi + αi ≥ 0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='14) is sufficient but not necessary to assure this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In the SSISS model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9) the physical triangle stays forward invariant for all parameters (αi, βi, γi, θi) ∈ Aphys, also including the border case α1 = α2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' □ We are now ready to state a main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Assuming ν1 = ν2 the normaliza- tion procedure to be introduced in Section 3 will further reduce the number of essential parameters from seven to five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This means, SSISS models fall into isomorphy classes map- ping to the same normalized system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' It turns out, that these isomorphy classes coincide with orbits under a parameter symmetry group GS acting simultaneously on phase P and parameter space A, such that parameters for the normalized system are naturally identified as elements of A/GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' For y = (S1, S2, I)T ∈ R3 and parameter values a = (α, β, γ, θ) ∈ A denote ˙y = Fa(y) the dynamical system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9) with vector field Fa : R3 → R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Let GS ⊂ GL+(R2) be the subgroup acting on S ∈ R2 from the left and leaving S1 + S2 invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 12 SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I i) Then there exists a free right action ◁ : A × GS → A such that A becomes a principal GS-bundle and Fa ◦ Tg = Tg ◦ Fa◁ g , Tg := � � g 0 0 0 0 1 � � , ∀(a, g) ∈ A × GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='18) ii) Put j := ( 0 1 1 0 ) and for g ∈ GS denote ¯g := jgj ∈ GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Viewing α, γ, θ ∈ R2 as column vectors and β ∈ B as a row vector and writing a ◁ g = a′ = (α′, β′, γ′, θ′) we have α′ = ¯g−1α, θ′ = ¯g−1θ + ϑ γ′ = g−1γ, ϑ = 1 β′ 1 − β′ 2 � −(β1 − β′ 1)(β2 − β′ 1) (β1 − β′ 2)(β2 − β′ 2) � β′ = βg iii) The GS-right action B × GS ∋ (β, g) �→ βg ∈ B is free and transitive and A ∼= A/GS × B as trivial principal fiber bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' iv) Put S′ = g−1S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Then ⟨β|S⟩ = ⟨β′|S′⟩ ≡ Xrep and therefore ˙Xrep = fa(Xrep, I) where fa = fa◁ g is GS-invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' it only depends on A/GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' v) If θ1 ≥ θ2 or θ1θ2 > 08, then there exists g ∈ GS such that a′ := a ◁ g ∈ Aθ=0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' the parameters θi may be “gauged to zero”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' If in this case a ∈ Abio then also a′ ∈ Abio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' As we will see, although the linear transformation Tg preserves the condition S1 + S2 + I = 1, it does not necessarily leave R3 ≥0 (and hence Tphys) invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Since dim GS = 2 we have dim A/GS = dim A − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' So, using (Xrep, I) as independent coordinates in Tphys, the number of essential parameters of the SSISS dynamical system reduces from seven to five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Parts i)-iv) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6 will be proven in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='7 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8 and part v) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Before coming to this let me close this Section in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4 with shortly revisiting some bench-marking models in the literature within the present framework, in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='5 with proving absence of periodic solutions by optimizing the methods of (Busenberg and Driessche 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Examples from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' For simplicity, in this subsection let me assume a compartment independent birth rate δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Formulating the dynamics for fractional variables y = (S1, S2, I) there always remains an ambiguity by adding a vectorfield vanishing on Tphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9) the vector field F ≡ Fa has the special form F(y) = My + Γ(y ⊗ y), ⟨1|M = ⟨1|Γ = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='19) where M ∈ R3×3, 1 = (1, 1, 1) and Γ ∈ Hom (R3 ⊗ R3, R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' As is shown in Appendix A, n-compartment models with at most quadratic terms and population size varying only due to constant per capita birth and death rates may always be normalized in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Using different conventions bears the risk of overlooking redundancies in parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Moreover, it also makes it tedious to pin down the differences between (or equivalence of) various models in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Table 2 shows how the examples quoted at the beginning of this Section9 compare with each other when mapped to the present set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 8Actually these conditions are sufficient but not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' For a weaker condition see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 9Heth = (Hethcote 1974, 1976, 1989);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' SIRI = (Derrick and Driessche 1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' BuDr = (Busenberg and Driessche 1990);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' SI(R)S = 10-parameter mixed SIRS/SIS model with constant population size and SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I 13 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Mapping models in the literature9 expressed in non-normalized vari- ables (S1, S2, I) to the present choice of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The column # counts the number of free parameters in the original models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' After passing to fractional variables (S1, S2, I) and tilde parameters, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2) or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='7), and resetting time scale to ˜γ = 1, the column #eff counts the number of effectively independent parameters as determined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='20)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='α1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='β1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='β2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='γ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='µ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='µI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='q1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='#eff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='Heth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='✓ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='µ1 = µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='AABH2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='µ1 = µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='Applying the transformations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2) or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='7), respectively, maps the above 11-parameter set to the redundancy-free 6-parameter set (˜αi, ˜βi, ˜γi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' After resetting time scale to ˜γ ≡ ˜γ1 + ˜γ2 = 1 the classification of the above models looks as follows: AHeth = Abio ∩ Aθ=0 ∩ {˜α1 = 0 ∧ ˜γ2 > 0 ∧ ˜γ1 = ˜α2 ∧ ˜β2 = 0} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='20) ASIRIi = Abio ∩ Aθ=0 ∩ {˜αi = 0 ∧ ˜γj > 0 ∧ ˜γi = ˜αj, j ̸= i} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='21) ABuDr = Abio ∩ Aθ=0 ∩ {˜α1 = 0 ∧ ˜γ2 > 0 ∧ ˜β2 < 0}11 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='22) ASIRS = Abio ∩ Aθ2=0 ∩ {˜β2 = 0} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='23) AKZVH = Abio ∩ Aθ=0 ∩ {˜β2 > 0} = AHaCa (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='24) ALM = Abio ∩ Aθ=0 ∩ {˜β2 < 0 ∧ ˜γ1 > 0} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='25) AAABHi = Abio ∩ Aθ=0 ∩ {˜γj > 0, j ̸= i} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='26) The dimensions of these parameter spaces are displayed in the last column of Table 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' To verify Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='20)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='26) the following explanations should suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The SIRI model of (Derrick and Driessche 1993) with varying population requires αi = γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Since for βR > βS the mapping to the SISS model permutes 1 ↔ 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' maps R → S1 and S → S2), if βR < βS we get ˜α1 = 0, ˜α2 = ˜γ1 = δ and ˜γ2 = γ2 > 0, and if βR > βS we get ˜α2 = 0, ˜α1 = ˜γ2 = δ and ˜γ1 = γ1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The SIRS model of (Busenberg and Driessche 1990) differs from SIRI by allowing α2 > 0 and µ1 < µ2, but in turn it requires βS > βR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Thus, we have ˜α1 = 0 θ2 = β2 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' HaCa = core system in (Hadeler and Castillo-Chavez 1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' KZVH = (Kribs-Zaleta and Velasco-Hernandez 2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' LM = (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Li and Ma 2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' AABH = (Avram, Adenane, Bianchin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' BuDr and AABH come in two versions, the subscript 1 refers to βS > βR and 2 to βS < βR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 10 The bulk of results in Section 5 and 6 of (Avram, Adenane, Bianchin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2022) assumes µ1 = µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 11To be comparable Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='22) refers to the sub-case µ1 = µ2 in (Busenberg and Driessche 1990), so dim ABuDr = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Allowing also an excess mortality µ2 − µ1 > 0 gives #eff = 5 in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 14 SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I and ˜γ1 = δ as in SIRI1, but ˜α2 = α2 + δ becomes independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' If, for comparison, we restrict to µ1 = µ2 = µ then β2 = 0 implies ˜β2 = −∆µI ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' If q1 > 0 then one of the three parameters (γ1, α2, δ) always becomes redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' So the models of (Hadeler and Castillo-Chavez 1995) and (Kribs-Zaleta and Velasco- Hernandez 2000) are isomorphic, in spite of the latter containing the additional im- munity waning rate α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Also, they both satisfy ˜β2 = β2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Putting q2 = 1 in the SIS-type model of (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Li and Ma 2002) the mapping (α1, α2, γ1, δ) �→ (˜αi, ˜γi) is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Also, the authors have defined µi = f(N) and µI = f(N) + ∆µI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Hence, the only restrictions in this model are ˜β2 = −∆µI < 0 and ˜γ1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In summary we get the following conclusions, which apparently have not yet been realized in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Assume µ1 = µ2 =: µ and put ∆µI := µI − µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' i) For β1 > β2 = ∆µI the SIRI model of (Derrick and Driessche 1993) is isomorphic to Hethcote’s classic endemic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Moreover, restricting to ˜γ1 > 0 and β2 ̸= ∆µI we have ii) The SIRS-type model of (Busenberg and Driessche 1990) reduces to a sub-case of the SIS-type model of (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Li and Ma 2002), which in turn covers Sectors III-VII of the SSISS model at θi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' iii) The models of (Hadeler and Castillo-Chavez 1995) and (Kribs-Zaleta and Velasco- Hernandez 2000) are isomorphic and cover Sector I of the SSISS model at θi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' iv) The models of (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Li and Ma 2002) and (Hadeler and Castillo-Chavez 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Kribs- Zaleta and Velasco-Hernandez 2000) only differ by the sign of ˜β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' v) Their disjoint union covers the SIRI model of (Derrick and Driessche 1993) and co- incides with the model of (Avram, Adenane, Bianchin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' An equivalent formulation of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9 based on normalized parameters and vari- ables is given in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='19 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Absence of periodic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this subsection I will specify parameter ranges guaranteeing absence of periodic solutions by optimizing methods from (Busenberg and Driessche 1990) (see also (Busenberg and Driessche 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Derrick and Driessche 1993)) for the present situation, including θi ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' To start with, the Busenberg-Driessche version of the classical Bendixson–Dulac Theorem may be given the following alternative formulation Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (Busenberg and Driessche 1990) Let F : R3 → R3 be smooth in a neigh- borhood of Tphys and assume Tphys forward invariant under the flow of ˙y = F(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Assume there exists a smooth scalar function u(y) defined in a neighborhood of Tphys such that Ψ(y) := ∇ · (uF)(y) − (y · ∇)(u � i Fi)(y) ≤ 0 , ∀y ∈ Tphys (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='27) and Ψ(y) < 0 for some y ∈ Tphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Then in Tphys \\ ∂Tphys there exist no periodic solutions, homoclinic loops or oriented phase polygons of the dynamical system ˙y = F(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Put 1 := (1, 1, 1) and g := y × uF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Then g · F = 0 and ⟨1 | ∇ × g⟩|Tphys = Ψ|Tphys, where the second identity easily follows from ⟨1 | F⟩|Tphys = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Now the claim follows by Stoke’s Theorem as in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1 of (Busenberg and Driessche 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1 in Appendix A it is shown that for models with constant per capita birth and death rates one may always replace F by ˜F obeying F|Tphys = ˜F|Tphys SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I 15 and ⟨1 | ˜F⟩ = 0 also outside Tphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' So in this case the second term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='27) vanishes and the condition ∇(u˜F) ≤ 0 looks like in the classical Bendixson-Dulac theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' As in (Busenberg and Driessche 1990) putting y = (S1, S2, I) and u = 1/(S1S2I) we now apply this to the dynamical system Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' We have uF(y) = uMy + uf(y) where M = � � −˜α1 ˜α2 ˜γ1 ˜α1 −˜α2 ˜γ2 0 0 −1 � � , (uf)(y) = � � −( ˜β1 + θ1)/S2 + θ2/S1 + (ν1 − ν2)/I −( ˜β2 + θ2)/S1 + θ1/S2 + (ν2 − ν1)/I ˜β1/S2 + ˜β2/S1 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='28) Here the time scale normalization ˜γ1 + ˜γ2 = 1 is understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Under the following conditions there exist no periodic solutions, homo- clinic loops or oriented phase polygons of the SSISS system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='10) in Tphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' i) (˜αi, ˜γi, θi) ∈ R6 ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' ii) (˜αi, ˜γi, θi) ∈ Cbio and ν1 = ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' First note that ˜γ1 + ˜γ2 = 1 implies that the boundary lines {S1 = 0} and {S2 = 0} cannot both be forward invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Hence, ∂Tphys cannot be a phase polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Next, the second term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='27) vanishes, because we have ⟨1 | F⟩ = 0 also outside of Tphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' We are left to compute ∇·(u(y)My) = − � i̸=j Mi,jyj/yi < 0 and ∇·f = −θ2/S2 1 −θ1/S2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Part i) follows by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='5 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The proof of part ii) relies on the normalization formalism of Section 3 and follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Note that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='12ii) doesn’t follow directly from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6, because there the equivalence transformation Tg need not preserve Tphys, see also Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Usually in the literature on models with constant per capita birth and death rates the vector field F appears in the form F = FM + f, where FM = My − ⟨1 | My⟩y, the second term being nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This makes computations more involved but still yields ΨM|Tphys ≡ ∇ · (uFM)|Tphys − (y · ∇)⟨1 | uFM⟩|Tphys = − � i̸=j Mi,jyj/yi, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8) in (Derrick and Driessche 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The fact that M may be chosen to satisfy ⟨1|M = 0 (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1 in Appendix A, see also remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='11) is rarely noticed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Normalization 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' From now on we drop again the tilde above parameters and also require ν1 = ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' To proceed one has to choose suitable coordinates (X, Y ) on a phase space P ⊃ Tphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Let’s first do some linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Put V = R2 and consider S ≡ |S⟩ = � S1 S2 � , α ≡ |α⟩ = ( α1 α2 ), γ ≡ |γ⟩ = ( γ1 γ2 ), θ ≡ |θ⟩ = � θ1 θ2 � as elements of V (“ket-” or “column-” vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Denote e ≡ ⟨e| := (1, 1) , β ≡ ⟨β| := (β1, β2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1) as a basis in the dual space V ∗ (“bra-” or “row-” vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Putting L(β, θ) := D(β)+E(θ) we then have ⟨e|E(α) = 0, ⟨e|L(β, θ) = ⟨β|, ⟨e | γ⟩ = 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2) where ⟨· | ·⟩ denotes the dual pairing V ∗ ⊗ V → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Generalizing this setting, pick (e, β) any oriented12 basis in V ∗ and γ ∈ V satisfying ⟨e | γ⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Denote E ⊂ End V the right 12The requirement of being oriented (with respect to a given orientation in V ) is a coordinate free version of the condition β2 < β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 16 SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I ideal anihilated by ⟨e| and L := {L ∈ End V | ⟨e|L = ⟨β|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' On V × R = R3 consider the dynamical system ˙S = − [E + IL] S + Iγ, S ∈ V, E ∈ E, L ∈ L, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3) ˙I = (X − 1)I, I ∈ R, X := ⟨β | S⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4) Fixing e and varying (β, E, L, γ) under the above constraints defines a 7-parameter dy- namical system which in fact provides a coordinate free reformulation of the SSISS model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Note that the conditions imply ⟨e | ˙S⟩ + ˙I ≡ ˙S1 + ˙S2 + ˙I = 0, so the dynamics (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4) leaves the cosets {⟨e | S⟩ + I = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='} ⊂ R3 invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Since I = 0 implies ˙I = 0 also the half spaces {I ∈ R±} as well as the plane {I = 0} stay invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The dynamical system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4) on phase space P = {(S, I) ∈ V ×R≥0 | ⟨e | S⟩ + I = 1} with parameter space A = C × B is called the extended SSISS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The extension to negative values of variables Si and parameters a is needed to construct the symmetry operation of GS in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Canonical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Putting I := 1 − ⟨e | S⟩ and using S as independent coordinates on P Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4) becomes redundant and we end up with a two-dimensional system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' However, based on the coordinate free formulation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4), there is another natural set of canonical coordinates for this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Put X := ⟨β | S⟩, Y := ⟨e | S⟩, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='5) or equivalently choose the basis dual to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1) in V e⊥ ≡ |e⊥⟩ := 1 β1 − β2 � 1 −1 � , β⊥ ≡ |β⊥⟩ := 1 β1 − β2 � −β2 β1 � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6) Hence we have X ≡ Xrep, Y ≡ S1 + S2 and S = Xe⊥ + Y β⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='7) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In canonical coordinates the extended SSISS model becomes ˙X = (−aX + b) + (−cX + d)I − ϵI2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8) ˙Y = (1 − X)I = − ˙I , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9) where I = 1 − Y and where the new parameters are given by a := α1 + α2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='10) b := α2β1 + α1β2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='11) c := β1 + β2 + θ1 + θ2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='12) d := γ1β1 + γ2β2 − b + ϵ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='13) ϵ := β1β2 + β1θ2 + β2θ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='14) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In canonical coordinates the matrices E(α) and L(β, θ) := D(β) + E(θ) take the normal form E(α) = � a −b 0 0 � , L(β, θ) = � c −ϵ 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='15) Using |γ⟩ = (β1γ1 +β2γ2)|e⊥⟩+|β⊥⟩ the claim follows by straightforward calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' □ SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I 17 The canonical form of the SSISS dynamical system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9) will also be called the RN- dynamical system (RN = replacement number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Beware that unless β2 ≥ 0 the “would-be” replacement number X may take negative values even for Si ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In fact, in canonical coordinates the physical triangle takes the form Tphys(β) = {(X, Y ) ∈ R × [0, 1] | β2Y ≤ X ≤ β1Y } = {(X, I) ∈ R × [0, 1] | β2(1 − I) ≤ X ≤ β1(1 − I)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='16) So in (X, I)-space Tphys is given by the corners T< = (β2, 0), T> = (β1, 0) and T∧ = (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' To stay with epidemiological conventions, from now on I will use X ≡ Xrep and I ≡ 1−Y as independent variables, in terms of which phase space is now given by P = {(X, I) ∈ R × R≥0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Also note that in canonical coordinates the dynamics is reduced from seven to five pa- rameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' the system no longer depends on β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' So, the role of β is reduced to fixing the image of physical triangles Tphys in canonical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Equivalently this means that fixing x = (a, b, c, d, ϵ) and varying β ∈ B we get an equivalence class of isomorphic dynamical systems, albeit physical triangles are not mapped onto each other under these isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' For a, a′ ∈ A, a = (α, β, γ, θ) and a′ = (α′, β′, γ′, θ′), assume x(a) = x(a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='7) put S := Xe⊥(β) + (1 − I)β⊥ , S′ := Xe⊥(β′) + (1 − I)β′⊥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='17) Then S1 + S2 = S′ 1 + S′ 2 = 1 − I and S = gS′ where g ∈ GL+(R2) is uniquely defined by g = |β⊥⟩⟨e| + |e⊥(β)⟩⟨β′| = 1 β1 − β2 � β′ 1 − β2 β′ 2 − β2 β1 − β′ 1 β1 − β′ 2 , � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='18) implying det g = (β′ 1 − β′ 2)/(β1 − β2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Moreover, (S, I) satisfies the SSISS dynamics (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4) at parameter values a iff (S′, I) satisfies it at parameter values a′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='17) implies ⟨e|S⟩ = ⟨e|S′⟩ = 1−I and ⟨β|S⟩ = ⟨β′|S′⟩ = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Hence, g must satisfy ⟨e|g = ⟨e| and ⟨β|g = ⟨β′| with unique solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Apparently we have g ∈ GS := {g ∈ GL+(R2) | ⟨e|g = ⟨e|} and by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='18) β �→ βg defines a transitive and free right action of GS on B13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='7 below this action will be transported to a free GS-action on A, thus proving parts i)-iv) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this subsection we study the constraints on the new parameters x := (a, b, c, d, ϵ) and admissible ranges of β - or equivalently Tphys(β) - for given values of x, which will finally lead to a proof of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Recalling A ≡ C × B denote φ : A ∋ a �→ (x(a), β) ∈ D × B, D := R+ × R4 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='19) where x(a) is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='10)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The proof of the following Lemma is by straight forward calculation and hence omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 13Note that dim GS = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The parametrization of g in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='18) is redundant by invariance under (β1, β2) �→ (β1 + λ, β2 + λ) and (β1, β2) �→ (χβ1, χβ2), (λ, χ) ∈ R × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 18 SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The map φ : A → D × B provides a diffeomorphism with φ−1 given by αi = b − aβi βj − βi , γi = d + b − ϵ − βj βi − βj , θi = β2 i − cβi + ϵ βj − βi , j ̸= i (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='20) □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Consider D×B as a trivial principal GS-bundle with fiber B and GS right action (x, β) ◁ g := (x, βg), see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Defining a ◁ g := φ−1(x(a), βg) we get an isomorphic GS-bundle structure on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Putting y := (S1, S2, I) and writing the dynamical system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4) with parameters a ∈ A as ˙y = Fa(y), Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4 becomes Fa ◦ Tg = Tg ◦ Fa◁ g , Tg := g ⊕ id , g ∈ GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This proves parts i), iii) and iv) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' □ The remaining transformation rules in part ii) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6 now boil down to an exercise in linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Let D(β) and E(α) be given as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3) and ϑ(β, β′) as in part ii) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Then for all g ∈ GS, α ∈ R2 and β′ = βg ∈ B E(¯gα)g = gE(α), D(β)g = g [D(β′) + E(ϑ(β, β′))] Applying these identities to the dynamical system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4) proves Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Beware that the transformation matrix g preserves S1+S2 but not necessarily R2 ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Also, if a ∈ Aphys (or Abio) and x(a) = x(a′) then it depends on β′ whether a′ ∈ Aphys (or Abio), see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='15 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Hence, the above equivalencies may produce scenarios where a ∈ Aphys and a′ = a ◁ g ̸∈ Aphys and T−1 g Tphys ̸∈ R3 ≥0 but still T−1 g Tphys is forward invariant under the flow of Fa′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Next, on D define the functions R0(x) := b/a ≡ α2β1 + α1β2 α1 + α2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='21) R1(x) := d + b − ϵ ≡ γ1β1 + γ2β2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='22) Obviously we may also use x ≡ (a, R0, R1, c, ϵ) ∈ R+ × R4 as independent parameters in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Moreover we clearly have φ(A+) = {(x, β) ∈ D × B | β2 ≤ Ri ≤ β1 , i = 1, 2} , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='23) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' on A+ the functions Ri may be interpreted as two kinds of mean values of β1 and β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Again beware that for β2 < 0 we may have Ri < 0 even on A+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' To explain the meaning of R0 note that for a > 0 the value of the replacement number X at the disease-free equilibrium (DFE) of the RN-dynamical system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9) is precisely given by X∗ 0 = R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Following results of (Driessche and Watmough 2002) this leads to Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' R0 is called the reduced reproduction number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' As has been shown by (Driessche and Watmough 2002, 2008), in models with just one infectious compartment the more general notion of R0 as the spectral radius of the next generation matrix ((Diekmann, Heesterbeek, and Metz 1990), see also (Diekmann and Heesterbeek 2000)) reduces to the above definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Denoting the values SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I 19 of Si at the DFE by S∗ i we have R0 = β1S∗ 1 + β2S∗ 2, which is the usual formula, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (Kribs-Zaleta and Velasco-Hernandez 2000) or (Avram, Adenane, Bianchin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Mostly in the literature R0 is called the basic reproduction number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' But in case β2 = 0 this terminology is already occupied by r0 := β1/γ as the expected number of secondary cases produced by a typical infectious individual in a totally susceptible population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' So to avoid confusion I prefer to call R0 the reduced reproduction number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Next put Dx := πD(φ(Ax)), x = phys or x = bio, where πD : D × B → D denotes the canonical projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' We look for suitable coordinates describing Dx and then derive additional bounds on β to describe φ(Ax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Consider the following functions on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' A±(x) := 1 2 � a + c ± � (a + c)2 − 4(b + ϵ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='24) B±(x) := 1 2 � c ± √ c2 − 4ϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='25) Then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='15) and the trace-det formula A± and B± provide the eigenvalues of E + L and L, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The meaning of these eigenvalues becomes clear by looking at (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='20) β1 = A+ ⇔ α1 + θ1 = 0 β1 = B+ ⇔ θ1 = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='26) β2 = A− ⇔ α2 + θ2 = 0 β2 = B− ⇔ θ2 = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='27) βi = R0 ⇔ αi = 0 βi = R1 ⇔ γj = 0, j ̸= i (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='28) More generally from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='20) we get θi = (βi − B−)(βi − B+) βj − βi , j ̸= i , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='29) αi + θi = (βi − A−)(βi − A+) βj − βi , j ̸= i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='30) Hence A± will serve to fix the constraints on (x, β) ∈ φ(Aphys) and B± (B ≡ “bio”) to fix constraints on (x, β) ∈ φ(Abio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' First we gather some trivial identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' c = B+ + B− = A+ + A− − a , ϵ = B+B− = A+A− − b , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='31) a = A+ + A− − B+ − B− , aR0 ≡ b = A+A− − B+B− (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='32) From these one immediately computes a(A± − R0) = (A± − B+)(A± − B−) = A2 ± − cA± + ϵ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='33) a(R0 − B±) = (B± − A+)(B± − A−) = B2 ± − (a + c)B± + (b + ϵ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='34) Now let’s introduce the notation DA := D ∩ {A± ∈ R} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='35) DB := D ∩ {B± ∈ R ∧ B− < B+} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='36) DAB := DA ∩ DB ∩ {B− ≤ A− ≤ B+ ≤ A+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='37) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The following identities hold DAB = {x ∈ DA | A− ≤ R0 ≤ A+ ∧ c2 ̸= 4ϵ} = {x ∈ DB | B− ≤ R0 ≤ B+} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='38) Hence in DAB we always have the additional bound B− ≤ A− ≤ R0 ≤ B+ ≤ A+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='39) 20 SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' By Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='33) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='34) on DAB we always have A− ≤ R0 ≤ A+ and B− ≤ R0 ≤ B+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Conversely, if A± ∈ R, c2 ̸= ϵ and A− ≤ R0 ≤ A+ then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='33) implies A2 − − cA− + ϵ ≤ 0 ≤ A2 + − cA+ + ϵ and therefore c2 > 4ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Hence B− < B+ ∈ R and again by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='33) B− ≤ A− ≤ B+ ≤ A+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The second identity follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Denoting iB : DB ∋ x �→ (x, B+, B−) ∈ DB × B the following identities hold φ(Asplit) = {(x, β) ∈ DB × B | B− ≤ β2 < β1 ≤ B+} , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='40) φ(Aθ+α≥0) = {(x, β) ∈ DA × B | β2 ≤ A− ≤ β1 ≤ A+} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='41) φ(Aθ=0) = iB(DB) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='42) φ(Aθ=0 ∩ Aα≥0) = iB(DAB) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='43) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' We have B± ∈ R iff there exists β ∈ R such that β2 − cβ + ϵ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Hence, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='20), if θ1 ≥ 0 and θ2 ≤ 0 then B± ∈ R and B− ≤ β2 < β1 ≤ B+, proving the “⊂”-part in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The opposite direction follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Similarly, A± ∈ R iff there exists β ∈ R such that β2 − (a + c)β + b + ϵ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Hence, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='20), if α1 + θ1 ≥ 0 then A± ∈ R and A− ≤ β1 ≤ A+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' If in addition α2 + θ2 ≥ 0 then also β2 ≤ A−, proving the “⊂”-part in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The opposite direction follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='42) follows since in Aθ=0 we have β1 = B+ and β2 = B−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' If in addition αi ≥ 0 then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='30) implies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' □ We are now in the position to summarize the constraints describing φ(Aphys) and φ(Abio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' For Ax = Cx × B as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='14) - (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='15) we have φ(Aphys) ≡ φ(A+) ∩ φ(Aθ+α≥0) = (DA × B) ∩ {β2 ≤ {A−, R0, R1} ≤ β1 ≤ A+} , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='44) Dphys = DA ∩ {R0,1 ≤ A+} , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='45) φ(Abio) ≡ φ(Asplit) ∩ φ(Aphys) = (DAB × B) ∩ {B− ≤ β2 ≤ A− ≤ R0 ≤ β1 ≤ B+} ∩ {R1 ∈ [β2, β1]} , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='46) Dbio = DAB ∩ {R1 ∈ [B−, B+]} ⊂ Dphys .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='47) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This is a summary of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='23) and Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='13 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='15 motivates the following notation and definition Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' For x ∈ Dbio put βmax 2 (x) := min{A−, R1}, βmin 1 (x) := max{R0, R1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='48) Then β ∈ B is called bio-compatible with x if B− ≤ β2 ≤ βmax 2 and βmin 1 ≤ β1 ≤ B+, equivalently if φ−1(x, β) ∈ Abio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Similarly, β is called compatible if β2 ≤ βmax 2 and βmin 1 ≤ β1 ≤ A+, equivalently if φ−1(x, β) ∈ Aphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' A physical triangle Tphys(β) is called (bio)-compatible, if β is (bio)-compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Hence, (bio-)compatible physical triangles are always forward invariant under the RN- dynamics (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9) and the smallest one is just Tphys(βmin 1 , βmax 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The following Corollary also proves part ii) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I 21 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Let x ∈ Dbio and let β ∈ B be compatible with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Then there exist no periodic solutions, homoclinic loops or oriented phase polygons of the RN-dynamical system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9) in Tphys(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Let Z ⊂ Tphys(β) be a solution cycle (image of a periodic solution, a homoclinic loop or an oriented phase polygon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' As argued in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='12, we must have Z ̸= ∂Tphys(β′) for all β′ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Hence, by forward invariance, Z must lie inside the smallest compatible triangle, Z ⊂ Tphys(βmin 1 , βmax 2 ) ⊂ Tphys(B+, B−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' But, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='15 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='43), φ−1(x, B+, B−) ∈ Abio ∩ Aθ=0 and we get a contradiction with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='12i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' □ Finally, to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6v), note that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='14 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='15 in particular imply (use that GS acts transitively on B) Aθ=0 ◁ GS = Asplit ◁ GS = φ−1(DB × B) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='49) Aθ+α≥0 ◁ GS = φ−1(DA × B) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='50) (Aθ=0 ∩ Aα≥0) ◁ GS = φ−1(DAB × B) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='51) Aphys ◁ GS = φ−1(Dphys × B) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='52) Abio ◁ GS = φ−1(Dbio × B) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='53) Aθ=0 ∩ Abio ◁ GS ⊂ Abio (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='54) where the last equation follows from (Dbio × B) ∩ iB(DB) = iB(Dbio) ⊂ φ(Abio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Part v) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6 now follows from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='49), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='54) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='18 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Put AB := φ−1(DB × B) = Aθ=0 ◁ GS, then AB ⊃ A ∩ {θ1 ≥ θ2 ∨ θ1θ2 > 0} ⊃ Asplit ⊃ Abio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The second and third inclusions are obvious from the definitions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='13) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='15) and the first inclusion follows from DB = D ∩ {c2 > 4ϵ} and c2 − 4ϵ = (β1 − β2)2 + (θ1 + θ2)2 + 2(β1 − β2)(θ1 − θ2) = (β1 − β2 + θ1 − θ2)2 + 4θ1θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' □ Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Sector classification in Abio generalizing Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Sector c = B− + B+ ϵ = B−B+ Interval [B−, B+] I + + 0 < B− < B+ II (SIRS) + 0 0 = B− < B+ III + − 0 < −B− < B+ IV 0 − 0 < −B− = B+ V − − B− < −B+ < 0 VI − 0 B− < B+ = 0 VII − + B− < B+ < 0 22 SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I Let me close by mentioning that the parametrizations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='31) can now be used to generalize the Sector classification of Table 1 from the special case Aθ=0 to all of Abio (more generally to AB := φ−1(DB × B) ⊃ Abio) as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Examples revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' For completeness let us revisit the examples in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4 within the present setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='20)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='26) translate into14 DHeth = Dbio ∩ {R0 = B+ ∧ a < 1 ∧ d = B− = 0} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='55) DSIRI1,2 = Dbio ∩ {R0 = B± ∧ a < 1 ∧ d = B∓(B± + 1 − a)} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='56) DBuDr = Dbio ∩ {R1 < R0 = B+ ∧ B− < 0}15 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='57) DSIRS = Dbio ∩ {B− = 0} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='58) DLM = Dbio ∩ {B− < min{0, R1}} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='59) DKZVH = Dbio ∩ {B− > 0} = DHaCa (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='60) DAABH1 = Dbio ∩ {B− ≤ R1 < B+}15 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='61) DAABH2 = Dbio ∩ {B− < R1 ≤ B+}15 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='62) Note that all models except SI(R)S already satisfy θi = 0 whence ˜β1 = B+, ˜β2 = B− by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='26)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In the SI(R)S model we have instead 0 = ˜β2 = B− < ˜β1 ≤ B+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9 may now be reformulated as follows Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Referring to the sub-cases µ1 = µ2 in (Avram, Adenane, Bianchin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Busenberg and Driessche 1990) and putting DAABH := DAABH1 ∪ DAABH2 we have DHeth = DSIRI1 ∩ {B− = 0} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='63) = DSIRS ∩ {a < 1 ∧ R0 = c ∧ d = 0} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='64) DLM ⊃ DBuDr ∩ {B− ̸= R1} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='65) DLM = DAABH2 ∩ {B− < 0} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='66) DKZVH = DAABH ∩ {B− > 0} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='67) Finally, we are now in the position to generalize the scaling symmetry for SI(R)S models of (Nill 2022) to the present setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' First note that having started from the 10-parameter extended SI(R)S model we now have arrived at dim DSIRS = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Also, dim DHeth = 2 with independent parameters a ∈ (0, 1) and c = R0 = B+ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In particular, if x ∈ DHeth then putting (u, v) := (X, cI) the RN-dynamical system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9) reduces to the classic endemic model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In a second normalization step the number of parameters in the SI(R)S case may now be reduced again by two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this way, for c > d,16 the normalized SI(R)S model also looks like the classic endemic model ˙u = −uv − c1u + c2 , ˙v = uv − v , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='68) the difference being that coming from DHeth we have c1 = a ∈ (0, 1) and c2 = aR0 ≥ 0, whereas coming from DSIRS gives (c1, c2) ∈ R+ × R17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' However, since endemic bifurcation 14Heth = (Hethcote 1974, 1976, 1989);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' SIRI = (Derrick and Driessche 1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' BuDr = (Busenberg and Driessche 1990);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' SIRS = 10-parameter mixed SIRS/SIS model with constant population size and θ2 = β2 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' HaCa = core system in (Hadeler and Castillo-Chavez 1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' KZVH = (Kribs-Zaleta and Velasco-Hernandez 2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' LM = (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Li and Ma 2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' AABH = (Avram, Adenane, Bianchin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' BuDr and AABH come in two versions, the subscript 1 refers to βS > βR and 2 to βS < βR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 15Referring to the sub-case µ1 = µ2 in these models, see Footnotes 10 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 16Note that in DSIRS we have c = B+ ≥ R1 − aR0 = d where equality implies R0 = 0 and R1 = B+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 17In case a = 0 we would get c1 = c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I 23 in the model (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='68) occurs at R0 = c2/c1 = 1, extending this model to the SI(R)S case by including also values c2 < 0 and c1 ≥ 1 doesn’t change its characteristic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In particular, various proofs in the literature on variants of constant population SI(R)S models with standard incidence become obsolete, it’s all contained in Hethcote’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='68) is proven in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In principle, the proof relies on the same structure as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6, with the symmetry group GS acting on A replaced by a dilatation group Gdil = R2 + acting on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Since these dilatations may blow up physical triangles to arbitrary size, we also get the following Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' For x ∈ DSIRS the forward flow of the RN-dynamical system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9) stays bounded for all initial conditions (X0, I0) ∈ R × R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This result may be used to prove, that SI(R)S models as above are always Hamiltonian (Nill n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='[a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='20 is also proven in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Summary and outlook In summary we have seen, that in canonical coordinates the 14-parameter SSISS model, constraint by ν1 = ν2, effectively depends on at most five parameters x = (a, b, c, d, ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' De- pending on natural model restrictions like “phys” or “bio” these parameters obey various relations which can be encoded by further reparametrizations like x = (a, R0, R1, B+, B−), see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='21), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='22), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='31) and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The incidence rates βi have disap- peared from the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Their role is reduced to fixing physical triangles Tphys(β) in (X, I)-space, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' If x ∈ Dbio, then for all compatible values β = (β1, β2) the triangles Tphys(β) stay forward invariant under the RN-dynamics (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8)- (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Independence of β also means that SSISS models at parameter values φ−1(x, β) for fixed x ∈ D and varying β ∈ B are all isomorphic to each other18 (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The isomorphisms are provided by a parameter symmetry group GS ⊂ GL+(R2) acting simultaneously on phase space P and parameter space A (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6i-iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' If x ∈ DB then a representative in A of the equivalence class x may always be chosen by putting β1 = B+ and β2 = B− and hence θi = 0 (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In combination with methods from (Busenberg and Driessche 1990) this also leads to a proof of absence of periodic solutions for all a ∈ Abio (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In part III of this work it will be shown, that the model also admits an additional scaling symmetry leading to a second normalization step, similar as described for the SI(R)S model in Appendix B, see also (Nill 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this way the number of essential parameters will further reduce from five to three (respectively two in Sectors II and VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Part II of this work will reanalyze equilibrium points and their stability properties in all Sectors of Abio, thereby recovering and extending the results of (Avram, Adenane, Bianchin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Hadeler and Castillo-Chavez 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Kribs-Zaleta and Velasco-Hernandez 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Li and Ma 2002), which had been obtained for θi = 0 and some more parameter restrictions, see Table 2 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This approach will differ from previ- ous papers by relying on the normalization formalism and sector classification of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this way the search for endemic equilibria (X∗, I∗) simplifies consider- ably, since always X∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' So one is left with analyzing roots of the quadratic equation h(I∗) := ˙X(X∗ = 1, I∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This will also uncover an exceptional scenario in Sectors III-V, which apparently has been overlooked in the literature so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 18By Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9, physical triangles are not mapped onto each other under these isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 24 SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Normalizing linear vital dynamics This Appendix gives a normalization prescription for the dynamics of fractional vari- ables in an n-compartment model with linear vital dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Let the vectorfield V : Rn → Rn be homogeneous of degree one and assume there exists ν = (ν1, · · · , νn) such that ⟨1|V(Y)⟩ ≡ � i Vi(Y) = ⟨ν|Y⟩ for all Y ∈ Rn, where 1 := (1, · · · , 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Call N(Y) := ⟨1|Y⟩ the total population and y := N−1Y the fractional compartment vari- ables, then the dynamical system ˙Y = V(Y) implies ˙y = V(y) − ⟨ν | y⟩y =: F(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Denote S := {y ∈ Rn | ⟨1|y⟩ = 1}, then clearly ⟨1|F⟩|S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The aim is to substitute F by ˜F such that F|S = ˜F|S and ⟨1|˜F⟩ = 0 holds as an identity on all of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The following Lemma holds by straight forward calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Put Λijk := (δij − δik)(νk − νj) and Λi(y) := � j,k Λijkyjyk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' i) For all y ∈ Rn and i = 1, · · · , n we have 1 2Λi(y) = � k (νk − νi)yiyk ≡ yi⟨ν|y⟩ − νiyi⟨1|y⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1) ii) Put ˜F := V − diag(ν) − 1 2Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2) Then F|S = ˜F|S and ⟨1|˜F⟩ = 0 as an identity on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' By this method we also get conditions guaranteeing that constant per capita birth and death rates become redundant as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Let V(Y) be of the form Vi(Y) = � j MijYj + 1 2 � j,k ΓijkYjYk/N + � j LijYj where without loss Γijk = Γikj and where � i Mij = � i Γijk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Hence, all vital dynamics parameters are encoded in (Lij) and νj := � i Lij satisfies ⟨1|V = ⟨ν|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' If in this case Lij ̸= νiδij ⇒ Mij ̸= 0 and νj ̸= νk ⇒ (Γjjk ̸= 0 ∧ Γkkj ̸= 0), then for the dynamics of fractional variables all parameters Lij are redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Applying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2) we have ˜Fi(y) = � j ˜ Mijyj+ 1 2 ˜Γijkyjyk, where ˜ Mij = Mij+Lij−νiδij and ˜Γijk = Γijk − Λijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The claim follows since Λijk = Λikj, Λjjk = −Λkkj and Λijk = 0 if νj = νk or if j ̸= i ̸= k, which also yields � i Λijk = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' □ Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Scaling the SI(R)S model In this appendix we extend the dilatation symmetry as proposed for a 6-parameter SI(R)S model in (Nill 2022) to the 10-parameter extended SI(R)S model as classified in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Denote Sector II in DB by DII := DB ∩ {B− = 0} and DSIRS := DII ∩ Dbio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Recall that in DII we have c = B+ > 0 and in DSIRS we have 0 ≤ Ri ≤ B+ and hence d − c = R1 − aR0 − B+ ≤ 0, where equality implies R0 = 0 and R1 = B+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Hence the following Lemma in particular includes Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I 25 Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Consider the RN-dynamical system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8) - (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9) on phase space P ≡ R×R≥0 for parameter values x = (a, b, c, d, ϵ = 0) ∈ DII ∩ {d ≤ c ∧ d = c ⇒ R0 < 1} ⊃ DSIRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Let T ⊂ P be a rectangular triangle with corners T◁ = (X◁, 0), T▷ = (X▷, 0) and T△ = (X◁, I△), where X◁ < X▷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Call T compatible with x if I△ = (X▷ − X◁)/c X◁ ≤ min{R0, d/c} R0 − X◁ ≤ I△ min{c, (c − d)/a} i) Then every x-compatible triangle T is forward invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' ii) The forward flow for arbitrary initial conditions (X0, I0) ∈ P stays bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' To prove part i), the upper bounds on X◁ imply ˙X > 0 on the line {X = X◁}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' We are left to show ˙X + c ˙I ≤ 0 on the hypotenuse X(I) = X◁ + c(I△ − I), 0 ≤ I ≤ I△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' ˙X + c ˙I = a(R0 − X(I)) + (d − c)I = a(R0 − X◁ − c(I△ − I)) + (d − c)I ≤ I△ min{ac, c − d} − ac(I△ − I) + (d − c)I ≤ 0 Part ii) follows since for d < c we may always choose X◁ < X0 and X▷ large enough, such T is x-compatible and (X0, I0) ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' For d = c and R0 < 1 x-compatibility requires X◁ = R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' If in this case X0 < R0 glue the rectangle R = [X0, R0]×[0, I△] to the left of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Then (X0, I0) ∈ R ∪ T for X▷ large enough and R ∪ T is forward invariant, since ˙I < 0 and ˙X > 0 for (X, I) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' □ Given x ∈ DII ∩ {d ≤ c ∧ d = c ⇒ R0 < 1} as above and T compatible with x we now show that the RN-dynamical system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8) - (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9) may always be rescaled to an isomorphic system with parameters x′ ∈ DSIRS such that T maps to the physical triangle Tphys(B′ +, 0) of the SI(R)S system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Following (Nill 2022) the dilatation symmetry group Gdil ≡ GX × GI ≡ R2 + is defined by rescaling (X, I) variables according to X(ξ,λ)(t) − 1 := ξ(X(ξt) − 1), I(ξ,λ)(t) := λI(ξt), (ξ, λ) ∈ R2 + The following Lemma is easily verified by straightforward calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Let the group action ▷ : Gdil × D ∋ (ξ, λ, x) �→ (ξ, λ) ▷ x ∈ D be given by (ξ, λ) ▷ (a, R0 − 1, c, d − c, ϵ) := (ξa, ξ(R0 − 1), ξc/λ, ξ2(d − c)/λ, ξ2ϵ/λ2) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1) and for x ∈ D let fx(X, I) denote the vector field of the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8) - (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Then ( ˙X, ˙I) = fx(X, I) ⇐⇒ ( ˙X(ξ,λ), ˙I(ξ,λ)) = fx′(X(ξ,λ), I(ξ,λ)), x′ = (ξ, λ) ▷ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' □ Note that this action leaves all Sectors in DB invariant, but in general not Dbio ⊂ DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' We now determine Gdil ▷ DSIRS, thereby also providing an alternative proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' i) Let T be compatible with x ∈ DII ∩ {d ≤ c ∧ d = c ⇒ R0 < 1} in the sense of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Then there exists a unique dilatation transformation (ξ, λ) ∈ Gdil such that x′ := (ξ, λ) ▷ x ∈ Dbio and such that the rescaled triangle satisfies T(ξ,λ) = Tphys(B′ +, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' ii) Gdil ▷ DSIRS = DII ∩ {d ≤ c ∧ d = c ⇒ R0 < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' 26 SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' To prove part i) denote transformed quantities by a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The requirements T′ ◁ = (0, 0) and T′ △ = (0, 1) fix ξ = (1 − X◁)−1 and λ = I−1 △ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Hence X▷ maps to ξcI△ = c′ = B′ + and therefore T(ξ,λ) = Tphys(B′ +, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' To show 0 ≤ R′ i ≤ B′ + use R′ 0 = ξ(R0 − 1) + 1 = ξ(R0 − X◁) and therefore 0 ≤ R′ 0 ≤ ξ λ min{c, (c − d)/a} = min{c′, (c′ − d′)/a′} ≤ B′ + By the above we also have R′ 1 = a′R′ 0 + d′ ≤ c′ = B′ + and we are left to show R′ 1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Sufficient is d′ ≥ 0 which follows from 1−d′/c′ = ξ(1−d/c) ≤ ξ(1−X◁) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' This proves part i) and therefore also the “⊃”-direction of part ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' To prove the “⊂”-direction use that the action of Gdil on D preserves the sign of d − c and in case d = c we have R0 = 0 and therefore R′ 0 = ξ(R0 − 1) + 1 = 1 − ξ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' □ As in (Nill 2022), the above dilatation symmetry leads to a second normalization step for the SIRS-Sector, thus further reducing its number of essential parameters from four to two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Equivalently this means, that equivalence classes of Gdil-isomorphic systems with parameters in Gdil ▷ DSIRS are naturally parametrized by KSIRS := (Gdil ▷ DSIRS)/Gdil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' A convenient realization of the normalized system on phase space P = {(q, p) ∈ R × R≥0} is given by putting q(t) := 1 a(X(t/a) − 1) , p(t) := c aI(t/a) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2) In terms of these variables the RN-dynamical system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8) - (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9) becomes ˙q = −q(p + 1) + κ0 − κ1p , ˙p = qp , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3) where the new Gdil-invariant parameters are given by κ0 := R0 − 1 a , κ1 := c − d ac .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4) The only remaining constraint on the reduced parameter space says KSIRS = {(κ0, κ1) ∈ R × R≥0 | κ1 = 0 ⇒ κ0 < 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='5) Thus, after normalization the whole SIRS Sector just looks like Hethcote’s classic endemic model except for a somewhat less restricted parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In fact, by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='55), DHeth ⊂ DSIRS is already two-dimensional with independent parameters a ∈ (0, 1) and c = R0 = B+ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' These map injectively to KSIRS via κ0 = (c − 1)/a and κ1 = 1/a, whence DHeth ∼= KHeth = KSIRS ∩ {κ1 > 1 ∧ κ0 + κ1 > 0} (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6) The normalization convention in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='68) is obtained under the restriction c > d or equivalently κ1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this case one may alternatively use u(t) − 1 := c c − d(X(ct/(c − d)) − 1) = 1 κ1 q(t/κ1) , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='7) v(t) := c2 c − dI(ct/(c − d)) = 1 κ1 p(t/κ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='8) In terms of these variables we recover the normalization convention (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='68) ˙u = −uv − c1u + c2 , ˙v = uv − v , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='9) where c1 = 1/κ1 and c2 = 1/κ1 + κ0/κ2 1, which is also the version given in (Nill 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In part III of this work the above normalization step will be generalized to all Sectors of Dbio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this way the equation for ˙q in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3) gets an additional term −κ2p2, and so our REFERENCES 27 initial 14-parameter19 SSISS model boils down to a much simpler 3-parameter dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' The case α1 = α2 = 0 This Appendix shortly discusses the border case α1 = α2 = 020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In this case define parameter spaces C0 x as in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='11)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='15) with αi = 0 and A0 x := C0 x ×B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' In particular, in A0 bio we have θ1 ≥ 0, θ2 = 0, γi ≥ 0 and γ1 + γ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3 still holds with a = b = 0 and d = R1 + ϵ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' the replacement number dynamics becomes ˙X = (d − cX)I − ϵI2 , ˙I = (X − 1)I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='1) In this case R0 is undefined and there is a continuum of disease free equilibria at I = 0, which are locally stable for X < 1 and unstable for X > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4 remains unchanged provided a = a′ ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Putting D0 = {(c, d, ϵ) ∈ R3} Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6 still holds with A replaced by A0 and D replaced by D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Moreover, in A0 bio we get A+ = B+ = β1+θ1, A− = B− = β2, c = β1+β2+θ1, ϵ = β2(β1+θ1) and putting D0 A = D0 B = D0 AB := D0∩{c2 > 4ϵ} Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='15 becomes φ(A0 phys) = (D0 B × B) ∩ {β2 ≤ {B−, R1} ≤ β1 ≤ B+} , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2) D0 phys = D0 B ∩ {R1 ≤ B+} , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='3) φ(A0 bio) = (D0 B × B) ∩ {B− = β2 ≤ R1 ≤ β1 ≤ B+} , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='4) D0 bio = D0 B ∩ {B− ≤ R1 ≤ B+} ⊂ D0 phys .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='5) So, for x ∈ D0 phys physical triangles Tphys(β1, β2) are forward invariant provided (β1, β2) satisfy the bounds C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Finally, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='42) becomes φ−1(iB(D0 B)) = φ(Aθ=0 ∩ Aα=0) and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='12, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='6 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='17 stay valid also for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' References Arino, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=', C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='[b]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' “Symmetries and normalization in 3-compartment epidemic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' II: Equilibria and stability.” paper to be written up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content=' Nill, Florian (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAyT4oBgHgl3EQfWPdu/content/2301.00159v1.pdf'} +page_content='[c]).' metadata={'source': 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b/XtFJT4oBgHgl3EQf5y0l/content/tmp_files/2301.11671v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b5226f574b054b916198c1bb2de92004a893fdba --- /dev/null +++ b/XtFJT4oBgHgl3EQf5y0l/content/tmp_files/2301.11671v1.pdf.txt @@ -0,0 +1,1895 @@ +arXiv:2301.11671v1 [math.LO] 27 Jan 2023 +PAC STRUCTURES AS INVARIANTS OF FINITE GROUP +ACTIONS +DANIEL MAX HOFFMANN† AND PIOTR KOWALSKI‡ +Abstract. We study model theory of actions of finite groups on substructures +of a stable structure. We give an abstract description of existentially closed +actions as above in terms of invariants and PAC structures. We show that if +the corresponding PAC property is first order, then the theory of such actions +has a model companion. Then, we analyze some particular theories of interest +(mostly various theories of fields of positive characteristic) and show that in +all the cases considered the PAC property is first order. +1. Introduction +In this paper, we consider the notion of a pseudo algebraically closed (PAC) +substructure of a stable structure. This notion originates from the theory of pseudo +algebraically closed fields, which were first considered by Ax in 1960’s while he +worked on pseudofinite fields ([2]). Studying PAC structures beyond the case of +fields was initiated by Hrushovski ([27]) in the strongly minimal context. Pillay +and Polkowska considered the PAC property in the stable case ([41]), there are +slight differences with the approach we take here. PAC structures also appeared in +Afshordel’s thesis ([1]). Recently, PAC structures were analized by the first author +([21], [22]) and also by Dobrowolski, the first author, and Lee ([15]). +Here, we are working with a (complete) stable theory T which admits quantifier +elimination and then focus on its universal part T∀. +In other words, a typical +situation looks as follows. +We have a universal theory T∀ with a stable model +completion T , so T has quantifier elimination and T axiomatizes existentially closed +models of T∀. Then, intuitively, the class of PAC structures in T lies in between +the class of existentially closed structures (models of T ) and the class of all the +structures considered (models of T∀). There are several possible definitions of the +notion of PAC, we adopt here the definition from [21] (expressed in terms involving +stationary types), which is a slight modification of the definition from [41], and +which is equivalent to Afshordel’s definition from [1] in the case of stable theories. +To define the notion of a PAC structure, one needs to use an appropriate notion +of irreducibility. In the classical case of PAC fields, a topological notion is used +coming from the Zariski topology. Hrushovski used in [27] “Morley irreducibility”, +that is he considered definable sets of Morely degree one. Pillay and Polkowska +used [41] stationary types and we proceed similarly here (however, we avoid any +†SDG. Supported by the Narodowe Centrum Nauki grant no. 2021/43/B/ST1/00405. +‡ Supported by the Narodowe Centrum Nauki grant no. 2021/43/B/ST1/00405 and by the +T¨ubitak 1001 grant no. 119F397. +2020 Mathematics Subject Classification Primary 03C60, 03C45 Secondary 12H10. +Key words and phrases. Finite group action, Model companion, PAC structure. +1 + +2 +D. M. HOFFMANN AND P. KOWALSKI +saturation requirements as given in [41]). We say that a structure F |= T∀ is PAC +in T (see Definition 2.3) if all stationary types (in the sense of the theory T ) over +F are finitely satisfiable in F. +Let us point out that in the case of the theory +of algebraically closed fields, all the irreducibility notions mentioned above are +essentially the same. However, this is not the case for other theories of interest as +the theory of differentially closed fields of characteristic 0 or the theory of compact +complex manifolds (see Section 4.1.2). Nevertheless, we show in Section 4.1 that +all these irreducibility notions lead to the same notion of a PAC structure. +For an extension F ⊆ K of models of T∀, we obtain relative notions of K-strongly +PAC and algebraically K-strongly PAC (see Definition 2.4). They are meaningful +and can be though of as measuring the distance between being PAC and being a +model of T (K-strongly PAC) or between being definably closed and algebraically +closed (algebraically K-strongly PAC), see Remark 2.5. +Our main motivation for considering PAC structures comes from model theory +of group actions. In the set-up above, we consider actions of a fixed group G on +models of T∀ by automorphisms. Clearly, such actions are first-order expressible +in an appropriate language and we aim to describe existentially closed actions and +check whether a model companion of the theory of such actions exists. The result +below may be considered as an abstract generalization of our theorem about finite +group actions on fields (see [24, Theorem 3.29]) and as a continuation of studies +from [21]. +Theorem 3.13. Let G be a finite group and T be a stable theory coding finite +sets, which has quantifier elimination and eliminates strong types (that is: types +over algebraically closed sets are stationary). Assume that G acts faithfully on +K = dcl(K) |= T∀. Then, the following are equivalent. +(1) The action of G on K is existentially closed. +(2) The structure of invariants KG is K-strongly PAC. +(3) The structure of invariants KG is PAC and algebraically K-strongly PAC. +The above theorem gives a description of existentially closed finite group actions, +but it is not clear whether this description is first-order, so this theorem does not +settle the question of the existence of a model companion of the theory of finite +actions. We can show the following implication. +Theorem 3.23. Let G be a finite group and T be as in the statement of Theorem +3.13. If the class of T -PAC structures is elementary, then the model companion of +the theory of G-actions on models of T∀ exists. +After the abstract description of existentially closed actions (Theorem 3.13) and +giving a criterion for existence of a model companion of the theory of finite actions +(Theorem 3.23), we focus on particular examples of theories. We discuss the fol- +lowing three stable theories of fields of positive characteristic (p is a prime and e is +a positive integer): +(1) The theory SCFp,e of separably closed fields of characteristic p and insep- +arability degree e. +(2) The theory SCFp,∞ of separably closed fields of characteristic p and infinite +inseparability degree. +(3) The theory DCFp of differentially closed fields of characteristic p. +In the most interesting cases of the theories SCFp,∞ and DCFp, we do not have +elimination of imaginaries, however we still have its weaker versions (coding finite + +PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS +3 +sets and eliminating strong types), which are enough for the set-up from Theorems +3.13 and 3.23. +For these theories, we describe PAC structures in a first-order +way using a result of Tamagawa (see Theorem 4.21) about positive characteristic +PAC fields. We finish with some general questions regarding the PAC property +and existence of a model companion of the theory of finite actions. It should be +mentioned that after replacing a finite group G with the infinite cyclic group (Z, +), +then the model theory of actions of (Z, +) has been thoroughly studied (see e.g. +[13] and [11]). We compare these two situation in Section 4.5. +This paper is organized as follows. In Section 2, we introduce several versions of +the notion of a PAC structure and show the basic results about them. In Section +3, we put the group action to the picture and prove the main two abstract results +stated above (Theorems 3.13 and 3.23). In Section 4, we consider some particular +theories (mostly theories of fields of positive characteristic) and give a first order +characterization of PAC structures with respect to these theories. +2. Preliminaries +2.1. Set-up. Let T be a complete first order theory with a monster model C |= T +(i.e. a strongly ¯κ-homogeneuos and ¯κ-saturated model of T for a very big cardinal +¯κ). Usually, x stands for a (finite) tuple of variables. Moreover, for the rest of this +paper, let G be a group such that |G| < ¯κ. +Bearing in mind any future applications, we try in this paper to formulate each +result with a minimal list of assumptions. +Therefore, we organize our general +model-theoretic assumptions in the following list (we are aware that there are some +overlaps, but we preferred more transparent exposition): +(QE) T has quantifier elimination. +(FS) T codes finite tuples (i.e. eliminates finite imaginaries). +(♠) T has (FS) and for every k < ω, for every variable x corresponding to a +real sort and the 0-definable equivalence relation E on Sk +x given by +E(¯x, ¯x′) +⇐⇒ +{x1, . . . , xk} = {x′ +1, . . . , x′ +k}, +there exists a 0-definable in L function f : Sk +x → Sw such that E is a +fibration of f. +(♥) T is stable and types over algebraically closed sets are stationary (elimina- +tion of strong types). +Convention: if a statement starts with any combination of the above properties, +it means that we assume the properties given in this particular combination. For +example, the following remark assumes property (FS): +Remark 2.1. (FS) The condition (♠) is equivalent to: +• on each sort there is at least one 0-definable element and +• there is a sort with at least two 0-definable elements. +Proof. Similarly as in the proof of Lemma 8.4.7 from [49], but, here, we allow many +sorted structures. +□ +Remark 2.2. Let us discuss what one can do to meet the above requirements +if starting from arbitrary stable L0-theory T0. As we would like to work under +assumptions of quantifier elimination and elimination of imaginaries, we pass to the +language L := (Leq +0 )m and L-theory T := (T eq +0 )m (we add imaginary sorts and then +do the Morleysation). This new theory T is stable, has quantifier elimination and + +4 +D. M. HOFFMANN AND P. KOWALSKI +elimination of imaginaries. On top of that, every 0-definable equivalence relation +E on Cn is the fibration of the canonical projection πE : Cn → Cn /E which is +build-in in the language (Leq +0 )m, thus a 0-definable function. Strong types in any +stable theory are stationary, and b |⌣A A for any b and A. Therefore T enjoys all +the properties: (QE), (FS), (♠) and (♥). +2.2. Notion of PAC structure and auxiliary facts. In this subsection, we +recall several definitions and useful facts from [21] and [22]. We also provide a few +new notions closely related to the old definitions. The reader may also consult [41] +and [43] for more on PAC structures in general model theoretic framework. Also [1] +provides a nice of exposition of the notion of a PAC structure and related topics. A +well-written survey on different variants of the notion of elimination of imaginaries +and related concepts from the Galois theory is [10]. +Definition 2.3. (Let T be stable.) A substructure F of C is pseudo-algebraically +closed (PAC) if every stationary type over F (in the sense of the L(F)-theory of +C) is finitely satisfiable in F. +The above definition appears in [21] (see also Definition 5.29 in [1]). In subsection +3.1 of [21], there is a discussion on possible choices of the definition of a PAC +substructure and a comparison of Definition 2.3 to definitions of PAC structures +given in [27] and in [41]. In short, Definition 2.3 coincides with the definition of a +PAC substructure in the strongly minimal context of [27] and relaxes the saturation +assumption from the definition of a PAC substructure from [41]. Note that every +PAC substructure is automatically definably closed. Thus PAC substructures for +T = ACF coincide with perfect pseudo-algebraically closed fields (as defined in +e.g. [18]). +Definition 2.4. Let F = dcl(F) ⊆ K ⊆ C. +(1) We say that F is K-strongly PAC if each type p(x) ∈ S(F), which has a +unique non-forking extension over K, is finitely satisfiable in F. +(2) We say that F is algebraically K-strongly PAC if each algebraic type p(x) ∈ +S(F), which has a unique non-forking extension over K, is finitely satisfiable +(thus realized) in F. +Note that being K-strongly PAC for F ⊆ K implies being algebraically K- +strongly PAC for F. Moreover, being K-strongly PAC for F implies being a PAC +substructure for F. +Remark 2.5. It should help to understand the relative notions of (algebraically) +K-strongly PAC by considering the ultimate cases of K = F and K |= T . It is +quite easy to see the following. +(1) A structure F is F-strongly PAC if and only if F |= T . +(2) (T is stable) A structure F is K-strongly PAC for K |= T if and only if F +is PAC. +(3) A structure F is algebraically F-strongly PAC if and only if F = acl(F). +(4) A structure F is algebraically K-strongly PAC for K |= T if and only if +F = dcl(F). +Definition 2.6. +(1) Let F ⊆ K be small subsets of C. We say that F ⊆ K is +primary if +dcl(K) ∩ acl(F) = dcl(F). + +PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS +5 +(2) Let F ⊆ K be small subsets of C. We say that F ⊆ K is regular if F ⊆ K +is primary and F = dcl(F). +(3) Let F be a small definably closed substructure of C. We say that F is +regularly closed if for every small substructure F ′ of C, which is a regular +extension of F, it follows F ⪯1 F ′ (i.e. F is existentially closed in F ′). +The above notion of a primary extension was previously (e.g. +[21], [22]) called +“regular”. It corresponds to regular extensions in T =ACF provided the smaller +field is perfect (equivalently, definably closed). Here, we decided to follow closer +the terminology from the theory of fields and distinguish between “primary” and +“regular” extensions. We plan to refine even more the notion of the model-theoretic +“regular” extension after studying a possible notion of the model-theoretic separable +extension in the future. +Now, we will sharpen facts from earlier articles which lead to the main results +in this manuscript. The majority of [21] was written under the assumption of (full) +elimination of imaginaries, elimination of quantifiers and stability. This is fine if we +are interested in an abstract approach to the subject. However, as we are interested +in applications of our results to particular theories, which do not enjoy elimination +of imaginaries (see Section 4), we need to relax this assumption. Moreover, the +assumption on stability was not crucial in several useful facts from [21], making +them applicable in a broader context. Therefore we take the opportunity to provide +the following results with minimal assumptions. The proofs of the following facts +remain almost the same as the proofs of their counterparts from [21]. Recall that +“regular” extensions from [21] are now “primary” extensions. +All the proper subsets, substructures and tuples of the monster model C are, +if not stated otherwise, small in comparison to the saturation of C. Here, upper +case letters, like E or A, are denoting proper subsets, and lower case letters, like a, +stand for tuples. +Fact 2.7 (Fact 3.32 in [21]). (FS) If E ⊆ A is primary then for every a ∈ acl(E) +there exists a unique extension of tp(a/E) over A. +Fact 2.8 (Fact 3.33 in [21]). (FS) If E ⊆ A is primary , f1, f2 ∈ Aut(C) and +f1|E = f2|E, then there exists h ∈ Aut(C) such that h|A = f1|A and h|acl(E) = +f2|acl(E). +Fact 2.9 (Corollary 3.34 in [21]). (FS) If E ⊆ A is primary and A0 ⊆ A then +tp(A0/E) has a unique extension over acl(E). +The following definition is taken from page 21. of [1]. +Definition 2.10. We say that a type p(x) ∈ S(A) is acl-stationary if it has a +unique extension over acl(A). +Lemma 2.11. (FS) Consider p ∈ S(E). The following are equivalent: +(1) p is acl-stationary, +(2) E ⊆ dcl(Ea) is primary for some a |= p, +(3) E ⊆ dcl(Ea) is primary for every a |= p. +Proof. The proof is similar to the proof of Lemma 3.35 in [21], but a few steps +require sharper reasoning, thus we include it here. +The equivalence (2) ⇐⇒ (3) follows by definition. First, we argue for (1)⇒(2): +assume (1) and suppose that (2) does not hold. As p is acl-stationary, there exists + +6 +D. M. HOFFMANN AND P. KOWALSKI +a unique extension p|acl(E) of p over E. Let a |= p|acl(E), then a |= p and E ⊆ Ea +is not primary. Take +c ∈ dcl(Ea) ∩ acl(E) \ dcl(E). +Since c ̸∈ dcl(E), there exists f ∈ Aut(C /E) such that f(c) ̸= c. We see that +f(a) |= p|acl(E), so there exists h ∈ Aut(C / acl(E)) such that h(a) = f(a). Note +that h−1f ∈ Aut(C /Ea) and, because c ∈ dcl(Ea) and c ∈ acl(E), +c = h−1f(c) = f(c) ̸= c, +so a contradiction. The implication (2)⇒(1) is contained in Fact 2.9. +□ +Fact 2.12 (Lemma 3.35 in [21]). (FS, ♥) Consider p ∈ S(E). The following are +equivalent: +(1) p is stationary, +(2) p is acl-stationary, +(3) E ⊆ Ea is primary for some a |= p, +(4) E ⊆ Ea is primary for every a |= p. +Fact 2.13 (Corollary 3.36 in [21]). (QE, FS, ♥) For any small substructure N +there exists a non-algebraic stationary type over N in any finitely many variables. +Fact 2.14 (Corollary 3.38 in [21]). (FS, ♥) Assume that A, B ⊆ C, E ⊆ A is +primary, f1, f2 ∈ Aut(C) and f1|E = f2|E. If A |⌣E B and f1(A) |⌣f1(E) f2(B) +then there exists h ∈ Aut(C) such that h|A = f1|A and h|B = f2|B. +Fact 2.15 (Lemma 3.39 in [21]). (FS, ♥) If E ⊆ A ∩ B, E ⊆ A is primary and +B |⌣E A then B ⊆ BA is primary. +Fact 2.16 (Corollary 3.40 in [21]). (FS, ♥) If E ⊆ A and E ⊆ B are primary, and +B |⌣E A then also E ⊆ BA is primary. +Remark 2.17. +(1) (FS, ♥) F ⊆ K is primary if and only if for every tuple b +from dcl(K), the type tp(b/F) is stationary (Fact 2.12). +(2) (QE, FS, ♥) Using the item (1), a substructure F is PAC if and only if it +is definably closed and regularly closed. +Definition 2.18. +(1) Assume that F ⊆ K are substructures of C. +We say +that K is normal over F (or we say that F ⊆ K is a normal extension) if +σ(K) ⊆ K for every σ ∈ Aut(C /K). (Note that if K is small and F ⊆ K +is normal, then it must be K ⊆ acl(F).) +(2) Assume that F ⊆ K ⊆ acl(F) are small substructures of C such that +F = dcl(F), K = dcl(K) and K is normal over F. In this situation we say +that F ⊆ K is a Galois extension. +Definition 2.19. Assume that F ⊆ K is an extension of substructures in C. We +define the Galois group of the extension F ⊆ K as +G(K/F) := Aut(K/F) = {f|K | f ∈ Aut(C /F), f(K) = K}. +Moreover B is any subset of C, then the extension dcl(B) ⊆ acl(B) is Galois and we +speak about the absolute Galois group of B which is the following profinite group: +G(B) := G(acl(B)/ dcl(B)). + +PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS +7 +Note that the above definition of G(K/F) is often expressed in terms of the +automorphisms of K as an L-structure on its own, but as we will work under the +assumption of the quantifier elimination, both variants of the definition coincide +and it just the matter of taste. +The following useful fact is standard and its proof is straightforward. +Lemma 2.20. Assume that F ⊆ K is a Galois extension and p(x) ∈ S(F). Then +the Galois group G(K/F) acts transitively on the set of extensions of p over K. +The following definition and example are taken from [15] and [41]. +A more +detailed discussion of examples of PAC structures and the property from Definition +2.21 will be given in Section 4. +Definition 2.21. (Let T be stable.) We say that PAC is a first order property in +T (= Th(C)) if there exists a set Σ of L-sentences such that for any P ⊆ C +P |= Σ +⇐⇒ +P is PAC. +Example 2.22. +(1) PAC is a first order property in ACFp for p = 0 and for p +being a prime number, see Proposition 11.3.2 in [18]. +(2) The axioms given in Proposition 5.6 from [41] show that PAC is a first order +property (in the above sense) in DCF0 which is formulated in a different +way than the condition “PAC is a first order property” appearing in [41]. +3. Finite group actions +The main goal of this section is to describe existentially closed substructures +with a finite group action in first order terms. The general strategy is as follows. +First, characterize their structure by the structure of the invariants of the group +action, then answer which properties of the invariants correspond to the existential +closedeness of the whole substructure with group action. +Finally, express these +properties as first order statements. +3.1. Basic facts. We introduce the language LG being the language L extended +by a unary function symbol σg for each g ∈ G, i.e. LG = L ∪ {σg | g ∈ G}. Often, +“σg” will denote also the interpretation of the symbol σg in a given LG-structure. +Moreover, we set ¯σ := (σg)g∈G. +We consider the collection of sentences in the +language LG, say AG, which precisely expresses the following +• σg is an automorphism of the L-structure for every g ∈ G, +• σg ◦ σh = σg·h for all g, h ∈ G. +In other words, if K is an L-structure, and there exists an LG-structure (K, ¯σ) +living on K, we have that (K, ¯σ) |= AG if and only if for each g ∈ G we have that +σg ∈ Aut(K) and the map +G ∋ g �→ σg ∈ Aut(K) +is a group homomorphism. +Definition 3.1. +(1) Let (K, ¯σ) be an LG-structure. We say that ¯σ is a G- +action on K if (K, ¯σ) |= AG. +(2) If T ′ is an L-theory, then by (T ′)G we denote the set of consequences of +T ′ ∪ AG. + +8 +D. M. HOFFMANN AND P. KOWALSKI +(3) If (K, ¯σ) |= (T∀)G, where K is of cardinality smaller than the saturation of +C, then we call it a substructure with G-action. Note that, without loss of +generality, K ⊆ C, thus the name “substructure”. +(4) We say that a substructure with G-action (K, ¯σ) is existentially closed if +(K, ¯σ) is an existentially closed model of the theory (T∀)G. +(5) If the existentially closed models of the theory (T∀)G form an elementary +class, we denote the theory of this class by G − T . +Definition 3.2. Assume that (K, ¯σ) is a substructure with G-action. Then we +denote +KG := {a ∈ K | (∀g ∈ G) (σg(a) = a) } +and call it the substructure of invariants. +Remark 3.3. (QE) Let (K, ¯σ) be a substructure with G-action. If (K, ¯σ) is ex- +istentially closed then K = dcl(K). If K = dcl(K) then KG = dcl(KG). For the +standard proofs, the reader may consult Remark 3.24 and Remark 3.26 in [21]. +Lemma 3.4. (QE) Let (K, ¯σ) be a substructure with G-action and let p(x) ∈ S(K) +be a G-invariant type (i.e. σg(p) = p for every g ∈ G). Then for any a |= p the set +dcl(K, a) might be equipped with a G-action extending (K, ¯σ) and acting trivially +on a. +Proof. Let a |= p and let ¯k be some enumeration of K. Then ¯ka ≡ σg(¯k)a for +any g ∈ G. This implies that, for each g ∈ G, there exists σ′ +g ∈ Aut(C) such that +σ′ +g|K = σg and σ′ +g(a) = a. Naturally, (K, (σg)g∈G) ⊆ (dcl(K, a), (σ′ +g)g∈G). +□ +Fact 3.5 (Lemma 2.10 from [22]). (QE, FS) If G is finite and (K, ¯σ) is a substruc- +ture with G-action such that dcl(K) = K and the action of G on K is faithful (i.e. +if g ̸= h then there is a ∈ K such that σg(a) ̸= σh(a)), then +• K ⊆ acl(KG), +• KG ⊆ K is a Galois extension, +• G(K/KG) ∼= G. +Proof. By Lemma 2.10 from [22], Fact 3.7 and Proposition 4.7 from [10]. Being +more precise, we obtain the two first bullets as in Lemma 3.23 from [21] and then +we repeat the proof of Lemma 2.10(4) from [21] using a variant of the finite Galois +correspondence stated in Proposition 4.7 in [10]. +□ +Lemma 3.6. (QE, FS, ♥) If (K, ¯σ) is an existentially closed substructure with +G-action, then the group action if faithful. +Proof. Consider any enumeration of G, say (gi)i∈I where (I, <) is a linear order. +Let p(x) ∈ S(KG) be a non-algebraic stationary type (existing by Fact 2.13), and +let ¯b = (bi)i∈I |= p⊗I|KG be such that K |⌣KG ¯b. Let F denote dcl(KG,¯b), and let +F ′ denote dcl(K,¯b). +As the type p⊗I|K is also stationary, the extension KG ⊆ F is regular. For each +g ∈ G, let θg be a bijection of I such that g · gi = gθg(i) holds for each i ∈ I. As the +set {bi | i ∈ I} is KG-indiscernible, for each g ∈ G there exists τg ∈ Aut(C /KG) +such that τg(bi) = bθg(i). +Now, Corollary 3.38 from [21], allows us to simultaneously extend each σg (over +K) and τg (over F) to an automorphism σ′ +g ∈ Aut(C), for each g ∈ G. +We +have that (K, (σg)g∈G) ⊆ (F ′, (σ′ +g)g∈G), thus (K, (σg)g∈G) is existentially closed + +PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS +9 +in (F ′, (σ′ +g)g∈G). If g ̸= h, then σ′ +g(b) ̸= σ′ +h(b) for some b ∈ F ′, and so there will be +a ∈ K such that σg(a) ̸= σh(a). +□ +Lemma 3.7. (QE) If G is finitely generated and (K, ¯σ) is an existentially closed +substructure with G-action, then KG is K-strongly PAC. +Proof. Consider p(x) ∈ S(KG) which has a unique non-forking extension over K, +say ˜p(x) ∈ S(K). +As p(x) is invariant under action of automorphisms σg|KG, +where g ∈ G, we have that ˜p(x) is invariant under action of automorphisms σg, +where g ∈ G (otherwise, we would get distinct non-forking extensions of p over K). +Let b |= ˜p, by Lemma 3.4 there exists an extension of substructures with G- +action, +(K, (σg)g∈G) ⊆ (K′, (σ′ +g)g∈G) +such that b ∈ (K′)G. By our assumption, we have that (K, (σg)g∈G) is existentially +closed in (K′, (σ′ +g)g∈G). +Now, let ϕ(a, x) ∈ p(x). As T has quantifier elimination, we may assume that +ϕ(y, x) is quantifier free, what we do. Of course |= ϕ(a, b) and so +(K′, (σ′ +g)g∈G) |= (∃x) (ϕ(a, x) ∧ +� +g∈X +σg(x) = x), +where X denotes the finite set of generators of G. Hence +(K, (σg)g∈G) |= (∃x) (ϕ(a, x) ∧ +� +g∈X +σg(x) = x) +and for some b0 ∈ KG we have that |= ϕ(a, b0). +□ +Therefore we see that an existentially closed substructure with G-action has a +quite tame substructure of invariants. +The next subsection is dedicated to the +converse of this implication, so we would like to show that “if the substructure +of invariants is tame then the whole substructure with G-action is existentially +closed”. +Remark 3.8. In Proposition 3.56 from [21], it was shown that if (K, ¯σ) is an ex- +istentially closed substructure with G-action, then K is PAC. However, the afore- +mentioned proposition assumes quantifier elimination, elimination of imaginaries +and stability (but G there can be arbitrary). +3.2. Invariants of existentially closed actions. +Lemma 3.9. (QE, FS) Assume that G is finite, (K, (σg)g∈G) ⊆ (K′, (σ′ +g)g∈G) is +an extension of substructures with G-action, the group action of G on K is faithful +and dcl(K) = K. If KG is algebraically K-strongly PAC, then KG ⊆ (K′)G is +regular. +Proof. If dcl(K) = K then also dcl(KG) = KG. Moreover, KG ⊆ (K′)G is regular +if and only if KG ⊆ dcl((K′)G) is regular and there is a unique way of extending G- +action from K′ over dcl(K′). Therefore, without loss of generality, we assume that +K′ = dcl(K”) and so dcl((K′)G) = (K′)G. We need to show that (K′)G∩acl(KG) = +KG. +Let a ∈ (K′)G ∩ acl(KG) \ KG. +Because for every g ∈ G, we have that +σg +� +tp(a/K) +� += tp +� +σg(a)/K +� +and a ∈ (K′)G, we see that tp(a/K) is a G-invariant + +10 +D. M. HOFFMANN AND P. KOWALSKI +type. By Fact 3.5 and Lemma 2.20, we see that tp(a/K) is a unique extension of +tp(a/KG) over K. +As a ∈ acl(KG) and acl(KG) |⌣KG K (e.g. Remark 5.3 in [9]), tp(a/KG) ⊆ +tp(a/K) is a non-forking extension. Because KG is algebraically K-strongly PAC, +tp(a/KG) is finitely satisfiable in KG. As a ∈ acl(KG), this means that it must be +a ∈ KG. +□ +Definition 3.10. Assume that C ⊆ K ⊆ C and that G is finite. We call the pair +(C, K) G-closed if C ⊆ K is a Galois extension, G(K/C) ∼= G and there is no +K′ ⊆ acl(K), K ⊊ K′, such that the action of G(K/C) extends over K′. +Lemma 3.11. (QE, FS) Assume that G is finite, (K, ¯σ) is a substructure with +G-action such that action of G on K is faithful and dcl(K) = K. Then (KG, K) +is G-closed if and only if KG is algebraically K-strongly PAC. +Proof. By Fact 3.5, K ⊆ acl(KG), KG ⊆ K is Galois and G(K/KG) ∼= G. +Assume that (KG, K) is G-closed and let p(x) ∈ S(KG) be algebraic with a +unique extension ˜p(x) over K (being a non-forking extension follows naturally from +acl(KG) |⌣KG K, e.g. Remark 5.3 in [9]). We have that ˜p is G-invariant and so, by +Lemma 3.4, if b |= ˜p then there exists an extension of substructures with a G-action, +(K, (σg)g∈G) ⊆ (K′, (σ′ +g)g∈G) +such that K′ = dcl(K, b) and b ∈ (K′)G. As K′ = dcl(K, b) ⊆ acl(KG) = acl(K), +it must be that K = K′, so b ∈ K and finally b ∈ KG. +Now, we show the right-to-left implication. Assume that K′ ⊆ acl(K) and there +is an extension of substructures with G-action: +(K, (σg)g∈G) ⊆ (K′, (σ′ +g)g∈G). +By Lemma 3.9, KG ⊆ (K′)G is regular. As (K′)G ⊆ K′ ⊆ acl(K) = acl(KG) it +must be (K′)G ⊆ dcl(KG) = KG, so KG = (K′)G. By the proof of Proposition 4.1 +from [15] and the Galois correspondence for finite extensions (e.g. Theorem 12 in +[32]), there exists a finite tuple b from K such that K = dcl(KG, b). Moreover, by +the same proof of Proposition 4.1 from [15], we also have that K′ = dcl((K′)G, b). +Because KG = (K′)G, we have that K = dcl(KG, b) = dcl((K′)G, b) = K′. +□ +The following remark is not important for the main results of this paper and its +purpose is mainly to generalize Theorem 3.25 from [24]. As we use in its proof the +Elementary Equivalence for PAC structures ([15]), we need to add more assump- +tions. +Remark 3.12. Let T be stable with elimination of quantifiers and elimination of +imaginaries. Assume that PAC is a first order property. Suppose that (C, K) ⊆ +(C′, K′) is an extension of G-closed substructures such that C and C′ are PAC. +Then C ⪯ C′. +Proof. It is enough to reproduce the proof of Theorem 3.25 from [24], but in this +more general context. +By the proof of Theorem 3.22 from [24] or more similar +Lemma 3.54 from [21], we have that C and C′ are bounded PAC structures. Thus, +by Corollary 3.11 from [15], it is enough to show that the restriction map r : +G(C′) → G(C) is an isomorphism. After combining Lemma 3.11 and Lemma 3.9, +we obtain that C ⊆ C′ is regular, so r is an epimorphism. + +PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS +11 +By Theorem 4.4 from [22], G(C) is projective, which means that there exists +embedding h as in the following diagram +G(C) += +� +h +�● +● +● +● +● +G(C) +G(C′) +r +� +But then G0 := h[G(C)] ⩽ G(C′) is a closed subgroup such that r|G0 : G0 → G(C) +is an isomorphism. +Because K ⊆ acl(C) and K′ ⊆ acl(C′), the restriction maps G(C) → G and +G(C′) → G lead to the following commutative diagram +G(C′) +�❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +r +� G(C) +�③③③③③③③③ +G +and so G0N = G(C′) for N := ker +� +G(C′) → G +� +. By Lemma 3.31 from [21], this +implies that G0 = G(C′) as expected. +□ +Theorem 3.13. (QE, FS, ♥) Assume that G is finite, say |G| = l. Let (K, ¯σ) be a +substructure with G-action such that G acts faithfully on K and dcl(K) = K. The +following are equivalent: +(1) (K, ¯σ) is existentially closed. +(2) KG is K-strongly PAC. +(3) KG is PAC and algebraically K-strongly PAC. +(4) KG is PAC and (KG, K) is G-closed. +Proof. By Lemma 3.11, (3) ⇐⇒ (4). (1)⇒(2) follows by Lemma 3.7. The im- +plication (2)⇒(3) follows by definitions. To get the theorem, we will show that +(3)⇒(1). +Assume that dcl(K) = K, the group action is faithful and that KG is PAC and +algebraically K-strongly PAC. Using Fact 3.5, we obtain the following +• K ⊆ acl(KG), +• KG ⊆ K is a Galois extension, +• G(K/KG) ∼= G. +The proof of Proposition 4.1 from [15] gives us existence of a finite tuple ¯b = +(b0, . . . , bl−1) from K such that K = dcl(KG,¯b). +Consider (K, (σg)g∈G) ⊆ (K′, (σ′ +g)g∈G). +Without loss of generality, we may +assume that (K′, (σ′ +g)g∈G) is existentially closed, in particular dcl(K′) = K′. We +have that the group action of G on K′ is faithful, thus by Fact 3.5, we have that +K′ ⊆ acl((K′)G), (K′)G ⊆ K′ is Galois, and G(K′/(K′)G) ∼= G. Moreover, by the +proof of Proposition 4.1 in [15], it holds also that +K′ = dcl((K′)G,¯b). +Lemma 3.9 gives us that KG ⊆ (K′)G is regular. +Let ¯B be some enumeration of {σg(bi) | g ∈ G, i < l}. We have that K′ = +dcl((K′)G,¯b) = dcl((K′)G, ¯B). Assume that +(K′, (σ′ +g)g∈G) |= φ(a) + +12 +D. M. HOFFMANN AND P. KOWALSKI +for some tuple a from K′ and some quantifier-free formula φ(x) ∈ LG(K). First, +we may present φ(a) as ϕ0(σ′ +g0(a), . . . , σ′ +gl−1(a)), where ϕ0(x0, . . . , xl−1) ∈ L(K) is +quantifier-free. Second, since K = dcl(KG, ¯B), we may present ϕ0(σ′ +g0(a), . . . , σ′ +gl−1(a)) +as ϕ(σ′ +g0(a), . . . , σ′ +gl−1(a), ¯B), where ϕ(x0, . . . , xl−1, ¯y) ∈ L(KG) is quantifier-free. +Let σ′ +g0 = idL, so σ′ +g0(a) = a. Because a ∈ K′ = dcl((K′)G, ¯B), there exists a +finite tuple ¯c ⊆ (K′)G and a quantifier-free formula ψ0(¯z, ¯y, x) ∈ L such that +• ψ0(¯c, ¯B, C) = {a}, +• |= (∀¯z, ¯y, x, x′) +� +ψ0(¯z, ¯y, x) ∧ ψ0(¯z, ¯y, x′) −→ x = x′� +. +Because σgi permutes ¯B, there exists a permutation si such that σgi( ¯B) = si( ¯B). +We define ψi(¯z, ¯y, x) as ψ0(¯z, si(¯y), x). Note that ψi(¯c, ¯B, C) = {σ′ +gi(a)} and +(K′, (σ′ +g)g∈G) |= +(∀¯z, x, x′) +� � +g∈G +σg(¯z) = ¯z ∧ ψ0(¯z, ¯B, x) ∧ ψi(¯z, ¯B, x′) → σgi(x) = x′� +. +To see the last line, let ¯d ⊆ (K′)G, m, m′ ∈ K′ be such that +|= ψ0( ¯d, ¯B, m) ∧ ψi( ¯d, ¯B, m′). +We do know that ψ0( ¯d, ¯B, C) = {m}, which after applying an extension ˜σgi ∈ +Aut(C) of σ′ +gi changes it into ψ0( ¯d, si( ¯B), C) = {σ′ +gi(m)}. We have that +m′ ∈ ψi( ¯d, ¯B, C) = {σ′ +gi(m)}. +Since the whole formula is universal and has only parameters from K, it follows +that +(K, (σg)g∈G) |= (∀¯z, x, x′) +� � +g∈G +σg(¯z) = ¯z ∧ ψ0(¯z, ¯B, x) ∧ ψi(¯z, ¯B, x′) → σgi(x) = x′� +, +where i < l. +Consider p(¯z) := tp(¯c/KG). Because KG ⊆ (K′)G is regular (thus also primary) +and ¯c ⊆ (K′)G, Fact 2.12 implies that p(¯z) is stationary. As KG is PAC, the type +p(¯z) is finitely satisfiable in KG. The tuple ¯B ⊆ K is algebraic over KG, hence there +exists a quantifier-free θ(¯y) ∈ L(KG) such that θ(¯y) ⊢ tp( ¯B/KG). The following +formula +(∃ ¯y, x0, . . . , xl−1) +� � +i 0. There ex- +ists a quantifier-free L-formula ψϕ(y) equivalent modulo T to the formula (∃=n x) (ϕ(y, x)). +We are in situation of Remark 3.20, so we can involve the formula ξϕ,n(y, w). Our +axiom scheme may be written as: +(∀ y) +� � +g∈G +σg(y) = y ∧ ψϕ(y) ∧ ¬(∃ w ∈ (Sx)n−1/E) (ξϕ,n(y, w)) +→ (∃ x) +� � +g∈G +σg(x) = x ∧ ϕ(y, x) +�� +. +□ +Question 3.24. (QE, ♠, ♥) Can we obtain a converse of Theorem 3.23? More +precisely, does the following equivalence hold: the model companion of the theory +of substructures with G-action exists for every finite group G if and only if PAC is +a first order property? +Remark 3.25. After writing the proofs of Theorems 3.13 and 3.23, we have noticed +(but we have not checked all the details) that this result holds in a much greater +generality, that is: if in the definition of PAC we replace “stationary” with “acl- +stationary” (a unique extension over algebraic closure of the parameters), then the + +16 +D. M. HOFFMANN AND P. KOWALSKI +assumptions of stability, coding finite sets, and eliminating strong types may be +skipped in Theorem 3.23. However, in this case it is unclear how useful such a +result would be in terms of axiomatizing existentially closed finite group actions in +this case, since there is no guarantee that faithful actions of a finite groups exist at +all in general (consider for example the theory of linear orders) and the faithfulness +in the stable was guaranteed by Lemma 3.6. +4. PAC structures in particular theories +In this section, we discuss the PAC property in some specific cases as well as +some general methods for understanding PAC structures with respect to a given +theory. As we are going to consider the notions of a regular extension and of a PAC +structure in different theories, we plan to write “T -regular” and “T -PAC” instead +of “regular in T ” and “PAC in T ” respectively. +We will often refer to several particular stable theories as: the theory of compact +complex manifolds CCM (for background, the reader is referred to [35]) and the +theories of differentially closed fields of characteristic 0 denoted DCF0 (see e.g. +[30]) and its positive characteristic version DCFp (see e.g. [50] and [51]), and the +theories of separably closed fields of positive characteristic SCFp,e and SCFp,∞ (see +e.g. [33]). +4.1. General methods. In this subsection, we focus on two general contexts in +which the PAC property is well understood. However, in both these cases showing +that PAC is a first-order property requires some extra work. +4.1.1. Totally transcendental theories. In this part, we assume that the theory T +is ω-stable. As before, let us fix for convenience a monster model C of T and an +arbitrary small substructure K ⊂ C. It is well-known that stationary types in ω- +stable theories are determined by the formulas of Morley degree one belonging to +them. In particular, we have the following result, which actually coincides with +Hrushovski’s definition of the PAC property in the strongly minimal case (see [27, +Definition 1.2] and [21, Proposition 3.10]). +Proposition 4.1. If T is a ω-stable theory, then K is T -PAC if and only if for +any formula ϕ ∈ L(K) of multiplicity (Morley degree) one, we have that ϕ(K) ̸= ∅. +We recall that “DMP” stands for “Definable Multiplicity Property” and it says +that for any formula φ(x; a) ∈ L(K), there is a formula θ(y) ∈ tp(a) such that +whenever C |= θ(a′) then we have: +RM (φ(x; a′)) = RM (φ(x; a)) , +degM (φ(x; a′)) = degM (φ(x; a)) +(see e.g. [28, Definition 1.1]). Some ω-stable theories have DMP and some do not +(see Remark 4.3 below). We get the following obvious conclusion, which was also +stated in [1] under the assumption of finiteness of the Morley rank. +Proposition 4.2. If T is ω-stable with quantifier elimination and has DMP, then +being T -PAC is first-order. +Proof. Since T has DMP, for each φ(x; y) ∈ L, there is θφ(y) such that for all +c ∈ C|y|, we have: +C |= θφ(c) +if and only if +degM (φ(x; c)) = 1. + +PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS +17 +Therefore, it is easy to write down a first-order axiom scheme expressing the T -PAC +property. +□ +Remark 4.3. We comment here on several particular ω-stable theories. +(1) Proposition 4.2 applies to the case of T = ACFp, that is to the classical +notion of PAC. +(2) It is known that Morley degree is not definable in the theory DCF0 (see +[16, Question 1.2] and [17]). However, DCF0-PAC is still first-order as it +was shown in [41]. +(3) It is open whether the theory of compact complex manifolds has DMP, +however another approach towards the PAC property works here, which +will be discussed in the next part. Partial results towards DMP for the +theory CCM were obtained in [45]. +4.1.2. Noetherian theories. In this part, we assume that models of T are naturally +equipped with an extra topological structure. This assumptions is modelled on the +case of T = ACFp and the Zariski topology. Such issues were thoroughly discussed +in [53]. We diverge here a bit from the set-up of [53] to cover the case of the theory +DCF0 as well. +We start from a purely topological context. +Assume that S is a Noetherian +topological space and let B be the Boolean algebra of constructible sets in S. In +this part, “irreducible” always refers to the topological irreducibility with respect +to a given Noetherian topology. The following properties are folklore and they can +be easily checked. +• If V is a non-empty closed irreducible subset of S, then +pV := {C ∈ B | intV (C ∩ V ) ̸= ∅} +is an ultrafilter on B. +• The map V �→ pV is a bijection between the set of closed irreducible subsets +of S and the set of ultrafilters on B. +We specify now our model-theoretic context. +Definition 4.4. By a Noetherian theory, we mean a pair (T, �), where T is a +complete L-theory and � consists of L-formulas of the form ϕ(x; y), where the +variables x, y vary, such that for any M |= T and any A ⊆ M, we have the following. +• A subset V ⊆ M |x| is said to be A-closed if and only if there is a ⊂ A and +ϕ(x; y) ∈ � such that V = ϕ(M; a). +• The family of A-closed sets constitutes the family of closed sets of a Noe- +therian topology, which we call the A-topology. +• Constructible sets with respect to the A-topology coincide with A-definable +subsets (in Cartesian powers of M). +Remark 4.5. It should automatically follow (possibly after adding some light +assumptions such as the equality being in �) that models of our Noetherian theories +are topological structures in the sense of [4, Definition 5.1] and [53, Section 2]. +Example 4.6. We discuss several examples and non-examples of the above situa- +tion. +(1) The theory of algebraically closed fields (of a given characteristic) is Noe- +therian by considering the Zariski topology. + +18 +D. M. HOFFMANN AND P. KOWALSKI +(2) The theory of compact complex manifolds (CCM) is also Noetherian, where +the (Zariski) Noetherian topology is given by closed analytic subsets (see +[53, Section 3.4.2]). +(3) In the case of differential fields, we have the Kolchin topology. +• The theory DCF0 is Noetherian by [30, Theorem 2.4]. +• More generally, the theory DCF0,m is Noetherian by [31, Theorem +3.1.7]. +• The theory DCFp is not an example, since there is no quantifier elim- +ination down to Kolchin constructible sets (see [50, Section 3]). +(4) The theory SCFp,e with the λ-topology is not an example, since the λ- +topology is not Noetherian (see [33, Section 4.6]). +For a fixed A |= T∀ and n > 0, it is clear that the map V �→ pV is a bijection +between the set of appropriate A-closed A-irreducible sets and the Stone space +Sn(A) of n-types over A. In particular, any Noetherian theory is ω-stable. We still +need to have a connection between the topology and forking, which is given by the +following. +Proposition 4.7. Assume that A ⊆ M |= T and pV ∈ Sn(M). Then, the type pV +does not fork over A if and only if V is definable over acl(A). +Proof. Since pV does not fork over A if and only if it does not fork over acl(A), we +can and will assume that A = acl(A). +(⇒) Let V = Vb and assume that Vb is not definable over A. Let us define +V0 := +� +tp(c/A)=tp(b/A) +Vc. +Since Vb is not definable over A, we get that V0 ⊊ V . By Noetherianity, V ) is +definable and closed. Since V0 is A-invariant, we get that V0 is A-definable. In +particular, the formula “x ∈ V \ V0” belongs to pV . Since the formula “x ∈ V \ V0” +forks over A (see e.g. the characterization of forking from [40, Lemma 2.16(c)]), +the type pV forks over A. +(⇐) We assume that V is A-definable. It is enough to show that for any proper +M-closed W = Wb ⊂ V , we have that the formula “x ∈ V \ W’ does not fork over +A. If this formula forks over A, then by (the logical) compactness, there is a finite +set of A-conjugates b = b1, . . . , bn such that: +(V \ Wb1) ∩ . . . ∩ (V \ Wbn) = ∅. +But then V = Wb1 ∪ . . . ∪ Wbn and each Wbi is a proper M-closed subset of V , +which contradicts the M-irreducibility of V . +□ +We obtain the expected description of stationary types. +Corollary 4.8. Let A |= T∀, and pW ∈ Sn(A). Then, pW is stationary if and +only if W is absolutely irreducible, that is for any M |= T containing A as a +substructure, W is irreducible in the M-topology. +Proof. Let W = W1∪. . .∪Wn be the decomposition of W into the M-irreducible M- +closed components. By uniqueness, each Wi is defined over acl(A). By Proposition +4.7, each type pWi does not fork over A. Since for each i, we have clA(Wi) = W, we +get that each pWi extends pW . It is easy to see now that Wi’s correspond exactly +to non-forking extensions of pW , which concludes the proof. +□ + +PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS +19 +Similarly as in the case of Proposition 4.1, we get the following result. +Proposition 4.9. For any K |= T∀, we have that K is T -PAC if and only if for any +absolutely irreducible K-closed set V and any non-empty relatively K-open U ⊆ V , +we have that U(K) ̸= ∅. +Remark 4.10. +(1) In the cases of T = ACFp and T = DCF0,m, we can just +consider the condition “V (K) ̸= ∅” in Proposition 4.9, since these topologies +have basis of open sets being definably isomorphic to affine closed sets. It +looks like there is no similar simplification for the theory CCM, since (at +least in the category of complex manifolds) being isomorphic to a compact +complex manifold would imply being closed. +(2) Proposition 4.9 together with Item (1) above directly generalizes the clas- +sical case of T = ACFp. +(3) For T = DCF0,m the description from Proposition 4.9 (together with Item +(1) above) coincides with the definition taken in [46, Section 5.16] (see also +[26, Remark 4.7(1)]). +(4) For T = CCM, we believe that this notion has not been considered before. +Similarly as in the previous part, we get the following result. +Proposition 4.11. Assume that T is a Noetherian theory. If the topological irre- +ducibility is definable in T , then T -PAC is first-order. +We would like to single out one important case below. +Corollary 4.12. CCM-PAC is first-order. +Proof. By Proposition 4.11, it is enough to show that the topological irreducibility +is definable in the theory CCM, which was shown in [44]. +□ +Remark 4.13. We would like to mention that Rahim Moosa pointed out to us +that an argument for the definability of the topological irreducibility in the case +of compact complex spaces can be also found in an earlier work of Campana. The +reader is advised to consult §.3.B of Premiere Partie of [8]. +For the terminology used in the next theorem, we advise the reader to recall +Definition 3.1(5). +Theorem 4.14. If G is finite, then the theory G-CCM, exists and it is super- +simple with geometric elimination of imaginaries, codes finite sets and has “semi” +quantifier elimination (in the same way as the theory ACFA). +Proof. The existence of G-CCM follows by Theorem 3.23 and Corollary 4.12. The +properties of G-CCM, listed in the statement, follow by Corollary 4.28, Theorem +4.36, Lemma 4.37 and Remark 4.13 from [21] after noticing that CCM is superstable +with elimination of quantifiers and elimination of imaginaries. +□ +Remark 4.15. All the properties stated in Theorem 4.14 hold also in the case of +the theory CCMA ([5]). One difference between G-CCM, for finite G, and CCMA +are the values of the SU-rank. As ACFA is stably embedded in CCMA, there is +a sort in CCMA on which the SU-rank is not finite. On the other hand, one can +show that the SU-rank of G-CCM is finite (sort-by-sort). +Remark 4.16. We finish this part with some comments on the theories of differ- +ential fields. + +20 +D. M. HOFFMANN AND P. KOWALSKI +(1) Let us recall here that for the theory DCF0,m the definability of Kolchin +topological irreducibility is equivalent to the (generalized) Ritt problem (see +[17]). However, the PAC property for DCF0,m is still first-order, which was +shown in [46]. +(2) It may be a good moment to point out that the methods of this paper +cover that DCF0-PAC is first-order, but fail to generalize it to the case of +DCF0,m for m > 1. In [46], the general case is shown in the following way. +• First the authors of [46] show that “differential largeness” is a first- +order property (see [46, Proposition 4.7]). +• Then they show that DCF0,m-PAC is equivalent with the classical field +PAC together with the “differential largeness” (see [46, Section 5.16]). +The above scheme of a proof looks like a possible another general approach +(at least for the theories of fields with operators). We will discuss it further +at the end of this paper (see Remark 4.45). +4.1.3. Equational theories. In [42], the notion of an equational theory is introduced. +Briefly, a theory T is equational if any formula is equivalent (modulo T ) to a Boolean +combination of instances of equations, where a formula ϕ(x, y) is an equation (mod- +ulo T ) if the family of definable sets given by finite intersections of its instances (in +any model of T ) has the DCC (Descending Chain Condition). +The set-up of equation theories generalizes, in some sense, the set-up of Noe- +therian theories from Section 4.1.2, since Noetherian theories are equational “in a +strong sense” that is the DCC condition holds not only in the case of instances of +one formula but for all closed sets with respect to the given Noetherian topology. +Not all the equational theories are Noetherian, since any Noetherian theory is ω- +stable and, for example, Th(Z, +) is equational and not ω-stable (see Remark at +the end of Section 2 in [42]). +There is a natural notion of irreducibility in the case of equational theories and it +is possible that for an equation theory T , if this notion of irreducibility is definable, +then T -PAC is first order. +4.2. Fields of positive characteristic. In this subsection, we focus on three +stable theories of fields of positive characteristic: SCFp,e (e finite), SCFp,∞, and +DCFp. There are several possible languages to consider for these fields of positive +characteristic and we will actually use three different options here. +4.3. Fields of positive characteristic. In this subsection, we focus on three +stable theories of fields of positive characteristic: SCFp,e (e finite), SCFp,∞, and +DCFp. There are several possible languages to consider for these fields of positive +characteristic and we will actually use three different options here. +4.3.1. Separably closed fields of finite imperfection degree. We consider the theory +SCFp,e (e finite) in the language Lλ,b, where b stands for an e-tuple of constant +symbols corresponding to a fixed p-basis and we also have symbols for unary λ- +functions defined with respect to b (see [12, Section 1.8]). Then, the theory SCFp,e +has quantifier elimination and elimination of imaginaries (cf. [12, Section 1.8]). +Let us fix as usual a monster model C |= SCFp,e. We make the following identi- +fication b = bC. A subfield K ⊆ C is an Lλ,b-substructure of C if and only if b ⊂ K +and b is a p-basis of K (in such a case the field extension K ⊆ C is separable and +even ´etale). Each Lλ,b-substructure is definably closed and for Lλ,b-substructures, + +PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS +21 +the model-theoretic algebraic closure coincides with the field theoretic separable clo- +sure (see [14, Section 1.6]) and similarly with forking (see [14, Section 1.8]). Using +the above, we immediately get the following description of regular extensions. +Fact 4.17. Let K0 ⊆ K1 be an extension of Lλ,b-substructures of C. Then, the +extension K0 ⊆ K1 is SCFp,e-regular if and only if K0 ⊆ K1 is a regular extension +of pure fields. +We describe now PAC structures in the theory SCFp,e. These description ap- +peared in [1], but the proof is only sketched there. +Theorem 4.18. Let K be a Lλ,b-substructure of C. Then, K is SCFp,e-PAC if +and only if K is a PAC field. +Proof. In this proof, K is a SCFp,e-substructure of C. +(⇒) Let us assume that K is SCFp,e-PAC and let K ⊆ N be a regular field exten- +sion. Then, there is a field extension N ⊂ N ′ such that b is a p-basis of N ′ (we +recall that now b is a fixed p-basis of C). Therefore, we can assume that K ⊂ N ′ +is an extension of Lλ,b-substructures of C. By Fact 4.17, K ⊆ N ′ is also a SCFp,e- +regular extension and K is Lλ,b-existentially closed in N ′. Therefore, K is also is +Lλ,b-existentially closed in N and K is existentially closed in N in the field sense, +hence K is a PAC field. +(⇐) Let us assume that K is a PAC field. Let b = (b1, . . . , be) be a p-basis of K, +which is also a p-basis of C. We will consider the unary λ-functions +λ1,e, . . . , λpe,e : C → C +(see [12, Section 1.8]) with respect to this p-basis (they preserve K). +We also +take a ∈ Cn such that p(x) := tp(a/K) is stationary and a quantifier-free Lλ,b(K)- +formula φ(x) ∈ p(x). We need to show that φ has a realization in K. We inductively +unravel all the terms appearing in the formula φ. For example, if we have: +φ(x): λi,n +� +λj,m(x) + x2� ++ x = 0, +then we set: +¯a := +� +a, λj,m(a), λi,n +� +λj,m(a) + a2�� +. +Then, for any +(b1, b2, b3) ∈ locusK(¯a), +we obtain that: +b2 = λj,m(b1), +b3 = λi,n +� +λj,m(b1) + b2 +2 +� +. +As usual when we deal with fields, we can assume that there are only equalities +in the formula φ (by replacing negations of equalities with equalities in “higher +dimensions”). Using the above procedure, we obtain a tuple ¯a = (a1, . . . , at) such +that a = a1 and if we define: +V := locusK(¯a), +then for any ¯b ∈ V (C), we get that C |= φ(¯b). Since the type p is stationary, V is +absolutely irreducible. Therefore, we obtain ¯b ∈ V (K) such that C |= φ(¯b). +□ +There are three possible languages such that the theory SCFp,e (e finite) con- +sidered in each of these languages has quantifier elimination: Lλ,b, Lλ, and the +Hasse-Schmidt language (see [52]). One could wonder whether such a choice of the +language affects the corresponding notion of a PAC-substructure. We address these +issues in general below. + +22 +D. M. HOFFMANN AND P. KOWALSKI +Remark 4.19. Assume that T is a stable L-theory and T ′ is an L′-theory being +an extension by definitions of T . Consider a model M of T and its counterpart M ′ +as an L′-structure (i.e. M equipped with the natural L′-structure), and a subset +K of M. +(1) If K is a PAC substructure in the sense of M, then K is an L′-substructure +of M ′ which is also a PAC substructure in the sense of M ′. +(2) If K is a PAC substructure in the sense of M ′, then K is a PAC substructure +in the sense of M. +For a finite group G, by Theorem 3.23 we get existence of the theory G−SCFp,e, +which is the model companion of the theory of actions of G on characteristic p fields +of inseparability degree e. This theory was already analyzed in [23] using different +methods. +4.3.2. Separably closed fields of infinite imperfection degree. . We consider the the- +ory SCFp,∞ in the language Lλ, where the λ-functions are multi-variable (see [14, +Section 1.4]), this definition is recalled below (we follow [12, Section 1.8] here). For +each e > 0, we fix an enumeration (mi,e)1⩽i⩽pe of the monomials Xi1 +1 . . . Xie +e where +0 ⩽ i1, . . . , ie ⩽ p − 1, and define the functions λi,e : Ke × K → K by considering +the following three cases. Let b1, . . . , be, c ∈ K and 1 ⩽ i ⩽ pn. +Case 1 b1, . . . , be are p-dependent. +We set λi,e(b1, . . . , be; c) = 0. +Case 2 b1, . . . , be, c are p-independent. +We set λi,e(b1, . . . , be; c) = 0. +Case 3 b1, . . . , be are p-independent and b1, . . . , be, c are p-dependent. +We use the following defining formula: +c = +pn +� +j=1 +λj,e(b1, . . . , be; c)pmj,e(b1, . . . , be). +Then, the theory SCFp,∞ has quantifier elimination in the language Lλ, but it +does not have elimination of imaginaries (see [12, Section 1.8]). As for any theory +of fields, SCFp,∞ eliminates finite imaginaries. Each Lλ-substructure is definably +closed and for Lλ-substructures, the model-theoretic algebraic closure coincides +with the field theoretic separable closure (see [14, Section 1.6]) and similarly with +forking (see [14, Section 1.8]). +As in the previous part, we get the following description. +Fact 4.20. Let M |= SCFp,∞ and K0 ⊆ K1 be an extension of Lλ-substructures +of M. Then, K0 ⊆ K1 is SCFp,∞-regular if and only if K0 ⊆ K1 is a regular +extension of pure fields. +We also need the following result of Tamagawa, which we phrase in geometric +terms. +Theorem 4.21 (Tamagawa, Proposition 11.4.1 in [18]). Let V be an absolutely +irreducible affine variety over a PAC field K of characteristic p > 0. +Suppose +that f1, . . . , fm ∈ K[V ] are p-independent in K(V ) and m is not greater than the +imperfection degree of K. Then, there is a ∈ V (K) such that f1(a), . . . , fm(a) are +p-independent in K. + +PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS +23 +We need a slight enhancement of Tamagawa’s Theorem, which we state and show +below. +Corollary 4.22. Let V be an absolutely irreducible affine variety over a PAC field +K of characteristic p > 0 and infinite imperfection degree. Suppose that we have a +finite matrix (fi,j)i,j of elements of K(V ) whose rows are p-independent in K(V ). +Then, there is a ∈ V (K) such that each fi,j is defined at a and the rows of the +matrix (fi,j(a))i,j are p-independent in K. +Proof. We deal first with the case when fi,j ∈ K[V ]. For simplicity, let us assume +that there are only three rows in our matrix and that each row has the same length +m. +Let us denote these rows by (fi)i, (gi)i, (hi)i. +After permutation, there are +k, l ⩽ m such that: +(1) f1, . . . , fm, g1, . . . , gk, h1, . . . , hl are p-independent in K(V ); +(2) f1, g1, . . . , fm, gm ∈ clK(V ) +p +(f1, . . . , fm, g1, . . . , gk); +(3) f1, g1, h1, . . . , fm, gm, hm ∈ clK(V ) +p +(f1, . . . , fm, g1, . . . , gk, h1, . . . , hl) +By Theorem 4.21 and Item (1) above, there is a ∈ V (K) such that +(∗) +f1(a), . . . , fm(a), g1(a), . . . , gk(a), h1(a), . . . , hl(a) are p-independent in K. +Using Item (3) above, we obtain that for every l + 1 ⩽ t ⩽ m: +(∗∗) +ht(a) ∈ clK +p (f1(a), . . . , fm(a), g1(a), . . . , gk(a), h1(a), . . . , hl(a)) . +We will show that this choice of a works. Suppose not, and let us consider the +most complicated case when h1(a), . . . , hm(a) are p-dependent in K (the case +when g1(a), . . . , gm(a) are p-dependent is indeed easier by Item (2) above). Since +h1, . . . , hm are p-independent in K(V ), there are (after a permutation) v ⩽ m, w ⩽ +k such that f1, . . . , fv, g1, . . . , gw, h1, . . . , hm are p-independent in K(V ) and +f1, g1, h1, . . . , fm, gm, hm ∈ clK(V ) +p +(f1, . . . , fv, g1, . . . , gw, h1, . . . , hm) +(we use all the time the fact that the p-independence satisfies the Steinitz Exchange +Principle, see [48, Remark C.1.1.3]). Since h1(a), . . . , hm(a) are p-dependent in K, +by (∗) we get that l < m, and therefore v < m or w < k. Let us assume that v < m +(the case of w < k can be handled analogously). Therefore, we obtain that: +fm ∈ clK(V ) +p +(f1, . . . , fm−1, g1, . . . , gk, h1, . . . , hm). +Hence, we get: +fm(a) ∈ clK +p (f1(a), . . . , fm−1(a), g1(a), . . . , gk(a), h1(a), . . . , hm(a)). +Using (∗∗), we get that +fm(a) ∈ clK +p (f1(a), . . . , fm−1(a), g1(a), . . . , gk(a), h1(a), . . . , hl(a)), +which contradicts (∗). +We consider now that case when fi,j ∈ K(V ). +Let U ⊆ V be an open K- +subvariety which is K-isomorphic to an affine variety such that is contained in the +intersection of all dom(fi,j). Then U is absolutely irreducible as well, and it is +enough to apply the previously shown case of fi,j ∈ K[V ]. +□ +Theorem 4.23. K be an Lλ-substructure of a monster model C |= SCF∞,e. Then, +K is SCFp,∞-PAC if and only if K is PAC and [K : Kp] = ∞. + +24 +D. M. HOFFMANN AND P. KOWALSKI +Proof. For the left-to-right implication, we take K which is SCFp,∞-PAC. By [14, +Section 1.7], the properties of the generic 1-type in the theory SCFp,∞ imply +that [K : Kp] = ∞. As in the previous part, we notice that extensions of Lλ- +substructures of models of SCFp,∞ are SCFp,∞-regular if and only if they are a +regular extension of pure fields. Now, the proof is identical to the proof of the +corresponding implication in Theorem 4.18. +For the right-to-left implication, assume that K is PAC and [K : Kp] = ∞. +Let us take a ∈ Cm such that p(x) := tp(b/K) is stationary and a quantifier-free +Lλ-formula φ(x) ∈ p(x) with parameters from K. We need to show that φ has a +realization in K. +Claim +There is ¯b = (b, b′) ∈ CN such that V := locusK(¯b) is absolutely irreducible, and +there is a finite matrix of rational functions (fi,j ∈ K(V ))i,j such that for each i, +fi,1, . . . , fi,mi are p-independent in K(V ) and such that for all ¯c = (c, c′) ∈ V (C), +we have: +IF for each i, fi,1(¯c), . . . , fi,mi(¯c) are p-independent in C, THEN C |= φ(c). +Proof of Claim. By [12, Lemma 2.9], the formula φ(x) is equivalent in C to an +L(K)-formula of the form: +∃y α(x, y) ∧ β(x, y), +where α is quantifier-free in the language of fields and β is a finite conjunction of +universal formulas expressing that some subtuples of xy are p-independent. Since +φ(x) ∈ tp(b/K), there is b′ ⊂ C such that: +C |= α(b, b′) ∧ β(b, b′). +This is our choice of b′ as in the statement of this claim and the matrix of rational +functions is given just by the coordinate functions expressing that β is a “conjunc- +tion of universal formulas expressing that some subtuples of xy are p-independent” +(each row in this matrix corresponds to one formula from the finite conjunction +giving the formula β). +□ +By Corollary 4.22, there is ¯a = (a, a′) ∈ V (K) such that for each i, fi,1(¯a), . . . , fi,mi(¯a) +are p-independent in K. +Since the field extension K ⊆ C is separable, each +fi,1(¯a), . . . , fi,mi(¯a) is also p-independent in C. By Claim, we get that C |= φ(a), +which we needed to show. +□ +Theorem 4.24. If G is finite, then the model companion of the theory +� +(SCFp,∞)∀ +� +G, +denoted by G − SCFp,∞, exists. +Proof. We want to use Theorem 3.23, so we need that SCFp,∞ has QE, ♠, ♥ and +that PAC is a first order property in SCFp,∞. After Theorem 4.23, the only thing +which needs to be checked is ♥, which might be a well-known fact, but as we could +not find a proof of it, we noticed that one can adapt the proof pf Lemma 4.16 from +[6]. +□ +4.3.3. Differentially closed fields. We consider the theory DCFp in the language +Lλ0,D, where λ0 is the inverse of Frobenius on p-th powers and identically 0 else- +where. Then, DCFp has quantifier elimination [50, Theorem 11], but it does not +have elimination of imaginaries [34, Remark 4.3]. It was shown in [47] that the +theory DCFp is stable. + +PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS +25 +Remark 4.25. The above “λ0-notation” was introduced by the second author in +[29] and perhaps it was not a very good choice, since: +• it does not follow the original “r-notation” of Wood (see [51, Section 2]); +• formally, λ∅ is the identity function (see [33, Section 4]). +However, this λ0-notation was already used in several other papers, so we stick with +it. +We think that the result below is a folklore, but we could not find a reference, +so we give a proof instead. +Fact 4.26. Let (C, D) |= DCFp be a monster model and K be an Lλ0,D-substructure +of (C, D). Then we have the following. +(1) The model-theoretic algebraic closure of K coincides with its field theoretic +separable closure. +(2) K = dcl(K). +Proof. Since K is an Lλ0,D-substructure of (C, D) |= DCFp, the field extension +K ⊆ C is separable (see e.g. the beginning of the proof of [6, Proposition 4.10], +where this separability appears in a much more general context). Therefore, K +is a also a Lλ,D-substructure of (C, D), where Lλ,D is the language with function +symbols for all λ-functions. +For Item (1), we note that the separable closure of K is still a Lλ,D-substructure +of (C, D). +By (a more general) [6, Lemma 4.14], we get our description of the +model-theoretic algebraic closure. +For Item (2), if a ∈ dcl(K), then by Item (1), we get that a is separably algebraic +over K. Since D on K extends uniquely to Ksep, by quantifier elimination of DCFp +in Lλ0,D (or Lλ,D), we get that the type tpDCFp(a/K) is isolated by fa: the minimal +polynomial of a over K. Since a ∈ dcl(K), we get that deg(fa) = 1, so a ∈ K, +which we needed to show. +□ +Again, we need the following description of regular extensions with respect to +the theory we consider. It follows immediately from Fact 4.26. +Fact 4.27. Let (C, D) |= DCFp be a monster model and (K0, D) ⊆ (K1, D) be an +extension of Lλ0,D-substructures of C. Then, K0 ⊆ K1 is SCFp,e-regular if and +only if K0 ⊆ K1 is a regular extension of pure fields. +We specify now an Lλ0,D-theory of some differential fields in positive character- +istic. We need the following working definition first. +Definition 4.28. Let K be a field of characteristic p > 0. A tuple (V ; f1, . . . , fn) is +admissible, if V is a K-irreducible affine K-variety and f1, . . . , fn ∈ K(V ) \ K(V )p. +We note the following obvious property. +Remark 4.29. Let K ⊆ M be a field extension. For any f ∈ K(V ) and any +a ∈ V (M) generic of V over K, we have that f ∈ K(V )p if and only if f(a) ∈ K(a)p. +Lemma 4.30. Assume that K is PAC of infinite imperfection degree (actually, +non-perfect would be enough). Then, the above notion of an admissible tuple is +first-order in parameters of this tuple. +Proof. Let us take f ∈ K(V ). By Corollary 4.22, we obtain that f ∈ K(V )p if and +only if f(V (K)) ⊆ Kp. Since the second condition is clearly first-order, the result +follows. +□ + +26 +D. M. HOFFMANN AND P. KOWALSKI +The next question is not related to our results and we find it a bit amusing. The +answer may be simple, but we were unable to find it. +Question 4.31. Is the property “f ∈ K(V )p” first-order in parameters of f and +V for an algebraically closed K (that is: modulo the theory ACFp)? +To state our axioms for PAC-DCFp differential fields, we need to recall some +notions. We decided to work here with the case of differential fields for the clarity +of presentation, however, as we will see in Section 4.4, these results hold in a much +greater generality. Still, our references here come from this more general context, +since we do not know any source where they are stated exactly for the differential +case. +Let (K, D) be a differential field (for a while the characteristic of K does not +matter) and V be a K-variety. Then, τ D(V ) denotes the prolongation of V with +respect to D, which in this case can be described as a torsor of the tangent bundle +of V (see [38, Definition 1.4] and [36, Definition 4.1]). We have a natural map (see +e.g. [6, Remark 2.13]): +DV : V (K) −→ τ D(V )(K). +Let K ⊆ Ω be a field extension and a, a′ ⊂ Ω be such that: +V = locusK(a), +W = locusK(a, a′). +For reader’s convenience, we recall now two results from [6] which we will use. +Lemma 4.32 (Lemma 3.5 in [6]). The following are equivalent. +(1) There is a derivation D′ : K[a] ⊆ K[a, a′] extending D such that D′(a) = a′. +(2) W ⊆ τ D(V ). +Assume that V, W are K-varieties as in the statement of Lemma 4.32, that is: +W ⊆ τ D(V ). Let ι : W → τ D(V ) denote the inclusion morphism and +α := πV +D ◦ ι : W → V. +Consider the following (not necessarily commutative!) diagram: +τ D(W) +πW +D +�♦♦♦♦♦♦♦♦♦♦♦♦♦ +τ D(α) +�◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +W +ι +� τ D(V ). +Using this diagram, we define the following K-subvariety of τD(W): +E := Equalizer +� +τ D(α), ι ◦ πW +D +� += +� +a ∈ τ D(W) | τ D(α)(a) = ι ◦ πW +D (a) +� +. +The following result is crucial. +Theorem 4.33 (Proposition 3.6 in [6]). The following are equivalent. +(1) The morphism πE : E → W is dominant. +(2) There is a B-operator on K(a, D(a)) extending D : K[a] → K[a, D(a)]. +We state our axioms below. +Axioms for D − PAC +Let (K, D) be a differential field of characteristic p > 0 and for each pair of affine +K-varieties (V, W) and each tuple f1, . . . , fn ∈ K(V ) such that + +PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS +27 +• W is absolutely irreducible, +• W ⊆ τD(V ), +• the projection π : W → V is dominant, +• E projects dominantly on W, +• the tuple (W; f1 ◦ π, . . . , fn ◦ π) is admissible; +there is x ∈ V (K) such that f1(x), . . . , fk(x) are not p-th powers in K and DV (x) ∈ +W(K). +By standard arguments (see e.g. [25, Remark 2.7(1)]) and Lemma 4.30, the above +axiom scheme is first-order. +Theorem 4.34. Let (K, D) be a differential field of characteristic p > 0 considered +as an Lλ0,D-structure. Then, (K, D) is DCFp-PAC if and only if (K, D) is D-PAC +(as defined above). +Proof. For the left-to-right implication, we assume that (K, D) is DCFp-PAC and +((V ; f1, . . . , fn), W) is as in the assumptions of the axioms of D-PAC. By Lemma +4.32, there is a derivation D′ : K(V ) → K(W) of the inclusion K(V ) ⊆ K(W) +(given by the dominant morphism π : W → V ). By Theorem 4.33, D′ extends to a +derivation D′′ : K(W) → K(W). Since W is absolutely irreducible, the extension +K ⊆ K(W) is regular and (K, D) ⊆ (K(W), D′′) is also a differential field extension. +By Fact 4.27, (K, D) ⊆ (K(W), D′′) is a DCFp-regular extension. Since (K, D) +is DCFp-PAC, we get that (K, D) is existentially closed in (K(W), D′′) (in the +language Lλ0,D). +Let us choose: +a = idK[V ] ∈ V (K(V )) ⊆ V (K(W)). +Then, as usual, we have D′′ +V (a) ∈ W(K(W)). Since a is a generic point of V over +K, by Remark 4.29, we get that f1(a), . . . , fk(a) are not p-th powers in K(W). +Since (K, D) is existentially closed in (K(W), D′′) (in the language Lλ0,D), there is +α ∈ V (K) such that f1(α), . . . , fk(α) are not p-th powers in K and DV (α) ∈ W(K). +For the right-to-left implication, we assume that (K, D) is D-PAC. Let us take +a ∈ M n such that p(x) := tp(a/K) is stationary and a quantifier-free Lλ0,D(K)- +formula φ(x) ∈ p(x). We need to show that φ has a realization in K. As usual, we +can assume that φ does not contain negations of equalities. We will “correct” now +the formula φ(x) (at the cost of adding extra variables, some fixed terms, and a new +tuple including a) into a new formula ϕ(¯x) ∈ LD over K such that ¯x = (x1, . . . , xl) +and x1 = x. +We illustrate this “correction” using an example first. Assume that the formula +φ(x) has the following form: +D [λ0 (D (λ0(x)) + D(x))] + x = 0. +We consider two cases, where each of them has two subcases. For the new variables, +we will use y, z rather than x2, x3. +Case I: λ0(a) = 0. +We fix the term t1(x) = x. +Subcase I.1: λ0(D(a)) = 0. +The “correction” is ϕ(x): x = 0 (no extra variables), the fixed terms are t1(x) = +x, t2(x) = D(x), and ¯a = a. + +28 +D. M. HOFFMANN AND P. KOWALSKI +Subcase I.2: λ0(D(a)) ̸= 0. +The “correction” is +ϕ(x, y): yp = D(x) ∧ D(y) + x = 0, +the fixed term is t1(x) = x, and ¯a = (a, D(a)1/p). +Case II: λ0(a) ̸= 0. +Subcase II.1: λ0 (D (λ0(a))) = 0. +The “correction” is ϕ(x): yp = x ∧ x = 0, the fixed term is t1(x, y) = D(y) + D(x), +and ¯a = (a, a1/p). +Subcase II.2: λ0 (D (λ0(a))) ̸= 0. +The “correction” is +ϕ(x, y, z): yp = x ∧ zp = D(y) + D(x) ∧ D(z) + x = 0, +there are no fixed terms, and +¯a = +� +a, a1/p, +� +D(a) + D(a)1/p�1/p� +. +The general procedure can be explained using induction on the complexity of Lλ0,D- +terms over K. We obtain a quantifier-free LD formula ϕ(¯x) over K, LD-terms +t1(¯x), . . . , tk(¯x) over K such that: +(∗) +if (K, D) |= ϕ(¯α) and t1(¯α) /∈ Kp, . . . , tk(¯α) /∈ Kp, then K |= φ(α), +and ¯a such that (M, D) |= ϕ(¯α). +Let us take now a quantifier-free L formula ψ(˜x) over K such that: +ϕ(¯x): +ψ (¯x, D(¯x), . . . , Dm(¯x)) +for some m ∈ N. Let us define: +˜a := (¯a, D(¯a), . . . , Dm(¯a)) , +V := locusK (˜a) , +W := locusK (˜a, D(˜a)) . +Let π : W → V denote the dominant projection on the “˜a-coordinates”. There are +rational function symbols f1(˜x), . . . , fk(˜x) over K such that for each i, we have: +ti(¯x) = fi (¯x, D(¯x), . . . , Dm(¯x)) . +Therefore, we obtain that for each i: +fi (˜a) = ti (¯a) /∈ M p ⊇ K (˜a, D(˜a))p = K(W)p, +so (W; f1 ◦ π, . . . , fn ◦ π) is an admissible tuple. +Let us take ˜α ∈ V (K) such that f1(˜α), . . . , fk(˜α) are not p-th powers in K and +DV (x) ∈ W(K). By the construction we get that: +˜α = (¯α, D(¯α), . . . , Dm(¯α)) . +Therefore, we obtain (K, D) |= ϕ(¯α) and t1(¯α), . . . , fk(¯α) are not p-th powers in +K. By (∗) above, we obtain that K |= φ(α). +□ +Remark 4.35. It was shown in [19] that the “equalizer condition” on the dominant +map E → W can be replaced with the easier condition of separability of the map +W → V and then we still get geometric axioms of DCFp (it also applies to the case +of derivations of the Frobenius map). However, we do not know whether such a +replacement would also work for the PAC-axioms, since we will not have Theorem +4.33 after such a replacement. + +PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS +29 +Theorem 4.36. If G is finite, then the model companion of +� +(DCFp)∀ +� +G, denoted +by G − DCFp, exists. +Proof. Once again, we would like to use Theorem 3.23. We have that DCFp satisfies +♠ and it will be shown in a greater generality (see again Section 4.4) that types +over algebraically closed sets are stationary (so ♥ follows). Theorem 4.34 assures +us that the PAC property is first order in DCFp. +□ +We would like to include here the following general result which will be immedi- +ately useful. +Remark 4.37. Let T be a L-theory with quantifier elimination and G be an +arbitrary group. +(1) If the theory G−T exists, then the theory G−(T eq)m exists as well, where +the superscript “m” denotes the Morleyization. +(2) If G is finite, T is stable, and the theory G − T exists, then the theory +G − (T eq)m is simple. +Proof. The proof of Item (1) is straightforward and we leave it to the reader. Item +(2) follows from [21, Cor 4.28]. +□ +Using Remark 4.37, we obtain the following. +Corollary 4.38. Let G be a finite group and p be a prime number. +(1) The theory G − SCFp,∞ is strictly simple, that is simple, not stable, and +not supersimple. +(2) The theory G − DCFp is strictly simple as well. +4.4. Fields with operators. In this subsection, we briefly explain how to gener- +alize Theorem 4.34 beyond the case of differential fields. We recall below some of +the set-up from [6]. +Let k be a field and B be a finite local k-algebra of dimension e. Assume that +we have a k-algebra map πB : B → k. Let {b0, . . . , be−1} be a fixed k-basis of B +such that b0 = 1 and πB(bi) = 0 for i > 0. For convenience, we also set d := e − 1. +Definition 4.39. Let ∂ = (∂0, . . . , ∂d) where ∂0, . . . , ∂d : R → T . +(1) If R = T and ∂0 = id, then we say that ∂ is a B-operator on R if the +corresponding map +R ∋ r �→ ∂0(r) ⊗ b0 + . . . + ∂d(r) ⊗ bd ∈ R ⊗k B +is a k-algebra homomorphism. We will also denote the map above by the +same symbol ∂. +(2) More generally, if the corresponding map +R ∋ r �→ ∂0(r) ⊗ b0 + . . . + ∂d(r) ⊗ bd ∈ T ⊗k B +is a k-algebra homomorphism, then we say that ∂ is a B-operator from R +to T . Note that if ∂ is a B-operator from R to T , then ∂0 : R → T is a +k-algebra homomorphism. +Assume that (K, ∂) is a field with a B-operator and V is an affine K-variety. +The notion of a prolongation τ∂(V ) was defined in this generality (see [37]) Under +the additional assumption of FrB(ker(πB)) = 0 (see [6, Remark 3.3]), we get the +versions of Lemma 4.32 and Theorem 4.33 (actually, the references from Section + +30 +D. M. HOFFMANN AND P. KOWALSKI +4.3.3 are coming exactly from the B-operator context). We get the corresponding +“Axioms for ∂-PAC (with the identical formulation) and a generalization of Theo- +rem 4.34. The proof of this generalization is conceptually the same, but would be +more cumbersome to write comparing to the proof of Theorem 4.34. Therefore we +decided to include the general case of B-operators as this comment only. +Remark 4.40. The argument above works in the even more general case of B- +operators (replacing B-operators) as considered in [20]. The main example of a +B-operator which is not a B-operator is a derivation of the Frobenius map. +Remark 4.41. Using Theorem 3.23, we obtain that for a finite group G, the theory +of G-actions on B-fields (and also on B-fields) has a model companion. It may be +shown more directly and without going through λ-functions. The axiomatization +is as follows. +Axioms for G-B-DCF +The structure (K, ∂, σ) is a G-B-field such that for each pair (V, W) of KG-varieties, +IF +• the action of G on K is faithful, +• V and W are K-irreducible, +• W ⊆ τ∂(V ), +• W projects generically on V , +• E projects generically on W; +THEN there is x ∈ V (KG) such that ∂V (x) ∈ W(KG). +4.5. Other examples and questions. We discuss now some other examples and +ask some questions. In [41], an example of a stable theory T is given such that the +class of T -PAC structures is not elementary. However, the theory T in this example +does not have quantifier elimination, so from our perspective it is not a good theory +to test whether the PAC property is first-order. +More precisely, in [41, Example 5.1] the theory of an equivalence relation with +exactly one finite class of n elements for each n > 0 appears. +This theory is +(implicitly) considered in the natural language with one unary relation symbol. +Then, this theory is not even model complete, since finite classes in a model may +become infinite in its extension. +Similarly, one can see that this theory is not +inductive. Therefore, to have any hopes for quantifier elimination, one needs to +add to the language the unary predicates (Rn)n>0 naming all finite equivalence +classes (see also [41, Example 5.2], where each element of each finite equivalence +class is named). Using Robinson’s Test, it is not difficult to check that with such a +choice of the language this theory becomes model complete and also substructure +complete, so it has quantifier elimination. Then, taking algebraic closure is the +same as adding “missing points” in all named finite classes. Therefore, if M ⊆ M ′ +and M ′ is a subset of a model, then acl(M) ∩ M ′ = M if and only if for all n, +we have Rn(M) = Rn(M ′). Thus, M is PAC if and only if M = dcl(M) and M +is infinite, so PAC is a first-order property in this case (note that definably closed +substructures are such ones that each finite class does not have “co-size one”). +As a conclusion, we do not know any stable theory T with quantifier elimination +such that the class of T -PAC structures is not elementary. We formulate the relevant +question below. +Question 4.42. Assume that T is stable and has quantifier elimination. Is the +class of T -PAC structures elementary? + +PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS +31 +Positive answer to the question above implies (using Theorem 3.23) that for a +finite group G and for T as above eliminating strong types and coding finite sets, +the theory of actions of G on models of T∀ has a model companion. +Remark 4.43. This is related to a general conjecture of the first author, see [21, +Conjecture 5.2]: +“Assume that T0 is theory with a model companion and G is a finite group. Does +the theory of G-actions on models of T0 have a model companion?” +This conjecture was meanwhile refuted in [7, Remark 3.9(2)], where T0 is the theory +of difference fields. In this example, the model companion of T0 (the theory ACFA) +is neither stable nor it has quantifier elimination. +To pursue the answer for the above question one can start with somehow stronger +assumption: +Question 4.44. Assume that T has nfcp (no finite cover property) and has quan- +tifier elimination. Is the class of T -PAC structures elementary? +The above assumption on not having the finite cover property is related to the +PAC property a little bit in Remark 3.6 in [41], but the main point here is that +it was shown in general that a stronger variant of the notion of nfcp (i.e. T does +not admit obstructions) implies the model companion of the theory of models of T +with a group action of Z exists ([3]). +Let T be a stable theory with quantifier elimination. If we replace a finite group +G with the cyclic infinite group Z, then the model theory of actions of Z on models +of T (we do not have distinguish between T∀ and T in this case) has been thoroughly +studied (see e.g. [13] and [11]). An analogue of our Question 3.24 was asked in +before Lemma 4.2 in [41], that is it is asked there whether the existence of the +theory T A (which is called Z − T in this paper) implies that T -PAC is first order. +The main result of this paper, Theorem 3.23, gives the opposite implication in the +case of finite groups. Such an implication is not true in the case of the actions of +Z (see [41, Example 5.2]). +Remark 4.45. To keep this paper reasonably sized, we have not checked all the +known stable theories. However, there is one theory which we would not mind to +analyze but the methods of this paper do not suffice to do that. This is the theory +DCFp,m for m > 1, that is the theory of differentially closed fields of characteristic +p > 0 with m commuting derivations, see [39] where the theory of differentially +closed fields with m commuting derivations is considered in arbitrary characteristic +(it is called m-DCF in [39]). Similarly as in the case of DCF0,m for m > 1, we +can not repeat our argument from the proof of Theorem 4.34, since we do not have +a version of Theorem 4.33 in the case of several commuting derivations. It looks +natural here to apply the approach of [46] described briefly in Remark 4.16(3) that +is: +• develop the appropriate notion of largeness for differential fields of positive +characteristic; +• show that the above notion is first-order; +• show that DCFp,m-PAC is the same as the largeness above together with +the PAC in the sense of fields. +We plan to pick it up in a further research. + +32 +D. M. HOFFMANN AND P. KOWALSKI +References +[1] Bijan Afshordel. Generic Automorphisms with Prescribed Fixed Fields. PhD thesis, Albert- +Ludwigs-Universitaet Freiburg, 2009. +[2] James Ax. The elementary theory of finite fields. Annals of Mathematics, 88:239–271, 1968. +[3] John T. Baldwin and Saharon Shelah. Model companions of Taut for stable T. Notre Dame +J. 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The model theory of differential fields of characteristic p ̸= 0. Proceedings of the +American Mathematical Society, 40(2):577–584, 1973. +[51] Carol Wood. The model theory of differential fields revisited. Israel Journal of Mathematics, +25(3):331–352, 1976. +[52] M. Ziegler. Separably closed fields with Hasse derivations. Journal of Symbolic Logic, 68:311– +318, 2003. +[53] B. Zilber. Zariski Geometries: Geometry from the Logician’s Point of View. London Math- +ematical Society Lecture Note Series. Cambridge University Press, 2010. + +34 +D. M. HOFFMANN AND P. KOWALSKI +† Instytut Matematyki, Uniwersytet Warszawski, Warszawa, Poland +Email address: daniel.max.hoffmann@gmail.com +URL: https://sites.google.com/site/danielmaxhoffmann/home +‡Instytut Matematyczny, Uniwersytet Wroc�lawski, Wroc�law, Poland +Email address: pkowa@math.uni.wroc.pl +URL: http://www.math.uni.wroc.pl/~pkowa/ + diff --git a/XtFJT4oBgHgl3EQf5y0l/content/tmp_files/load_file.txt b/XtFJT4oBgHgl3EQf5y0l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..773687903c9fec481497f7a71f920341d1fe3c1d --- /dev/null +++ b/XtFJT4oBgHgl3EQf5y0l/content/tmp_files/load_file.txt @@ -0,0 +1,1956 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf,len=1955 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='11671v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='LO] 27 Jan 2023 PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS DANIEL MAX HOFFMANN† AND PIOTR KOWALSKI‡ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We study model theory of actions of finite groups on substructures of a stable structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered the PAC property is first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Introduction In this paper, we consider the notion of a pseudo algebraically closed (PAC) substructure of a stable structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' This notion originates from the theory of pseudo algebraically closed fields, which were first considered by Ax in 1960’s while he worked on pseudofinite fields ([2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Studying PAC structures beyond the case of fields was initiated by Hrushovski ([27]) in the strongly minimal context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Pillay and Polkowska considered the PAC property in the stable case ([41]), there are slight differences with the approach we take here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' PAC structures also appeared in Afshordel’s thesis ([1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Recently, PAC structures were analized by the first author ([21], [22]) and also by Dobrowolski, the first author, and Lee ([15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Here, we are working with a (complete) stable theory T which admits quantifier elimination and then focus on its universal part T∀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' In other words, a typical situation looks as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We have a universal theory T∀ with a stable model completion T , so T has quantifier elimination and T axiomatizes existentially closed models of T∀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Then, intuitively, the class of PAC structures in T lies in between the class of existentially closed structures (models of T ) and the class of all the structures considered (models of T∀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' There are several possible definitions of the notion of PAC, we adopt here the definition from [21] (expressed in terms involving stationary types), which is a slight modification of the definition from [41], and which is equivalent to Afshordel’s definition from [1] in the case of stable theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' To define the notion of a PAC structure, one needs to use an appropriate notion of irreducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' In the classical case of PAC fields, a topological notion is used coming from the Zariski topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Hrushovski used in [27] “Morley irreducibility”, that is he considered definable sets of Morely degree one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Pillay and Polkowska used [41] stationary types and we proceed similarly here (however, we avoid any †SDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Supported by the Narodowe Centrum Nauki grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' 2021/43/B/ST1/00405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' ‡ Supported by the Narodowe Centrum Nauki grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' 2021/43/B/ST1/00405 and by the T¨ubitak 1001 grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' 119F397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' 2020 Mathematics Subject Classification Primary 03C60, 03C45 Secondary 12H10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Finite group action, Model companion, PAC structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' 1 2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' HOFFMANN AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' KOWALSKI saturation requirements as given in [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We say that a structure F |= T∀ is PAC in T (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='3) if all stationary types (in the sense of the theory T ) over F are finitely satisfiable in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Let us point out that in the case of the theory of algebraically closed fields, all the irreducibility notions mentioned above are essentially the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' However, this is not the case for other theories of interest as the theory of differentially closed fields of characteristic 0 or the theory of compact complex manifolds (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Nevertheless, we show in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='1 that all these irreducibility notions lead to the same notion of a PAC structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' For an extension F ⊆ K of models of T∀, we obtain relative notions of K-strongly PAC and algebraically K-strongly PAC (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' They are meaningful and can be though of as measuring the distance between being PAC and being a model of T (K-strongly PAC) or between being definably closed and algebraically closed (algebraically K-strongly PAC), see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Our main motivation for considering PAC structures comes from model theory of group actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' In the set-up above, we consider actions of a fixed group G on models of T∀ by automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Clearly, such actions are first-order expressible in an appropriate language and we aim to describe existentially closed actions and check whether a model companion of the theory of such actions exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The result below may be considered as an abstract generalization of our theorem about finite group actions on fields (see [24, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='29]) and as a continuation of studies from [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Let G be a finite group and T be a stable theory coding finite sets, which has quantifier elimination and eliminates strong types (that is: types over algebraically closed sets are stationary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Assume that G acts faithfully on K = dcl(K) |= T∀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Then, the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (1) The action of G on K is existentially closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (2) The structure of invariants KG is K-strongly PAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (3) The structure of invariants KG is PAC and algebraically K-strongly PAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The above theorem gives a description of existentially closed finite group actions, but it is not clear whether this description is first-order, so this theorem does not settle the question of the existence of a model companion of the theory of finite actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We can show the following implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Let G be a finite group and T be as in the statement of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' If the class of T -PAC structures is elementary, then the model companion of the theory of G-actions on models of T∀ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' After the abstract description of existentially closed actions (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='13) and giving a criterion for existence of a model companion of the theory of finite actions (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='23), we focus on particular examples of theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We discuss the fol- lowing three stable theories of fields of positive characteristic (p is a prime and e is a positive integer): (1) The theory SCFp,e of separably closed fields of characteristic p and insep- arability degree e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (2) The theory SCFp,∞ of separably closed fields of characteristic p and infinite inseparability degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (3) The theory DCFp of differentially closed fields of characteristic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' In the most interesting cases of the theories SCFp,∞ and DCFp, we do not have elimination of imaginaries, however we still have its weaker versions (coding finite PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS 3 sets and eliminating strong types), which are enough for the set-up from Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='13 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' For these theories, we describe PAC structures in a first-order way using a result of Tamagawa (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='21) about positive characteristic PAC fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We finish with some general questions regarding the PAC property and existence of a model companion of the theory of finite actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' It should be mentioned that after replacing a finite group G with the infinite cyclic group (Z, +), then the model theory of actions of (Z, +) has been thoroughly studied (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' [13] and [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We compare these two situation in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' In Section 2, we introduce several versions of the notion of a PAC structure and show the basic results about them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' In Section 3, we put the group action to the picture and prove the main two abstract results stated above (Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='13 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' In Section 4, we consider some particular theories (mostly theories of fields of positive characteristic) and give a first order characterization of PAC structures with respect to these theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Let T be a complete first order theory with a monster model C |= T (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' a strongly ¯κ-homogeneuos and ¯κ-saturated model of T for a very big cardinal ¯κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Usually, x stands for a (finite) tuple of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Moreover, for the rest of this paper, let G be a group such that |G| < ¯κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Bearing in mind any future applications, we try in this paper to formulate each result with a minimal list of assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Therefore, we organize our general model-theoretic assumptions in the following list (we are aware that there are some overlaps, but we preferred more transparent exposition): (QE) T has quantifier elimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (FS) T codes finite tuples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' eliminates finite imaginaries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (♠) T has (FS) and for every k < ω, for every variable x corresponding to a real sort and the 0-definable equivalence relation E on Sk x given by E(¯x, ¯x′) ⇐⇒ {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' , xk} = {x′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' , x′ k}, there exists a 0-definable in L function f : Sk x → Sw such that E is a fibration of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (♥) T is stable and types over algebraically closed sets are stationary (elimina- tion of strong types).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Convention: if a statement starts with any combination of the above properties, it means that we assume the properties given in this particular combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' For example, the following remark assumes property (FS): Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (FS) The condition (♠) is equivalent to: on each sort there is at least one 0-definable element and there is a sort with at least two 0-definable elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Similarly as in the proof of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='7 from [49], but, here, we allow many sorted structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Let us discuss what one can do to meet the above requirements if starting from arbitrary stable L0-theory T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' As we would like to work under assumptions of quantifier elimination and elimination of imaginaries, we pass to the language L := (Leq 0 )m and L-theory T := (T eq 0 )m (we add imaginary sorts and then do the Morleysation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' This new theory T is stable, has quantifier elimination and 4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' HOFFMANN AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' KOWALSKI elimination of imaginaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' On top of that, every 0-definable equivalence relation E on Cn is the fibration of the canonical projection πE : Cn → Cn /E which is build-in in the language (Leq 0 )m, thus a 0-definable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Strong types in any stable theory are stationary, and b |⌣A A for any b and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Therefore T enjoys all the properties: (QE), (FS), (♠) and (♥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Notion of PAC structure and auxiliary facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' In this subsection, we recall several definitions and useful facts from [21] and [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We also provide a few new notions closely related to the old definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The reader may also consult [41] and [43] for more on PAC structures in general model theoretic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Also [1] provides a nice of exposition of the notion of a PAC structure and related topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' A well-written survey on different variants of the notion of elimination of imaginaries and related concepts from the Galois theory is [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (Let T be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=') A substructure F of C is pseudo-algebraically closed (PAC) if every stationary type over F (in the sense of the L(F)-theory of C) is finitely satisfiable in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The above definition appears in [21] (see also Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='29 in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' In subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='1 of [21], there is a discussion on possible choices of the definition of a PAC substructure and a comparison of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='3 to definitions of PAC structures given in [27] and in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' In short, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='3 coincides with the definition of a PAC substructure in the strongly minimal context of [27] and relaxes the saturation assumption from the definition of a PAC substructure from [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Note that every PAC substructure is automatically definably closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Thus PAC substructures for T = ACF coincide with perfect pseudo-algebraically closed fields (as defined in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Let F = dcl(F) ⊆ K ⊆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (1) We say that F is K-strongly PAC if each type p(x) ∈ S(F), which has a unique non-forking extension over K, is finitely satisfiable in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (2) We say that F is algebraically K-strongly PAC if each algebraic type p(x) ∈ S(F), which has a unique non-forking extension over K, is finitely satisfiable (thus realized) in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Note that being K-strongly PAC for F ⊆ K implies being algebraically K- strongly PAC for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Moreover, being K-strongly PAC for F implies being a PAC substructure for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' It should help to understand the relative notions of (algebraically) K-strongly PAC by considering the ultimate cases of K = F and K |= T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' It is quite easy to see the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (1) A structure F is F-strongly PAC if and only if F |= T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (2) (T is stable) A structure F is K-strongly PAC for K |= T if and only if F is PAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (3) A structure F is algebraically F-strongly PAC if and only if F = acl(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (4) A structure F is algebraically K-strongly PAC for K |= T if and only if F = dcl(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (1) Let F ⊆ K be small subsets of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We say that F ⊆ K is primary if dcl(K) ∩ acl(F) = dcl(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS 5 (2) Let F ⊆ K be small subsets of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We say that F ⊆ K is regular if F ⊆ K is primary and F = dcl(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (3) Let F be a small definably closed substructure of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We say that F is regularly closed if for every small substructure F ′ of C, which is a regular extension of F, it follows F ⪯1 F ′ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' F is existentially closed in F ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The above notion of a primary extension was previously (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' [21], [22]) called “regular”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' It corresponds to regular extensions in T =ACF provided the smaller field is perfect (equivalently, definably closed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Here, we decided to follow closer the terminology from the theory of fields and distinguish between “primary” and “regular” extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We plan to refine even more the notion of the model-theoretic “regular” extension after studying a possible notion of the model-theoretic separable extension in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Now, we will sharpen facts from earlier articles which lead to the main results in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The majority of [21] was written under the assumption of (full) elimination of imaginaries, elimination of quantifiers and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' This is fine if we are interested in an abstract approach to the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' However, as we are interested in applications of our results to particular theories, which do not enjoy elimination of imaginaries (see Section 4), we need to relax this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Moreover, the assumption on stability was not crucial in several useful facts from [21], making them applicable in a broader context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Therefore we take the opportunity to provide the following results with minimal assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The proofs of the following facts remain almost the same as the proofs of their counterparts from [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Recall that “regular” extensions from [21] are now “primary” extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' All the proper subsets, substructures and tuples of the monster model C are, if not stated otherwise, small in comparison to the saturation of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Here, upper case letters, like E or A, are denoting proper subsets, and lower case letters, like a, stand for tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='7 (Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='32 in [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (FS) If E ⊆ A is primary then for every a ∈ acl(E) there exists a unique extension of tp(a/E) over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='8 (Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='33 in [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (FS) If E ⊆ A is primary , f1, f2 ∈ Aut(C) and f1|E = f2|E, then there exists h ∈ Aut(C) such that h|A = f1|A and h|acl(E) = f2|acl(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='9 (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='34 in [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (FS) If E ⊆ A is primary and A0 ⊆ A then tp(A0/E) has a unique extension over acl(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The following definition is taken from page 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We say that a type p(x) ∈ S(A) is acl-stationary if it has a unique extension over acl(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (FS) Consider p ∈ S(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The following are equivalent: (1) p is acl-stationary, (2) E ⊆ dcl(Ea) is primary for some a |= p, (3) E ⊆ dcl(Ea) is primary for every a |= p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The proof is similar to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='35 in [21], but a few steps require sharper reasoning, thus we include it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The equivalence (2) ⇐⇒ (3) follows by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' First, we argue for (1)⇒(2): assume (1) and suppose that (2) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' As p is acl-stationary, there exists 6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' HOFFMANN AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' KOWALSKI a unique extension p|acl(E) of p over E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Let a |= p|acl(E), then a |= p and E ⊆ Ea is not primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Take c ∈ dcl(Ea) ∩ acl(E) \\ dcl(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Since c ̸∈ dcl(E), there exists f ∈ Aut(C /E) such that f(c) ̸= c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We see that f(a) |= p|acl(E), so there exists h ∈ Aut(C / acl(E)) such that h(a) = f(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Note that h−1f ∈ Aut(C /Ea) and, because c ∈ dcl(Ea) and c ∈ acl(E), c = h−1f(c) = f(c) ̸= c, so a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The implication (2)⇒(1) is contained in Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' □ Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='12 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='35 in [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (FS, ♥) Consider p ∈ S(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The following are equivalent: (1) p is stationary, (2) p is acl-stationary, (3) E ⊆ Ea is primary for some a |= p, (4) E ⊆ Ea is primary for every a |= p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='13 (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='36 in [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (QE, FS, ♥) For any small substructure N there exists a non-algebraic stationary type over N in any finitely many variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='14 (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='38 in [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (FS, ♥) Assume that A, B ⊆ C, E ⊆ A is primary, f1, f2 ∈ Aut(C) and f1|E = f2|E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' If A |⌣E B and f1(A) |⌣f1(E) f2(B) then there exists h ∈ Aut(C) such that h|A = f1|A and h|B = f2|B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='15 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='39 in [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (FS, ♥) If E ⊆ A ∩ B, E ⊆ A is primary and B |⌣E A then B ⊆ BA is primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='16 (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='40 in [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (FS, ♥) If E ⊆ A and E ⊆ B are primary, and B |⌣E A then also E ⊆ BA is primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (1) (FS, ♥) F ⊆ K is primary if and only if for every tuple b from dcl(K), the type tp(b/F) is stationary (Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (2) (QE, FS, ♥) Using the item (1), a substructure F is PAC if and only if it is definably closed and regularly closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (1) Assume that F ⊆ K are substructures of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We say that K is normal over F (or we say that F ⊆ K is a normal extension) if σ(K) ⊆ K for every σ ∈ Aut(C /K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (Note that if K is small and F ⊆ K is normal, then it must be K ⊆ acl(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=') (2) Assume that F ⊆ K ⊆ acl(F) are small substructures of C such that F = dcl(F), K = dcl(K) and K is normal over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' In this situation we say that F ⊆ K is a Galois extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Assume that F ⊆ K is an extension of substructures in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We define the Galois group of the extension F ⊆ K as G(K/F) := Aut(K/F) = {f|K | f ∈ Aut(C /F), f(K) = K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Moreover B is any subset of C, then the extension dcl(B) ⊆ acl(B) is Galois and we speak about the absolute Galois group of B which is the following profinite group: G(B) := G(acl(B)/ dcl(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS 7 Note that the above definition of G(K/F) is often expressed in terms of the automorphisms of K as an L-structure on its own, but as we will work under the assumption of the quantifier elimination, both variants of the definition coincide and it just the matter of taste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The following useful fact is standard and its proof is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Assume that F ⊆ K is a Galois extension and p(x) ∈ S(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Then the Galois group G(K/F) acts transitively on the set of extensions of p over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The following definition and example are taken from [15] and [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' A more detailed discussion of examples of PAC structures and the property from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='21 will be given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (Let T be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=') We say that PAC is a first order property in T (= Th(C)) if there exists a set Σ of L-sentences such that for any P ⊆ C P |= Σ ⇐⇒ P is PAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (1) PAC is a first order property in ACFp for p = 0 and for p being a prime number, see Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='2 in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (2) The axioms given in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='6 from [41] show that PAC is a first order property (in the above sense) in DCF0 which is formulated in a different way than the condition “PAC is a first order property” appearing in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Finite group actions The main goal of this section is to describe existentially closed substructures with a finite group action in first order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The general strategy is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' First, characterize their structure by the structure of the invariants of the group action, then answer which properties of the invariants correspond to the existential closedeness of the whole substructure with group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Finally, express these properties as first order statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Basic facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We introduce the language LG being the language L extended by a unary function symbol σg for each g ∈ G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' LG = L ∪ {σg | g ∈ G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Often, “σg” will denote also the interpretation of the symbol σg in a given LG-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Moreover, we set ¯σ := (σg)g∈G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We consider the collection of sentences in the language LG, say AG, which precisely expresses the following σg is an automorphism of the L-structure for every g ∈ G, σg ◦ σh = σg·h for all g, h ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' In other words, if K is an L-structure, and there exists an LG-structure (K, ¯σ) living on K, we have that (K, ¯σ) |= AG if and only if for each g ∈ G we have that σg ∈ Aut(K) and the map G ∋ g �→ σg ∈ Aut(K) is a group homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (1) Let (K, ¯σ) be an LG-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We say that ¯σ is a G- action on K if (K, ¯σ) |= AG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (2) If T ′ is an L-theory, then by (T ′)G we denote the set of consequences of T ′ ∪ AG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' 8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' HOFFMANN AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' KOWALSKI (3) If (K, ¯σ) |= (T∀)G, where K is of cardinality smaller than the saturation of C, then we call it a substructure with G-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Note that, without loss of generality, K ⊆ C, thus the name “substructure”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (4) We say that a substructure with G-action (K, ¯σ) is existentially closed if (K, ¯σ) is an existentially closed model of the theory (T∀)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (5) If the existentially closed models of the theory (T∀)G form an elementary class, we denote the theory of this class by G − T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Assume that (K, ¯σ) is a substructure with G-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Then we denote KG := {a ∈ K | (∀g ∈ G) (σg(a) = a) } and call it the substructure of invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (QE) Let (K, ¯σ) be a substructure with G-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' If (K, ¯σ) is ex- istentially closed then K = dcl(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' If K = dcl(K) then KG = dcl(KG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' For the standard proofs, the reader may consult Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='24 and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='26 in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (QE) Let (K, ¯σ) be a substructure with G-action and let p(x) ∈ S(K) be a G-invariant type (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' σg(p) = p for every g ∈ G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Then for any a |= p the set dcl(K, a) might be equipped with a G-action extending (K, ¯σ) and acting trivially on a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Let a |= p and let ¯k be some enumeration of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Then ¯ka ≡ σg(¯k)a for any g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' This implies that, for each g ∈ G, there exists σ′ g ∈ Aut(C) such that σ′ g|K = σg and σ′ g(a) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Naturally, (K, (σg)g∈G) ⊆ (dcl(K, a), (σ′ g)g∈G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' □ Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='5 (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='10 from [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (QE, FS) If G is finite and (K, ¯σ) is a substruc- ture with G-action such that dcl(K) = K and the action of G on K is faithful (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' if g ̸= h then there is a ∈ K such that σg(a) ̸= σh(a)), then K ⊆ acl(KG), KG ⊆ K is a Galois extension, G(K/KG) ∼= G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='10 from [22], Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='7 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='7 from [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Being more precise, we obtain the two first bullets as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='23 from [21] and then we repeat the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='10(4) from [21] using a variant of the finite Galois correspondence stated in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='7 in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (QE, FS, ♥) If (K, ¯σ) is an existentially closed substructure with G-action, then the group action if faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Consider any enumeration of G, say (gi)i∈I where (I, <) is a linear order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Let p(x) ∈ S(KG) be a non-algebraic stationary type (existing by Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='13), and let ¯b = (bi)i∈I |= p⊗I|KG be such that K |⌣KG ¯b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Let F denote dcl(KG,¯b), and let F ′ denote dcl(K,¯b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' As the type p⊗I|K is also stationary, the extension KG ⊆ F is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' For each g ∈ G, let θg be a bijection of I such that g · gi = gθg(i) holds for each i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' As the set {bi | i ∈ I} is KG-indiscernible, for each g ∈ G there exists τg ∈ Aut(C /KG) such that τg(bi) = bθg(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Now, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='38 from [21], allows us to simultaneously extend each σg (over K) and τg (over F) to an automorphism σ′ g ∈ Aut(C), for each g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We have that (K, (σg)g∈G) ⊆ (F ′, (σ′ g)g∈G), thus (K, (σg)g∈G) is existentially closed PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS 9 in (F ′, (σ′ g)g∈G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' If g ̸= h, then σ′ g(b) ̸= σ′ h(b) for some b ∈ F ′, and so there will be a ∈ K such that σg(a) ̸= σh(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (QE) If G is finitely generated and (K, ¯σ) is an existentially closed substructure with G-action, then KG is K-strongly PAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Consider p(x) ∈ S(KG) which has a unique non-forking extension over K, say ˜p(x) ∈ S(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' As p(x) is invariant under action of automorphisms σg|KG, where g ∈ G, we have that ˜p(x) is invariant under action of automorphisms σg, where g ∈ G (otherwise, we would get distinct non-forking extensions of p over K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Let b |= ˜p, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='4 there exists an extension of substructures with G- action, (K, (σg)g∈G) ⊆ (K′, (σ′ g)g∈G) such that b ∈ (K′)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' By our assumption, we have that (K, (σg)g∈G) is existentially closed in (K′, (σ′ g)g∈G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Now, let ϕ(a, x) ∈ p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' As T has quantifier elimination, we may assume that ϕ(y, x) is quantifier free, what we do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Of course |= ϕ(a, b) and so (K′, (σ′ g)g∈G) |= (∃x) (ϕ(a, x) ∧ � g∈X σg(x) = x), where X denotes the finite set of generators of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Hence (K, (σg)g∈G) |= (∃x) (ϕ(a, x) ∧ � g∈X σg(x) = x) and for some b0 ∈ KG we have that |= ϕ(a, b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' □ Therefore we see that an existentially closed substructure with G-action has a quite tame substructure of invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The next subsection is dedicated to the converse of this implication, so we would like to show that “if the substructure of invariants is tame then the whole substructure with G-action is existentially closed”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' In Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='56 from [21], it was shown that if (K, ¯σ) is an ex- istentially closed substructure with G-action, then K is PAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' However, the afore- mentioned proposition assumes quantifier elimination, elimination of imaginaries and stability (but G there can be arbitrary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Invariants of existentially closed actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (QE, FS) Assume that G is finite, (K, (σg)g∈G) ⊆ (K′, (σ′ g)g∈G) is an extension of substructures with G-action, the group action of G on K is faithful and dcl(K) = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' If KG is algebraically K-strongly PAC, then KG ⊆ (K′)G is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' If dcl(K) = K then also dcl(KG) = KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Moreover, KG ⊆ (K′)G is regular if and only if KG ⊆ dcl((K′)G) is regular and there is a unique way of extending G- action from K′ over dcl(K′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Therefore, without loss of generality, we assume that K′ = dcl(K”) and so dcl((K′)G) = (K′)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We need to show that (K′)G∩acl(KG) = KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Let a ∈ (K′)G ∩ acl(KG) \\ KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Because for every g ∈ G, we have that σg � tp(a/K) � = tp � σg(a)/K � and a ∈ (K′)G, we see that tp(a/K) is a G-invariant 10 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' HOFFMANN AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' KOWALSKI type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' By Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='5 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='20, we see that tp(a/K) is a unique extension of tp(a/KG) over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' As a ∈ acl(KG) and acl(KG) |⌣KG K (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='3 in [9]), tp(a/KG) ⊆ tp(a/K) is a non-forking extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Because KG is algebraically K-strongly PAC, tp(a/KG) is finitely satisfiable in KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' As a ∈ acl(KG), this means that it must be a ∈ KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Assume that C ⊆ K ⊆ C and that G is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We call the pair (C, K) G-closed if C ⊆ K is a Galois extension, G(K/C) ∼= G and there is no K′ ⊆ acl(K), K ⊊ K′, such that the action of G(K/C) extends over K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (QE, FS) Assume that G is finite, (K, ¯σ) is a substructure with G-action such that action of G on K is faithful and dcl(K) = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Then (KG, K) is G-closed if and only if KG is algebraically K-strongly PAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' By Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='5, K ⊆ acl(KG), KG ⊆ K is Galois and G(K/KG) ∼= G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Assume that (KG, K) is G-closed and let p(x) ∈ S(KG) be algebraic with a unique extension ˜p(x) over K (being a non-forking extension follows naturally from acl(KG) |⌣KG K, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='3 in [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We have that ˜p is G-invariant and so, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='4, if b |= ˜p then there exists an extension of substructures with a G-action, (K, (σg)g∈G) ⊆ (K′, (σ′ g)g∈G) such that K′ = dcl(K, b) and b ∈ (K′)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' As K′ = dcl(K, b) ⊆ acl(KG) = acl(K), it must be that K = K′, so b ∈ K and finally b ∈ KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Now, we show the right-to-left implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Assume that K′ ⊆ acl(K) and there is an extension of substructures with G-action: (K, (σg)g∈G) ⊆ (K′, (σ′ g)g∈G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='9, KG ⊆ (K′)G is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' As (K′)G ⊆ K′ ⊆ acl(K) = acl(KG) it must be (K′)G ⊆ dcl(KG) = KG, so KG = (K′)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' By the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='1 from [15] and the Galois correspondence for finite extensions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Theorem 12 in [32]), there exists a finite tuple b from K such that K = dcl(KG, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Moreover, by the same proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='1 from [15], we also have that K′ = dcl((K′)G, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Because KG = (K′)G, we have that K = dcl(KG, b) = dcl((K′)G, b) = K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' □ The following remark is not important for the main results of this paper and its purpose is mainly to generalize Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='25 from [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' As we use in its proof the Elementary Equivalence for PAC structures ([15]), we need to add more assump- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Let T be stable with elimination of quantifiers and elimination of imaginaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Assume that PAC is a first order property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Suppose that (C, K) ⊆ (C′, K′) is an extension of G-closed substructures such that C and C′ are PAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Then C ⪯ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' It is enough to reproduce the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='25 from [24], but in this more general context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' By the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='22 from [24] or more similar Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='54 from [21], we have that C and C′ are bounded PAC structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Thus, by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='11 from [15], it is enough to show that the restriction map r : G(C′) → G(C) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' After combining Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='11 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='9, we obtain that C ⊆ C′ is regular, so r is an epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS 11 By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='4 from [22], G(C) is projective, which means that there exists embedding h as in the following diagram G(C) = � h �● G(C) G(C′) r � But then G0 := h[G(C)] ⩽ G(C′) is a closed subgroup such that r|G0 : G0 → G(C) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Because K ⊆ acl(C) and K′ ⊆ acl(C′), the restriction maps G(C) → G and G(C′) → G lead to the following commutative diagram G(C′) �❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ r � G(C) �③③③③③③③③ G and so G0N = G(C′) for N := ker � G(C′) → G � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='31 from [21], this implies that G0 = G(C′) as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (QE, FS, ♥) Assume that G is finite, say |G| = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Let (K, ¯σ) be a substructure with G-action such that G acts faithfully on K and dcl(K) = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The following are equivalent: (1) (K, ¯σ) is existentially closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (2) KG is K-strongly PAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (3) KG is PAC and algebraically K-strongly PAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (4) KG is PAC and (KG, K) is G-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='11, (3) ⇐⇒ (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' (1)⇒(2) follows by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The im- plication (2)⇒(3) follows by definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' To get the theorem, we will show that (3)⇒(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Assume that dcl(K) = K, the group action is faithful and that KG is PAC and algebraically K-strongly PAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Using Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='5, we obtain the following K ⊆ acl(KG), KG ⊆ K is a Galois extension, G(K/KG) ∼= G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='1 from [15] gives us existence of a finite tuple ¯b = (b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' , bl−1) from K such that K = dcl(KG,¯b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Consider (K, (σg)g∈G) ⊆ (K′, (σ′ g)g∈G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Without loss of generality, we may assume that (K′, (σ′ g)g∈G) is existentially closed, in particular dcl(K′) = K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We have that the group action of G on K′ is faithful, thus by Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='5, we have that K′ ⊆ acl((K′)G), (K′)G ⊆ K′ is Galois, and G(K′/(K′)G) ∼= G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Moreover, by the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='1 in [15], it holds also that K′ = dcl((K′)G,¯b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='9 gives us that KG ⊆ (K′)G is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Let ¯B be some enumeration of {σg(bi) | g ∈ G, i < l}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We have that K′ = dcl((K′)G,¯b) = dcl((K′)G, ¯B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Assume that (K′, (σ′ g)g∈G) |= φ(a) 12 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' HOFFMANN AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' KOWALSKI for some tuple a from K′ and some quantifier-free formula φ(x) ∈ LG(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' First, we may present φ(a) as ϕ0(σ′ g0(a), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' , σ′ gl−1(a)), where ϕ0(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' , xl−1) ∈ L(K) is quantifier-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Second, since K = dcl(KG, ¯B), we may present ϕ0(σ′ g0(a), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' , σ′ gl−1(a)) as ϕ(σ′ g0(a), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' , σ′ gl−1(a), ¯B), where ϕ(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' , xl−1, ¯y) ∈ L(KG) is quantifier-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Let σ′ g0 = idL, so σ′ g0(a) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Because a ∈ K′ = dcl((K′)G, ¯B), there exists a finite tuple ¯c ⊆ (K′)G and a quantifier-free formula ψ0(¯z, ¯y, x) ∈ L such that ψ0(¯c, ¯B, C) = {a}, |= (∀¯z, ¯y, x, x′) � ψ0(¯z, ¯y, x) ∧ ψ0(¯z, ¯y, x′) −→ x = x′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Because σgi permutes ¯B, there exists a permutation si such that σgi( ¯B) = si( ¯B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We define ψi(¯z, ¯y, x) as ψ0(¯z, si(¯y), x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Note that ψi(¯c, ¯B, C) = {σ′ gi(a)} and (K′, (σ′ g)g∈G) |= (∀¯z, x, x′) � � g∈G σg(¯z) = ¯z ∧ ψ0(¯z, ¯B, x) ∧ ψi(¯z, ¯B, x′) → σgi(x) = x′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' To see the last line, let ¯d ⊆ (K′)G, m, m′ ∈ K′ be such that |= ψ0( ¯d, ¯B, m) ∧ ψi( ¯d, ¯B, m′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We do know that ψ0( ¯d, ¯B, C) = {m}, which after applying an extension ˜σgi ∈ Aut(C) of σ′ gi changes it into ψ0( ¯d, si( ¯B), C) = {σ′ gi(m)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' We have that m′ ∈ ψi( ¯d, ¯B, C) = {σ′ gi(m)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Since the whole formula is universal and has only parameters from K, it follows that (K, (σg)g∈G) |= (∀¯z, x, x′) � � g∈G σg(¯z) = ¯z ∧ ψ0(¯z, ¯B, x) ∧ ψi(¯z, ¯B, x′) → σgi(x) = x′� , where i < l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Consider p(¯z) := tp(¯c/KG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' Because KG ⊆ (K′)G is regular (thus also primary) and ¯c ⊆ (K′)G, Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content='12 implies that p(¯z) is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' As KG is PAC, the type p(¯z) is finitely satisfiable in KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The tuple ¯B ⊆ K is algebraic over KG, hence there exists a quantifier-free θ(¯y) ∈ L(KG) such that θ(¯y) ⊢ tp( ¯B/KG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' The following formula (∃ ¯y, x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFJT4oBgHgl3EQf5y0l/content/2301.11671v1.pdf'} +page_content=' , xl−1) � � i0{x ∈ M| lim inf +k→∞ r2−m +� +Br(x) +| ▽ uk|2dy ≥ ǫ0}. +We can always assume that uk ⇀ u weakly in W 1,2(M, N) and that +| ▽ uk|2dx ⇀ | ▽ u|2dx + ν +in the sense of measure as k → ∞. Here ν is a nonnegative Radon measure on M +with support in Σ. It is known that Σ is a Hm−2-rectifiable set, and we may write +ν = θ(x)Hm−2⌊Σ. It is clear that strongly convergence in H1,2(M, N) preserves the +identity (2). In this paper we mainly prove the following blow-up formula for weakly +convergence sequence of stationary quaternionic maps. +2 + +Theorem 1.2 Let uk be a sequence of stationary quaternionic map with E(uk) ≤ Λ. +Assume that uk → u weakly in H1(M, N). Then there exist +(a1, a2, a3) ∈ R3 with +�3 +α=1(aα)2 = 1 such that, for any smooth (m − 3)-form η with compact support in +M, +lim +k→∞ +3 +� +α=1 +aα +� +M +dη ∧ u∗ +kJ α = +3 +� +α=1 +aα +� +M +dη ∧ u∗J α + +� +Σ +θdη|Σ +(5) +and for any (b1, b2, b3) ⊥ (a1, a2, a3), there holds +lim +k→∞ +3 +� +α=1 +bα +� +M +dη ∧ u∗ +kJ α = +3 +� +α=1 +bα +� +M +dη ∧ u∗J α. +As a corollary of the theorem, the maps constructed by Chen-Li [CL2] and by Fos- +colo [F] can not be tangent maps (c.f [LT], Theorem 3.1) of a stationary quaternionic +map satisfing d(u∗J α) = 0. +2 +The proof of the blow-up formula +If u is a strong limit of a sequence of stationary quaternionic maps in H1,2(M, N), +then it’s easy to see that u satisfies (2). If u is just a weak limit, i.e. there exists a +sequence of stationary quaternionic maps uk satisfying uk → u weakly in H1,2(M, N) +and |∇uk|2dV → |∇u|2dV +θHm−2|Σ in the sense of measure, we prove in this section +a formula for the blow-up set θHm−2|Σ and the limiting map u. +Without loss of generality, we may assume that m = 4. Because Σ is a Hm−2- +rectifiable set, so we may assume that Σ = ∪∞ +i=0Σi, Σi ∩Σi′ = φ if i ̸= i′, Hm−2(Σ0) = +0, Σi ⊂ Ni and Ni (i = 1, 2, · · ·) is an (m − 2)-dimensional embedded C1 submanifold +of M. It is important that (see p. 61 in [Si]) TxΣ = TxNi for Hm−2-a.e. x ∈ Σi. +It is known that ν = θ(x)Hm−2⌊Σ, where θ(x) is upper semi-continuous with +ǫ0 ≤ θ(x) ≤ C(l1) for Hm−2-a.e. x ∈ Σ, C(l1) is a positive constant depending only +on M and l1 (cf. [Lin], Lemma 1.6). Since Hm−2(Σ) < +∞, for any 1. > 0, there +exist Σ1. ⊂ Σ and i0 such that Hm−2(Σ1. ) < 1., Σc +1. = Σ\Σ1. = ∪i0 +i=1Σ1. +i where Σ1. +i ⊂ Σi +(i = 1, · · ·, i0) is a bounded closed set. We choose a covering {Brn|n = 1, 2, · · ·} of +Σ1. such that � +n rm−2 +n +< C1.. Here and in the sequel, C always denotes a uniform +constant depending only on M and N. +Suppose that (x1, ..., x4) is a local normal coordinate system in Bǫ(Σδ +i), and that +(x3, x4) is the corresponding coordinate system in Σi, and the matrix expressions of +the complex structures are given by (6), (7) and (8). +J1 = + + + + +0 +0 +0 +−1 +0 +0 +1 +0 +0 +−1 +0 +0 +1 +0 +0 +0 + + + + , +A1βJ β = + + + + +J1 +· +· +J1 + + + + +(6) +3 + +J2 = + + + + +0 +−1 +0 +0 +1 +0 +0 +0 +0 +0 +0 +1 +0 +0 +−1 +0 + + + + , +A2βJ β = + + + + +J2 +· +· +J2 + + + + +(7) +J3 = + + + + +0 +0 +1 +0 +0 +0 +0 +1 +−1 +0 +0 +0 +0 +−1 +0 +0 + + + + , +A3βJ β = + + + + +J3 +· +· +J3 + + + + +(8) +where AαβJ β are 4n×4n-matrices, Aαβ are the entries of a matrix A in SO(3). Then +the quaternionic equation is + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +u1 +1 + u2 +2 + u3 +3 + u4 +4 += +0 +u2 +1 − u1 +2 + u4 +3 − u3 +4 += +0 +u3 +1 − u1 +3 − u4 +2 + u2 +4 += +0 +u4 +1 − u1 +4 − u2 +3 + u3 +2 += +0 +u5 +1 + u6 +2 + u7 +3 + u8 +4 += +0 +u6 +1 − u5 +2 + u8 +3 − u7 +4 += +0 +u7 +1 − u5 +3 − u8 +2 + u6 +4 += +0 +u8 +1 − u5 +4 − u6 +3 + u7 +2 += +0 +· · · +u4n−3 +1 ++ u4n−2 +2 ++ u4n−1 +3 ++ u4n +4 += +0 +u4n−2 +1 +− u4n−3 +2 ++ u4n +3 − u4n−1 +4 += +0 +u4n−1 +1 +− u4n−3 +3 +− u4n +2 + u4n−2 +4 += +0 +u4n +1 − u4n−3 +4 +− u4n−2 +3 ++ u4n−1 +2 += +0. +(9) +Theorem 2.1 For any smooth (m − 3)-form η with compact support in M, we have +lim +k→∞ +3 +� +α=1 +Aαβ +� +M +dη ∧ u∗ +kJ β = +3 +� +α=1 +Aαβ +� +M +dη ∧ u∗J β + +� +Σ +θdη|Σ +and +lim +k→∞ A1β +� +M +dη ∧ u∗ +kJ β = A1β +� +M +dη ∧ u∗J β, +lim +k→∞ A3β +� +M +dη ∧ u∗ +kJ β = A3β +� +M +dη ∧ u∗J β, +Proof. Assume that η = � +I ηIdxI. We have +lim +k→∞ +� +M +dη ∧ u∗ +k(AαβJ β) = +� +M +dη ∧ u∗(AαβJ β) ++ +lim +δ→0 lim +ǫ→0 lim +k→∞ +� +Bǫ(∪i0 +i=1Σδ +i ) +dη ∧ u∗ +k(AαβJ β) ++ +lim +δ→0 lim +ǫ→0 lim +k→∞ +� +∪nBrn\Bǫ(∪i0 +i=1Σδ +i ) +dη ∧ u∗ +k(AαβJ β). +(10) +4 + +It’s easy to see that +lim +δ→0 lim +ǫ→0 lim +k→∞ +� +∪nBrn +dη ∧ u∗ +k(J β) = 0 +(11) +By Lemma 2.2 in [LT], we get +lim +δ→0 lim +ǫ→0 lim +k→∞ +� +Bǫ(Σδ +i ) +dη ∧ u∗ +k(AαβJ β) += +lim +δ→0 lim +ǫ→0 lim +k→∞ +� +Bǫ(Σδ +i ) +2∂ηI +∂xl +∂uσ +k +∂x1 (AαβJ β)σγ +∂uγ +k +∂x2 dxl ∧ dxI ∧ dx1 ∧ dx2 +(12) +Substituting (9) to (12) and applying Lemma 2.2 in [LT], we have +lim +δ→0 lim +ǫ→0 lim +k→∞ +� +Bǫ(Σδ +i ) +dη ∧ u∗ +k(A1βJ β) = lim +δ→0 lim +ǫ→0 lim +k→∞ +� +Bǫ(Σδ +i ) +dη ∧ u∗ +k(A3βJ β) = 0 +and +lim +δ→0 lim +ǫ→0 lim +k→∞ +� +Bǫ(Σδ +i ) +dη ∧ u∗ +k(A2βJ β) = lim +δ→0 lim +ǫ→0 lim +k→∞ +� +Bǫ(Σδ +i ) +|∇uk|2dη ∧ dx1 ∧ dx2 += +lim +δ→0 lim +ǫ→0( +� +Bǫ(Σδ +i ) +|∇u|2dη ∧ dx1 ∧ dx2 + +� +Bǫ(Σδ +i )∩Σ +θdη|Σ) = +� +Σi +θdη|Σ +(13) +Then the proof of the theorem is completed. +Q.E.D. +Remark 2.2 From this theorem, we see that if uk satisfies (2), the weak limit u still +satisfies (2) if and only if θ = constant. +As a corollary, we can derive that θ(x) is locally constant. Precisely, +Corollary 2.3 Under the assumption of Theorem 1.2, and assume that there is an +open ball Bm ⊂ M \ Singu with Hm−2(Σ ∩ Bm) > 0. We have θ(x) is constant on +Σ ∩ Bm. +Proof. +In (5), we choose cutoff function η such that suppη ⊂ Bm. +Since Bm ⊂ +M \ Singu, we have u is smooth on Bm. Then du∗J β = 0 on Bm for β = 1, 2, 3. In +view of (5), we conclude that θ is constant on Σ ∩ Bm. +Q.E.D. +Let φ : S2 → N be a nonconstant smooth map satisfying (3) and (4). Set +u(x, x4) = φ( x +|x|) for any x ∈ R3\{0} x4 ∈ Rm−3 +(14) +as Chen-Li ([CL2]) did. Then we have +5 + +Proposition 2.4 For any smooth (m − 3)-form η with compact support in Rm, we +have +� +Rm dη ∧ u∗J α = −Eα +T (φ) +� +Rm−3 η(0, x4), +(15) +where +ET(φ) = +� +S2⟨Jα +S2, u∗J α⟩dσ. +Proof. We choose a spherical coordinate system (r, ϕ, θ) in R3, because u is smooth +for any r > 0, we have +� +Rm dη ∧ u∗J α += +� +Rm−3 +� ∞ +0 +∂ηI +∂r dr ∧ dxI +� +S2 φ∗J α += +− +� +Rm−3 η(0, x4) +� +S2 φ∗J α += +−Eα +T (φ) +� +Rm−3 η(0, x4) +Q.E.D. +By Theorem 2.1 and Proposition 2.4, we have the following corollary. +Corollary 2.5 The map u defined in (14) can not be a tangent map (c.f [LT], The- +orem 3.1) of a stationary quaternionic map with the property (2) at a singular point. +Proof. Suppose that u is defined as in (14). If it is a tangent map, then we have +by Theorem 2.1, +3 +� +α=1 +Aαβ +� +M +dη ∧ u∗J β + +� +Σ +θdη|Σ = 0. +By Proposition 2.4, we obtain +3 +� +α=1 +AαβEβ +T(φ) +� +Rm−3 η(0, x4) = +� +Σ +θdη|Σ. +Since u is stationary, by the blow-up formula of Li-Tian [LT], we have Σ is station- +ary. Using the constancy theorem (Theorem 41.1 in [Si]), it follows that the density +function θ is constant in every connected component of Σ, which implies that φ is +homotopy to a constant map. We therefore get a contradiction. +Q.E.D. +REFERENCES +[BT] C. Bellettini and G. Tian, Compactness results for triholomorphic maps, J. Eur. Math. Soc., 2(2019), +1271-1317. +6 + +[Ch] J. Chen, Complex anti-self-dual connections on product of Calabi-Yau surfaces and triholomorphic +curves, Commun. Math. Phys. 201(1999), 201-247. +[CL1] J. Chen and J. Li, Quaternionic maps between Hyperk¨ahler manifolds, J. Diff. Geom. 55(2000), no. +2, 355-384. +[CL2] J. Chen and J. Li, Quarternionic maps and minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. +4 (2005), no. 3, 375-388. +[FKS] J.M. Figuroa-O’Farrill, C. K¨ohl and B. Spence, Supersymmetric Yang-Mills, octonionic instantons +and triholomorphic curves, Nucl. Phys. B 521 (1998) no. 3, 419-443. +[F] L. Foscolo, ALF gravitational instantons and collapsing Ricci-flat metrics on the K3 surface, J. Diff. +Geom., 112(2019), 79-120. +[LT] J. Li, and G. Tian, A blow-up formula for stationary harmonic maps, IMRN, 14(1998), 735-755. +[Lin] F.-H. Lin, Gradient estimates and blow-up analysis for stationary harmonic maps I, Ann. of Math. +149(1999), 785-829. +[Si] L. Simon, Lectures on Geometric Measure Theory, Proc. Center Math. Anal. 3(1983), Australian +National Univ. Press. +[W] C. Wang, Energy quantization for triholomorphic maps, Calc. Var. PDE 18(2003), 145-158. +7 + diff --git a/b9AyT4oBgHgl3EQfXPer/content/tmp_files/load_file.txt b/b9AyT4oBgHgl3EQfXPer/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a8e7b5f6fbb2b5a52dc5afa6861aad3ea2b2ea37 --- /dev/null +++ b/b9AyT4oBgHgl3EQfXPer/content/tmp_files/load_file.txt @@ -0,0 +1,337 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf,len=336 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='00180v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='DG] 31 Dec 2022 A blow-up formula for stationary quaternionic maps ∗† Jiayu Li‡ Chaona Zhu§ Abstract Let (M, Jα, α = 1, 2, 3) and (N, J α, α = 1, 2, 3) be Hyperk¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Suppose that uk is a sequence of stationary quaternionic maps and converges weakly to u in H1,2(M, N), we derive a blow-up formula for limk→∞ d(u∗ kJ α), for α = 1, 2, 3, in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' As a corollary, we show that the maps constructed by Chen-Li [CL2] and by Foscolo [F] can not be tangent maps (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='f [LT], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='1) of a stationary quaternionic map satisfing d(u∗J α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' 1 Introduction and the main result A hyperk¨ahler manifold is a Riemannian manifold (M, g) with three parallel com- plex structures {J1, J2, J3} compatible with the metric g such that (J1)2 = (J2)2 = (J3)2 = J1J2J3 = −id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' The simplest hyperk¨ahler manifold is the Euclidean space R4m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' It is well-known that the only compact hyperk¨ahler manifolds of dimension 4 are K3 surfaces and complex tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Let (M, g, Jα, α = 1, 2, 3) and (N, h, J α, α = 1, 2, 3) be hyperk¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Let ωα(·, ·) = g(·, Jα·) and Ωα(·, ·) = h(·, J α·), (α = 1, 2, 3) be the K¨ahler forms on M and N respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' A smooth map u : M → N is called a quaternionic map (triholomorphic map) if AαβJ β ◦ du ◦ Jα = du (1) where Aαβ denote the entries of a matrix A in SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' For simplicity, we choose Aαβ = δαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' The quaternionic maps (triholomorphic maps) between Hyperk¨ahler manifolds has been studied by many aothors (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' [BT], [Ch], [CL1, [CL2], [FKS], [W]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Quater- nionic maps automatically minimize the energy functional in their homotopy classes (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' [Ch], [CL1] and [FKS]) and hence they are harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' It can be verified that holomorphic and anti-holomorphic maps with respect to some complex structures on M and N are quaternionic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' However, Chen-Li constructed quaternionic maps which are not holomorphic with respect to any complex structures on M and N (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' [CL1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' ∗This work is supported by NSF grant 11721101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' †MSC (2000): 53C26, 53C43, 58E12, 58E20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Keywords: Stationary harmonic maps, quaternionic maps, blow-up formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' ‡jiayuli@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='cn §zcn1991@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='cn 1 Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='1 A map u from M to N is called a stationary quaternionic map if it is a stationary harmonic map and it is a quaternionic map outside its singular set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' It is clear that (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' [BT]), if u satisfies (1) almost everywhere, and d(u∗J α) = 0, for α = 1, 2, 3, (2) then u is a stationary quaternionic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Chen-Li ([CL2]) proved that, if there is a harmonic sphere φ : S2 → N which satisfies dφ JS2 = − 3 � k=1 akJ k dφ, (3) where ⃗a = (a1, a2, a3) : S2 → S2, and � S2 xi|∇φ|2dσ = 0, i = 1, 2, 3, (x1, x2, x3) ∈ S2, (4) then u(x, x4) = φ( x |x|) for any x ∈ R3\\{0} is a stationary quaternionic map with the x4-axis as its singular set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Chen-Li ([CL2]) showed that there does exist a complete noncompact hyperk¨ahler manifold, into which there is a harmonic S2 which satisfies (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Recently, Foscolo [F] showed that there exists a compact K3 surface with the above property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' However, the map u constructed by Chen-Li or by Foscolo does not satisfy (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Now the question is whether the maps constructed by Chen-Li or by Foscolo could be a tangent map of a stationary quaternionic map with identity (2), if not the singular set of a stationary quaternionic map with identity (2) might be of codimensional 4 (Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='2 in [BT]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Suppose that uk is a sequence of stationary quaternionic maps with bounded energies E(uk) ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' The blow-up set of uk can be defined as Σ = ∩r>0{x ∈ M| lim inf k→∞ r2−m � Br(x) | ▽ uk|2dy ≥ ǫ0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' We can always assume that uk ⇀ u weakly in W 1,2(M, N) and that | ▽ uk|2dx ⇀ | ▽ u|2dx + ν in the sense of measure as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Here ν is a nonnegative Radon measure on M with support in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' It is known that Σ is a Hm−2-rectifiable set, and we may write ν = θ(x)Hm−2⌊Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' It is clear that strongly convergence in H1,2(M, N) preserves the identity (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' In this paper we mainly prove the following blow-up formula for weakly convergence sequence of stationary quaternionic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' 2 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='2 Let uk be a sequence of stationary quaternionic map with E(uk) ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Assume that uk → u weakly in H1(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Then there exist (a1, a2, a3) ∈ R3 with �3 α=1(aα)2 = 1 such that, for any smooth (m − 3)-form η with compact support in M, lim k→∞ 3 � α=1 aα � M dη ∧ u∗ kJ α = 3 � α=1 aα � M dη ∧ u∗J α + � Σ θdη|Σ (5) and for any (b1, b2, b3) ⊥ (a1, a2, a3), there holds lim k→∞ 3 � α=1 bα � M dη ∧ u∗ kJ α = 3 � α=1 bα � M dη ∧ u∗J α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' As a corollary of the theorem, the maps constructed by Chen-Li [CL2] and by Fos- colo [F] can not be tangent maps (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='f [LT], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='1) of a stationary quaternionic map satisfing d(u∗J α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' 2 The proof of the blow-up formula If u is a strong limit of a sequence of stationary quaternionic maps in H1,2(M, N), then it’s easy to see that u satisfies (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' If u is just a weak limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' there exists a sequence of stationary quaternionic maps uk satisfying uk → u weakly in H1,2(M, N) and |∇uk|2dV → |∇u|2dV +θHm−2|Σ in the sense of measure, we prove in this section a formula for the blow-up set θHm−2|Σ and the limiting map u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Without loss of generality, we may assume that m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Because Σ is a Hm−2- rectifiable set, so we may assume that Σ = ∪∞ i=0Σi, Σi ∩Σi′ = φ if i ̸= i′, Hm−2(Σ0) = 0, Σi ⊂ Ni and Ni (i = 1, 2, · · ·) is an (m − 2)-dimensional embedded C1 submanifold of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' It is important that (see p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' 61 in [Si]) TxΣ = TxNi for Hm−2-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' x ∈ Σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' It is known that ν = θ(x)Hm−2⌊Σ, where θ(x) is upper semi-continuous with ǫ0 ≤ θ(x) ≤ C(l1) for Hm−2-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' x ∈ Σ, C(l1) is a positive constant depending only on M and l1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' [Lin], Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Since Hm−2(Σ) < +∞, for any 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' > 0, there exist Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' ⊂ Σ and i0 such that Hm−2(Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' ) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=', Σc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' = Σ\\Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' = ∪i0 i=1Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' i where Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' i ⊂ Σi (i = 1, · · ·, i0) is a bounded closed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' We choose a covering {Brn|n = 1, 2, · · ·} of Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' such that � n rm−2 n < C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='. Here and in the sequel, C always denotes a uniform constant depending only on M and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Suppose that (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=', x4) is a local normal coordinate system in Bǫ(Σδ i), and that (x3, x4) is the corresponding coordinate system in Σi, and the matrix expressions of the complex structures are given by (6), (7) and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' J1 = \uf8eb \uf8ec \uf8ec \uf8ed 0 0 0 −1 0 0 1 0 0 −1 0 0 1 0 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 , A1βJ β = \uf8eb \uf8ec \uf8ec \uf8ed J1 J1 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (6) 3 J2 = \uf8eb \uf8ec \uf8ec \uf8ed 0 −1 0 0 1 0 0 0 0 0 0 1 0 0 −1 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 , A2βJ β = \uf8eb \uf8ec \uf8ec \uf8ed J2 J2 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (7) J3 = \uf8eb \uf8ec \uf8ec \uf8ed 0 0 1 0 0 0 0 1 −1 0 0 0 0 −1 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 , A3βJ β = \uf8eb \uf8ec \uf8ec \uf8ed J3 J3 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (8) where AαβJ β are 4n×4n-matrices, Aαβ are the entries of a matrix A in SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='the quaternionic equation is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='\uf8f1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='+ u4n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='+ u4n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='+ u4n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='u4n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='− u4n−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='+ u4n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='3 − u4n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='u4n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='− u4n−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='− u4n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='2 + u4n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='u4n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='1 − u4n−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='− u4n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='+ u4n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' (9) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='1 For any smooth (m − 3)-form η with compact support in M, we have lim k→∞ 3 � α=1 Aαβ � M dη ∧ u∗ kJ β = 3 � α=1 Aαβ � M dη ∧ u∗J β + � Σ θdη|Σ and lim k→∞ A1β � M dη ∧ u∗ kJ β = A1β � M dη ∧ u∗J β, lim k→∞ A3β � M dη ∧ u∗ kJ β = A3β � M dη ∧ u∗J β, Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Assume that η = � I ηIdxI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' We have lim k→∞ � M dη ∧ u∗ k(AαβJ β) = � M dη ∧ u∗(AαβJ β) + lim δ→0 lim ǫ→0 lim k→∞ � Bǫ(∪i0 i=1Σδ i ) dη ∧ u∗ k(AαβJ β) + lim δ→0 lim ǫ→0 lim k→∞ � ∪nBrn\\Bǫ(∪i0 i=1Σδ i ) dη ∧ u∗ k(AαβJ β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' (10) 4 It’s easy to see that lim δ→0 lim ǫ→0 lim k→∞ � ∪nBrn dη ∧ u∗ k(J β) = 0 (11) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='2 in [LT], we get lim δ→0 lim ǫ→0 lim k→∞ � Bǫ(Σδ i ) dη ∧ u∗ k(AαβJ β) = lim δ→0 lim ǫ→0 lim k→∞ � Bǫ(Σδ i ) 2∂ηI ∂xl ∂uσ k ∂x1 (AαβJ β)σγ ∂uγ k ∂x2 dxl ∧ dxI ∧ dx1 ∧ dx2 (12) Substituting (9) to (12) and applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='2 in [LT], we have lim δ→0 lim ǫ→0 lim k→∞ � Bǫ(Σδ i ) dη ∧ u∗ k(A1βJ β) = lim δ→0 lim ǫ→0 lim k→∞ � Bǫ(Σδ i ) dη ∧ u∗ k(A3βJ β) = 0 and lim δ→0 lim ǫ→0 lim k→∞ � Bǫ(Σδ i ) dη ∧ u∗ k(A2βJ β) = lim δ→0 lim ǫ→0 lim k→∞ � Bǫ(Σδ i ) |∇uk|2dη ∧ dx1 ∧ dx2 = lim δ→0 lim ǫ→0( � Bǫ(Σδ i ) |∇u|2dη ∧ dx1 ∧ dx2 + � Bǫ(Σδ i )∩Σ θdη|Σ) = � Σi θdη|Σ (13) Then the proof of the theorem is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='2 From this theorem, we see that if uk satisfies (2), the weak limit u still satisfies (2) if and only if θ = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' As a corollary, we can derive that θ(x) is locally constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Precisely, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='3 Under the assumption of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='2, and assume that there is an open ball Bm ⊂ M \\ Singu with Hm−2(Σ ∩ Bm) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' We have θ(x) is constant on Σ ∩ Bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' In (5), we choose cutoff function η such that suppη ⊂ Bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Since Bm ⊂ M \\ Singu, we have u is smooth on Bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Then du∗J β = 0 on Bm for β = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' In view of (5), we conclude that θ is constant on Σ ∩ Bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Let φ : S2 → N be a nonconstant smooth map satisfying (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Set u(x, x4) = φ( x |x|) for any x ∈ R3\\{0} x4 ∈ Rm−3 (14) as Chen-Li ([CL2]) did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Then we have 5 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='4 For any smooth (m − 3)-form η with compact support in Rm, we have � Rm dη ∧ u∗J α = −Eα T (φ) � Rm−3 η(0, x4), (15) where ET(φ) = � S2⟨Jα S2, u∗J α⟩dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' We choose a spherical coordinate system (r, ϕ, θ) in R3, because u is smooth for any r > 0, we have � Rm dη ∧ u∗J α = � Rm−3 � ∞ 0 ∂ηI ∂r dr ∧ dxI � S2 φ∗J α = − � Rm−3 η(0, x4) � S2 φ∗J α = −Eα T (φ) � Rm−3 η(0, x4) Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='4, we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='5 The map u defined in (14) can not be a tangent map (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='f [LT], The- orem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='1) of a stationary quaternionic map with the property (2) at a singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Suppose that u is defined as in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' If it is a tangent map, then we have by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='1, 3 � α=1 Aαβ � M dη ∧ u∗J β + � Σ θdη|Σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='4, we obtain 3 � α=1 AαβEβ T(φ) � Rm−3 η(0, x4) = � Σ θdη|Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Since u is stationary, by the blow-up formula of Li-Tian [LT], we have Σ is station- ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Using the constancy theorem (Theorem 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='1 in [Si]), it follows that the density function θ is constant in every connected component of Σ, which implies that φ is homotopy to a constant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' We therefore get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' REFERENCES [BT] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Bellettini and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Tian, Compactness results for triholomorphic maps, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=', 2(2019), 1271-1317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' 6 [Ch] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'} +page_content=' Chen, Complex anti-self-dual connections on product of Calabi-Yau surfaces and triholomorphic curves, 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0000000000000000000000000000000000000000..fdc5dab2d0e78c941d40e436d50dd3236e2f9fd4 --- /dev/null +++ b/d9FIT4oBgHgl3EQfoiuQ/content/tmp_files/2301.11319v1.pdf.txt @@ -0,0 +1,3523 @@ +arXiv:2301.11319v1 [math.CO] 26 Jan 2023 +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS +TO GEOMETRIC RAMSEY THEORY +NEIL LYALL +´AKOS MAGYAR +Abstract. Let ∆ = ∆1 ×. . .×∆d ⊆ Rn, where Rn = Rn1 ×· · ·×Rnd with each ∆i ⊆ Rni a non-degenerate +simplex of ni points. We prove that any set S ⊆ Rn, with n = n1 + · · · + nd of positive upper Banach +density necessarily contains an isometric copy of all sufficiently large dilates of the configuration ∆. +In +particular any such set S ⊆ R2d contains a d-dimensional cube of side length λ, for all λ ≥ λ0(S). We +also prove analogous results with the underlying space being the integer lattice. The proof is based on a +weak hypergraph regularity lemma and an associated counting lemma developed in the context of Euclidean +spaces and the integer lattice. +1. Introduction +1.1. Existing Results I: Distances and Simplices in Subsets of Rn. Recall that the upper Banach +density of a measurable set S ⊆ Rn is defined by +(1.1) +δ∗(S) = lim +N→∞ sup +t∈Rn +|S ∩ (t + Q(N))| +|Q(N)| +, +where | · | denotes Lebesgue measure on Rn and Q(N) denotes the cube [−N/2, N/2]n. +A result of Furstenberg, Katznelson, and Weiss [6] states that if S ⊆ R2 has positive upper Banach density, +then its distance set {|x − x′| : x, x′ ∈ S} contains all sufficiently large numbers. Note that the distance set +of any set of positive Lebesgue measure in Rn automatically contains all sufficiently small numbers (by the +Lebesgue density theorem) and that it is easy to construct a set of positive upper density which does not +contain a fixed distance by placing small balls centered on an appropriate square grid. +Theorem A (Furstenberg, Katznelson, and Weiss [6]). If S ⊆ R2 with δ∗(S) > 0, then there exists a +λ0 = λ0(S) such that S is guaranteed to contain pairs of points {x1, x2} with |x2 − x1| = λ for all λ ≥ λ0. +This result was later reproved using Fourier analytic techniques by Bourgain in [1] where he established +the following more general result for all configurations of n points in Rn whose affine span is n−1 dimensional, +namely for all non-degenerate simplices. +Theorem B (Bourgain [1]). Let ∆ ⊆ Rn be a non-degenerate simplex of n points. If S ⊆ Rn with δ∗(S) > 0, +then there exists a threshold λ0 = λ0(S, ∆) such that S contains an isometric copy of λ∆ for all λ ≥ λ0. +Recall that a finite point configuration ∆′ is said to be an isometric copy of λ∆ if there exists a bijection +φ : ∆ → ∆′ such that |φ(v) − φ(w)| = λ |v − w| for all v, w ∈ ∆, i.e. if ∆′ is obtained from λ∆ (the dilation +of ∆ by a factor λ) via a rotation and translation. +Bourgain deduced Theorem B as an immediate consequence of the following stronger quantitative result +for measurable subsets of the unit cube of positive measure. In the proposition below, and throughout this +article, we shall refer to a decreasing sequence {λj}J +j=1 as lacunary if λj+1 ≤ λj/2 for all 1 ≤ j < J. +Proposition B (Bourgain [1]). Let ∆ ⊆ Rn be a non-degenerate simplex of n points. For any 0 < δ ≤ 1 +there exists a constant J = O∆(δ−3n) such that if 1 ≥ λ1 ≥ · · · ≥ λJ is any lacunary sequence and S ⊆ [0, 1]n +with |S| ≥ δ, then there exists 1 ≤ j < J such that S contains an isometric copy of λ∆ for all λ ∈ [λj+1, λj]. +2010 Mathematics Subject Classification. 11B30. +The first and second authors were partially supported by grants NSF-DMS 1702411 and NSF-DMS 1600840, respectively. +1 + +2 +NEIL LYALL +´AKOS MAGYAR +In [12] the authors provided a short direct proof of Theorem B without using Proposition B. It is based on +the observation that uniformly distributed sets S ⊆ Rd contain the expected “number” of isometric copies +of dilates λ∆ and that all sets of positive upper density become uniformly distributed at sufficiently large +scales. However, for the purposes of this paper it will be important to recall Bourgain’s indirect approach. +To see that Proposition B implies Theorem B notice that if Theorem B were not to hold for some set +S ⊆ Rn of upper Banach density δ∗(S) > δ > 0, then there must exist a lacunary sequence λ1 ≥ · · · ≥ λJ ≥ 1, +with J the constant in Proposition B, such that S does not contain an isometric copy of λj∆ for any 1 ≤ j ≤ J. +Taking a sufficiently large cube Q with side length N ≥ λ1 and |S ∩ Q| ≥ δ|Q| and scaling back Q → [0, 1]n +contradicts Proposition B. +We further note that by taking λj = 2−j in Proposition B we obtain the following “Falconer-type” result +for subsets of [0, 1]n of positive Lebesgue measure. +Corollary B. If ∆ ⊆ Rn is a non-degenerate simplex of n points, then any S ⊆ [0, 1]n with |S| > 0 will +necessarily contain an isometric copy of λ∆ for all λ in some interval of length at least exp(−C∆|S|−3n). +Bourgain further demonstrated in [1] that no result along the lines of Theorem B can hold for configurations +that contain any three points in arithmetic progression along a line, specifically showing that for any n ≥ 1 +there are sets of positive upper Banach density in Rn which do not contain an isometric copy of configurations +of the form {0, y, 2y} with |y| = λ for all sufficiently large λ. This should be contrasted with the following +remarkable result of Tamar Ziegler. +Theorem C (Ziegler [25]). Let F be any configuration of k points in Rn with n ≥ 2. +If S ⊆ Rn has positive upper density, then there exists a threshold λ0 = λ0(S, F) such that Sε contains +an isometric copy of λF for all λ ≥ λ0 and any ε > 0, where Sε denotes the ε-neighborhood of S. +Bourgain’s example was later generalized by Graham [9] to establish that the condition that ε > 0 in +Theorem C is necessary and cannot be strengthened to ε = 0 for any given non-spherical configuration F +in Rn for any n ≥ 1, that is for any finite configuration of points that cannot be inscribed in some sphere. +We note that the sets constructed by Bourgain and Graham have the property that for any ε > 0 their +ε-neighborhoods will contain arbitrarily large cubes and hence trivially satisfy Theorem C with λ0 = 0. +It is natural to ask if any spherical configuration F, beyond the known example of simplices, has the +property that every positive upper Banach density subset of Rn, for some sufficiently large n, contains +an isometric copy of λF for all sufficiently large λ, and even to conjecture that this ought to hold for all +spherical configurations. The first breakthrough in this direction came in [12] when the authors established +this for configurations of four points forming a 2-dimensional rectangle in R4 and more generally for any +configuration that is the direct product of two non-degenerate simplices in Rn for suitably large n. +The purpose of this article is to present a strengthening of the results in [12] and to extend them to cover +configurations with a higher dimensional product structure in both the Euclidean and discrete settings. +1.2. New Results I: Rectangles and Products of Simplices in Subsets of Rn. +The first main result of this article is the following +Theorem 1.1. Let R be 2d points forming the vertices of a fixed d-dimensional rectangle in R2d. +(i) If S ⊆ R2d has positive upper Banach density, then there exists a threshold λ0 = λ0(S, R) such that +S contains an isometric copy of λR for all λ ≥ λ0. +(ii) For any 0 < δ ≤ 1 there exists a constant c = c(δ, R) > 0 such that any S ⊆ [0, 1]2d with |S| ≥ δ is +guaranteed to contain an isometric copy of λR for all λ in some interval of length at least c. +Moreover, if R has sidelengths given by t1, . . . , td, then the isometric copies of λR in both (i) and (ii) above +can all be realized in the special form {x11, x12} × · · ·× {xd1, xd2} ⊆ R2 × · · ·× R2 with each |xj2 − xj1| = λtj. +The multi-dimensional extension of Szemer´edi’s theorem on arithmetic progressions in sets of positive +density due to Furstenberg and Katznelson [5] implies, and is equivalent to the fact, that there are isometric +copies of λR in S for arbitrarily large λ, with sides parallel to the coordinate axis. Theorem 1.1 states that +there is an isometric copy of λR in S for every sufficiently large λ, but only with sides parallel to given + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +3 +2-dimensional coordinate subspaces which provides an extra degree of freedom for each side vector of the +rectangle R. +A weaker version of Theorem 1.1, with R2d replaced with R5d, was later established by Durcik and Kovaˇc +in [4] using an adaptation of arguments of the second author with Cook and Pramanik in [3]. This approach +also makes direct use of the full strength of the multi-dimensional Szemer´edi theorem and as such leads to +quantitatively weaker results. +Our arguments work for more general patterns where d-dimensional rectangles are replaced with direct +products of non-degenerate simplices. +Theorem 1.2. Let ∆ = ∆1 × · · · × ∆d ⊆ Rn, where Rn = Rn1 × · · · × Rnd and each ∆j ⊆ Rnj is a +non-degenerate simplex of nj points. +(i) If S ⊆ Rn has positive upper Banach density, then there exists a threshold λ0 = λ0(S, ∆) such that +S contains an isometric copy of λ∆ for all λ ≥ λ0. +(ii) For any 0 < δ ≤ 1 there exists a constant c = c(δ, ∆) > 0 such that any S ⊆ [0, 1]n with |S| ≥ δ is +guaranteed to contain an isometric copy of λ∆ for all λ in some interval of length at least c. +Moreover the isometric copies of λ∆ in both (i) and (ii) above can all be realized in the special form ∆′ +1 × +· · · × ∆′ +d with each ∆′ +j ⊆ Rnj an isometric copy of λ∆j. +Quantitative Remark. +A careful analysis of our proof reveals that the constant c(δ, ∆) can be taken +greater than Wd(C′ +∆δ−3n1···nd)−1 where Wk(m) is a tower of exponentials defined by W1(m) = exp(m) and +Wk+1(m) = exp(Wk(m)) for k ≥ 1. +1.3. Existing Results II: Distances and Simplices in Subsets of Zn. The problem of counting isomet- +ric copies of a given non-degenerate simplex in Zn (with one vertex fixed) has been extensively studied via +its equivalent formulation as the number of ways a quadratic form can be represented as a sum of squares of +linear forms, see [11] and [19]. This was exploited by the second author in [16] and [17] to establish analogous +results to those described in Section 1.1 above for subsets of the integer lattice Zn of positive upper density. +Recall that the upper Banach density of a set S ⊆ Zn is analogously defined by +(1.2) +δ∗(S) = lim +N→∞ sup +t∈Rn +|S ∩ (t + Q(N))| +|Q(N)| +, +where | · | now denotes counting measure on Zn and Q(N) the discrete cube [−N/2, N/2]n ∩ Zn. +In light of the fact that any pairs of distinct points {x1, x2} in Zn has the property that the square of the +distance between them |x2 − x1|2 is always a positive integer we introduce the convenient notation +√ +N := {λ : λ > 0 and λ2 ∈ Z}. +Theorem A′ (Magyar [16]). Let 0 < δ ≤ 1. +If S ⊆ Z5 has upper Banach density at least δ, then there exists an integer q0 = q0(δ) and λ0 = λ0(S) +such that S contains pairs of points {x1, x2} with |x2 − x1| = q0λ for all λ ∈ +√ +N with λ ≥ λ0. +Theorem B′ (Magyar [17]). Let 0 < δ ≤ 1 and ∆ ⊆ Z2n+3 be a non-degenerate simplex of n points. +(i) If S ⊆ Z2n+3 has upper Banach density at least δ, then there exists an integer q0 = O(exp(C∆δ−13n)) +and λ0 = λ0(S, ∆) such that S contains an isometric copy of q0λ∆ for all λ ∈ +√ +N with λ ≥ λ0. +(ii) If N ≥ exp(2C∆δ−13n), then any S ⊆ {1, . . . , N}2n+3 with cardinality |S| ≥ δN 2n+3 will necessarily +contain an isometric copy of λ∆ for some λ ∈ +√ +N with 1 ≤ λ ≤ N. +Note that the fact that S ⊆ Zn could fall entirely into a fixed congruence class of some integer 1 ≤ q ≤ +δ−1/n ensures that the q0 that appears in Theorems A′ and B′ above must be divisible by the least common +multiple of all integers 1 ≤ q ≤ δ−1/n. Indeed if S = (qZ)n with 1 ≤ q ≤ δ−1/n then S has upper Banach +density at least δ, however the distance between any two points x, y ∈ S is of the form |x − y| = qλ for some +λ ∈ +√ +N. + +4 +NEIL LYALL +´AKOS MAGYAR +However, in both Theorems A′ and Part (i) of Theorem B′, one can take q0 = 1 if the sets S are assumed +to be suitably uniformly distributed on congruence classes of small modulus. This leads via an easy density +increment strategy to short new proofs, see [14] for Theorem A′ and Section 8 for Part (i) of Theorem B′. +The original argument in [17] deduced Theorem B′ from the following discrete analogue of Proposition B. +Proposition B′ (Magyar [17]). Let ∆ ⊆ Z2n+3 be a non-degenerate simplex of n points. +For any 0 < δ ≤ 1 there exist constants J = O∆(δ−3n) and q0 = O(exp(C∆δ−13n)) such that if N ≥ λ1 ≥ +· · · ≥ λJ ≥ 1 is any lacunary sequence in q0 +√ +N and S ⊆ {1, . . . , N}2n+3 with cardinality |S| ≥ δN 2n+3, then +S will necessarily contain an isometric copy of λj∆ for some 1 ≤ j ≤ J. +To see that Proposition B′ implies Theorem B′ notice that if Part (i) of Theorem B′ were not to hold +for some set S ⊆ Z2n+3 of upper Banach density δ∗(S) > δ > 0 with q0 from Proposition B′, then there +must exist a lacunary sequence λ1 ≥ · · · ≥ λJ ≥ 1 in q0 +√ +N, with J the constant from Proposition B′, such +that S does not contain an isometric copy of λj∆ for any 1 ≤ j ≤ J. Since we can find a sufficiently large +cube Q with integer side length N that is divisible by q0 and greater than λ1 such that |S ∩ Q| ≥ δ|Q| , this +contradicts Proposition B′. Part (ii) of Theorem B′ follows from Proposition B′ by taking λj = 2J−jq0. +1.4. New Results II: Rectangles and Products of Simplices in Subsets of Zn. +We will also establish the following discrete analogues of Theorem 1.1 and 1.2. +Theorem 1.3. Let 0 < δ ≤ 1 and R be 2d points forming the vertices of a d-dimensional rectangle in Z5d. +(i) If S ⊆ Z5d has upper Banach density at least δ, then there exist integers q0 = q0(δ, R) and λ0 = +λ0(S, R) such that S contains an isometric copy of q0λR for all λ ∈ +√ +N with λ ≥ λ0. +(ii) There exists a constant N(δ, R) such that if N ≥ N(δ, R), then any S ⊆ {1, . . . , N}5d with cardinality +|S| ≥ δN 5d will necessarily contain an isometric copy of λR for some λ ∈ +√ +N with 1 ≤ λ ≤ N. +If R has side lengths given by t1, . . . , td, then each of the isometric copies in (i) and (ii) above can be realized +in the form {x11, x12} × · · · × {xd1, xd2} ⊆ Z5 × · · · × Z5 with each |xj2 − xj1| = q0λtj and λtj, respectively. +Our arguments again work for more general patterns where d-dimensional rectangles are replaced with +direct products of non-degenerate simplices. +Theorem 1.4. Let 0 < δ ≤ 1 and ∆ = ∆1 × · · · × ∆d ⊆ Zn, where Zn = Z2n1+3 × · · · × Z2nd+3 and each +∆i ⊆ Z2ni+3 is a non-degenerate simplex of ni points. +(i) If S ⊆ Zn has upper Banach density at least δ, then there exist integers q0 = q0(δ, ∆) and λ0 = +λ0(S, ∆) such that S contains an isometric copy of q0λ∆ for all λ ∈ +√ +N with λ ≥ λ0. +(ii) There exists a constant N(δ, ∆) such that if N ≥ N(δ, ∆), then any S ⊆ {1, . . . , N}n with cardinality +|S| ≥ δN n will necessarily contain an isometric copy of λ∆ for some λ ∈ +√ +N with 1 ≤ λ ≤ N. +Moreover, each of the isometric copies in (i) and (ii) above can be realized in the special form ∆′ +1 × · · · × ∆′ +d +with each ∆′ +i ⊆ Z2ni+3 an isometric copy of q0λ∆j and λ∆j, respectively. +Quantitative Remark. A careful analysis of our proof reveals that the constant q0(δ, ∆) (and consequently +also N(δ, ∆)) can be taken less than Wd(C′ +∆δ−13n1···nd) where Wk(m) is a tower of exponentials defined by +W1(m) = exp(m) and Wk+1(m) = exp(Wk(m)) for k ≥ 1. +1.5. Notations and Outline. We will consider the parameters d, n1, . . . , nd fixed and will not indicate the +dependence on them. Thus we will write f = O(g) if |f| ≤ C(n1, . . . , nd)g. If the implicit constants in our +estimates depend on additional parameters ε, δ, K, . . . the we will write f = Oε,δ,K,...(g). We will use the +notation f ≪ g to indicate that |f| ≤ c g for some constant c > 0 sufficiently small for our purposes. +Given an ε > 0 and a (finite or infinite) sequence L0 ≥ L1 ≥ · · · > 0, we will say that the sequence is +ε-admissible if Lj/Lj+1 ∈ N and Lj+1 ≪ ε2Lj for all j ≥ 1. Moreover, if q ∈ N is given and Lj ∈ N for +all 1 ≤ j ≤ J, then we will call the sequence L0 ≥ L1 ≥ · · · ≥ LJ (ε, q)-admissible if in addition LJ/q ∈ N. +Such sequences of scales will often appear in our statements both in the continuous and the discrete case. +Our proofs are based on a weak hypergraph regularity lemma and an associated counting lemma developed +in the context of Euclidean spaces and the integer lattice. In Section 2 we introduce our approach in the + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +5 +model case of finite fields and prove an analogue of Theorem 1.1 in this setting. In Section 3 we review +Theorem 1.2 for a single simplex and ultimately establish the base case of our general inductive approach +to Theorem 1.2. In Section 4 we address Theorem 1.2 for the direct product of two simplices, this provides +a new proof (and strengthening) of the main result of [12] and serves as a gentle preparation for the more +complicated general case which we present in the Section 5. The proof of Theorem 1.4 is outlined in Sections +6 and 7, while a short direct proof of Part (i) of Theorem B′ is presented in Section 8. +2. Model case: vector spaces over finite fields. +In this section we will illustrate our general method by giving a complete proof of Theorem 1.1 in the +model setting of Fn +q where Fq denotes the finite field of q elements. We do this as the notation and arguments +are more transparent in this setting yet many of the main ideas are still present. +We say that two vectors u, v ∈ Fn +q are orthogonal, if x·y = 0, where “·” stands for the usual dot product. A +rectangle in Fn +q is then a set R = {x1, y1} × · · ·× {xn, yn} with side vectors yi − xi being pairwise orthogonal. +The finite field analogue of Theorem 1.1 is the following +Proposition 2.1. For any 0 < δ ≤ 1 there exists an integer q0 = q0(δ) with the following property: +If q ≥ q0 and t1, . . . , td ∈ F∗ +q, then any S ⊆ F2d +q +with |S| ≥ δ q2d will contain points +{x11, x12} × · · · × {xd1, xd2} ⊆ V1 × · · · × Vd +with |xj2 − xj1|2 = tj for 1 ≤ j ≤ d +where we have written F2d +q = V1 × · · · × Vd with Vj ≃ F2 +q pairwise orthogonal coordinate subspaces. +2.1. Overview of the proof of Proposition 2.1. Write F2d +q += V1 × . . . × Vd with Vj ≃ F2 +q pairwise +orthogonal coordinate subspaces. For any t := (t1, . . . , td) ∈ F∗ +q and S ⊆ F2d +q +we define +Nt(1S) := Ex1∈V 2 +1 ,...,xd∈V 2 +d +� +(ℓ1,...,ℓd)∈{1,2}d +1S(x1ℓ1, . . . , xdℓd) +d +� +j=1 +σtj(xj2 − xj1) +where we used the shorthand notation xj := (xj1, xj2) for each 1 ≤ j ≤ d and the averaging notation: +Ex∈Af(x) := +1 +|A| +� +x∈A +f(x) +for a finite set A ̸= ∅. We have also used the notation +σt(x) = +� +q +if |x|2 = t +0 +otherwise +for each t ∈ F∗ +q. Note that the function σt may be viewed as the discrete analogue of the normalized surface +area measure on the sphere of radius +√ +t. It is well-known, see [10], that +Ex∈F2q σt(x) = 1 + O(q−1/2) +and for all ξ ̸= 0 one has +ˆσt(ξ) := Ex∈F2q σt(x) e2πi x·ξ +q = O(q−1/2). +Note that if Nt(1S) > 0, then this implies that S contains a rectangle of the form {x11, x12}×· · ·×{xd1, xd2} +with xj1, xj2 ∈ Vj and |xj2 − xj1|2 = tj for 1 ≤ j ≤ d. +Our approach to Proposition 2.1 in fact establishes the following quantitatively stronger result. +Proposition 2.2. For any 0 < ε ≤ 1 there exists an integer q0 = q0(ε) with the following property: +If q ≥ q0, then for any S ⊆ F2d +q +and t1, . . . , td ∈ F∗ +q one has +Nt(1S) > +� |S| +q2d +�2d +− ε +where we have written F2d +q = V1 × . . . × Vd with Vj ≃ F2 +q pairwise orthogonal coordinate subspaces. + +6 +NEIL LYALL +´AKOS MAGYAR +A crucial observation in the proof of Proposition 2.2 is that the averages Nt(1S) can be compared to ones +which can be easily estimated from below. We define, for any S ⊆ F2d +q , the (unrestricted) count +M(1S) := Ex1∈V 2 +1 ,...,xd∈V 2 +d +� +(ℓ1,...,ℓd)∈{1,2}d +1S(x1ℓ1, . . . , xdℓd). +It is easy to see, by carefully applying Cauchy-Schwarz d times to Ex11∈V1,...,xd1∈Vd1S(x11, . . . , xd1), that +(2.1) +M(1S) ≥ +� |S| +q2d +�2d +. +Our approach to Proposition 2.2 therefore reduces to establishing that for any ε > 0 one has +(2.2) +Nt(1S) = M(1S) + O(ε) + Oε(q−1/2). +The validity of (2.2) will follow immediately from the d = k case of Proposition 2.3 below. However, +before we can state this counting lemma we need to introduce some further notation from the theory of +hypergraphs, notation that we shall ultimately make use of throughout the paper. +2.2. Hypergraph Notation and a Counting Lemma. +In order to streamline our notation we will make use the language of hypergraphs. For J := {1, . . ., d} +and 1 ≤ k ≤ d, we let Hd,k = {e ⊆ J; |e| = k} denote the full k-regular hypergraph on the vertex set J. For +K := {jl; j ∈ J, l ∈ {1, 2}} we define the projection π : K → J as π(jl) := j and use this in turn to define +the hypergraph bundle +H2 +d,k := {e ⊆ K; |e| = |π(e)| = k} +using the shorthand notation 2 = (2, 2, . . . , 2) to indicate that |π−1(j)| = 2 for all j ∈ J. +Notice when k = d then Hd,d consists of one element, the set e = {1, . . ., d}, and +H2 +d,d = { {1l1, . . . , dld}; (l1, . . . , ld) ∈ {1, 2}d}. +Let V := F2d +q +and V = V1 × . . . × Vd with Vj ≃ F2 +q pairwise orthogonal coordinate subspaces. For a given +x = (x11, x12, . . . , xd1, xd2) ∈ V 2 with xj1, xj2 ∈ Vj and a given edge e = {1l1, . . . , dld}, we write +xe := (x1l1, . . . , xdld). +Note that the map x → xe defines a projection πe : V 2 → V . With this notation, we can clearly now write +Nt(1S) = Ex∈V 2 +� +e∈H2 +d,d +1S(xe) +d +� +j=1 +σtj(xj2 − xj1) +M(1S) = Ex∈V 2 +� +e∈H2 +d,d +1S(xe). +Now for any 1 ≤ k ≤ d and any edge e′ ∈ Hd,k, i.e. e′ ⊆ {1, . . . , d}, |e′| = k, we let Ve′ := � +j∈e′ Vj. For +every x ∈ V 2 and e ∈ H2 +d,k, we define xe := πe(x) where πe : V 2 → Vπ(e) is the natural projection map. +Our key counting lemma, Proposition 2.3 below, which we will establish by induction on 1 ≤ k ≤ d +below, is then the statement that given a family of functions fe : Vπ(e) → [−1, 1], e ∈ H2 +d,k, the averages +(generalizing those discussed above) which are defined by +(2.3) +Nt(fe; e ∈ H2 +d,k) := Ex∈V 2 +� +e∈H2 +d,k +fe(xe) +d +� +j=1 +σtj(xj2 − xj1) +(2.4) +M(fe; e ∈ H2 +d,k) := Ex∈V 2 +� +e∈H2 +d,k +fe(xe). +are approximately equal. Specifically, one has + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +7 +Proposition 2.3 (Counting Lemma). Let 1 ≤ k ≤ d and 0 < ε ≤ 1. For any collection of functions +fe : Vπ(e) → [−1, 1] with e ∈ H2 +d,k +one has +(2.5) +Nt(fe; e ∈ H2 +d,k) = M(fe; e ∈ H2 +d,k) + O(ε) + Oε(q−1/2). +If we apply this Proposition with d = k and fe = 1S for all e ∈ H2 +d,d, then Theorem 2.1 clearly follows +given the lower bound (2.1). +2.3. Proof of Proposition 2.3. We will establish Proposition 2.3 by inducting on 1 ≤ k ≤ d. +For k = 1 the result follows from the basic observation that if f1, f2 : F2 +q → [−1, 1] and let t ∈ F∗ +q, then +Ex1,x2∈F2q f1(x1)f2(x2) σt(x2 − x1) = +� +ξ∈F2q +ˆf1(ξ) ˆf2(ξ)ˆσt(ξ) += ˆf1(0) ˆf2(0) + O(q−1/2) +(2.6) += Ex1,x2∈F2q f1(x1)f2(x2) + O(q−1/2) +by the properties of the function ˆσ given above. +To see how this implies Proposition 2.3 for k = 1 we note that since H2 +d,1 = {jl : 1 ≤ j ≤ d, 1 ≤ l ≤ 2} it +follows that +Nt(fe; e ∈ H2 +d,1) = +d +� +j=1 +Exj1,xj2∈F2q fj1(xj1)fj2(xj2) σt(xj2 − xj1) += +d +� +j=1 +Exj1,xj2∈F2q fj1(xj1)fj2(xj2) + O(q−1/2) = M(fe; e ∈ H2 +d,1) + O(q−1/2). +The induction step has two main ingredients, the first is an estimate of the type which is often referred +to as a generalized von-Neumann inequality, namely +Lemma 2.1. Let 1 ≤ k ≤ d. For any collection of functions fe : Vπ(e) → [−1, 1] with e ∈ H2 +d,k one has +(2.7) +Nt(fe; e ∈ H2 +d,k) ≤ +min +e∈H2 +d,k +∥fe∥□(Vπ(e)) + O(q−1/2) +where for any e ∈ H2 +d,k and f : Vπ(e) → [−1, 1] we define +(2.8) +∥f∥2k +□(Vπ(e)) := Ex∈V 2 +π(e) +� +e∈H2 +d,k +f(xe). +The corresponding inequality for the multilinear expression M(fe; e ∈ H2 +d,k), namely the fact that +M(fe; e ∈ H2 +d,k) ≤ +� +e∈H2 +d,k +∥fe∥□(Vπ(e)) ≤ min +e∈H2 +d,k +∥fe∥□(Vπ(e)) +is well-known and is referred to as the Gowers-Cauchy-Schwarz inequality [8]. +The second and main ingredient is an approximate decomposition of a graph to simpler ones, and is +essentially the so-called weak (hypergraph) regularity lemma of Frieze and Kannan [7]. We choose to state +this from a somewhat more abstract/probabilistic point of view, a perspective that will be particularly helpful +when we consider our general results in the continuous and discrete settings. +We will first introduce this in the case d = 2. A bipartite graph with (finite) vertex sets V1, V2 is a +set S ⊆ V1 × V2 and a function f : V1 × V2 → R may be viewed as weighted bipartite graph with weights +f(x1, x2) on the edges (x1, x2). If P1 and P2 are partitions of V1 and V2 respectively then P = P1 × P2 is +a partition V1 × V2 and we let E(f|P) denote the function that is constant and equal to Ex∈Af(x) on each + +8 +NEIL LYALL +´AKOS MAGYAR +atom A = A1 × A2 of P. The weak regularity lemma states that for any ε > 0 and for any weighted graph +f : V1 × V2 → [−1, 1] there exist partitions Pi of Vi with |Pi| ≤ 2O(ε−2) for i = 1, 2, so that +(2.9) +|Ex1∈V1Ex2∈V2(f − E(f|P))(x1, x2) 1U1(x1)1U2(x2)| ≤ ε +for all U1 ⊆ V1 and U2 ⊆ V2. Informally this means that the graph f can be approximated with precision ε +with the “low complexity” graph E(f, P). If we consider the σ-algebras Bi generated by the partitions Pi and +the σ-algebra B = B1 ∨ B2 generated by P1 × P2 then we have E(f|B), the so-called conditional expectation +function of f. Moreover it is easy to see, using Cauchy-Schwarz, that estimate (2.9) follows from +(2.10) +∥f − E(f|B1 ∨ B2)∥□(V1×V2) ≤ ε. +With this more probabilistic point of view the weak regularity lemma says that the function f can be +approximated with precision ε by a low complexity function E(f|B1 +� B2), corresponding to σ-algebras Bi +on Vi generated by O(ε−2) sets. This formulation is also referred to as a Koopman- von Neumann type +decomposition, see Corollary 6.3 in [23]. +We will need a natural extension to k-regular hypergraphs. See [22, 8], and also [2] for extension to sparse +hypergraphs. Given an edge e′ ∈ Hd,k of k elements we define its boundary ∂e′ := {f′ ∈ Hd,k−1; f′ ⊆ e′}. +For each f′ = e′\{j} ∈ ∂e′ let B′ +f be a σ-algebra on Vf′ := � +j∈f′ Vj and ¯Bf′ := {U × Vj; U ∈ Bf′} denote its +pull-back over the space Ve′. The σ-algebra B = � +f′∈∂e′ Bf′ is the smallest σ-algebra on ∂e′ containing ¯Bf′ +for all f′ ∈ ∂e′. Note that the atoms of B are of the form A = � +f′∈∂e′ Af′ where Af′ is an atom of ¯Bf′. We +say that the complexity of a σ-algebra Bf′ is at most m, and write complex(Bf′) ≤ m, if it is generated by +m sets. +Lemma 2.2 (Weak hypergraph regularity lemma). Let 1 ≤ k ≤ d and fe : Vπ(e) → [−1, 1] be a given +function for each e ∈ H2 +d,k. For any ε > 0 there exists σ-algebras Bf′ on Vf′ for each f′ ∈ Hd,k−1 such that +(2.11) +complex(Bf′) = O(ε−2k+1) +and +(2.12) +∥fe − E(fe| +� +f′∈∂π(e) +Bf′)∥□(Vπ(e)) ≤ ε +for all e ∈ H2 +d,k. +The proof of Lemmas 2.1 and 2.2 are presented in Section 2.4 below. We close this subsection by demon- +strating how these lemmas can be combined to establish Proposition 2.3. +Proof of Proposition 2.3. +Let ε > 0, 2 ≤ k ≤ d and assume that the lemma holds for k − 1. It follows from Lemma 2.2 that +there exists σ-algebras Bf′ of complexity O(ε−2k+1) on Vf′ for each f′ ∈ Hd,k−1 for which (2.12) holds for all +e ∈ H2 +d,k. For each e ∈ H2 +d,k we let ¯fe := E(fe| � +f′∈∂π(e) Bf′) and write fe = ¯fe + he. By Lemma 2.1 and +multi-linearity we have that +(2.13) +Nt(fe; e ∈ H2 +d,k) = Nt( ¯fe; e ∈ H2 +d,k) + O(ε) + O(q−1/2) +and also by the Gowers-Cauchy-Schwarz inequality +(2.14) +M(fe; e ∈ H2 +d,k) = M( ¯fe; e ∈ H2 +d,k) + O(ε). +The conditional expectation functions ¯fe are linear combinations of the indicator functions 1Ae of the +atoms Ae of the σ-algebras Be := � +f′∈∂π(e) Bf′. Since the number of terms in this linear combination is at most +2Cε−2k+1 +, with coefficients at most 1 in modulus, plugging these into the multi-linear expressions Nt( ¯fe; e ∈ +H2 +d,k) and M( ¯fe; e ∈ H2 +d,k) one obtains a linear combination of expressions of the form Nt(1Ae; e ∈ H2 +d,k) +and M(1Ae; e ∈ H2 +d,k) respectively with each Ae being an atoms of Be for all e ∈ H2 +d,k. +The key observation is that these expressions are at level k − 1 instead of k. Indeed, 1Ae = � +f′∈∂π(e) 1Aef′ +where Aef′ = A′ +ef′ × Vj, with A′ +ef′ being an atom of Bf′ when f′ = π(e)\{j}. If e = (j1l1, . . . , jl, . . . , jklk), + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +9 +let pf′(e) := (j1l1, . . . , jklk) ∈ H2 +d,k−1, obtained from e by removing the jl-entry. Then we have 1Aef′ (xe) = +1A′ +ef′(xp′ +f(e)) since xjl ∈ Vj, and hence +1Ae(xe) = +� +f′∈∂π(e) +1A′ +ef′ (xp′ +f(e)). +It therefore follows that +Nt(1Ae; e ∈ H2 +d,k) = Ex∈V 2 +� +e∈H2 +d,k +� +f′∈∂π(e) +1A′ +ef′(xpf′ (e)) +d +� +j=1 +σtj(xj2 − xj1) += Ex∈V 2 +� +f∈H2 +d,k−1 +� +e∈H2 +d,k, f′∈∂π(e) +pf′ (e)=f +1A′ +ef′(xpf′ (e)) +� +�� +� +=:gf +d +� +j=1 +σtj(xj2 − xj1) = Nt(gf; f ∈ H2 +d,k−1) +and similarly that +M(1Ae; e ∈ H2 +d,k) = M(gf; f ∈ H2 +d,k−1). +It then follows from the induction hypotheses that +Nt(1Ae; e ∈ H2 +d,k) = M(1Ae; e ∈ H2 +d,k) + O(ε1) + Oε1(q−1/2) +for any ε1 > 0. If we choose ε1 := 2−C1 ε−2k+1 +, with C1 ≫ 1 sufficiently large, then ε1 2Cε−2k+1 += O(ε) and it +follows that +Nt( ¯fe; e ∈ H2 +d,k) = M( ¯fe; e ∈ H2 +d,k) + O(ε) + Oε(q−1/2). +This, together with (2.13) and (2.14), establishes that (2.5) hold for d = k as required. +□ +2.4. Proof of Lemmas 2.1 and 2.2. +Proof of Lemma 2.1. We start by observing the following consequence of (2.6), namely that +(2.15) +���Ex1,x2∈F2qf1(x1)f2(x2)σt(x2 − x1) +��� +2 +≤ Ex1,x2∈F2qf1(x1)f1(x2) + O(q−1/2) +for any f1, f2 : F2 +q → [−1, 1] and t ∈ F∗ +q. +Now, fix an edge, say e0 = (11, 21, . . ., k1). Partition the edges e ∈ H2 +d,k into three groups; the first group +consisting of edges e for which 1 /∈ π(e), the second where 11 ∈ e and write e = (11, e′) with e′ ∈ H2 +d−1,k−1 +and the third when 12 ∈ e, using the notation H2 +d−1,k−1 := {(j2l2, . . . , jklk)}. Accordingly we can write +(2.16) Nt(fe; e ∈ H2 +d,k) = Ex∈V 2 +� +1/∈π(e) +fe(xe) +� +e′∈H2 +d−1,k−1 +f(11,e′)(x11, xe′) +� +e′∈H2 +d−1,k−1 +f(12,e′)(x12, xe′) +d +� +j=1 +σtj(xj2−xj1). +If for given x ∈ V1 and x′ = (x21, x22, . . . , xd1, xd2) ∈ V 2 +2 × . . . × V 2 +d we define +g1(x, x′) := +� +e′∈H2 +d−1,k−1 +f(11,e′)(x, xe′) +and +g2(x, x′) := +� +e′∈H2 +d−1,k−1 +f(12,e′)(x, xe′) +then we can write +Nt(fe; e ∈ H2 +d,k) = Ex21,x22,...,xd1,xd2 +� +1/∈π(e) +fe(xe) +d +� +j=2 +σtj(xj2 − xj1) +(2.17) +× Ex11,x12 g1(x11, x′)g2(x12, x′) σt1(x12 − x11). +By (2.15) we can estimate the inner sum in (2.17) by the square root of +Ex11,x12 g1(x11, x′)g1(x12, x′) + O(q−1/2). + +10 +NEIL LYALL +´AKOS MAGYAR +Thus by Cauchy-Schwarz, and the fact that fe : Vπ(e) → [−1, 1] for all e ∈ H2 +d,k, we can conclude that +(2.18) +Nt(fe; e ∈ H2 +d,k)2 ≤ Ex11,x12,...,xd1,xd2 +� +e′∈H2 +d−1,k−1 +f(11,e′)(x11, xe′)f(11,e′)(x12, xe′) +d +� +j=2 +σtj(xj2 − xj2). +The expression on the right hand side of the inequality above is similar to that in (2.16) except for the +following changes. The functions fe for 1 /∈ e are eliminated i.e. replaced by 1, as well as the factor σt1. The +functions f(12,e′), are replaced by f(11,e′) for all e′ ∈ H2 +d−1,k−1. Repeating the same procedure for j = 2, . . . , k +one eliminates all the factors σtj for 1 ≤ j ≤ k, moreover all the functions fe for edges e such that j /∈ π(e) +for some 1 ≤ j ≤ k, which leaves only the edges e so that π(e) = (1, 2, . . . , k), moreover for such edges the +functions fe are eventually replaced by fe0 = f11,21,...,k1. The factors σtj(xj2 −xj1) are not changed for j > k +however as the function fe0 does not depend on the variables xjl for j > k, averaging over these variables +gives rise to a factor of 1 + O(q−1/2). Thus one obtains the following final estimate +(2.19) +Nt(fe; e ∈ H2 +d,k)2k ≤ Ex11,x12,...,xk1,xk2 +� +π(e)=(1,...,k) +fe0(xe) + O(q−1/2) = ∥fe0∥2k +□(Vπ(e0)) + O(q−1/2). +This proves the lemma, as it is clear that the above procedure can be applied to any edge in place of +e0 = (11, 21, . . ., k1). +□ +Proof of Lemma 2.2. For a function fe : Vπ(e) → [−1, 1] and a σ-algebra Bπ(e) on Vπ(e) define the energy of +fe with respect to Bπ(e) as +E(fe, Bπ(e)) := ∥E(fe|Bπ(e))∥2 +2 = Ex∈Vπ(e) |E(fe|Bπ(e))(x)|2, +and for a family of functions fe and σ-algebras Bπ(e), e ∈ H2 +d,k its total energy as +E(fe, Bπ(e); e ∈ H2 +d,k) := +� +e∈H2 +d,k +E(fe, Bπ(e)). +We will show that if (2.12) does not hold for a family of σ-algebras Bπ(e) = � +f′∈∂π(e) Bf′ , then the σ-algebras +Bf′ can be refined so that the total energy of the system increases by a quantity depending only on ε. Since +the functions fe are bounded the total energy of the system is O(1), the energy increment process must stop +in Oε(1) steps, and (2.12) must hold. The idea of this procedure appears already in the proof of Szemer´edi’s +regularity lemma [20], and have been used since in various places [7, 22, 8]. +Initially set Bf′ := {∅, Vf′} and hence Bπ(e) = {∅, Vπ(e)} to be the trivial σ-algebras. Assume that in +general (2.12) does not hold for a family of σ-algebras Bf′, with f′ ∈ Hd,k−1. Then there exists an edge +e ∈ H2 +d,k so that ∥ge∥□(Vπ(e)) ≥ ε, with ge := fe − E(fe|Bπ(e)). Let e = (11, . . . , k1) for simplicity of notation, +hence π(e) = (1, . . . , k). Then, with notation x′ = (x12, . . . , xk2), one has +ε2k ≤ ∥ge∥2k +□(Vπ(e)) = Ex11,x12,...,xk1,xk2 +� +l1,...,lk=1,2 +ge(x1l1, . . . , xklk) +≤ Ex12,...,xk2 +���Ex11,...,xk1ge(x11, . . . , xk1) +k +� +j=1 +hj,x′(x11, . . . , xj−1 1, xj+1 1, . . . , xk1) +��� +for some functions hj,x′ that are bounded by 1 in magnitude. Indeed if and edge e ̸= (11, . . . , k1) then xe +does not depend at least one of the variables xj1. Thus there must be an x′ for which the inner sum in +the above expression is at least ε2k. Fix such an x′. Decomposing the functions hj,x′ into their positive +and negative parts and then writing them as an average of indicator functions, one obtains that there sets +Bj ⊆ Vπ(e)\{j} such that +���Ex11,...,xk1ge(x11, . . . , xk1) +k +� +j=1 +1Bj(x11, . . . , xj−1 1, xj+1 1, . . . , xk1) +��� ≥ 2−k ε2k + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +11 +which can be written more succinctly, using the inner product notation, as +(2.20) +���⟨fe − E(fe|Bπ(e)), +k +� +j=1 +1Bj⟩ +��� ≥ 2−k ε2k. +For f′ = ∂π(e)\{j} let B′ +f′ be the σ-algebra generated by Bf′ and the set Bj and let B′ +π(e) := � +f′∈∂π(e) B′ +f′ . +Since the functions 1Bj are measurable with respect to the σ-algebra B′ +π(e) for all 1 ≤ j ≤ k, we have that +(2.21) +⟨fe − E(fe|B′ +π(e)), +k +� +j=1 +1Bj⟩ = 0 +and hence, by Cauchy-Schwarz, that +(2.22) +∥E(fe|B′ +π(e)) − E(fe|Bπ(e))∥2 +2 = ∥E(fe|B′ +π(e))∥2 +2 − ∥E(fe|Bπ(e))∥2 +2 ≥ 2−2k ε2k+1. +Note that the first equality above follows from the fact that conditional expectation function E(f|B) is the +orthogonal projection of f to the subspace of B-measurable functions in L2. This also implies that energy of +a function is always increasing when the underlying σ-algebra is refined, and (2.22) tells us that the energy +of fe is increased by at least ck ε2k+1. +For f′ /∈ ∂π(e) we set B′ +f′ := Bf′. Then the total energy of the family fe with respect to the system +B′ +π(e) = � +f′∈∂π(e) B′ +f′, e ∈ H2 +d,k is also increased by at least ck ε2k+1. +It is clear that the complexity of the σ-algebras Bf′ are increased by at most 1, hence, as explained above, +the lemma follows by applying this energy increment process at most O(ε−2k+1) times. +□ +3. The base case of an inductive strategy to establish Theorem 1.2 +In this section we will ultimately establish the base case of our more general inductive argument. We +however start by giving a quick review of the proof of Theorem 1.2 when d = 1 (which contains both Theorem +B and Corollary B as stated in Section 1.1), namely the case of a single simplex. This was originally addressed +in [1] and revisited in [12] and [13]. +3.1. A Single Simplex in Rn. Let Q ⊆ Rn be a fixed cube and let l(Q) denotes its side length. +Let ∆0 = {v1 = 0, v2, . . . , vn} ⊆ Rn be a fixed non-degenerate simplex and define tkl := vk · vl for +2 ≤ k, l ≤ n where “ · ” is the dot product on Rn. Given λ > 0, a simplex ∆ = {x1 = 0, x2, . . . , xn} ⊆ Rn +is isometric to λ∆0 if and only if xk · xl = λ2tkl for all 2 ≤ k, l ≤ n. Thus the configuration space Sλ∆0 +of isometric copies of λ∆0 is a non-singular real variety given by the above equations. Let σλ∆0 be natural +normalized surface area measure on Sλ∆0, described in [1], [12], and [13]. It is clear that the variable x1 can +be replaced by any of the variables xi by redefining the constants tkl. +For any family of functions f1, . . . , fn : Q → [−1, 1] and 0 < λ ≪ l(Q) we define the multi-linear expression +(3.1) +N 1 +λ∆0,Q(f1, . . . , fn) := + +x1∈Q +ˆ +x2,...,xn +f1(x1) . . . fn(xn) dσλ∆0(x2 − x1, . . . , xn − x1) dx1. +We note that all of our functions are 1-bounded and both integrals, in fact all integrals in this paper, are +normalized. Recall that we are using the normalized integral notation +ffl +A f := +1 +|A| +´ +A f. Since the normalized +measure σλ∆0 is supported on Sλ∆0 we will not indicate the support of the variables (x2, . . . , xn) explicitly. +Note also that if S ⊆ Q is a measurable set and N 1 +λ∆0,Q(1S, . . . , 1S) > 0 then S must contain an isometric +copy of λ∆0. The following proposition (with Q = [0, 1]n) is a quantitatively stronger version of Proposition +B that appeared in Section 1.1 and hence immediately establishes Theorem 1.2 for d = 1. +Proposition 3.1. For any 0 < ε ≤ 1 there exists an integer J = O(ε−2 log ε−1) with the following property: +Given any lacunary sequence l(Q) ≥ λ1 ≥ · · · ≥ λJ and S ⊆ Q, there is some 1 ≤ j < J such that +(3.2) +N 1 +λ∆0,Q(1S, . . . , 1S) > +� |S| +|Q| +�n +− ε +for all λ ∈ [λj+1, λj]. + +12 +NEIL LYALL +´AKOS MAGYAR +Our approach to establishing Proposition 3.1 is to compare the above expressions to simpler ones for +which it is easy to obtain lower bounds. Given a scale 0 < λ ≪ l(Q) we define the multi-linear expression +(3.3) +M1 +λ,Q(f1, . . . , fn) := + +t∈Q + +x1,x2,...,xn∈t+Q(λ) +f1(x1) . . . fn(xn) dx1 . . . dxn dt +where Q(λ) = [− λ +2 , λ +2 ]n and t + Q(λ) is the shift of the cube Q(λ) by the vector t. Note that if S ⊆ Q is a +set of measure |S| ≥ δ|Q| for some δ > 0, then for a given ε > 0, H¨older implies +(3.4) M1 +λ,Q(1S, . . . , 1S) = + +t∈Q +� +x∈t+Q(λ) +1S(x) dx +�n +dt ≥ +� +t∈Q + +x∈t+Q(λ) +1S(x) dx dt +�n +≥ δn − O(ε), +for all scales 0 < λ ≪ ε l(Q). +Recall that for any ε > 0 we call a sequence L1 ≥ · · · ≥ LJ ε-admissible if Lj/Lj+1 ∈ N and Lj+1 ≪ ε2Lj +for all 1 ≤ j < J. Note that given any lacunary sequence l(Q) ≥ λ1 ≥ · · · ≥ λJ′ with J′ ≫ (log ε−1) J, one +can always finds an ε-admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJ such that for each 1 ≤ j < J the +interval [Lj+1, Lj] contains at least two consecutive elements from the original lacunary sequence. +In light of this observation, and the one above regarding a lower bound for M1 +λ,Q(1S, . . . , 1S), our proof +of Proposition 3.1 reduces to establishing the following “counting lemma”. +Proposition 3.2. Let 0 < ε < 1. There exists an integer J1 = O(ε−2) such that for any ε-admissible +sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJ1 and S ⊆ Q there is some 1 ≤ j < J1 such that +(3.5) +N 1 +λ∆0,Q(1S, . . . , 1S) = M1 +λ,Q(1S, . . . , 1S) + O(ε) +for all λ ∈ [Lj+1, Lj]. +There are two main ingredients in the proof of Proposition 3.2, this will be typical to all of our arguments. +The first ingredient is a result which establishes that the our multi-linear forms N 1 +λ∆0,Q(f1, . . . , fn) are +controlled by an appropriate norm which measures the uniformity of distribution of functions f : Q → [−1, 1] +with respect to particular scales L. This is analogous to estimates in additive combinatorics [8] which are +often referred to as generalized von-Neumann inequalities. +The result below was proved in [12] for Q = [0, 1]n, however a simple scaling of the variables xi transfers +the result to an arbitrary cube Q. +Lemma 3.1 (A Generalized von-Neumann inequality [12]). Let ε > 0, 0 < λ ≪ l(Q), and 0 < L ≪ ε6λ. +For any collections of functions f1, . . . , fn : Q → [−1, 1] we have +(3.6) +|N 1 +λ∆0,Q(f1, . . . , fn)| ≤ +min +i=1,...,n ∥fi∥U1 +L(Q) + O(ε) +where for any f : Q → [−1, 1] we define +(3.7) +∥f∥2 +U1 +L(Q) := + +t∈Q +��� + +x∈t+Q(L) +f(x) dx +��� +2 +dt +with t + Q(L) denoting the shift of the cube Q(L) = [− L +2 , L +2 ]n by the vector t. +The corresponding inequality for the multilinear expression M1 +λ,Q(f1, . . . , fn), namely the fact that +M1 +λ,Q(f1, . . . , fn) ≤ +min +i=1,...,n ∥fi∥U1 +L(Q) + O(ε) +whenever 0 < L ≪ ε6λ follows easily from Cauchy-Schwarz together with the simple observation that +∥f∥U1 +L(Q) ≤ ∥f∥U1 +L′(Q) + O(ε) +whenever L′ ≪ εL. +The second key ingredient, proved in [13] and generalized in Lemma 3.3 below, is a Koopman-von Neumann +type decomposition of functions where the underlying σ-algebras are generated by cubes of a fixed length. +To recall it, let Q ⊆ Rn be a cube, L > 0 be scale that divides l(Q), Q(L) = [− L +2 , L +2 ]n, and GL,Q denote +the collection of cubes t + Q(L) partitioning the cube Q and ΓL,Q denote the grids corresponding to the +centers of the cubes. By a slightly abuse of notation we also write GL,Q for the σ-algebra generated by the + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +13 +grid. Recall that the conditional expectation function E(f|GL,Q) is constant and equal to +ffl +A f on each cube +A ∈ GL,Q. +Lemma 3.2 (A Koopman-von Neumann type decomposition [13]). Let 0 < ε ≤ 1 and Q ⊆ Rn be a cube. +There exists an integer ¯J1 = O(ε−2) such that for any ε-admissible sequence l(Q) ≥ L1 ≥ · · · ≥ L ¯ +J1 and +function f : Q → [−1, 1] there is some 1 ≤ j < ¯J1 such that +(3.8) +∥f − E(f|GLj,Q)∥U1 +Lj+1 (Q) ≤ ε +Proof of Proposition 3.2. Let GLj,Q be the grid obtained from Lemma 3.2 for the functions f = 1S for some +fixed ε > 0. Let ¯f := E(f|GLj,Q), then by (3.6) and multi-linearity, we have +N 1 +λ∆0,Q(f, . . . , f) = N 1 +λ∆0,Q( ¯f, . . . , ¯f) + O(ε), +and also +M1 +λ,Q(f, . . . , f) = M1 +λ,Q( ¯f, . . . , ¯f) + O(ε) +provided for ε−6Lj+1 ≪ λ. Thus in showing (6.4) one can replace the functions f with ¯f. If we make the +additional assumption that λ ≪ εLj then it is easy to see, using the fact that the function ¯f is constant on +the cubes Qt(Lj) ∈ GLj,Q, that +N 1 +λ∆0,Q( ¯f, . . . , ¯f) = M1 +λ,Q( ¯f, . . . , ¯f) + O(ε). +Since the condition ε−6Lj+1 ≪ λ ≪ εLj can be replaced with Lj+1 ≪ λ ≪ Lj if one passes to a +subsequence of scales, for example L′ +j = L5j, this completes the proof of Proposition 3.2. +□ +3.2. The base case of a general inductive strategy. +In this section, as preparation to handle the case of products of simplices, we prove a parametric version of +Proposition 3.2, namely Proposition 3.3 below, which will serve as the base case for later inductive arguments. +Let Q = Q1 × · · · × Qd with Qi ⊆ Rni be cubes of equal side length l(Q). Let L be a scale dividing +l(Q) and for each t = (t1, . . . , td) ∈ ΓL,Q let Qt(L) = t + Q(L) and Qti(L) = ti + Qi(L). +Note that +Qt(L) = Qt1(L) × · · · × Qtd(L). Here Q(L) = [− L +2 , L +2 ]n and Qi(L) = [− L +2 , L +2 ]ni for each 1 ≤ i ≤ d. +Let ∆0 +i = {vi +1, . . . , vi +ni} ⊆ Rni be a non-degenerate simplex for each 1 ≤ i ≤ d. +Proposition 3.3 (Parametric Counting Lemma on Rn for Simplices). +Let 0 < ε ≤ 1 and R ≥ 1. There exists an integer J1 = J1(ε, R) = O(R ε−4) such that for any ε-admissible +sequence of scales L0 ≥ L1 ≥ · · · ≥ LJ1 with the property that L0 divides l(Q) and collection of functions +f i,r +k,t : Qti(L0) → [−1, 1] with 1 ≤ i ≤ d, 1 ≤ k ≤ ni, 1 ≤ r ≤ R and t ∈ ΓL0,Q +there exists 1 ≤ j < J1 and a set Tε ⊆ ΓL0,Q of size |Tε| ≤ ε|ΓL0,Q| such that +(3.9) +N 1 +λ∆0 +i ,Qti (L0)(f i,r +1,t, . . . , f i,r +ni,t) = M1 +λ,Qti (L0)(f i,r +1,t, . . . , f i,r +ni,t) + O(ε) +for all λ ∈ [Lj+1, Lj] and t /∈ Tε uniformly in 1 ≤ i ≤ d and 1 ≤ r ≤ R. +The proof of Proposition 3.3 will follow from Lemma 3.1 and the following generalization of Lemma 3.2 +in which we simultaneously consider a family of functions supported on the subcubes in a partition of an +original cube Q. +Lemma 3.3 (A simultaneous Koopman-von Neumann type decomposition). +Let 0 < ε ≤ 1, m ≥ 1, and Q ⊆ Rn be a cube. There exists an integer ¯J1 = O(mε−3) such that for any +ε-admissible sequence L0 ≥ L1 ≥ · · · ≥ L ¯ +J1 with the property that L0 divides l(Q), and collection of functions +f1,t, . . . , fm,t : Qt(L0) → [−1, 1] +defined for each t ∈ ΓL0,Q, there is some 1 ≤ j < ¯J1 and a set Tε ⊆ ΓL0,Q of size |Tε| ≤ ε|ΓL0,Q| such that +(3.10) +∥fi,t − E(fi,t|GLj,Qt(L0))∥U1 +Lj+1(Qt(L0)) ≤ ε +for all 1 ≤ i ≤ m and t /∈ Tε. + +14 +NEIL LYALL +´AKOS MAGYAR +Proof of Proposition 3.3. Fix 1 ≤ i ≤ d. For 1 ≤ k ≤ ni and t = (t1, . . . , td) ∈ ΓL0,Q , we will abuse notation +and write +f i,r +k,t(x1, . . . , xd) := f i,r +k,t(xi) +for (x1, . . . , xd) ∈ Qt(L0). +If we apply Lemma 3.3 to the family of functions f i,r +k,t on Qt(L0) for 1 ≤ i ≤ d, 1 ≤ k ≤ ni, and 1 ≤ r ≤ R, +with m = (n1 + . . . + nd)R, then this produces a grid GLj,Q for some 1 ≤ j ≤ ¯J1 = O(ε−3R), and a set +Tε ⊆ ΓL0,Q of size |Tε| ≤ ε|ΓL0,Q|, such that +∥f i,r +k,t − E(f i,r +k,t|GLj,Q)∥U1 +Lj+1 (Qt(L0)) ≤ ε +uniformly for 1 ≤ i ≤ d, 1 ≤ k ≤ ni and 1 ≤ r ≤ R for t /∈ Tε. +Since f i,r +k,t(x1, . . . , xd) = f i,r +k,t(xi) for (x1, . . . , xd) ∈ Qt(L0) it is easy to see that +∥f i,r +k,t − E(f i,r +k,t|GLj,Q)∥U1 +Lj+1 (Qt(L0)) = ∥f i,r +k,t − E(f i,r +k,t|GLj,Qi)∥U1 +Lj+1(Qti (L0)). +Let ¯f i,r +k,t := E(f i,r +k,t|GLj,Qi) , then by Lemma 3.1, one has +N 1 +λ∆0 +i ,Qti(L0)(f i,r +1,t, . . . , f i,r +ni,t) = N 1 +λ∆0 +i ,Qti (L0)( ¯f i,r +1,t, . . . , ¯f i,r +ni,t) + O(ε), +and +M1 +λ,Qti (L0)(f i,r +1,t, . . . , f i,r +ni,t) = M1 +λ,Qti (L0)( ¯f i,r +1,t, . . . , ¯f i,r +ni,t) + O(ε) +for all t /∈ Tε provided ε−6Lj+1 ≪ λ. Finally, if we also have λ ≪ εLj then it is easy to see that +N 1 +λ∆0 +i ,Qti (L0)( ¯f i,r +1,t, . . . , ¯f i,r +ni,t) = M1 +λ,Qti (L0)( ¯f i,r +1,t, . . . , ¯f i,r +ni,t) + O(ε) +as the functions ¯f i,r +k,t are constant on cubes Qti(Lj) of GLj,Qi, which are of size Lj ≪ εL0. +Passing first to a subsequence of scales, for example L′ +j = L5j, the condition ε−6Lj+1 ≪ λ ≪ εLj can be +replaced with Lj+1 ≪ λ ≪ Lj so this completes the proof of the Proposition. +□ +We conclude this section with a sketch of the proof of Lemma 3.3. These arguments are standard, see for +example the proof of Lemma 3.2 given in [12]. +Proof of Lemma 3.3. First we make an observation about the U 1 +L(Q)-norm. Suppose 0 < L′ ≪ ε2L with L′ +dividing L. If s ∈ ΓL′,Q and t ∈ Qs(L′) then |t − s| = O(L′) and hence + +x∈Qt(L) +g(x) dx = + +x∈Qs(L) +g(x) dx + O(L′/L) +for any function g : Q → [−1, 1]. Moreover, since the cube Qs(L) is partitioned into the smaller cubes Qt(L′), +we have by Cauchy-Schwarz +��� + +x∈Qs(L) +g(x) dx +��� +2 +≤ Et∈ΓL′,Qs(L) +��� + +x∈Qt(L′) +g(x) dx +��� +2 +. +From these observations it is easy to see that +∥g∥2 +U1 +L(Q) = + +t∈Q +��� + +x∈Qt(L) +g(x) dx +��� +2 +dt ≤ Et∈ΓL′,Q +��� + +x∈Qt(L′) +g(x) dx +��� +2 ++ O(L′/L) +and we note that the right side of the above expression is ∥E(g|GL′,Q)∥2 +L2(Q) since the conditional expectation +function E(g|GL′,Q) is constant and equal to +ffl +x∈Qt(L′) g(x) dx on the cubes Qt(L′). +Suppose that (3.10) does not hold for some 1 ≤ i ≤ m for every t in some set Tε ⊆ ΓL0,Q of size +|Tε| > ε |ΓL0,Q|. If we apply the above observation to g := fi,t − E(fi,t|GLj,Qt(L0)), for every t ∈ Tε, we +obtain by orthogonality that +m +� +i=1 +∥E(fi,t|GLj+2,Qt(L0))∥2 +L2(Qt(L0)) ≥ +m +� +i=1 +∥E(fi,t|GLj,Qt(L0))∥2 +L2(Qt(L0)) + cε2 +for some constant c > 0. + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +15 +If we now define fi : Q → [−1, 1] such that fi|(Qt(L0)) = fi,t, for 1 ≤ i ≤ m, average over t ∈ ΓL0,Q, and +use the fact ∥fi∥2 +L2(Q) = Et∈ΓL0,Q∥fi,t∥2 +L2(Qt(L0)), we obtain +(3.11) +m +� +i=1 +∥E(fi|GLj+2,Q)∥2 +L2(Q) ≥ +m +� +i=1 +∥E(fi|GLj,Q)∥2 +L2(Q) + cε3. +It is clear that the sums in the above expressions are bounded by m for all j ≥ 1, thus (3.11) cannot hold +for some 1 ≤ j ≤ ¯J1 for ¯J1 := C m ε−3. This implies that (3.10) must hold for some 1 ≤ j ≤ ¯J1, for all +1 ≤ i ≤ m and all t /∈ Tε for a set Tε ⊆ ΓL0,Q of size |Tε| ≤ ε |ΓL0,Q|. +□ +4. Product of two simplices in Rn +Although not strictly necessary, we discuss in this section the special case d = 2 of Theorem 1.2. This +already gives an improvement of the main results of [12], but more importantly serves as a gentle preparation +for the more complicated general case, presented in the Section 5, which involve both a plethora of different +scales and the hypergraph bundle notation introduced in Section 2.2. +4.1. Proof of Theorem 1.2 with d = 2. +Let Q = Q1 × Q2 with Q1 ⊆ Rn1 and Q2 ⊆ Rn2 be cubes of equal side length l(Q) and ∆0 = ∆0 +1 × ∆0 +2 +with ∆0 +1 = {v11, . . . , v1n1} ⊆ Rn1 and ∆0 +2 = {v11, . . . , v2n2} ⊆ Rn2 two non-degenerate simplices. +In order to “count” configurations of the form ∆ = ∆1 × ∆2 ⊆ Rn1+n2 with ∆1 and ∆2 isometric copies +of λ∆0 +1 and λ∆0 +2 respectively for some 0 < λ ≪ l(Q) in a set S ⊆ Q we introduce the multi-linear expression +N 2 +λ∆0,Q({fkl}) := + +x11∈Q1 + +x21∈Q2 +ˆ +x12,...,x1n1 +ˆ +x22,...,x2n2 +n1 +� +k=1 +n2 +� +l=1 +fkl(x1k, x2l) +dσλ∆0 +1(x12 − x11, . . . , x1n1 − x11) dσλ∆0 +2(x22 − x21, . . . , x2n2 − x21) dx21 dx11 +for any family of functions fkl : Q1 × Q2 → [−1, 1] with 1 ≤ k ≤ n1 and 1 ≤ l ≤ n2. +Indeed, if fkl = 1S for all 1 ≤ k ≤ n1 and 1 ≤ l ≤ n2 then the above expression is 0 unless there exists a +configuration ∆ ⊆ S of the form ∆1 × ∆2 with ∆1 and ∆2 isometric copies of λ∆0 +1 and λ∆0 +2 respectively. +The short argument presented in Section 1.1 demonstrating how both Theorem B and Corollary B follow +from Proposition B, and hence from Proposition 3.1, applies equally well to each of our main theorems. This +reduces our main theorems to analogous quantitative results involving an arbitrary lacunary sequence of +scales. In the case d = 2 of Theorem 1.2 this stronger quantitative result takes the following form: +Proposition 4.1. For any 0 < ε ≪ 1 there exists an integer J = O(exp(Cε−13)) with the following property: +Given any lacunary sequence l(Q) ≥ λ1 ≥ · · · ≥ λJ and S ⊆ Q, there is some 1 ≤ j < J such that +(4.1) +N 2 +λ∆0,Q({1S}) > +� |S| +|Q| +�n1n2 +− ε +for all λ ∈ [λj+1, λj]. +Our approach to establishing Proposition 4.1 is again to compare the above expressions to simpler ones for +which it is easy to obtain lower bounds. For any 0 < λ ≪ l(Q) and family of functions fkl : Q1×Q2 → [−1, 1] +with 1 ≤ k ≤ n1 and 1 ≤ l ≤ n2 we consider +M2 +λ,Q({fkl}) := + +t∈Q + +x1∈(t1+Q1(λ))n1 + +x2∈(t2+Q2(λ))n2 +n1 +� +k=1 +n2 +� +l=2 +fkl(x1k, x2l) dx2 dx1 dt +where t = (t1, t2) ∈ Q1 × Q2, xi = (xi1, . . . , xini) and Qi(λ) = [− λ +2 , λ +2 ]ni for i = 1, 2. +Note that if S ⊆ Q is a set of measure |S| ≥ δ|Q| for some δ > 0, then careful applications of H¨older’s +inequality give +M2 +λ,Q({1S}) ≥ + +t∈Q +� +(x1,x2)∈t+Q(λ) +1S(x1, x2) dx1dx2 +�n1n2 +dt ≥ δn1n2 − O(ε) +for all scales 0 < λ ≪ ε l(Q). + +16 +NEIL LYALL +´AKOS MAGYAR +In light of the observation above, and the discussion preceding Proposition 3.2, we see that Proposition +4.1, and hence Theorem 1.2 when d = 2, will follows as a consequence of the following +Proposition 4.2. Let 0 < ε ≪ 1. There exists an integer J2 = O(exp(Cε−12)) such that for any ε-admissible +sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJ2 and S ⊆ Q there is some 1 ≤ j < J2 such that +(4.2) +N 2 +λ∆0,Q({1S}) = M2 +λ,Q({1S}) + O(ε) +for all λ ∈ [Lj+1, Lj]. +There are again two main ingredients in the proof of Proposition 4.2. The first establishes that the our +multi-linear forms N 2 +λ∆0,Q({fkl}) are controlled by an appropriate box-type norm attached to a scale L. +Let Q = Q1 × Q2 be a cube. For any scale 0 < L ≪ l(Q) and function f : Q → R we define its local box +norm at scale L to be +(4.3) +∥f∥4 +□L(Q1×Q2) := + +t∈Q +∥f∥4 +□(t+Q(L)) dt +where Q(L) = [− L +2 , L +2 ]n1+n2 and +(4.4) +∥f∥4 +□( � +Q) := + +x11,x12∈ � +Q1 + +x21,x22∈ � +Q2 +f(x11, x21)f(x12, x21)f(x11, x22)f(x12, x22) dx11 . . . dx22 +for any cube �Q ⊆ Q of the form �Q = �Q1 × �Q2 with �Qj ⊆ Qj for j = 1, 2. +Lemma 4.1 (A Generalized von-Neumann inequality [12]). Let ε > 0, 0 < λ ≪ l(Q), and 0 < L ≪ ε24λ. +For any collections of functions fkl : Q1 × Q2 → [−1, 1] with 1 ≤ k ≤ n1 and 1 ≤ l ≤ n2 we have both +(4.5) +|N 2 +λ∆0,Q({fkl})| ≤ +min +1≤k≤n1, 1≤l≤n2 ∥fkl∥□L(Q1×Q2) + O(ε) +(4.6) +|M2 +λ,Q({fkl})| ≤ +min +1≤k≤n1, 1≤l≤n2 ∥fkl∥□L(Q1×Q2). +The result above was essentially proved in [12] for the multi-linear forms N 2 +λ∆0,Q when Q = [0, 1]n1+n2, +however a simple scaling argument transfers the result to an arbitrary cube Q. For completeness we include +its short proof in Section 4.2 below. +The second and main ingredient is an analogue of a weak form of Szemer´edi’s regularity lemma due to +Frieze and Kannan [7]. The more probabilistic formulation, we will use below, can be found for example in +[21], [22], and [23], and is also sometimes referred to as a Koopman-von Neumann type decomposition. +For any cube Q ⊆ Rn and scale L > 0 that divides l(Q) we will let Q(L) = [− L +2 , L +2 ]n and GL,Q denote +the collection of cubes Qt(L) = t + Q(L) partitioning the cube Q and let ΓL,Q denote grid corresponding to +the centers of these cubes. We will say that a finite σ-algebra B on Q is of scale L if it contains GL,Q and +for simplicity of notation will write Bt for B|Qt(L). +Recall that if we have two σ-algebras B1 on a cube Q1 and B2 on Q2 then by B1 ∨ B2 we mean the +σ-algebra on Q = Q1 × Q2 generated by the sets B1 × B2 with B1 ∈ B1 and B2 ∈ B2. Recall also that we +say the complexity of a σ-algebra B is at most m, and write complex(B) ≤ m, if it is generated by m sets. +Lemma 4.2 (Weak regularity lemma in Rn). +Let 0 < ε ≪ 1 and Q = Q1 × Q2 with Q1 ⊆ Rn1 and Q2 ⊆ Rn2 be cubes of equal side length l(Q). +There exists an integer ¯J2 = O(ε−12) such that for any ε4-admissible sequence l(Q) ≥ L1 ≥ · · · ≥ L ¯ +J2 and +function f : Q → [−1, 1] there is some 1 ≤ j ≤ ¯J2 and a σ-algebra B of scale Lj on Q such that +(4.7) +∥f − E(f|B)∥□Lj+1(Q1×Q2) ≤ ε +which has the additional local structure that for each t = (t1, t2) ∈ ΓLj,Q there exist σ-algebras B1,t on Qt1(Lj) +and B2,t on Qt2(Lj) with complex(Bi,t) = O(j) for i = 1, 2 such that Bt = B1,t ∨ B2,t. +Comparing the above statement to Lemma 2.2 for d = 2, i.e to the weak regularity lemma, note that the +σ-algebra B of scale Lj has a direct product structure only locally, inside each cube Qt(Lj). Moreover this +product structure varies with t ∈ ΓLj,Q, however the “local complexity” remains uniformly bounded. + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +17 +Assuming for now the validity of Lemmas 4.1 and 4.2 we prove Proposition 4.2. We will make crucial use +of Proposition 3.3, namely our parametric counting lemma on Rn for simplices. +Proof of Proposition 4.2. Let 0 < ε ≪ 1, ε1 := exp(−C1ε−12) for some C1 ≫ 1, and {Lj}j≥1 be an ε1- +admissible sequence of scales. Set R = ε ε−1 +1 +and J1(ε1, R) be the parameter appearing in Proposition 3.3, +noting that J1(ε1, R) = O(ε−5 +1 ). +For L ∈ {Lj}j≥1 write index(L) = j if L = Lj. We now choose a subsequence {L′ +j} ⊆ {Lj} so that +L′ +1 = L1 and index(L′ +j+1) ≥ index(L′ +j) + J1(ε1, R) + 2. Applying Lemma 4.2, with fkl = f := 1S for all +1 ≤ k ≤ n1 and 1 ≤ l ≤ n2, guarantees the existence of a σ-algebra B of scale L′ +j on Q such that +(4.8) +∥f − E(f|B)∥□L′ +j+1(Q1×Q2) ≤ ε +for some 1 ≤ j ≤ Cε−12. +Moreover, we know that B has the additional local structure that for each +t = (t1, t2) ∈ ΓL′ +j,Q there exist σ-algebras B1,t on Qt1(L′ +j) and B2,t on Qt2(L′ +j) with complex(Bi,t) = O(ε−12) +for i = 1, 2 such that Bt = B1,t ∨ B2,t. Thus, if we let R1,t and R2,t denote the number of atoms in B1,t +and B2,t respectively, then we can assume, by formally adding the empty set to these collections of atoms if +necessary, that R1,t = R2,t = R′ := exp(Cε−12) for all t ∈ ΓL′ +j,Q. +If we let ¯f := E(f|B1 ∨ B2), then by Lemma 4.1 and multi-linearity we have +(4.9) +N 2 +λ∆0,Q({f}) = N 2 +λ∆0,Q({ ¯f}) + O(ε) +and +M2 +λ,Q({f}) = M2 +λ,Q({ ¯f}) + O(ε) +provided for ε−24L′ +j+1 ≪ λ. For a given t ∈ ΓQ,L′ +j write ¯ft for the restriction of ¯f to the cube Qt(L′ +j). By +localization, one then has +(4.10) +N 2 +λ∆0,Q({ ¯f}) = Et∈ΓL′ +j ,Q N 2 +λ∆0,Qt(L′ +j)({ ¯ft}) + O(ε), +and +(4.11) +M2 +λ,Q({ ¯f}) = Et∈ΓL′ +j,Q M2 +λ,Qt(L′ +j)({ ¯ft}) + O(ε) +provided one also insists that λ ≪ ε L′ +j. +For given t ∈ ΓL′ +j,Q, the functions ¯ft(x1, x2) are linear combinations of functions of the form 1Ar1 +1,t(x1)1Ar2 +2,t(x2), +where {Ar1 +1,t}1≤r1≤R′ and {Ar2 +2,t}1≤r2≤R′ are the collections of the atoms of the σ-algebras B1,t and B2,t defined +on the cubes Qt1(L′ +j) and Qt2(L′ +j). Thus for each t ∈ ΓL′ +j,Q one has +¯ft = +R′ +� +r1=1 +R′ +� +r2=1 +αr,t1Ar1 +1,t × 1Ar2 +2,t +where r = (r1, r2). Plugging these linear expansions into the multi-linear expressions in above one obtains +N 2 +λ∆0,Qt(L′ +j)({ ¯ft}) = +� +r={rkl}kl +αr,t N 2 +λ∆0,Qt(L′ +j)({1A +r1,kl +1,t +× 1A +r2,kl +2,t +}) +using the notations rkl = (r1,kl, r2,kl), αr,t = � +kl αrkl,t. Notice that the product +n1 +� +k=1 +n2 +� +l=1 +1A +r1,kl +1,t +(x1k)1A +r2,kl +2,t +(x2l) +is nonzero only if Ar1,kl +1,t += Ar1,k +1,t , that is if r1,kl = r1,k for all 1 ≤ l ≤ n2, as the atoms Ar +1,t are all disjoint. +Similarly, one has that r2,kl = r2,l for all 1 ≤ k ≤ n1. Thus, in fact +(4.12) +N 2 +λ∆0,Qt(L′ +j)({ ¯ft}) = +� +r={rkl}kl +αr,t N 2 +λ∆0,Qt(L′ +j)({1A +r1,k +1,t +× 1A +r2,l +2,t }) +and similarly +(4.13) +M2 +λ,Qt(L′ +j)({ ¯ft}) = +� +r={rkl}kl +αr,t M2 +λ,Qt(L′ +j)({1A +r1,k +1,t +× 1A +r2,l +2,t }). +Note, that indices r are running through the index set [1, R′]n1 × [1, R′]n2 of size at most R if C1 ≫ 1. + +18 +NEIL LYALL +´AKOS MAGYAR +The key observation is that +(4.14) +N 2 +λ∆0,Qt(L′ +j)(1A +r1,k +1,t +× 1A +r2,l +2,t ) = N 1 +λ∆0 +1,Qt1 (L′ +j)(1A +r1,1 +1,t , . . . , 1A +r1,n1 +1,t +) N 1 +λ∆0 +2,Qt2 (L′ +j)(1A +r2,1 +2,t , . . . , 1A +r2,n2 +2,t +) +and +(4.15) +M2 +λ,Qt(L′ +j)(1A +r1,k +1,t +× 1A +r2,l +2,t ) = M1 +λ,Qt1 (L′ +j)(1A +r1,1 +1,t , . . . , 1A +r1,n1 +1,t +) M1 +λ,Qt2 (L′ +j)(1A +r2,1 +2,t , . . . , 1A +r2,n2 +2,t +). +Let r = {(r1,k, r2,l)}kl and g1,r +k,t := 1A +r1,k +1,t , g2,r +l,t := 1A +r2,l +2,t . Writing j′ := index(L′ +j) and J′ := index(L′ +j+1), +one may apply Proposition 3.3 for the families of functions g1,r +k,t, g2,r +l,t , where 1 ≤ k ≤ n1, 1 ≤ l ≤ n2 and +r = (r1,k, r2,l)kl ∈ [1, R′]n1 × [1, R′]n2, with respect to the ε1-admissible sequence of scales +Lj′+1 ≥ Lj′+2 ≥ · · · ≥ LJ′−1. +This is possible as J′ − j′ = J1(ε1, R). Then there is a scale Lj with j′ ≤ j < J′ so that +(4.16) +N 1 +λ∆0 +1,Qt1 (L′ +j)(g1,r +1,t , . . . , g1,r +n1t) = M1 +λ,Qt1 (L′ +j)(g1,r +1,t , . . . , g1,r +n1,t) + O(ε1) +and +(4.17) +N 1 +λ∆0 +2,Qt2 (L′ +j)(g2,r +1,t , . . . , g2,r +n2,t) = M1 +λ,Qt2 (L′ +j)(g2,r +1,t , . . . , g2,r +n2,t) + O(ε1), +for all λ ∈ [Lj+1, Lj] uniformly in r = {(r1,k, r2,l)}kl and t /∈ Tε1 ⊆ ΓL′ +j,Q, for a set of size |Tε1| ≤ ε1|ΓL′ +j,Q|. +Then, by (4.14)-(4.15) and (4.12)-(4.13), we have +(4.18) +N 2 +λ∆0,Qt(L′ +j)({ ¯ft}) = M2 +λ,Qt(L′ +j) ({ ¯ft}) + O(ε) +for t /∈ Tε1, as |αr,t| ≤ 1 and Rε1 ≤ ε. Finally, since |Tε1| ≤ ε1|ΓL′ +j,Q|, by averaging in t ∈ ΓL′ +j,Q, one has +N 2 +λ∆0,Q({ ¯f}) = M2 +λ,Q ({ ¯f}) + O(ε) +using (4.10)-(4.11) and the Proposition follows by (4.9) with an index 1 ≤ j < J2 = O(ε−12ε−5 +1 ). +□ +4.2. Proof of Lemmas 4.1 and 4.2. +Proof of Lemma 4.1. First we note that if χL := L−n1[−L/2,L/2]n and ψL := χL ∗ χL, then +ψL(x2 − x1) = +ˆ +t +χL(x1 − t)χL(x2 − t) dt +and hence for any function f : Q → [−1, 1], with Q ⊆ Rn being a cube of side length l(Q), one has +∥f∥2 +U1 +L(Q) = + +x1∈Q +ˆ +x2 +f(x1)f(x2)ψL(x2 − x1) dx1dx2 + O(L/l(Q)). +Write x′ := (x21, . . . , x2n2) and let gk,x′(x) := �n2 +l=1 fkl(x, x2l). Then one may write +N 2 +λ∆0,Q({fkl}) = + +x21∈Q2 +ˆ +x22,...,x2n2 +N 1 +λ∆0 +1,Q1(g1,x′, . . . , gn1,x′) dσλ∆0 +2(x22 − x21, . . . , x2n2 − x21) dx21. +Using estimate (3.6), the above observation, and Cauchy-Schwarz one has +|N 2 +λ∆0,Q({fkl})|2 ≤ + +x11∈Q1 +ˆ +x12 +ψL(x12 − x11) N 1 +λ∆0 +2,Q2(h1,x11,x12, . . . , hn2,x11,x12) dx11dx12 + O(ε4) +provided 0 < λ ≪ l(Q) and 0 < L ≪ ε24λ where hl,x11,x12(x) = f1l(x11, x)f1l(x12, x) for 1 ≤ l ≤ n2. +Applying the same procedure again ultimately gives +|N 2 +λ∆0,Q({fkl})|4 ≤ ∥f11∥4 +□L(Q1×Q2) + O(ε4). +The same estimate can of course be given for any function fkl in place of f11. This establishes (4.5). +Estimate (4.6) is established similarly. +□ + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +19 +Proof of Lemma 4.2. For each t = (t1, t2) ∈ ΓL1,Q we will let B1,t(L1) := {∅, Qt1(L1)} and B2,t(L1) := +{∅, Qt2(L1)}, in other words the trivial σ-algebras on Qt1(L1) and Qt2(L1) respectively. If (4.8) holds with +B(L1) = GL1,Q, noting that Bt(L1) := B1,t(L1) ∨ B2,t(L1) in this case, then we are done. +We now assume that we have developed, for each t = (t1, t2) ∈ ΓLj,Q, σ-algebras B1,t(Lj) on Qt1(Lj) +and B2,t(Lj) on Qt2(Lj) with complex(Bi,t(Lj)) ≤ j for i = 1, 2. Let B(Lj) be the σ-algebra such that +Bt(Lj) = B1,t(Lj) ∨ B2,t(Lj) for all t ∈ ΓLj,Q and assume that (4.8) does not hold, namely that +∥g∥□Lj+1(Q) ≥ ε +where g := f − E(f|B(Lj)). By the definition of the local box norm this means that + +t∈Q +∥g∥4 +□(t+Q(Lj+1)) dt ≥ ε4 +and hence, as Lj+2 ≪ ε4Lj+1, it is easy to see that +Es∈ΓLj+2,Q ∥g∥4 +□(s+Q(Lj+2)) ≥ ε4/2. +This implies that there is a set S ⊆ ΓLj+2,Q of size |S| ≥ (ε4/4)|ΓLj+2,Q| such that for all s = (s1, s2) ∈ S, +one has that ∥g∥4 +□(Qs(Lj+2)) ≥ ε4/4. It therefore follows, as is well-known see for example [12] or [23], that +there exist sets B1,s ⊆ Qs1(Lj+2) and B2,s ⊆ Qs2(Lj+2) such that +(4.19) + +x1∈Qs1 (Lj+2) + +x2∈Qs2 (Lj+2) +g(x1, x2) 1B1,s(x1)1B2,s(x2) dx1 dx2 ≥ ε4/16. +For a given s ∈ ΓLj+2,Q there is a unique t = t(s) such that Qs(Lj+2) ⊆ Qt(Lj). Let B′ +1,s(Lj+2) := +B1,t(Lj)|Qs1 (Lj+2) and B′ +2,s(Lj+2) := B2,t(Lj)|Qs2 (Lj+2) noting that complex(B′ +i,s(Lj+2)) ≤ j for i = 1, 2, as +the complexity of a σ-algebra does not increase when restricted to a set. If, for i = 1, 2, we let Bi,s(Lj+2) +denote the σ-algebra generated by B′ +i,s(Lj+2) and the set Bi,s if s ∈ S and let Bi,s(Lj+2) := B′ +i,s(Lj+2) +otherwise, then clearly complex(Bi,s(Lj+2)) is at most j + 1. We now define B(Lj+2) to the the sigma +algebra of scale Lj+2 with the property that Bs(Lj+2) = B1,s(Lj+2) ∨ B2,s(Lj+2) for all s ∈ ΓLj+2,Q. +Using the inner product notation ⟨f, g⟩Q = +ffl +Q f(x)g(x) dx we can rewrite (4.19) as +⟨f − E(f|B(Lj)) , 1B1,s × 1B2,s ⟩Qs(Lj+2) ≥ ε4/16 +for all s ∈ S. Since the function 1B1,s × 1B2,s is measurable with respect to B(Lj+2) one clearly has +⟨f − E(f|B(Lj+2)) , 1B1,s × 1B2,s⟩Qs(Lj+2) = 0 +and hence +⟨E(f|B(Lj+2)) − E(f|B(Lj)) , 1B1,s × 1B2,s⟩Qt(Lj+2) ≥ ε4/16. +It then follows from Cauchy-Schwarz and orthogonality that +∥E(f|B(Lj+2))∥2 +L2(Qs(Lj+2)) − ∥E(f|B1(Lj))∥2 +L2(Qs(Lj+2)) ≥ ε8/256. +Since |S| ≥ (ε4/4)|ΓLj+2,Q| averaging over all s ∈ ΓLj+2,Q gives +∥E(f|B(Lj+2))∥2 +L2(Q) ≥ ∥E(f|B(Lj))∥2 +L2(Q) + ε12/210. +Trivially both sides are at most 1 thus the process must stop at a step j = O(ε−12) where (4.8) holds for a +σ-algebra of “local complexity” at most j. This proves the Lemma. +□ +5. Proof of Theorem 1.2: The general case. +After these preparations we will now consider the general case of Theorem 1.2. Let Q = Q1×· · ·×Qd ⊆ Rn +with Qi ⊆ Rni cubes of equal side length l(Q) and ∆0 = ∆0 +1 ×· · ·×∆0 +d with each ∆i ⊆ Rni a non-degenerate +simplex of ni points for 1 ≤ i ≤ d. +We will use a generalized version of the hypergraph terminology introduced in Section 2. In particular, +for a vertex set I = {1, 2, . . ., d} and set K = {il; 1 ≤ i ≤ d, 1 ≤ l ≤ ni} we will let π : K → I denote the + +20 +NEIL LYALL +´AKOS MAGYAR +projection defined by π(il) := i. As before we will let Hd,k := {e ⊆ I; |e| = k} denote the complete k-regular +hypergraph with vertex set I, and for the multi-index n = (n1, . . . , nd) define the hypergraph bundle +Hn +d,k := {e ⊆ K; |e| = |π(e)| = k} +noting that |π−1(i)| = ni for all i ∈ I. +In order to parameterize the vertices of direct products of simplices, i.e. sets of the form ∆ = ∆1×· · ·×∆d +with ∆i ⊆ Qi, we consider points x = (x1, . . . , xd) with xi = (xi1, . . . , xini) ∈ Qni +i +for each i ∈ I. Now for +any 1 ≤ k ≤ d and any edge e′ ∈ Hd,k we will write Qe′ := � +i∈e′ Qi, and for every x ∈ Qn1 +1 × · · · × Qnd +d +and e ∈ Hn +d,k we define xe := πe(x), where πe : Qn1 +1 +× · · · × Qnd +d +→ Qπ(e) is the natural projection map. +Writing ∆i = {xi1, . . . , xini} we have that ∆1 × · · · × ∆d = {xe : e ∈ Hn +d,d} since every edge xe is of the +form (x1l1, . . . , xdld). We can therefore identify points x with configurations of the form ∆1 × · · · × ∆d. +For any 0 < λ ≪ l(Q) the measures dσλ∆0 +i , introduced in Section 3.1, are supported on points (y2, . . . , yni) +for which the simplex ∆i = {0, y2, . . . , yni} is isometric to λ∆0 +i . For simplicity of notation we will write +ˆ +xi +f(xi) dσλ +i (xi) := + +xi1∈Qi +ˆ +xi2,...,xini +f(xi) dσλ∆0 +i (xi2 − xi1, . . . , xini − xi1) dxi1 +Note that the support of the measure dσλ +i is the set of points xi so that the simplex ∆i := {xi1, . . . , xini} +is isometric to λ∆0 +i and xi1 ∈ Qi, moreover the measure is normalized. Thus if S ⊆ Q is a set then the +density of configurations ∆ in S of the form ∆ = ∆1 × . . . × ∆d with each ∆i ⊆ Qi an isometric copy of λ∆0 +i +is given by the expression +(5.1) +N d +λ∆0,Q(1S ; e ∈ Hn +d,d) := +ˆ +x1 +· · · +ˆ +xd +� +e∈Hn +d,d +1S(xe) dσλ +1 (x1) . . . dσλ +d (xd). +The proof of Theorem 1.2 reduces to establishing the following stronger quantitative result. +Proposition 5.1. For any 0 < ε ≪ 1 there exists an integer Jd = Jd(ε) with the following property: +Given any lacunary sequence l(Q) ≥ λ1 ≥ · · · ≥ λJd and S ⊆ Q, there is some 1 ≤ j < Jd such that +(5.2) +N d +λ∆0,Q(1S ; e ∈ Hn +d,d) > +� |S| +|Q| +�n1··· nd +− ε +for all λ ∈ [λj+1, λj]. +Quantitative Remark. A careful analysis of our proof reveals that there is a choice of Jd(ε) which is less +than Wd(log(C∆ε−3)), where Wk(m) is again the tower-exponential function defined by W1(m) = exp(m) +and Wk+1(m) = exp(Wk(m)) for k ≥ 1. +For any 0 < λ ≪ l(Q) and set S ⊆ Q we define the expression: +(5.3) +Md +λ,Q(1S ; e ∈ Hn +d,d) := + +t∈Q +Md +t+Q(λ)(1S ; e ∈ Hn +d,d) dt +where Q(λ) = [− λ +2 , λ +2 ]n and +(5.4) +Md +� +Q(1S ; e ∈ Hn +d,d) := + +x1∈ � +Qn1 +1 +· · · + +xd∈ � +Q +nd +d +� +e∈Hn +d,d +1S(xe) dx1 . . . dxd +for any cube �Q ⊆ Q of the form �Q = �Q1 × · · · × �Qd with �Qi ⊆ Qi for 1 ≤ i ≤ d. Note that if S ⊆ Q is a set +of measure |S| ≥ δ|Q| for some δ > 0, then careful applications of H¨older’s inequality give +Md +λ,Q(1S ; e ∈ Hn +d,d) ≥ + +t∈Q +� +(x1,...,xd)∈t+Q(λ) +1S(x1, . . . , xd) dx1 . . . dxd +�n1··· nd +dt ≥ δn1··· nd − O(ε) +for all scales 0 < λ ≪ ε l(Q). +In light of the discussion above, and that preceding Proposition 3.2, we see that Proposition 5.1, and +hence Theorem 1.2 in general, will follows as a consequence of the following + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +21 +Proposition 5.2. Let 0 < ε ≪ 1. There exists an integer Jd = Jd(ε) such that for any ε-admissible sequence +of scales l(Q) ≥ L1 ≥ · · · ≥ LJd and S ⊆ Q there is some 1 ≤ j < Jd such that +(5.5) +N d +λ∆0,Q(1S ; e ∈ Hn +d,d) = Md +λ,Q(1S ; e ∈ Hn +d,d) + O(ε) +for all λ ∈ [Lj+1, Lj]. +The validity of Proposition 5.2 will follow immediately from the d = k case of Proposition 5.3 below. +5.1. Reduction of Proposition 5.2 to a more general “local” counting lemma. +For any given 1 ≤ k ≤ d and collection of functions fe : Qπ(e) → [−1, 1] with e ∈ Hn +d,k we define the +following multi-linear expressions +(5.6) +N d +λ∆0,Q(fe; e ∈ Hn +d,k) := +ˆ +x1 +· · · +ˆ +xd +� +e∈Hn +d,k +fe(xe) dσλ +1 (x1) . . . .dσλ +d (xd) +and +(5.7) +Md +λ,Q(fe ; e ∈ Hn +d,k) := + +t∈Q +Md +t+Q(λ)(fe ; e ∈ Hn +d,k) dt +where Q(λ) = [− λ +2 , λ +2 ]n and +(5.8) +Md +� +Q(fe ; e ∈ Hn +d,k) := + +x1∈ � +Qn1 +1 +· · · + +xd∈ � +Q +nd +d +� +e∈Hn +d,k +fe(xe) dx1 . . . dxd +for any cube �Q ⊆ Q of the form �Q = �Q1 × · · · × �Qd with �Qi ⊆ Qi for 1 ≤ i ≤ d. +Our strategy to proving Proposition 5.2 is the same as illustrated in the finite field settings, that is we +would like to compare averages Nλ∆0,Q(fe; e ∈ Hn +d,k) to those of Md +λ,Q(fe ; e ∈ Hn +d,k), at certain scales +λ ∈ [Lj+1, Lj], inductively for 1 ≤ k ≤ d. However in the Euclidean case, an extra complication emerges due +to the fact the (hypergraph) regularity lemma, the analogue of Lemma 2.2, does not produce σ-algebras Bf, +for f ∈ Hn +d,k−1, on the cubes Qf. In a similar manner to the case for d = 2 discussed in the previous section, +we will only obtain σ-algebras “local” on cubes Qtf(L0) at some scale L0 > 0. This will have the effect that +the functions fe will be replaced by a family of functions fe,t, where t runs through a grid ΓL0,Q. +To be more precise, let L > 0 be a scale dividing the side-length l(Q). For t ∈ ΓL,Q and e′ ∈ Hd,k we will +use te′ to denote the projection of t onto Qe′ and Qte′ (L) := te′ + Qe′(L) to denote the projection of the +cube Qt(L) centered at t onto Qe′. It is then easy to see that for any ε > 0 we have +(5.9) +N d +λ∆0,Q(fe; e ∈ Hn +d,k) = Et∈ΓL,Q N d +λ∆0,Qt(L)(fe,t ; e ∈ Hn +d,k) + O(ε) +and +(5.10) +Md +λ,Q(fe; e ∈ Hn +d,k) = Et∈ΓL,Q Md +λ,Qt(L)(fe,t ; e ∈ Hn +d,k) + O(ε) +provided 0 < λ ≪ εL where fe,t denotes the restriction of a function fe to the cube Qt(L). +At this point the proof of Proposition 5.2 reduces to showing that the expressions in (7.8) and (7.9) only +differ by O(ε) at some scales λ ∈ [Lj+1, Lj], given an ε-admissible sequence L0 ≥ L1 ≥ · · · ≥ LJ, for any +collection of bounded functions fe,t, e ∈ Hn +d,k, t ∈ ΓL0,Q. Indeed, our crucial result will be the following +Proposition 5.3 (Local Counting Lemma). Let 0 < ε ≪ 1 and M ≥ 1. There exists an integer Jk = +Jk(ε, M) such that for any ε-admissible sequence of scales L0 ≥ L1 ≥ · · · ≥ LJk with the property that L0 +divides l(Q), and collection of functions +f m +e,t : Qtπ(e)(L0) :→ [−1, 1] with e ∈ Hn +d,k, 1 ≤ m ≤ M and t ∈ ΓL0,Q +there exists 1 ≤ j < Jk and a set Tε ⊆ ΓL0,Q of size |Tε| ≤ ε|ΓL0,Q| such that +(5.11) +N d +λ∆0,Qt(L0)(f m +e,t; e ∈ Hn +d,k) = Mλ,Qt(L0)(f m +e,t; e ∈ Hn +d,k) + O(ε) +for all λ ∈ [Lj+1, Lj] and t /∈ Tε uniformly in e ∈ Hn +d,k and 1 ≤ m ≤ M. + +22 +NEIL LYALL +´AKOS MAGYAR +5.2. Proof of Proposition 5.3. +We will prove Proposition 5.3 by induction on 1 ≤ k ≤ d. For k = 1 this is basically Proposition 3.3. +Indeed, in this case for a given t = (t1, . . . , td) ∈ ΓL0,Q and edge e ∈ Hn +d,1 = {il : 1 ≤ i ≤ d, 1 ≤ l ≤ ni} +we have that f m +e,t(xe) = f m +il,t(xil) with xil ∈ Qti(L0) and hence both +N d +λ∆0,Qt(L0)(f m +e,t; e ∈ Hn +d,1) = +d +� +i=1 +N 1 +λ∆0 +i ,Qti (L0)(f m +i1,t, . . . , f m +ini,t) +Md +λ,Qt(L0)(f m +e,t; e ∈ Hn +d,1) = +d +� +i=1 +M1 +λ,Qti (L0)(f m +i1,t, . . . , f m +ini,t). +By Proposition 3.3 there exists an 1 ≤ j < J1 = O(Mε−4) and an exceptional set Tε ⊆ ΓL0,Q of size +|Tε| ≤ ε|ΓL0,Q|, such that uniformly for t /∈ Tε and for 1 ≤ i ≤ d, one has +N 1 +λ∆0 +i ,Qti(L0)(f m +i1,t, . . . , f m +ini,t) = M1 +λ,Qti (L0)(f m +i1,t, . . . , f m +ini,t) + O(ε) +hence +N d +λ∆0,Qt(L0)(f m +e,t; e ∈ Hn +d,1) = Md +λ,Qt(L0)(f m +e,t; e ∈ Hn +d,1) + O(ε) +as the all factors are trivially bounded by 1 in magnitude. This implies (5.11) for k = 1. +For the induction step we again need two main ingredients. The first establishes that the our multi-linear +forms N d +λ∆0,Q(fe; e ∈ Hn +d,k) are controlled by an appropriate box-type norm attached to a scale L. +Let Q = Q1 × · · · × Qd and 1 ≤ k ≤ d. For any scale 0 < L ≪ l(Q) and function f : Qe′ → [−1, 1] with +e′ ∈ Hd,k we define its local box norm at scale L by +(5.12) +∥f∥2k +□L(Qe′ ) := + +s∈Qe′ +∥f∥2k +□(s+Q(L)) ds +where +(5.13) +∥f∥2k +□( � +Q) := + +x11,x12∈ � +Q1 +· · · + +xk1,xk2∈ � +Qk +� +(ℓ1,...,ℓk)∈{1,2}k +f(x1ℓ1, . . . , xkℓk) dx11 dx12 . . . dxk1 dxk2 +for any cube �Q of the form �Q = �Q1 × · · · × �Qk. +Lemma 5.1 (Generalized von-Neumann inequality). Let ε > 0, 0 < λ ≪ l(Q) and 0 < L ≪ (ε2k)6λ. +For any 1 ≤ k ≤ d and collection of functions fe : Qπ(e) → [−1, 1] with e ∈ Hn +d,k we have both +(5.14) +|N d +λ∆0,Q(fe; e ∈ Hn +d,k)| ≤ min +e∈Hn +d,k +∥fe∥□L(Qπ(e)) + O(ε) +(5.15) +|Md +λ,Q(fe; e ∈ Hn +d,k)| ≤ min +e∈Hn +d,k +∥fe∥□L(Qπ(e)). +The crucial ingredient is the following analogue of the weak hypergraph regularity lemma. +Lemma 5.2 (Parametric weak hypergraph regularity lemma for Rn). Let 0 < ε ≪ 1, M ≥ 1, and 1 ≤ k ≤ d. +There exists ¯Jk = O(Mε−2k+3) such that for any ε2k-admissible sequence L0 ≥ L1 ≥ · · · ≥ L ¯ +Jk with the +property that L0 divides l(Q) and collection of functions +f m +e,t : Qtπ(e)(L0) → [−1, 1] with e ∈ Hn +d,k, 1 ≤ m ≤ M, and t ∈ ΓL0,Q +there is some 1 ≤ j < ¯Jk and σ-algebras Be′,t of scale Lj on Qte′ (L0) for each t ∈ ΓL0,Q and e′ ∈ Hd,k such +that +(5.16) +∥f m +e,t − E(f m +e,t|Bπ(e),t)∥□Lj+1(Qtπ(e)(L0)) ≤ ε +uniformly for all t /∈ Tε, e ∈ Hn +d,k, and 1 ≤ m ≤ M, where Tε ⊆ ΓL0,Q with |Tε| ≤ ε|ΓL0,Q|. + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +23 +Moreover, the σ-algebras Be′,t have the additional local structure that the exist σ-algebras Be′,f′,s on +Qsf′ (Lj) with complex(Be′,f′,s) = O(j) for each s ∈ ΓLj,Q, e′ ∈ Hd,k, and f′ ∈ ∂e′ such that if s ∈ Qt(L0), +then +(5.17) +Be′,t +�� +Qse′ (Lj) = +� +f′∈∂e′ +Be′,f′,s. +Lemma 5.2 is the parametric and simultaneous version of the extension of Lemma 3.7 to the product of +d simplices. The difference is that in the general case one has to deal with a parametric family of functions +f m +e,t as t is running through a grid ΓL0,Q. The essential new content of Lemma 5.2 is that one can develop +σ-algebras Be′,t on the cubes Qt(L0) with respect to the family of functions f m +e,t such that the local structure +described above and (5.16) hold simultaneously for almost all t ∈ ΓL0,Q. +Proof of Proposition 5.3. Assume the Proposition holds for k − 1. +Let ε > 0, ε1 := exp (−C1ε−2k+3) for some large constant C1 = C1(n, k, d) ≫ 1, and {Lj}j≥1 be an +ε1-admissible sequence of scales. Set F(ε) := Jk−1(ε1, M) with M = ε ε−1 +1 . +For L ∈ {Lj}j≥1 we again write index(L) = j if L = Lj. We now choose a subsequence {L′ +j} ⊆ {Lj} +so that L′ +0 = L0 and index(L′ +j+1) ≥ index(L′ +j) + F(ε) + 2. Lemma 5.2 then guarantees the existence of +σ-algebras Be′,t of scale L′ +j on Qte′ (L0) for each t ∈ ΓL0,Q and e′ ∈ Hd,k, with the local structure described +above, such that +(5.18) +∥f m +e,t − E(f m +e,t|Bπ(e),t)∥□L′ +j+1(Qtπ(e)(L0)) ≤ ε +uniformly for all t /∈ T ′ +ε, e ∈ Hn +d,k, and 1 ≤ m ≤ M, for some 1 ≤ j < ¯Jk(ε, M) = O(Mε−2k+3), where +T ′ +ε ⊆ ΓL0,Q with |T ′ +ε| ≤ ε|ΓL0,Q|. Let ¯f m +e,t := E(f m +e,t|Bπ(e),t) for t ∈ ΓL0,Q and e ∈ Hn +d,k. If t /∈ T ′ +ε, then by +(5.14), (5.15), and (5.16) we have both +(5.19) +N d +λ∆0,Qt(L0)(f m +e,t; e ∈ Hn +d,k) = N d +λ∆0,Qt(L0)( ¯f m +e,t; e ∈ Hn +d,k) + O(ε) +(5.20) +Md +λ,Qt(L0)(f m +e,t; e ∈ Hn +d,k) = Md +λ,Qt(L0)( ¯f m +e,t; e ∈ Hn +d,k) + O(ε). +provided (ε−2k)6L′ +j+1 ≪ λ. For given s ∈ ΓL′ +j,Qt(L0) one may write ¯f m +e,s for the restriction of ¯f m +e,t on the cube +Qs(L′ +j) ⊆ Qt(L0), as s uniquely determines t. By localization, provided λ ≪ εL′ +j, we then have both +(5.21) +N d +λ∆0,Qt(L0)( ¯f m +e,t; e ∈ Hn +d,k) = Es∈ΓL′ +j,Qt(L0)N d +λ∆0,Qs(L′ +j)( ¯f m +e,s; e ∈ Hn +d,k) + O(ε), +(5.22) +Md +λ,Qt(L0)( ¯f m +e,t; e ∈ Hn +d,k) = Es∈ΓL′ +j,Qt(L0)Md +λ,Qs(L′ +j)( ¯f m +e,s; e ∈ Hn +d,k) + O(ε). +For a fixed cube Qs(L′ +j) we have that +(5.23) +¯f m +e,s = +Re,s +� +re=1 +αs,re,m 1Are +π(e),s +where {Are +π(e),s}1≤r≤Re,s is the family of atoms of the σ-algebra Bπ(e),t restricted to the cube Qs(L′ +j). Note +that |αs,re| ≤ 1 and |Re,s| = O(exp (Cε−2k+3)). By adding the empty set to the collection of atoms one +may assume |Re,s| = R := exp (Cε−2k+3) for all e ∈ Hn +d,k and s ∈ ΓL′ +j,Q. Then, by multi-linearity, using the +notations r = (re)e∈Hn +d,k and αr,s = � +e αs,re, one has both +(5.24) +N d +λ∆0,Qs(L′ +j)( ¯f m +s,e; e ∈ Hn +d,k) = +� +r +αs,r,m N d +λ∆0,Qs(L′ +j)(1Are +π(e),s; e ∈ Hn +d,k) +(5.25) +Md +λ,Qs(L′ +j)( ¯f m +s,e; e ∈ Hn +d,k) = +� +r +αs,r,m Md +λ,Qs(L′ +j)(1Are +π(e),s; e ∈ Hn +d,k). +The key observation is that these expressions in the sum above are all at level k − 1 instead of k. To see +this let e = (i1l1, . . . , imlm, . . . , iklk) so e′ = π(e) = (i1, . . . , im, . . . , ik). If f′ = e′\{im} then recall that the + +24 +NEIL LYALL +´AKOS MAGYAR +edge pf′(e) = (i1l1, . . . , iklk) ∈ Hn +d,k−1 is obtained from e by removing the imlm-entry. Thus, for any atom +Ae′,s of Bs,e′(L′ +j) we have by (5.17), that +(5.26) +1Ae′,s(xe) = +� +f′∈∂e′ +1Ae′,f′,s,(xpf′ (e)) +where Ae′,f′,s is an atom of the σ-algebra Be′,f′,s. Thus +(5.27) +� +e∈Hn +d,k +1Are +π(e),s(xe) = +� +f∈Hn +d,k−1 +� +e∈Hn +d,k,f′∈∂π(e) +pf′ (e)=f +1Are +π(e),f′,s(xf) = +� +f∈Hn +d,k−1 +gr +f,s (xf). +It follows that +(5.28) +N d +λ∆0,Qs(L′ +j)(1Are +π(e),s; e ∈ Hn +d,k) = N d +λ∆0,Qs(L′ +j) (gr +f,s; f ∈ Hn +d,k−1) +and hence that +(5.29) +N d +λ∆0,Qs(L′ +j)( ¯f m +e,s; e ∈ Hn +d,k) = +� +r +αs,r,m N d +λ∆0,Qs(L′ +j) (gr +f,s; f ∈ Hn +d,k−1) +and similarly +(5.30) +Md +λ,Qs(L′ +j)( ¯f m +e,s; e ∈ Hn +d,k) = +� +r +αr,s,m Md +λ,Qs(L′ +j) (gr +f,s; f ∈ Hn +d,k−1). +Note that number of index vectors r = (re)e∈Hn +d,k is RD with D := |Hn +d,k| and hence RD ≤ M if C1 ≫ 1. +Writing j′ := index(L′ +j) and J′ := index(L′ +j+1) it then follows from our inductive hypothesis functions, +applied with respect to the ε1-admissible sequence of scales +Lj′+1 ≥ Lj′+2 ≥ · · · ≥ LJ′−1 +which is possible as J′ − j′ ≫ Jk−1(ε1, RD), that there is a scale Lj with j′ ≤ j < J′ so that +(5.31) +Nλ∆0,Qs(L′ +j) (gr +s,f; f ∈ Hn +d,k−1) = Mλ,Qs(L′ +j) (gr +s,f; f ∈ Hn +d,k−1) + O(ε1) +for all λ ∈ [Lj+1, Lj] uniformly in r for s /∈ Sε1, where Sε1 ⊆ ΓL′ +j,Q is a set of size |Sε1| ≤ ε1|ΓL′ +j,Q|. +Since the cubes Qt(L0) form a partition of Q as t runs through the grid ΓL0,Q the relative density of the +set Sε1 can substantially increase only of a few cubes Qt(L0). Indeed, it is easy to see that |T ′′ +ε1| ≤ ε1/2 +1 +|ΓL0,Q| +for the set +T ′′ +ε1 := {t ∈ ΓL0,Q : |Sε1 ∩ Qt(L0)| ≥ ε1/2 +1 +|ΓL′ +j,Q ∩ Qt(L0)|}. +We claim that (5.11) holds for λ ∈ [Lj+1, Lj] uniformly in t /∈ Tε := T ′ +ε ∪ T ′′ +ε1, e ∈ Hn +d,k, and 1 ≤ m ≤ M. +Indeed, from (7.17), (7.18), and (5.31) and the fact that |αs,r| ≤ 1, it follows +N d +λ∆0,Qs(L′ +j) ( ¯fe,s; e ∈ Hn +d,k) = Md +λ,Qs(L′ +j) ( ¯fe,s; e ∈ Hn +d,k) + O(ε) +for s /∈ Sε1 ∩ Qt(L0) since RDε1 ≪ ε. Finally, the fact that t /∈ T ′′ +ε1 together with localization, namely (5.21) +and (5.22), ensures that averaging over ΓL′ +j,Qt(L0) gives +N d +λ∆0,Qt(L0) ( ¯fe,t; e ∈ Hn +d,k) = Md +λ,Qt(L0) ( ¯fe,t; e ∈ Hn +d,k) + O(ε) + O(ε1/2 +1 +) +which in light of (5.19), (5.20), and the fact that ε1 ≪ ε2 complete the proof. +□ +5.3. Proof of Lemmas 5.1 and 5.2. +Proof of Lemma 5.1. The argument is similar to that of Lemma 2.1. Fix an edge, say e0 = (11, 12, . . ., 1k), +and partition the edges e ∈ Hn +d,k in to as follows. Let H0 be the set of those edges e for which 1 /∈ π(e), +and for l = 1, . . . , n1 let Hl denote the collection of edges of the form e = (1l, j2l2, . . . , jklk), in other words +e ∈ Hl if e = (1l, e′) for some edge e′ = (j2l2, . . . , jklk) ∈ Hn +d−1,k−1. Accordingly write +� +e∈Hn +d,k +fe(xe) = +� +e∈H0 +fe(xe) +n1 +� +l=1 +� +e′∈Hn +d−1,k−1 +f1l,e′(x1l, xe′). + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +25 +For x ∈ Q1 and x′ = (x2, . . . , xd) with xi ∈ Qni +i , define +(5.32) +gl(x, x′) := +� +e′∈Hn +d−1,k−1 +f1l,e′(x1l, xe′) +Then one may write +(5.33) +N d +λ∆0,Q(fe; e ∈ Hn +d,k) = + +x2 +. . . + +xd +� +e∈H0 +fe(xe) +� +x1 +n1 +� +l=1 +gl(x1l, x′) dσλ +1 (x1) +� +dσλ +d (xd) . . . dσλ +2 (x2). +For the inner integrals we have, using (3.6), the estimate +� +x1 +n1 +� +l=1 +gl(x1l, x′) dσλ +1 +�2 +≤ ∥g1∥2 +U1 +L(Q) + O(ε2k) = + +y11 +ˆ +y12 +g1(y11)g1(y12)ψ1 +L(y12 − y11) dy11 dy12 + O(ε2k). +provided 0 < L ≪ (ε2k)6λ, where as in the proof of Lemma 4.1 we use the notation +ψi +L(y2 − y1) = +ˆ +t +χi +L(y1 − t)χi +L(y2 − t) dt +with χi +L := L−ni1[−L/2,L/2]ni for 1 ≤ i ≤ k. By Cauchy-Schwarz we then have +���N d +λ∆0,Q(fe; e ∈ Hn +d,k +��� +2 +≤ +ˆ +y1 + +x2 +. . . + +xd +� +e′∈Hn +d−1,k−1 +f11,e′(x11, xe′)f11,e′(x12, xe′) dσλ +d . . . dσλ +2 dω1 +L(y1) + O(ε2k) +where dωi +L(yi) = |Qi|−1ψi +L(yi2 − yi1) dyi1 dyi2 with yi = (yi1, yi2) ∈ Q2 +i for 1 ≤ i ≤ k. +The expression we have obtained above is similar to the one in (5.6) except for the following changes. The +variable x1 ∈ Qn1 +1 +is replaced by y1 ∈ Q2 +1 and the measure dσλ +1 by dω1 +L. The functions f1l,e′ are replaced +by f11,e′, for 1 ≤ l ≤ n1, while the functions fe for all e ∈ Hn +d,k such that 1 /∈ π(e) are eliminated, that is +replaced by 1. Repeating the same procedure for i = 2, . . . , k replaces all variables xi with variables yi as +well as the measures dσλ +i with dωi +L. The procedure eliminates all functions fe when e is an edge such that +i /∈ π(e) for some 1 ≤ i ≤ k; for the remaining edges, when π(e) = (1, . . . , k), it replaces the functions fe +with fe0 = f11,21,...,1k. For k < i the variables xi and the measures dσλ +i are not changed, however integrating +in these variables will have no contribution as the measures are normalized. Thus one obtains the following +final estimate +(5.34) +���Nλ∆0,Q(fe; e ∈ Hn +d,k +��� +2k +≤ +1 +|Q1| +ˆ +y1 +. . . +1 +|Qk| +ˆ +yk +� +e∈H2 +k,k +fe0(ye) +k +� +i=1 +ψi +L(yi2 − yi1) dyi1 dyi2 + O(ε2k) +noting that these integrals are not normalized. Thus, one may write the expression in (5.34), using a change +of variables yi1 := yi1 − ti, yi2 := yi2 − ti, as +(5.35) +1 +|Q1| +ˆ +t1 + +y1∈t1+Q1 +. . . +1 +|Qk| +ˆ +tk + +yk∈tk+Qk +� +e∈H2 +k,k +fe0(ye) dy1 . . . dyk dt = ∥fe0∥2k +□L(Qπ(e0)) + O(ε2k) +where the last equality follows from the facts that the function fe0 is supported on the cube Qπ(e0) and hence +the integration in t is restricted to the cube Q + Q(L), giving rise an error of O(L/l(Q)). Estimate (5.14) +follows from (5.34) and (5.35) noting that the above procedure can be applied to any e ∈ Hn +d,k in place of +e0. Estimate (5.15) is established similarly. +□ +Proof of Lemma 5.2. For j = 0 we set Be′,t(L0) := {Qt(L0), ∅} and Be′,f′,s(L0) := {Qsf′ (L0), ∅} for e′ ∈ Hd,k, +f′ ∈ ∂e′, and t, s ∈ ΓL0,Q. We will develop σ-algebras Be′,t(Lj) of scale Lj such that (5.17) holds with +complex(Be′,f′,s(Lj)) ≤ j. +We define the total energy of a family of functions f m +e,t with respect to a family of σ-algebras Be′,t(Lj) as +(5.36) +E(f m +e,t|Be′,t(Lj)) := Et∈ΓL0,Q +M +� +m=1 +� +e∈Hn +d,k +∥E(f m +e,t|Bπ(e),t(Lj))∥2 +L2(Qtπ(e)(L0)). + +26 +NEIL LYALL +´AKOS MAGYAR +Since |f m +e,t| ≤ 1 for all e, m, and t it follows that the total energy is bounded by M · |Hn +d,k| = O(M). Our +strategy will be to show that if (5.16) does not hold then there exist a family of σ-algebras Be′,t(Lj+2) such +that the total energy of the family of functions f m +e,t is increased by at least ckε2k+3 with respect to this new +family of σ-algebras, and at the same time ensuring that (5.17) remains valid with complex(Be′,f′,s(Lj+2)) ≤ +j + 2. This iterative process must stop at some j = O(M ε−2k+3) proving the Lemma. +Assume that we have developed σ-algebras Be′,t(Lj) and Be′,f′,s(Lj) of scale Lj such that (5.17) holds +with complex(Be′,f′,s(Lj)) ≤ j. If (5.16) does not hold then |Tε| ≥ ε|ΓL0,Q| for the set +Tε := {t ∈ ΓL0,Q : ∥f m +e,t − E(f m +e,t|Bπ(e),t(Lj))∥□Lj+1 (Qtπ(e) (L0)) ≥ ε for some e ∈ Hn +d,k and 1 ≤ m ≤ M}. +Fix t ∈ Tε and let e ∈ Hn +d,k and 1 ≤ m ≤ M be such that +∥f m +e,t − E(f m +e,t|Bπ(e),t(Lj))∥□Lj+1(Qtπ(e)(L0)) ≥ ε +and write e′ := π(e). Consider the partition of the cube Qte′ (L0) into small cubes Qse′ (Lj+2) where se′ ∈ +ΓLj+2,Qe′ ∩Qte′ (L0). By the localization properties of the □Lj+1(Q)-norm, and the fact that Lj+2 ≪ ε2kLj+1 +we have that +∥f∥2k +□Lj+1(Qte′ (L0)) ≤ Ese′ ∈ΓLj+2,Qte′ (L0) ∥f∥2k +□(Qse′ (Lj+2)) + ε2k +2 +for any function f : Qte′(L0) → [−1, 1]. Thus there exists a set Sε,e,t ⊆ ΓLj+2,Qte′ (L0) of size +|Sε,e,t| ≥ ε2k +4 |ΓLj+2,Qte′ (L0)| +such that +(5.37) +∥f m +e,t − E(f m +e,t|Be′,t(Lj))∥2k +□(Qse′ (Lj+2) ≥ ε2k +4 +for all se′ ∈ Sε,e,t. +For a given cube Q and functions f, g : Q → R, define the normalized inner product of f and g as +⟨f, g⟩Q := + +Q +f(x)g(x) dx. +Then by the well-known property of the □-norm, see for example [23] or the proof of Lemma 2.2, it follows +from (5.37) that there exits sets +Bf′,se′ ,t ⊆ Qsf′ (Lj+2) +for f′ ∈ ∂e′ such that +(5.38) +� +f m +e,t − E(f m +e,t|Be′,t(Lj)) , +� +f′∈∂e′ +1Bf′,se′ ,t +� +Qse′ (Lj+2) ≥ ε2k +2k+2 . +If s ∈ ΓLj+2,Q then there is a unique t = t(s) ∈ ΓL0,Q such that s ∈ Qt(L0). If t ∈ Tε and se′ ∈ Sε,e,t +then we define the σ-algebras Bf′,e′,s(Lj+2) on Qsf′ (Lj+2) as follows. Write Bf′,e′,s = Bf′,se′ ,t where t = t(s) +and let Bf′,e′,s(Lj+2) be the σ-algebra generated by the set Bf′,e′,s and the σ-algebra Bf′,e′,s′(Lj) restricted +to Qsf′ (Lj+2) where s′ ∈ ΓLj,Q is the unique element so that s ∈ Qs′(Lj). Note that that the complexity +of the σ-algebra Bf′,e′,s(Lj+2) is at most one larger then the complexity of the σ-algebra Bf′,e′,s′(Lj) as +restricting a σ-algebra to a set does not increase its complexity. If t = t(s) /∈ Tε or se′ /∈ Sε,e,t then let +Bf′,e′,s(Lj+2) be simply the restriction of Bf′,e′,s′(Lj) to the cube Qsf′ (Lj+2), or equivalently define the sets +Bf′,e′,s := Qsf′ (Lj+2). Finally, let +(5.39) +Be′,s(Lj+2) := +� +f′∈∂e′ +Bf′,e′,s(Lj+2) +be the corresponding σ-algebra on the cube Qse′ (Lj+2). +Since the cubes Qse′ (Lj+2) partition the cube Qte′ (L0) as se′ runs through the grid ΓLj+2,Qe′ ∩ Qte′ (L0), +these σ-algebras define a σ-algebra Be′,t(Lj+2) on Qte′(L0), such that its restriction to the cubes Qse′ (Lj+2) +is equal to the σ-algebras Be′,s(Lj+2). + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +27 +Since the function � +f′∈∂e′ 1Bf′,e′,s is measurable with respect to the σ-algebra Be′,t(Lj+2) restricted to +the cube Qse′ (Lj+2) one clearly has +(5.40) +⟨ f m +e,t − E(f m +e,t|Be′,t(Lj+2)), +� +f′∈∂e′ +1Bf′,e′,s ⟩Qse′ (Lj+2) = 0. +and hence, by (5.38), that +(5.41) +⟨ E(f m +e,t|Be′,t(Lj+2)) − E(f m +e,t|Be′,t(Lj)), +� +f′∈∂e′ +1Bf′,e′,s ⟩Qse′ (Lj+2) ≥ ε2k +2k+2 . +It then follows from Cauchy-Schwarz and orthogonality, using the fact that the σ-algebra Be′,t(Lj+2)) is +a refinement of Be′,t(Lj+2), that +∥E(f m +e,t|Be′,t(Lj+2))−E(f m +e,t|Be′,t(Lj))∥2 +L2(Qse′ (Lj+2)) +(5.42) += ∥E(f m +e,t|Be′,t(Lj+2))∥2 +L2(Qse′ (Lj+2)) − ∥E(f m +e,t|Be′,t(Lj))∥2 +L2(Qse′ (Lj+2)) +≥ +� ε2k +2k+2 +�2 +for se′ ∈ Sε,e,t. Since |Sε,e,t| ≥ ε2k +4 |ΓLj+2,Qte′ (L0)| averaging over se′ ∈ ΓLj+2,Qte′ (L0) implies +(5.43) +∥E(f m +e,t|Be′,t(Lj+2))∥2 +L2(Qte′ (L0)) ≥ ∥E(f m +e,t|Be′,t(Lj))∥2 +L2(Qte′ (L0)) + ε2k+2 +22k+6 . +At this point we have shown that if t ∈ Tε then there exists an edge e ∈ Hn +d,k, 1 ≤ m ≤ M, and σ-algebras +Be′,t(Lj+2)) of scale Lj+2 on Qte′ (L0), with e′ = π(e), such that (5.43) holds. +For all e′′ ∈ Hd,k with e′′ ̸= e′ let Bf′,e′′,s(Lj+2) be the restriction of the σ-algebra Bf′,e′′,s′(Lj) to the +cube Qsf′ (Lj+2), where s′ is such that s ∈ Qs′(Lj). By (5.39) this implies that Be′′,s(Lj+2) is also the +restriction of Be′′,s′(Lj) to the cube Qse′′ (Lj+2), and hence the σ-algebra Be′′,t(Lj+2) is generated by the +grid GLj+2,Qte′′ (L0) and the σ-algebra Be′′,t(Lj). +We have therefore defined a family of the σ-algebras Be′,t(Lj+2) for e′ ∈ Hd,k, satisfying +M +� +m=1 +� +e∈Hn +d,k +∥E(f m +e,t|Bπ(e),t(Lj+2))∥2 +L2(Qtπ(e) (L0)) ≥ +M +� +m=1 +� +e′∈Hn +d,k +∥E(f m +e,t|Bπ(e),t(Lj))∥2 +L2(Qtπ(e)(L0)) + ε2k+2 +22k+6 . +Using the fact that |Tε| ≥ ε|ΓL0,Q| and averaging over t ∈ ΓL0,Q it follows using the notations of (5.36) that +E(f m +e,t|Be′,t(Lj+2)) ≥ E(f m +e,t|Be′,t(Lj)) + ε2k+3 +22k+6 . +As the total energy E(f m +e,t|Be′,t(Lj)) is bounded by O(M), the process must stop at a step j = O(M ε−2k+3) +where (5.16) holds for a σ-algebra of “local complexity” at most j, completing the proof of Lemma 5.2. +□ +6. The base case of an inductive strategy to establish Theorem 1.4 +In this section we will ultimately establish the base case of our more general inductive argument. We will +however start by giving a (new) proof of Theorem B′, namely the case d = 1 of Theorem 1.4. +6.1. A Single Simplex in Zn. Let ∆0 = {v1 = 0, v2, . . . , vn1} be a fixed non-degenerate simplex of n1 +points in Zn with n = 2n1 + 3 and define tkl := vk · vl for 2 ≤ k, l ≤ n1. Recall, see [17], that a simplex +∆ = {m1 = 0, . . . , mn1} ⊆ Zn is isometric to λ∆0 if and only if mk · ml = λ2tkl for all 2 ≤ k, l ≤ n1. +For any positive integer q and λ ∈ q +√ +N we define Sλ∆0,q(m2, . . . , mn1) : Zn(n1−1) → {0, 1} be the function +whose value is 1 if mk · ml = λ2tkl with both mk and ml in (qZ)n for all 2 ≤ k, l ≤ n1 and is equal to 0 +otherwise. It is a well-known fact in number theory, see [11] or [17], that for n ≥ 2n1 + 1 we have that +� +m2,...,mn1 +Sλ∆0,q(m2, . . . , mn1) = ρ(∆0) (λ/q)(n−n1)(n1−1)(1 + O(λ−τ)) + +28 +NEIL LYALL +´AKOS MAGYAR +for some absolute constant τ > 0 and some constant ρ(∆0) > 0, the so-called singular series, which can +be interpreted as the product of the densities of the solutions of the above system of equations among the +p-adics and among the reals. Thus if we define +σλ∆0,q := ρ(∆0)−1(λ/q)−(n−n1)(n1−1)Sλ∆0,q +then σλ∆0,q is normalized in so much that +� +m2,...,mn1 +σλ∆0,q(m2, . . . , mn1) = 1 + O(λ−τ) +for some absolute constant τ > 0. +Let Q ⊆ Zn be a fixed cube and let l(Q) denotes its side length. For any family of functions +f1, . . . , fn1 : Q → [−1, 1] +and 0 < λ ≪ l(Q) we define the following two multi-linear expressions +(6.1) +N 1 +λ∆0,q,Q(f1, . . . , fn1) := Em1∈Q +� +m2,...,mn1 +f1(m1) . . . fn1(mn1) σλ∆0,q(m2 − m1, . . . , mn1 − m1) +and +(6.2) +M1 +λ,q,Q(f1, . . . , fn1) := Et∈Q Em1,...,mn1∈t+Q(q,λ) f1(m1) . . . fn1(mn1) +where Q(q, λ) := [− λ +2 , λ +2 ]n ∩ (qZ)n. Note that if S ⊆ Q and N 1 +λ∆0,q,Q(1S, . . . , 1S) > 0 then S must contain +an isometric copy of λ∆0, while if |S| ≥ δ|Q| for some δ > 0 then as before H¨older implies that +(6.3) +M1 +λ,q,Q(1S, . . . , 1S) ≥ δn − O(ε) +for all scales λ ∈ q +√ +N with 0 < λ ≪ ε l(Q). +Recall that for any 0 < ε ≪ 1 and positive integer q we call a sequence L1 ≥ · · · ≥ LJ (ε, q)-admissible +if Lj/Lj+1 ∈ N and Lj+1 ≪ ε2Lj for all 1 ≤ j < J and LJ/q ∈ N. Note that if λ1 ≥ · · · ≥ λJ′ ≥ 1 is any +lacunary sequence in q +√ +N with J′ ≫ (log ε−1) J + log q, one can always finds an (ε, q)-admissible sequence +of scales L1 ≥ · · · ≥ LJ with the property that for each 1 ≤ j < J the interval [Lj+1, Lj] contains at least +two consecutive elements from the original lacunary sequence. +In light of these observations we see that the following “counting lemma” ultimately establishes a quanti- +tatively stronger version of Proposition B′ that appeared in Section 1.3 and hence immediately establishes +Theorem 1.4 for d = 1. +Proposition 6.1. Let 0 < ε ≪ 1 and qj := q1(ε)j for j ≥ 1 with q1(ε) := lcm{1 ≤ q ≤ Cε−10}. +There exists J1 = O(ε−2) such that for any (ε, qJ1)-admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJ1 +and S ⊆ Q there is some 1 ≤ j < J1 such that +(6.4) +N 1 +λ∆0,qj,Q(1S, . . . , 1S) = M1 +λ,qj,Q(1S, . . . , 1S) + O(ε) +for all λ ∈ qj +√ +N with Lj+1 ≤ λ ≤ Lj. +As in the continuous setting the proof of Proposition 6.1 has two main ingredients, namely Lemmas 6.1 +and 6.2 below. In these lemmas, and for the remainder Sections 6 and 7, we will continue to use the notation +q1(ε) := lcm{1 ≤ q ≤ Cε−10} +for any given ε > 0. +Lemma 6.1 (A Generalized von Neumann inequality). +Let 0 < ε ≪ 1, q, q′ ∈ N with qq1(ε)|q′, and λ ∈ q +√ +N with λ ≪ l(Q) and 1 ≪ L ≪ ε10λ. For any +collection of functions f1, . . . , fn1 : Q → [−1, 1] we have +(6.5) +|N 1 +λ∆0,q,Q(f1, . . . , fn1)| ≤ +min +1≤i≤n1 ∥fi∥U1 +q′,L(Q) + O(ε) +where for any function f : Q → [−1, 1] we define +(6.6) +∥f∥U1 +q,L(Q) := +� 1 +|Q| +� +t∈Q +|f ∗ χq,L(t)|2�1/2 + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +29 +with χq,L denoting the normalized characteristic function of the cubes Q(q, L) := [− L +2 , L +2 ]n ∩ (qZ)n. +For any cube Q ⊆ Zn of side length l(Q) and q, L ∈ N satisfying q ≪ L with L dividing l(Q), we shall +now partition Q into cubic grids Qt(q, L) = t + ((qZ)n ∩ Q(L)), with Q(L) = [− L +2 , L +2 ]n as usual. These grids +form the atoms of a σ-algebra Gq,L,Q. Note that if q|q′ and L′|L then Gq,L,Q ⊆ Gq′,L′,Q. +Lemma 6.2 (A Koopman-von Neumann type decomposition). +Let 0 < ε ≪ 1 and qj := q1(ε)j for all j ≥ 1. There exists an integer ¯J1 = O(ε−2) such that any (ε, q ¯ +J1)- +admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ L ¯ +J1 and function f : Q → [−1, 1] there is some 1 ≤ j < ¯J1 +such that +(6.7) +∥f − E(f|Gqj,Lj,Q)∥U1 +qj+1,Lj+1(Q) ≤ ε. +The reduction of Proposition 6.1 to these two lemmas is essentially identical to the analogous argument +in the continuous setting as presented at the end of Section 3.1, we choose to omit the details. +Proof of Lemma 6.1. We will rely on some prior exponential sum estimates, specifically Propositions 4.2 and +4.4 in [17]. First we deal with the case n1 ≥ 3. By the change of variables m1 := m1, mi := mi − m1 for +2 ≤ i ≤ n1, one may write +N 1 +λ∆0,q,Q(f1, . . . , fn1) := Em1∈QN +� +m2,...,mn1 +f1(m1)f2(m1 + m2) · · · fn1(m1 + mn1) σλ∆0,q(m2, . . . , mn1). +We now write +σλ∆0,q(m2, . . . , mn1) = σλ∆0′,q(m2, . . . , mn1−1) σ +m2,...,mn1−1 +λ,q +(mn1) +where ∆0′ = {v1 = 0, v2, . . . , vn1−1} and for each m2, . . . , mn1−1 ∈ (qZ)n we are using σ +m2,...,mn1−1 +λ,q +(m) +denote the (essentially) normalized indicator function of the subset of (qZ)n that contains m if and only if +m · mk = λ2tkn1 for all 2 ≤ k ≤ n1. +Using the fact that |fi| ≤ 1, together with Cauchy-Schwarz and Plancherel, one can then easily see that +(6.8) +|N 1 +λ∆0,q,Q(f1, . . . , fn1)|2 ≤ |Q|−1 +ˆ +ξ∈Tn | �fn1(ξ)|2Hλ,q(ξ) dξ +with +Hλ,q(ξ) = +� +m2,...,mn1 +σλ∆0′,q(m2, . . . , mn1−1) | +� +σ +m2,...,mn1−1 +λ,q +(ξ)|2. +It then follows by Propositions 4.2 and 4.4 in [17], with δ = ε4 and after rescaling by q, that in addition +to being non-negative and uniformly bounded in ξ we in fact have +(6.9) +Hλ,q(ξ) = O(ε) +whenever +����qξ − +l +q1(ε) +���� ≥ +q +ε4λ, +for all l ∈ Zn. +We note that the expression Hλ,q(ξ) may be interpreted as the Fourier transform of the indicator function +of the set of integer points on a certain variety, and estimate (6.9) indicates that this concentrates near +rational points of small denominator. It is this crucial fact from number theory which makes results like +Theorem B′ possible. +Since +�χq,L(ξ) = qn +Ln +� +m∈[− L +2 , L +2 )n, q|m +e−2πim·ξ +it is easy to see that �χq,L(l/q) = 1 for all l ∈ Zn and that there exists some absolute constant C > 0 such +that +(6.10) +0 ≤ 1 − �χq,L(ξ)2 ≤ C L |ξ − l/q| +for all ξ ∈ Tn and l ∈ Zn. It is then easy to see using our assumption that qq1(ε)|q′ that +(6.11) +0 ≤ Hλ,q(ξ)(1 − �χq′,L(ξ)2) ≤ Cε + +30 +NEIL LYALL +´AKOS MAGYAR +for some constant C > 0 uniformly in ξ ∈ Tn provided L ≪ ε5λ. Substituting inequality (6.7) into (6.8), we +obtain +|N 1 +λ∆0,q,Q(f1, . . . , fn1)|2 ≤ |Q|−1 +�ˆ +| ˆfn1(ξ)|2Hλ(ξ)�χq′,L(ξ)2 dξ + +ˆ +| ˆfn1(ξ)|2Hλ(ξ)(1 − �χq′,L(ξ)2) dξ +� +≤ ∥fn1∥2 +U1 +q′,L(Q) + O(ε) +provided L ≪ ε5λ. This proves Lemma 6.1 for k ≥ 3, as it is clear that by re-indexing the above estimate +holds for any of the functions fi in place of fn1. For n1 = 2 an easy modification of arguments in [14], +specifically the proof of Lemma 3 therein, establishes that +|N 1 +λ∆0,q,Q(f1, f2)|2 ≤ ∥fi∥2 +U1 +q′,L(Q) + O(ε) +for i = 1, 2 provided L ≪ ε5λ. +□ +Proof of Lemma 6.2. Let q, L ∈ N such that L|N, q|L. The “modulo q” grids Qt(q, L) = t+Q(q, L) partition +the cube Q with t running through the set Γq,L,Q = {1, . . ., q}n + ΓL,Q, where ΓL,Q denote the centers of +the “integer” grids t + Q(L) in an initial partition of Q. Let q′, L′ be positive integers so that q|q′, L′|L and +L′ ≪ ε2L. If s ∈ Γq′,L′,Q and t ∈ Qs(q′, L′) then |t − s| = O(L′) and hence +Ex∈Qt(q,L)g(x) = Ex∈Qs(q,L)g(x) + O(L′/L) +for any function g : Q → [−1, 1]. Moreover, since the cube Qs(q, L) is partitioned into the smaller cubes +Qt(q′, L′), we have by Cauchy-Schwarz +|Ex∈Qs(q,L) g(x)|2 ≤ Et∈Γq′,L′,Qs(q,L)|Ex∈Qt(q′,L′)g(x)|2. +From this it is easy to see that +∥g∥2 +U1 +q,L(Q) = Et∈Q|Ex∈Qt(q,L)g(x)|2 ≤ Et∈Γq′,L′,Q |Ex∈Qt(q′,L′)g(x)|2 + O(L′/L) +and we note that the right side of the above expression is ∥E(g|Gq′,L′,Q)∥2 +L2(Q) since the conditional expecta- +tion function E(g|Gq′,L′,Q) is constant and equal to Ex∈Qt(q′,L′)g(x) on the cubes Qt(q′, L′). +Now suppose (6.7) does not hold for some j ≥ 1, that is +∥f − E(f|Gqj,Lj,Q)∥2 +U1 +qj+1,Lj+1 (Q) ≥ ε2. +Since Lj+2 ≪ ε2Lj+1, Lj+2|Lj, and qj+1|qj+2 we can apply the above observations to g := f − E(f|Gqj,Lj,Q) +and obtain, by orthogonality, that +(6.12) +∥E(f|Gqj+2,Lj+2,Q)∥2 +L2(Q) ≥ ∥E(f|Gqj,Lj,Q)∥2 +L2(Q) + cε2 +for some constant c > 0. Since the above expressions are clearly bounded by 1, the above procedure must +stop in O(ε−2) steps at which (6.7) must hold for some 1 ≤ j ≤ ¯J1(ε) with ¯J1(ε) = O(ε−2). +□ +6.2. The base case of our general inductive strategy. +Let Q = Q1 × . . . × Qd with Qi ⊆ Z2ni+3 be cubes of equal side length l(Q) and ∆0 +i ⊆ Z2ni+3 be a +non-degenerate simplex of ni points for 1 ≤ i ≤ d. +We note that for any q0 ∈ N and scale L0 dividing l(Q) if t = (t1, . . . , td) ∈ Γq0,L0,Q, then the corresponding +grids Qt(q0, L0) in the partition of Q take the form Qt(q0, L0) = Qt1(q0, L0) × · · · × Qtd(q0, L0). +As in the continuous setting we will ultimately need a parametric version of Proposition 6.1, namely +Proposition 6.2 below. +Proposition 6.2 (Parametric Counting Lemma on Zn for Simplices). Let 0 < ε ≤ 1 and R ≥ 1. +There exists an integer J1 = J1(ε, R) = O(R ε−4) such that for any (ε, qJ1)-admissible sequence of scales +L0 ≥ L1 ≥ · · · ≥ LJ1 with L0 dividing l(Q) and qj := q0q1(ε)j for 0 ≤ j ≤ J1 with q0 ∈ N, and collection of +functions +f i,r +k,t : Qti(q0, L0) → [−1, 1] with 1 ≤ i ≤ d, 1 ≤ k ≤ ni, 1 ≤ r ≤ R and t ∈ Γq0,L0,Q +there exists 1 ≤ j < J1 and a set Tε ⊆ Γq0,L0,Q of size |Tε| ≤ ε|Γq0,L0,Q| such that +(6.13) +N 1 +λ∆0 +i ,qj,Qti (q0,L0)(f i,r +1,t, . . . , f i,r +ni,t) = M1 +λ,qj,Qti (q0,L0)(f i,r +1,t, . . . , f i,r +ni,t) + O(ε) + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +31 +for all λ ∈ qj +√ +N with Lj+1 ≤ λ ≤ Lj and t /∈ Tε uniformly in 1 ≤ i ≤ d and 1 ≤ r ≤ R. +This proposition follows, as the analogous result did in the continuous setting, from Lemma 6.1 and the +follow parametric version of Lemma 6.2. +Lemma 6.3 (A simultaneous Koopman-von Neumann type decomposition). +Let 0 < ε ≪ 1, m ≥ 1, and Q ⊆ Zn be a cube. There exists an integer ¯J1 = O(mε−3) such that for any +(ε, q ¯ +J1)-admissible sequence L0 ≥ L1 ≥ · · · ≥ L ¯ +J1 with L0 dividing l(Q) and qj := q0q1(ε)j for 0 ≤ j ≤ ¯J1 +with q0 ∈ N, and collection of functions +f1,t, . . . fm,t : Qt(q0, L0) → [−1, 1] +defined for each t ∈ Γq0,L0,Q, there is some 1 ≤ j < ¯J1 and a set Tε ⊆ Γq0,L0,Q of size |Tε| ≤ ε|Γq0,L0,Q| such +that +(6.14) +∥fi,t − E(fi,t|Gqj,Lj,Qt(q0,L0)∥U1 +qj+1,Lj+1 (Qt(q0,L0)) ≤ ε +for all 1 ≤ i ≤ m and t /∈ Tε. +Lemma 6.3 above is of course the discrete analogue of Lemma 3.2. Since the proofs of Proposition 6.2 and +Lemma 6.3 are almost identical to the arguments presented in Section 3.2 we choose to omit these details. +7. Proof of Theorem 1.4: The general case +After the preparations in Section 6 we can proceed very similarly as in Section 5 to prove our main result +in the discrete case, namely Theorem 1.4. The main difference will be that given 0 < ε ≪ 1 and 1 ≤ k ≤ d, +we construct a positive integer qk(ε) and assume that all our sequences of scales will be (ε, qk(ε))-admissible. +The cubes Qt(L) will be naturally now be replaced by the grids Qt(q, L) of the form that already appear in +Section 6 where we always assume q|L. +Let ∆0 = ∆0 +1 × . . . × ∆0 +d with each ∆0 +i ⊆ Z2ni+3 a non-degenerate simplex of ni points for 1 ≤ i ≤ d +and Q = Q1 × . . . × Qd ⊆ Zn with Qi ⊆ Z2ni+3 cubes of equal side length l(Q) (taken much larger than +the diameter of ∆0). We will use the same parameterizations in terms of hypergraph bundles Hn +d,k and +corresponding notations as in Section 5 to count the configurations ∆ = ∆1 × . . . × ∆d ⊆ Q with each +∆i ⊆ Qi an isometric copy of λ∆0 +i for some λ ∈ +√ +N. +Given any positive integer q and λ ∈ q +√ +N we will make use of the notation +(7.1) +� +xi +f(xi) σi +λ,q(xi) := Exi1∈Qi +� +xi2,...,xini +f(xi) σλ∆0 +i ,q(xi2 − xi1, . . . , xini − xi1) dxi1 +with σλ∆0 +i ,q as defined in the previous section and xi = (xi1, . . . , xini) ∈ Qni +i . +Note that if S ⊆ Q then the density of configurations ∆ in S, of the form ∆ = ∆1 × . . . × ∆d with each +∆i ⊆ Qi an isometric copy of λ∆0 +i for some λ ∈ q +√ +N is given by the expression +(7.2) +N d +λ∆0,q,Q(1S ; e ∈ Hn +d,d) := +� +x1 +· · · +� +xd +� +e∈Hn +d,d +1S(xe) σ1 +λ,q(x1) . . . σd +λ,q(xd). +More generally, for any given 1 ≤ k ≤ d and a family of functions fe : Qπ(e) → [−1, 1] with e ∈ Hn +d,k we +define the multi-linear expression +(7.3) +N d +λ∆0,q,Q(fe; e ∈ Hn +d,k) := +� +x1 +· · · +� +xd +� +e∈Hn +d,k +fe(xe) σ1 +λ,q(x1) . . . .σd +λ,q(xd). +as well as +(7.4) +Md +λ,q,Q(fe; e ∈ Hn +d,k) := Et∈Q Md +t+Q(q,L) (fe; e ∈ Hn +d,k) +where Q(q, L) = Q1(q, L) × · · · × Qd(q, L) with each Qi(q, L) = (qZ ∩ [− L +2 , L +2 ])2ni+3 and +(7.5) +Md +� +Q(fe; e ∈ Hn +d,k) := Ex1∈ � +Qn1 +1 · · · Exd∈ � +Q +nd +d +� +e∈Hn +d,k +fe(xe) +for any cube �Q ⊆ Q of the form �Q = �Q1 × · · · × �Qd with �Qi ⊆ Qi for 1 ≤ i ≤ d. + +32 +NEIL LYALL +´AKOS MAGYAR +We note that it is easy to show, as in the continuous, that if S ⊆ Q with |S| ≥ δ|Q| for some δ > 0 then +(7.6) +Md +λ,q,Q(1S; e ∈ Hn +d,d) ≥ δn1··· nd − O(ε) +for all scales λ ∈ q +√ +N with 0 < λ ≪ ε l(Q). In light of this observation and the discussion preceding +Proposition 6.1 the proof of Theorem 1.4 reduces, as it did in the continuous setting, to the following +Proposition 7.1. Let 0 < ε ≪ 1. There exist positive integers Jd = Jd(ε) and qd(ε) such that for any +(ε, qd(ε)Jd)-admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJ1 and S ⊆ Q there is some 1 ≤ j < Jd such +that +(7.7) +N d +λ∆0,qj,Q(1S ; e ∈ Hn +d,d) = Md +λ,qj,Q(1S; e ∈ Hn +d,d) + O(ε), +for all λ ∈ qj +√ +N with Lj+1 ≤ λ ≤ Lj with qj := qd(ε)j. +Quantitative Remark. +A careful analysis of our proof reveals that there exist choices of Jd(ε) and +qd(ε) which are less than Wd(log(C∆ε−3)) and Wd(C∆ε−13) respectively where Wk(m) is again the tower- +exponential function defined by W1(m) = exp(m) and Wk+1(m) = exp(Wk(m)) for k ≥ 1. +The proof of Proposition 7.1 follows along the same lines as the analogous result in the continuous setting. +As before we will compare the averages N d +λ∆0,q,Q(fe; e ∈ Hn +d,k) to those of Md +λ,q,Q(fe; e ∈ Hn +d,k), at certain +scales q and λ ∈ q +√ +N with with Lj+1 ≤ λ ≤ Lj, inductively for 1 ≤ k ≤ d. As the arguments closely follow +those given in Section 5 we will be brief and emphasize mainly just the additional features. +7.1. Reduction of Proposition 7.1 to a more general “local” counting lemma. +For any given 1 ≤ k ≤ d and a family of functions fe : Qπ(e) → [−1, 1] with e ∈ Hn +d,k it is easy to see that +for any ε > 0, scale L0 > 0 dividing the side-length l(Q), and q0|q we have +(7.8) +N d +λ∆0,q,Q(fe; e ∈ Hn +d,k) = Et∈Γq0,L0,Q N d +λ∆0,q,Qt(q0,L0)(fe,t; e ∈ Hn +d,k) + O(ε) +and +(7.9) +Md +λ,q,Q(fe; e ∈ Hn +d,k) = Et∈ΓL,Q Md +λ,q,Qt(q0,L0)(fe,t; e ∈ Hn +d,k) + O(ε) +provided 0 < λ ≪ εL0 where fe,t denotes the restriction of a function fe to the cube Qt(q0, L0). +Thus the proof of Proposition 7.1 reduces to showing that the expressions in (7.8) and (7.9) only differ +by O(ε) for all scales λ ∈ q +√ +N with Lj+1 ≤ λ ≤ Lj, given an (ε, q)-admissible sequence L0 ≥ L1 ≥ · · · ≥ LJ, +for any collection of bounded functions fe,t, e ∈ Hn +d,k, t ∈ Γq0,L0,Q. Indeed, our crucial result will be the +following +Proposition 7.2 (Local Counting Lemma in Zn). Let 0 < ε ≪ 1 and q0, M ∈ N. +There exist positive integers Jk = Jk(ε, M) and qk(ε) such that for any (ε, qJd)-admissible sequence of +scales L0 ≥ L1 ≥ · · · ≥ LJ1 with L0 dividing l(Q) and qj := q0 qk(ε)j for j ≥ 1, and collection of functions +f m +e,t : Qtπ(e)(q0, L0) :→ [−1, 1] with e ∈ Hn +d,k, 1 ≤ m ≤ M and t ∈ Γq0,L0,Q +there exists 1 ≤ j < Jk and a set Tε ⊆ Γq0,L0,Q of size |Tε| ≤ ε|Γq0,L0,Q| such that +(7.10) +N d +λ∆0,qj,Qt(q0,L0)(fe,t; e ∈ Hn +d,k) = Md +λ,qj,Qt(q0,L0)(fe,t; e ∈ Hn +d,k) + O(ε) +for all λ ∈ qj +√ +N with Lj+1 ≤ λ ≤ Lj and t /∈ Tε uniformly in e ∈ Hn +d,k and 1 ≤ m ≤ M. +Note that if k = d, L0 = l(Q), q0 = M = 1, then |Γq0,L0,Q| = 1, and moreover if fe,t = 1S for all e ∈ Hn +d,k +for a set S ⊆ Q, then Proposition 7.2 reduces to precisely Proposition 7.1. In fact, Proposition 7.2 is a +parametric, multi-linear and simultaneous extension of Proposition 7.1 which we need in the induction step, +i.e. when going from level k − 1 to level k. + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +33 +7.2. Proof of Proposition 7.2. We will prove Proposition 7.2 by induction on 1 ≤ k ≤ d. +For k = 1 this is basically Proposition 6.2, exactly as it was in the base case of the proof of Proposition +5.3. +For the induction step we will again need two main ingredients. The first establishes that the our multi- +linear forms N d +λ∆0,q,Q(fe; e ∈ Hn +d,k) are controlled by a box-type norm attached to scales q′ and L. +Let Q = Q1 × . . . × Qd with Qi ⊆ Z2ni+3 be cubes of equal side length l(Q) and 1 ≤ k ≤ d. For any scale +0 < L ≪ l(Q) and function f : Qe′ → [−1, 1] with e′ ∈ Hd,k we define its local box norm at scales q′ and L +by +(7.11) +∥f∥2k +□q′,L(Qe′ ) := Es∈Qe′ ∥f∥2k +□(Qs(q′,L)) +where +(7.12) +∥f∥2k +□( � +Q) := Ex11,x12∈ � +Q1 · · · Exk1,xk2∈ � +Qk +� +(ℓ1,...,ℓk)∈{1,2}k +f(x1ℓ1, . . . , xkℓk) +for any cube �Q of the form �Q = �Q1 × · · · × �Qk. We note that (7.4) and (7.5) are special cases of (7.11) and +(7.12) with k = d, n = (2, . . . , 2), and fe = f for all e ∈ Hn +d,d. +Lemma 7.1 (A Generalized von-Neumann inequality on Zn). Let 1 ≤ k ≤ d. +Let 0 < ε ≪ 1, q, q′ ∈ N with qq1(ε)|q′, and λ ∈ q +√ +N with λ ≪ l(Q) and 1 ≪ L ≪ (ε2k)10λ. For any +collection of functions fe : Qπ(e) → [−1, 1] with e ∈ Hn +d,k we have both +(7.13) +|N d +λ∆0,q,Q(fe; e ∈ Hn +d,k)| ≤ min +e∈Hn +d,k +∥fe∥□q′,L′ (Qπ(e)) + O(ε) +and +(7.14) +|Md +λ,q,Q(fe; e ∈ Hn +d,k)| ≤ min +e∈Hn +d,k +∥fe∥□q′,L′ (Qπ(e)). +The proof of inequalities (7.13) and (7.14) follow exactly as in the continuous case, see Lemma 5.1, using +Lemma 6.1 in place of Lemma 3.1. We omit the details. +The crucial ingredient is again a parametric weak hypergraph regularity lemma, i.e. Lemma 5.2 adapted +to the discrete settings. The proof is essentially the same as in the continuous case, with exception that +the □Lj-norms are replaced by □qj,Lj-norms where qj = q0qj is a given sequence of positive integers and +L0 ≥ L1 ≥ · · · ≥ LJ is an (ε, qJ)-admissible sequence of scales. To state it we say that a σ-algebra B on a +cube Q is of scale (q, L) if it is refinement of the grid Gq,L,Q, i.e. if its atoms partition each cube Qt(q, L) of +the grid. We will always assume that q|L and L|l(Q). Recall also that we say the complexity of a σ-algebra +B is at most m, and write complex(B) ≤ m, if it is generated by m sets. +Lemma 7.2 (Parametric weak hypergraph regularity lemma for Zn). +Let 0 < ε ≪ 1, 1 ≤ k ≤ d, q0, q, L0, M ∈ N, and let qj := q0qj for j ≥ 1. There exists ¯Jk = O(Mε−2k+3) +such that for any (ε2k, q ¯ +Jk)-admissible sequence L0 ≥ L1 ≥ · · · ≥ L ¯ +Jk with the property that L0 divides l(Q) +and collection of functions +f m +e,t : Qtπ(e)(q0, L0) → [−1, 1] with e ∈ Hn +d,k, 1 ≤ m ≤ M, and t ∈ Γq0,L0,Q +there is some 1 ≤ j < ¯Jk and σ-algebras Be′,t of scale (qj, Lj) on Qte′ (q0, L0) for each t ∈ Γq0,L0,Q and +e′ ∈ Hd,k such that +(7.15) +∥f m +e,t − E(f m +e,t|Bπ(e),t)∥□qj+1,Lj+1 (Qtπ(e) (L0)) ≤ ε +uniformly for all t /∈ Tε, e ∈ Hn +d,k, and 1 ≤ m ≤ M, where Tε ⊆ Γq0,L0,Q with |Tε| ≤ ε|Γq0,L0,Q|. +Moreover, the σ-algebras Be′,t have the additional local structure that the exist σ-algebras Be′,f′,s on +Qsf′ (qj, Lj) with complex(Be′,f′,s) = O(j) for each s ∈ Γqj,Lj,Q, e′ ∈ Hd,k, and f′ ∈ ∂e′ such that if +s ∈ Qt(q0, L0), then +(7.16) +Be′,t +�� +Qse′ (qj,Lj) = +� +f′∈∂e′ +Be′,f′,s. + +34 +NEIL LYALL +´AKOS MAGYAR +The proof of Lemma 7.2 follows exactly as the corresponding proof of Lemma 5.2 in the continuous setting, +so we will omit the details. We will however provide some details of how one deduces Proposition 7.2, from +Lemmas 7.1 and 7.2. The arguments are again very similar to those in the continuous setting, however one +needs to make a careful choice of the integers qk(ε), appearing in the statement of the Proposition. +Proof of Proposition 7.2. Let 2 ≤ k ≤ d and assume that the lemma holds for k − 1. +Let 0 < ε ≪ 1 and ε1 := exp (−C1ε−2k+3) for some large constant C1 = C1(n, k, d) ≫ 1. +We then define qk(ε) := qk−1(ε1) recalling that q1(ε) := lcm{1 ≤ q ≤ Cε−10} and note that it is easy +to see by induction that qk(ε)|qk(ε′) for 0 < ε′ ≤ ε and qk−1(ε)|qk(ε). +We further define the function +F(ε) := Jk−1(ε1, M) with M = ε ε−1 +1 +and recall that qj := q0 qk(ε)j for j ≥ 1. +We now proceed exactly as in the proof of Proposition 5.3 but with {Lj}j≥1 being a (ε1, q � +J)-admissible +sequence of scales, with �J ≫ F(ε) ¯Jk(ε, M). We again choose a subsequence {L′ +j} ⊆ {Lj} so that L′ +0 = L0 +and index(L′ +j+1) ≥ index(L′ +j) + F(ε) + 2, but also now set q′ +j = qj′, where j′ := index(L′ +j). Lemma 7.2 +then guarantees the existence of σ-algebras Be′,t of scale (q′ +j, L′ +j) on Qte′ (q0, L0) for each t ∈ Γq0,L0,Q and +e′ ∈ Hd,k, with the local structure described above, such that (7.15) holds uniformly for all t /∈ T ′ +ε, e ∈ Hn +d,k, +and 1 ≤ m ≤ M, for some 1 ≤ j < ¯Jk(ε, M) = O(Mε−2k+3), where T ′ +ε ⊆ Γq0,L0,Q with |T ′ +ε| ≤ ε|Γq0,L0,Q|. +Arguing as in the proof of Proposition 5.3 we can conclude from this that for each j′ ≤ l < J′ we have +(7.17) +N d +λ∆0,ql,Qs(q′ +j,L′ +j)(f m +e,s; e ∈ Hn +d,k) = +� +r +αs,r,m N d +λ∆0,ql,Qs(q′ +j,L′ +j) (gr +f,s; f ∈ Hn +d,k−1) + O(ε) +and +(7.18) +Md +λ,ql,Qs(q′ +j,L′ +j)(f m +e,s; e ∈ Hn +d,k) = +� +r +αr,s,m Md +λ,ql,Qs(q′ +j,L′ +j) (gr +f,s; f ∈ Hn +d,k−1) + O(ε) +provided (ε−2k)10L′ +j+1 ≪ λ with λ ∈ ql +√ +N, where each |αs,re| ≤ 1 and number of index vectors r = (re)e∈Hn +d,k +is RD with D := |Hn +d,k| and hence RD ≤ M if C1 ≫ 1. +By induction, we apply Proposition 7.2 to the sequence of scales L′ +j = Lj′ ≥ Lj′+1 ≥ · · · ≥ LJ′ = L′ +j+1 +with ε1 > 0 and for ql := q′ +j qk(ε)l−j′ = qj′ qk−1(ε1)l−j′ where j′ ≤ l ≤ J′ with respect to the family of +functions gr +s,f : Qsf(q′ +j, L′ +j) → [−1, 1] . This is possible as J′ − j′ ≫ Jk−1(ε1, RD) and our sequence of scales +is (ε1, qJ′)-admissible. Thus there exists an index j′ ≤ l < J′ such that for all λ ∈ ql +√ +N with Ll+1 ≤ λ ≤ Ll +we have +(7.19) +N d +λ∆0,ql,Qs(q′ +j,L′ +j) (gr +f,s; f ∈ Hn +d,k−1) = Md +λ,ql,Qs(q′ +j,L′ +j) (gr +f,s; f ∈ Hn +d,k−1) + O(ε1) +uniformly in r for s /∈ Sε1, where Sε1 ⊆ Γq′ +j,L′ +j,Q is a set of size |Sε1| ≤ ε1|Γq′ +j,L′ +j,Q|. +The remainder of the proof follows as just as it did for Proposition 5.3. +□ +8. Appendix: A short direct proof of Part (i) of Theorem B′ +We conclude by providing a short direct proof of Part (i) of Theorem B′, namely the following +Theorem 8.1 (Magyar [17]). Let 0 < δ ≤ 1 and ∆ ⊆ Z2k+3 be a non-degenerate simplex of k points. +If S ⊆ Z2k+3 has upper Banach density at least δ, then there exists an integer q0 = q0(δ) and λ0 = λ0(S, ∆) +such that S contains an isometric copy of q0λ∆ for all λ ∈ +√ +N with λ ≥ λ0. +For any ε > 0 we define +qε := lcm{1 ≤ q ≤ Cε−10} +with C > 0 a (sufficiently) large absolute constant. Following [14] we further define S ⊆ Zn to be ε-uniformly +distributed (modulo qε) if its relative upper Banach density on any residue class modulo qε never exceeds +(1 + ε2) times its density on Zn, namely if +δ∗(S | s + (qεZ)d) ≤ (1 + ε2) δ∗(S) + +WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY +35 +for all s ∈ {1, . . . , qε}d. It turns out that this notion is closely related to the U 1 +q,L(Q)-norm introduced in +Section 6. Recall that for any cube Q ⊆ Zn and function f : Q → [−1, 1] we define +(8.1) +∥f∥U1 +q,L(Q) := +� 1 +|Q| +� +t∈Q +|f ∗ χq,L(t)|2�1/2 +with χq,L denoting the normalized characteristic function of the cubes Q(q, L) := [− L +2 , L +2 ]n ∩ (qZ)n. Note +that the U 1 +q,L(Q)-norm measures the mean square oscillation of a function with respect to cubic grids of size +L and gap q. +The following observation from [14] (specifically Lemmas 1 and 2) is key to our short proof of Theorem +8.1. +Lemma 8.1. Let ε > 0. If S ⊆ Zn be ε-uniformly distributed with δ := δ∗(S) > 0, then there exists an +integer L = L(S, ε) > 0 and cubes Q of arbitrarily large side length l(Q) with l(Q) ≫ ε−4L such that +∥1S − δ1Q∥U1 +qε,L(Q) = O(ε). +Let ∆0 = {v1 = 0, v2, . . . , vk} be a fixed non-degenerate simplex of k points in Zn with n = 2k + 3 and +define tij := vi · vj for 2 ≤ i, j ≤ k. We now define a function which counts isometric copies of λ∆0. +Recall, see [17], that a simplex ∆ = {m1 = 0, . . . , mk} ⊆ Zn is isometric to λ∆0 if and only if mi · mj = +λ2tij for all 2 ≤ i, j ≤ k. For any λ ∈ +√ +N we define Sλ∆0(m2, . . . , mk) : Zn(k−1) → {0, 1} be the function +whose value is 1 if mi · mj = λ2tij for all 2 ≤ i, j ≤ k and is equal to 0 otherwise. It is a well-known fact in +number theory, see [11] or [17], that for n ≥ 2k + 1 we have that +� +m2,...,mk +Sλ∆0(m2, . . . , mk) = ρ(∆0) λ(n−k)(k−1)(1 + O(λ−τ)) +for some absolute constant τ > 0 and constant ρ(∆0) > 0, the so-called singular series, which can be +interpreted as the product of the densities of the solutions of the above system of equations among the +p-adics and among the reals. Thus if we define +σλ∆0 := ρ(∆0)−1λ−(n−k)(k−1)Sλ∆0 +then σλ∆0 is normalized in so much that +� +m2,...,mk +σλ∆0(m2, . . . , mk) = 1 + O(λ−τ) +for some absolute constant τ > 0. +Let Q ⊆ Zn be a fixed cube and let l(Q) denotes its side length. For any family of functions +f1, . . . , fk : Q → [−1, 1] +and 0 < λ ≪ l(Q) we define +(8.2) +N 1 +λ∆0,Q(f1, . . . , fk) := Em1∈Q +� +m2,...,mk +f1(m1) . . . fk(mk) σλ∆0(m2 − m1, . . . , mk − m1). +It is clear that if f1 = · · · = fk = 1S restricted to Q, then the above expression is a normalized count +of the isometric copies of λ∆0 in S ∩ Q. Thus, Theorem 8.1 will follow from Lemma 8.1 and the following +special case (with q = 1) of Lemma 6.1. +Lemma 8.2 (A Generalized von Neumann inequality). Let 0 < ε ≪ 1. +If λ ∈ +√ +N with λ ≪ l(Q) and 1 ≪ L ≪ ε10λ then for any collection of functions f1, . . . , fk : Q → [−1, 1] +we have +(8.3) +|N 1 +λ∆0,Q(f1, . . . , fk)| ≤ min +1≤j≤k ∥fj∥U1 +qε,L(Q) + O(ε). +This compares with the purely number theoretic fact that the number of simplices ∆ = {v1 = 0, v2, . . . , vk} ⊆ +Zn isometric to λ∆0 is asymptotic to ρ(∆0) λ(n−k)(k−1). Thus, under the same conditions as in Lemma 8.2, +we have +(8.4) +N 1 +λ∆0,Q(1Q, . . . , 1Q) = 1 + O(λ−τ) + O(ε) + +36 +NEIL LYALL +´AKOS MAGYAR +provided one also has λ ≪ εl(Q). +Proof of Theorem 8.1. Let 0 < ε ≪ δk and S ⊆ Zn be a set of upper Banach density δ. +We assume first that S is ε-uniformly distributed. Select a scale L = L(ε, S) and a sufficiently large cube +Q so that the conclusion of Lemma 8.1 holds. For a given λ ∈ +√ +N with λ ≪ εl(Q) and L ≪ ε10λ write +1S = δ1Q + g and substitute this decomposition into the multi-linear expression N 1 +λ∆0,Q(1S, . . . , 1S). Then +by Lemma 8.2 and (8.3)-(8.4), we have that +(8.5) +N 1 +λ∆0,Q(1S, . . . , 1S) ≥ δk − O(ε) +and we can conclude that S must contain an isometric copy of λ∆0. +If S is not ε-uniformly distributed, then its upper Banach density is increased to at least δ1 := (1 + ε2)δ +when restricted to a residue class s+(qεZ)n. Identify s+(qεZ)n with Zn and simultaneously the set S|s+(qεZ)n +with a set S1 ⊆ Zn, via the map y → q−1 +ε (y − s). Note that if S1 is ε-uniformly distributed then it contains +an isometric copy of λ∆0 for all sufficiently large λ ∈ +√ +N and hence S contains an isometric copy of qελ∆0. +Repeating the above procedure one arrives to a set Sj = q−j +ε (S − sj) ⊆ Zn for some sj ∈ Zn in j = +O(log ε−1) steps which contains an isometric copy of λ∆0 for all sufficiently large λ ∈ +√ +N. +□ +References +[1] J. Bourgain, A Szemer´edi type theorem for sets of positive density in Rk, Israel J. Math. 54 (1986), no. 3, 307–316. +[2] D. Conlon, J. Fox, Y. Zhao, A relative Szemeredi theorem, Geometric and Functional Analysis 25.3 (2015): 733-762. +[3] B. Cook, ´A. Magyar, M. Pramanik, A Roth-type theorem for dense subsets of Rd, Bull. Lond. Math. Soc. 49 (2017), no. +4, 676-689. +[4] P. Durcik, V. Kovaˇc, Boxes, extended boxes, and sets of positive upper density in the Euclidean space, arXiv 1809.08692 +[5] H. Furstenberg, Y. Katznelson, An ergodic Szemer´edi theorem for commuting trnasformations, J. Analyse Math. 31 +(1978), 275-291 +[6] H. Furstenberg, Y. Katznelson and B. Weiss, Ergodic theory and configurations in sets of positive density, Mathematics +of Ramsey theory, 184–198, Algorithms Combin., 5, Springer, Berlin, 1990. +[7] A. Frieze, R. Kannan, The regularity lemma and approximation schemes for dense problems, In Foundations of Computer +Science (1996) Proc. 37th Annual Symp. IEEE., 12-20 +[8] W. T. Gowers. Hypergraph regularity and the multidimensional Szemer´edi theorem, Annals of Mathematics (2007), 897- +946. +[9] R.L. Graham. Recent trends in Euclidean Ramsey theory, Discrete Mathematics 136, no. 1-3 (1994), 119-127. +[10] A. Iosevich and M. Rudnev, Erd˝os distance problem in vector spaces over finite fields, Trans. Amer. Math. Soc. 359, no. +12 (2007), 6127-6142. +[11] Y.Kitaoka, Siegel modular forms and representation by quadratic forms Lectures on Mathe- matics and Physics, Tata +Institute of Fundamental Research, Springer-Verlag, (1986) +[12] N. Lyall and ´A. Magyar, Product of simplices and sets of positive upper density in Rd, Math. Proc. of the Cambridge +Philos. Soc. 165. no. 1. (2018), 25-51 +[13] N. Lyall and ´A. Magyar, Distance Graphs and sets of positive upper density in Rd, to appear in Anal. PDE +[14] N. Lyall and ´A. Magyar, Distances and trees in dense subsets of Zd, arXiv 1509.09298 +[15] N. Lyall, ´A. Magyar, H. Parshall, Spherical configurations over finite fields, to appear in Amer. J. Math. +[16] ´A. Magyar, Distance sets of large sets of integer points, Israel J. Math., v (2008) pp. +[17] ´A. Magyar, k-point configurations in sets of positive density of Zn, Duke Math. J., v 146/1, (2009) pp. 1-34. +[18] H. Parshall, Simplices over finite fields, Proc. Amer. Math. Soc. 145.6 (2017), 2323-2334. +[19] C. L. Siegel, On the theory of indefinite quadratic forms, Ann. of Math. (2) 45 (1944), 577-622 +[20] E., Szemer´edi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 199-245. +[21] T. Tao, Szemer´edi’s regularity lemma revisited, arXiv preprint math/0504472 (2005) +[22] T. Tao, A variant of the hypergraph removal lemma, Journal of Combinatorial Theory, Series A 113.7 (2006): 1257-1280 +[23] T. Tao, The ergodic and combinatorial approaches to Szemer´edi’s theorem, Additive combinatorics, 145–193, CRM Proc. +Lecture Notes, 43, Amer. Math. Soc., Providence, RI, 2007. +[24] T. Tao and V. Vu, Additive combinatorics, Cambridge Studies in Advanced Mathematics, 105. Cambridge University +Press, Cambridge, 2006. xviii+512 pp. +[25] T. Ziegler, Nilfactors of Rm-actions and configurations in sets of positive upper density in Rm, J. Anal. Math. 99 (2006), +249-266. +Department of Mathematics, The University of Georgia, Athens, GA 30602, USA +Email address: lyall@math.uga.edu +Email address: magyar@math.uga.edu + diff --git a/d9FIT4oBgHgl3EQfoiuQ/content/tmp_files/load_file.txt b/d9FIT4oBgHgl3EQfoiuQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..78105ceb7cc2d4639ddb6c430c5856b8d0d0a8f0 --- /dev/null +++ b/d9FIT4oBgHgl3EQfoiuQ/content/tmp_files/load_file.txt @@ -0,0 +1,2177 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf,len=2176 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11319v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='CO] 26 Jan 2023 WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY NEIL LYALL ´AKOS MAGYAR Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆ = ∆1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='×∆d ⊆ Rn, where Rn = Rn1 ×· · ·×Rnd with each ∆i ⊆ Rni a non-degenerate simplex of ni points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We prove that any set S ⊆ Rn, with n = n1 + · · · + nd of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of the configuration ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In particular any such set S ⊆ R2d contains a d-dimensional cube of side length λ, for all λ ≥ λ0(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We also prove analogous results with the underlying space being the integer lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The proof is based on a weak hypergraph regularity lemma and an associated counting lemma developed in the context of Euclidean spaces and the integer lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Existing Results I: Distances and Simplices in Subsets of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that the upper Banach density of a measurable set S ⊆ Rn is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1) δ∗(S) = lim N→∞ sup t∈Rn |S ∩ (t + Q(N))| |Q(N)| , where | · | denotes Lebesgue measure on Rn and Q(N) denotes the cube [−N/2, N/2]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A result of Furstenberg, Katznelson, and Weiss [6] states that if S ⊆ R2 has positive upper Banach density, then its distance set {|x − x′| : x, x′ ∈ S} contains all sufficiently large numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that the distance set of any set of positive Lebesgue measure in Rn automatically contains all sufficiently small numbers (by the Lebesgue density theorem) and that it is easy to construct a set of positive upper density which does not contain a fixed distance by placing small balls centered on an appropriate square grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem A (Furstenberg, Katznelson, and Weiss [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If S ⊆ R2 with δ∗(S) > 0, then there exists a λ0 = λ0(S) such that S is guaranteed to contain pairs of points {x1, x2} with |x2 − x1| = λ for all λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This result was later reproved using Fourier analytic techniques by Bourgain in [1] where he established the following more general result for all configurations of n points in Rn whose affine span is n−1 dimensional, namely for all non-degenerate simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem B (Bourgain [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆ ⊆ Rn be a non-degenerate simplex of n points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If S ⊆ Rn with δ∗(S) > 0, then there exists a threshold λ0 = λ0(S, ∆) such that S contains an isometric copy of λ∆ for all λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that a finite point configuration ∆′ is said to be an isometric copy of λ∆ if there exists a bijection φ : ∆ → ∆′ such that |φ(v) − φ(w)| = λ |v − w| for all v, w ∈ ∆, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' if ∆′ is obtained from λ∆ (the dilation of ∆ by a factor λ) via a rotation and translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Bourgain deduced Theorem B as an immediate consequence of the following stronger quantitative result for measurable subsets of the unit cube of positive measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In the proposition below, and throughout this article, we shall refer to a decreasing sequence {λj}J j=1 as lacunary if λj+1 ≤ λj/2 for all 1 ≤ j < J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition B (Bourgain [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆ ⊆ Rn be a non-degenerate simplex of n points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < δ ≤ 1 there exists a constant J = O∆(δ−3n) such that if 1 ≥ λ1 ≥ · · · ≥ λJ is any lacunary sequence and S ⊆ [0, 1]n with |S| ≥ δ, then there exists 1 ≤ j < J such that S contains an isometric copy of λ∆ for all λ ∈ [λj+1, λj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 11B30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The first and second authors were partially supported by grants NSF-DMS 1702411 and NSF-DMS 1600840, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1 2 NEIL LYALL ´AKOS MAGYAR In [12] the authors provided a short direct proof of Theorem B without using Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is based on the observation that uniformly distributed sets S ⊆ Rd contain the expected “number” of isometric copies of dilates λ∆ and that all sets of positive upper density become uniformly distributed at sufficiently large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' However, for the purposes of this paper it will be important to recall Bourgain’s indirect approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' To see that Proposition B implies Theorem B notice that if Theorem B were not to hold for some set S ⊆ Rn of upper Banach density δ∗(S) > δ > 0, then there must exist a lacunary sequence λ1 ≥ · · · ≥ λJ ≥ 1, with J the constant in Proposition B, such that S does not contain an isometric copy of λj∆ for any 1 ≤ j ≤ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Taking a sufficiently large cube Q with side length N ≥ λ1 and |S ∩ Q| ≥ δ|Q| and scaling back Q → [0, 1]n contradicts Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We further note that by taking λj = 2−j in Proposition B we obtain the following “Falconer-type” result for subsets of [0, 1]n of positive Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If ∆ ⊆ Rn is a non-degenerate simplex of n points, then any S ⊆ [0, 1]n with |S| > 0 will necessarily contain an isometric copy of λ∆ for all λ in some interval of length at least exp(−C∆|S|−3n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Bourgain further demonstrated in [1] that no result along the lines of Theorem B can hold for configurations that contain any three points in arithmetic progression along a line, specifically showing that for any n ≥ 1 there are sets of positive upper Banach density in Rn which do not contain an isometric copy of configurations of the form {0, y, 2y} with |y| = λ for all sufficiently large λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This should be contrasted with the following remarkable result of Tamar Ziegler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem C (Ziegler [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let F be any configuration of k points in Rn with n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If S ⊆ Rn has positive upper density, then there exists a threshold λ0 = λ0(S, F) such that Sε contains an isometric copy of λF for all λ ≥ λ0 and any ε > 0, where Sε denotes the ε-neighborhood of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Bourgain’s example was later generalized by Graham [9] to establish that the condition that ε > 0 in Theorem C is necessary and cannot be strengthened to ε = 0 for any given non-spherical configuration F in Rn for any n ≥ 1, that is for any finite configuration of points that cannot be inscribed in some sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We note that the sets constructed by Bourgain and Graham have the property that for any ε > 0 their ε-neighborhoods will contain arbitrarily large cubes and hence trivially satisfy Theorem C with λ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is natural to ask if any spherical configuration F, beyond the known example of simplices, has the property that every positive upper Banach density subset of Rn, for some sufficiently large n, contains an isometric copy of λF for all sufficiently large λ, and even to conjecture that this ought to hold for all spherical configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The first breakthrough in this direction came in [12] when the authors established this for configurations of four points forming a 2-dimensional rectangle in R4 and more generally for any configuration that is the direct product of two non-degenerate simplices in Rn for suitably large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The purpose of this article is to present a strengthening of the results in [12] and to extend them to cover configurations with a higher dimensional product structure in both the Euclidean and discrete settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' New Results I: Rectangles and Products of Simplices in Subsets of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The first main result of this article is the following Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let R be 2d points forming the vertices of a fixed d-dimensional rectangle in R2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (i) If S ⊆ R2d has positive upper Banach density, then there exists a threshold λ0 = λ0(S, R) such that S contains an isometric copy of λR for all λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (ii) For any 0 < δ ≤ 1 there exists a constant c = c(δ, R) > 0 such that any S ⊆ [0, 1]2d with |S| ≥ δ is guaranteed to contain an isometric copy of λR for all λ in some interval of length at least c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover, if R has sidelengths given by t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td, then the isometric copies of λR in both (i) and (ii) above can all be realized in the special form {x11, x12} × · · ·× {xd1, xd2} ⊆ R2 × · · ·× R2 with each |xj2 − xj1| = λtj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The multi-dimensional extension of Szemer´edi’s theorem on arithmetic progressions in sets of positive density due to Furstenberg and Katznelson [5] implies, and is equivalent to the fact, that there are isometric copies of λR in S for arbitrarily large λ, with sides parallel to the coordinate axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 states that there is an isometric copy of λR in S for every sufficiently large λ, but only with sides parallel to given WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 3 2-dimensional coordinate subspaces which provides an extra degree of freedom for each side vector of the rectangle R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A weaker version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, with R2d replaced with R5d, was later established by Durcik and Kovaˇc in [4] using an adaptation of arguments of the second author with Cook and Pramanik in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This approach also makes direct use of the full strength of the multi-dimensional Szemer´edi theorem and as such leads to quantitatively weaker results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our arguments work for more general patterns where d-dimensional rectangles are replaced with direct products of non-degenerate simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆ = ∆1 × · · · × ∆d ⊆ Rn, where Rn = Rn1 × · · · × Rnd and each ∆j ⊆ Rnj is a non-degenerate simplex of nj points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (i) If S ⊆ Rn has positive upper Banach density, then there exists a threshold λ0 = λ0(S, ∆) such that S contains an isometric copy of λ∆ for all λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (ii) For any 0 < δ ≤ 1 there exists a constant c = c(δ, ∆) > 0 such that any S ⊆ [0, 1]n with |S| ≥ δ is guaranteed to contain an isometric copy of λ∆ for all λ in some interval of length at least c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover the isometric copies of λ∆ in both (i) and (ii) above can all be realized in the special form ∆′ 1 × · · × ∆′ d with each ∆′ j ⊆ Rnj an isometric copy of λ∆j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Quantitative Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A careful analysis of our proof reveals that the constant c(δ, ∆) can be taken greater than Wd(C′ ∆δ−3n1···nd)−1 where Wk(m) is a tower of exponentials defined by W1(m) = exp(m) and Wk+1(m) = exp(Wk(m)) for k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Existing Results II: Distances and Simplices in Subsets of Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The problem of counting isomet- ric copies of a given non-degenerate simplex in Zn (with one vertex fixed) has been extensively studied via its equivalent formulation as the number of ways a quadratic form can be represented as a sum of squares of linear forms, see [11] and [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This was exploited by the second author in [16] and [17] to establish analogous results to those described in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 above for subsets of the integer lattice Zn of positive upper density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that the upper Banach density of a set S ⊆ Zn is analogously defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) δ∗(S) = lim N→∞ sup t∈Rn |S ∩ (t + Q(N))| |Q(N)| , where | · | now denotes counting measure on Zn and Q(N) the discrete cube [−N/2, N/2]n ∩ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In light of the fact that any pairs of distinct points {x1, x2} in Zn has the property that the square of the distance between them |x2 − x1|2 is always a positive integer we introduce the convenient notation √ N := {λ : λ > 0 and λ2 ∈ Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem A′ (Magyar [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < δ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If S ⊆ Z5 has upper Banach density at least δ, then there exists an integer q0 = q0(δ) and λ0 = λ0(S) such that S contains pairs of points {x1, x2} with |x2 − x1| = q0λ for all λ ∈ √ N with λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem B′ (Magyar [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < δ ≤ 1 and ∆ ⊆ Z2n+3 be a non-degenerate simplex of n points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (i) If S ⊆ Z2n+3 has upper Banach density at least δ, then there exists an integer q0 = O(exp(C∆δ−13n)) and λ0 = λ0(S, ∆) such that S contains an isometric copy of q0λ∆ for all λ ∈ √ N with λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (ii) If N ≥ exp(2C∆δ−13n), then any S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , N}2n+3 with cardinality |S| ≥ δN 2n+3 will necessarily contain an isometric copy of λ∆ for some λ ∈ √ N with 1 ≤ λ ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that the fact that S ⊆ Zn could fall entirely into a fixed congruence class of some integer 1 ≤ q ≤ δ−1/n ensures that the q0 that appears in Theorems A′ and B′ above must be divisible by the least common multiple of all integers 1 ≤ q ≤ δ−1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed if S = (qZ)n with 1 ≤ q ≤ δ−1/n then S has upper Banach density at least δ, however the distance between any two points x, y ∈ S is of the form |x − y| = qλ for some λ ∈ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 4 NEIL LYALL ´AKOS MAGYAR However, in both Theorems A′ and Part (i) of Theorem B′, one can take q0 = 1 if the sets S are assumed to be suitably uniformly distributed on congruence classes of small modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This leads via an easy density increment strategy to short new proofs, see [14] for Theorem A′ and Section 8 for Part (i) of Theorem B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The original argument in [17] deduced Theorem B′ from the following discrete analogue of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition B′ (Magyar [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆ ⊆ Z2n+3 be a non-degenerate simplex of n points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < δ ≤ 1 there exist constants J = O∆(δ−3n) and q0 = O(exp(C∆δ−13n)) such that if N ≥ λ1 ≥ · · ≥ λJ ≥ 1 is any lacunary sequence in q0 √ N and S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , N}2n+3 with cardinality |S| ≥ δN 2n+3, then S will necessarily contain an isometric copy of λj∆ for some 1 ≤ j ≤ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' To see that Proposition B′ implies Theorem B′ notice that if Part (i) of Theorem B′ were not to hold for some set S ⊆ Z2n+3 of upper Banach density δ∗(S) > δ > 0 with q0 from Proposition B′, then there must exist a lacunary sequence λ1 ≥ · · · ≥ λJ ≥ 1 in q0 √ N, with J the constant from Proposition B′, such that S does not contain an isometric copy of λj∆ for any 1 ≤ j ≤ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since we can find a sufficiently large cube Q with integer side length N that is divisible by q0 and greater than λ1 such that |S ∩ Q| ≥ δ|Q| , this contradicts Proposition B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Part (ii) of Theorem B′ follows from Proposition B′ by taking λj = 2J−jq0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' New Results II: Rectangles and Products of Simplices in Subsets of Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will also establish the following discrete analogues of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < δ ≤ 1 and R be 2d points forming the vertices of a d-dimensional rectangle in Z5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (i) If S ⊆ Z5d has upper Banach density at least δ, then there exist integers q0 = q0(δ, R) and λ0 = λ0(S, R) such that S contains an isometric copy of q0λR for all λ ∈ √ N with λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (ii) There exists a constant N(δ, R) such that if N ≥ N(δ, R), then any S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , N}5d with cardinality |S| ≥ δN 5d will necessarily contain an isometric copy of λR for some λ ∈ √ N with 1 ≤ λ ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If R has side lengths given by t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td, then each of the isometric copies in (i) and (ii) above can be realized in the form {x11, x12} × · · · × {xd1, xd2} ⊆ Z5 × · · · × Z5 with each |xj2 − xj1| = q0λtj and λtj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our arguments again work for more general patterns where d-dimensional rectangles are replaced with direct products of non-degenerate simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < δ ≤ 1 and ∆ = ∆1 × · · · × ∆d ⊆ Zn, where Zn = Z2n1+3 × · · · × Z2nd+3 and each ∆i ⊆ Z2ni+3 is a non-degenerate simplex of ni points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (i) If S ⊆ Zn has upper Banach density at least δ, then there exist integers q0 = q0(δ, ∆) and λ0 = λ0(S, ∆) such that S contains an isometric copy of q0λ∆ for all λ ∈ √ N with λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (ii) There exists a constant N(δ, ∆) such that if N ≥ N(δ, ∆), then any S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , N}n with cardinality |S| ≥ δN n will necessarily contain an isometric copy of λ∆ for some λ ∈ √ N with 1 ≤ λ ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover, each of the isometric copies in (i) and (ii) above can be realized in the special form ∆′ 1 × · · · × ∆′ d with each ∆′ i ⊆ Z2ni+3 an isometric copy of q0λ∆j and λ∆j, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Quantitative Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A careful analysis of our proof reveals that the constant q0(δ, ∆) (and consequently also N(δ, ∆)) can be taken less than Wd(C′ ∆δ−13n1···nd) where Wk(m) is a tower of exponentials defined by W1(m) = exp(m) and Wk+1(m) = exp(Wk(m)) for k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Notations and Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will consider the parameters d, n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , nd fixed and will not indicate the dependence on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus we will write f = O(g) if |f| ≤ C(n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , nd)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If the implicit constants in our estimates depend on additional parameters ε, δ, K, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' the we will write f = Oε,δ,K,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will use the notation f ≪ g to indicate that |f| ≤ c g for some constant c > 0 sufficiently small for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Given an ε > 0 and a (finite or infinite) sequence L0 ≥ L1 ≥ · · · > 0, we will say that the sequence is ε-admissible if Lj/Lj+1 ∈ N and Lj+1 ≪ ε2Lj for all j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover, if q ∈ N is given and Lj ∈ N for all 1 ≤ j ≤ J, then we will call the sequence L0 ≥ L1 ≥ · · · ≥ LJ (ε, q)-admissible if in addition LJ/q ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Such sequences of scales will often appear in our statements both in the continuous and the discrete case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our proofs are based on a weak hypergraph regularity lemma and an associated counting lemma developed in the context of Euclidean spaces and the integer lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In Section 2 we introduce our approach in the WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 5 model case of finite fields and prove an analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In Section 3 we review Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 for a single simplex and ultimately establish the base case of our general inductive approach to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In Section 4 we address Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 for the direct product of two simplices, this provides a new proof (and strengthening) of the main result of [12] and serves as a gentle preparation for the more complicated general case which we present in the Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4 is outlined in Sections 6 and 7, while a short direct proof of Part (i) of Theorem B′ is presented in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Model case: vector spaces over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In this section we will illustrate our general method by giving a complete proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 in the model setting of Fn q where Fq denotes the finite field of q elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We do this as the notation and arguments are more transparent in this setting yet many of the main ideas are still present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We say that two vectors u, v ∈ Fn q are orthogonal, if x·y = 0, where “·” stands for the usual dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A rectangle in Fn q is then a set R = {x1, y1} × · · ·× {xn, yn} with side vectors yi − xi being pairwise orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The finite field analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 is the following Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < δ ≤ 1 there exists an integer q0 = q0(δ) with the following property: If q ≥ q0 and t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td ∈ F∗ q, then any S ⊆ F2d q with |S| ≥ δ q2d will contain points {x11, x12} × · · · × {xd1, xd2} ⊆ V1 × · · · × Vd with |xj2 − xj1|2 = tj for 1 ≤ j ≤ d where we have written F2d q = V1 × · · · × Vd with Vj ≃ F2 q pairwise orthogonal coordinate subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Overview of the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Write F2d q = V1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × Vd with Vj ≃ F2 q pairwise orthogonal coordinate subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any t := (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td) ∈ F∗ q and S ⊆ F2d q we define Nt(1S) := Ex1∈V 2 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xd∈V 2 d � (ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',ℓd)∈{1,2}d 1S(x1ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xdℓd) d � j=1 σtj(xj2 − xj1) where we used the shorthand notation xj := (xj1, xj2) for each 1 ≤ j ≤ d and the averaging notation: Ex∈Af(x) := 1 |A| � x∈A f(x) for a finite set A ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We have also used the notation σt(x) = � q if |x|2 = t 0 otherwise for each t ∈ F∗ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that the function σt may be viewed as the discrete analogue of the normalized surface area measure on the sphere of radius √ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is well-known, see [10], that Ex∈F2q σt(x) = 1 + O(q−1/2) and for all ξ ̸= 0 one has ˆσt(ξ) := Ex∈F2q σt(x) e2πi x·ξ q = O(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if Nt(1S) > 0, then this implies that S contains a rectangle of the form {x11, x12}×· · ·×{xd1, xd2} with xj1, xj2 ∈ Vj and |xj2 − xj1|2 = tj for 1 ≤ j ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our approach to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 in fact establishes the following quantitatively stronger result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < ε ≤ 1 there exists an integer q0 = q0(ε) with the following property: If q ≥ q0, then for any S ⊆ F2d q and t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td ∈ F∗ q one has Nt(1S) > � |S| q2d �2d − ε where we have written F2d q = V1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × Vd with Vj ≃ F2 q pairwise orthogonal coordinate subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 6 NEIL LYALL ´AKOS MAGYAR A crucial observation in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 is that the averages Nt(1S) can be compared to ones which can be easily estimated from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We define, for any S ⊆ F2d q , the (unrestricted) count M(1S) := Ex1∈V 2 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xd∈V 2 d � (ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',ℓd)∈{1,2}d 1S(x1ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xdℓd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is easy to see, by carefully applying Cauchy-Schwarz d times to Ex11∈V1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xd1∈Vd1S(x11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd1), that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1) M(1S) ≥ � |S| q2d �2d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our approach to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 therefore reduces to establishing that for any ε > 0 one has (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) Nt(1S) = M(1S) + O(ε) + Oε(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The validity of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) will follow immediately from the d = k case of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' However, before we can state this counting lemma we need to introduce some further notation from the theory of hypergraphs, notation that we shall ultimately make use of throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Hypergraph Notation and a Counting Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In order to streamline our notation we will make use the language of hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For J := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', d} and 1 ≤ k ≤ d, we let Hd,k = {e ⊆ J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' |e| = k} denote the full k-regular hypergraph on the vertex set J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For K := {jl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' j ∈ J, l ∈ {1, 2}} we define the projection π : K → J as π(jl) := j and use this in turn to define the hypergraph bundle H2 d,k := {e ⊆ K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' |e| = |π(e)| = k} using the shorthand notation 2 = (2, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 2) to indicate that |π−1(j)| = 2 for all j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Notice when k = d then Hd,d consists of one element, the set e = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', d}, and H2 d,d = { {1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , dld};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ld) ∈ {1, 2}d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let V := F2d q and V = V1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × Vd with Vj ≃ F2 q pairwise orthogonal coordinate subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For a given x = (x11, x12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd1, xd2) ∈ V 2 with xj1, xj2 ∈ Vj and a given edge e = {1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , dld}, we write xe := (x1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xdld).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that the map x → xe defines a projection πe : V 2 → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' With this notation, we can clearly now write Nt(1S) = Ex∈V 2 � e∈H2 d,d 1S(xe) d � j=1 σtj(xj2 − xj1) M(1S) = Ex∈V 2 � e∈H2 d,d 1S(xe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Now for any 1 ≤ k ≤ d and any edge e′ ∈ Hd,k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e′ ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , d}, |e′| = k, we let Ve′ := � j∈e′ Vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For every x ∈ V 2 and e ∈ H2 d,k, we define xe := πe(x) where πe : V 2 → Vπ(e) is the natural projection map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our key counting lemma, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 below, which we will establish by induction on 1 ≤ k ≤ d below, is then the statement that given a family of functions fe : Vπ(e) → [−1, 1], e ∈ H2 d,k, the averages (generalizing those discussed above) which are defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3) Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) := Ex∈V 2 � e∈H2 d,k fe(xe) d � j=1 σtj(xj2 − xj1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) M(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) := Ex∈V 2 � e∈H2 d,k fe(xe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' are approximately equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Specifically, one has WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 7 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 (Counting Lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 1 ≤ k ≤ d and 0 < ε ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any collection of functions fe : Vπ(e) → [−1, 1] with e ∈ H2 d,k one has (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = M(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) + O(ε) + Oε(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If we apply this Proposition with d = k and fe = 1S for all e ∈ H2 d,d, then Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 clearly follows given the lower bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will establish Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 by inducting on 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For k = 1 the result follows from the basic observation that if f1, f2 : F2 q → [−1, 1] and let t ∈ F∗ q, then Ex1,x2∈F2q f1(x1)f2(x2) σt(x2 − x1) = � ξ∈F2q ˆf1(ξ) ˆf2(ξ)ˆσt(ξ) = ˆf1(0) ˆf2(0) + O(q−1/2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) = Ex1,x2∈F2q f1(x1)f2(x2) + O(q−1/2) by the properties of the function ˆσ given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' To see how this implies Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 for k = 1 we note that since H2 d,1 = {jl : 1 ≤ j ≤ d, 1 ≤ l ≤ 2} it follows that Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,1) = d � j=1 Exj1,xj2∈F2q fj1(xj1)fj2(xj2) σt(xj2 − xj1) = d � j=1 Exj1,xj2∈F2q fj1(xj1)fj2(xj2) + O(q−1/2) = M(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,1) + O(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The induction step has two main ingredients, the first is an estimate of the type which is often referred to as a generalized von-Neumann inequality, namely Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any collection of functions fe : Vπ(e) → [−1, 1] with e ∈ H2 d,k one has (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) ≤ min e∈H2 d,k ∥fe∥□(Vπ(e)) + O(q−1/2) where for any e ∈ H2 d,k and f : Vπ(e) → [−1, 1] we define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) ∥f∥2k □(Vπ(e)) := Ex∈V 2 π(e) � e∈H2 d,k f(xe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The corresponding inequality for the multilinear expression M(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k), namely the fact that M(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) ≤ � e∈H2 d,k ∥fe∥□(Vπ(e)) ≤ min e∈H2 d,k ∥fe∥□(Vπ(e)) is well-known and is referred to as the Gowers-Cauchy-Schwarz inequality [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The second and main ingredient is an approximate decomposition of a graph to simpler ones, and is essentially the so-called weak (hypergraph) regularity lemma of Frieze and Kannan [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We choose to state this from a somewhat more abstract/probabilistic point of view, a perspective that will be particularly helpful when we consider our general results in the continuous and discrete settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will first introduce this in the case d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A bipartite graph with (finite) vertex sets V1, V2 is a set S ⊆ V1 × V2 and a function f : V1 × V2 → R may be viewed as weighted bipartite graph with weights f(x1, x2) on the edges (x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If P1 and P2 are partitions of V1 and V2 respectively then P = P1 × P2 is a partition V1 × V2 and we let E(f|P) denote the function that is constant and equal to Ex∈Af(x) on each 8 NEIL LYALL ´AKOS MAGYAR atom A = A1 × A2 of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The weak regularity lemma states that for any ε > 0 and for any weighted graph f : V1 × V2 → [−1, 1] there exist partitions Pi of Vi with |Pi| ≤ 2O(ε−2) for i = 1, 2, so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) |Ex1∈V1Ex2∈V2(f − E(f|P))(x1, x2) 1U1(x1)1U2(x2)| ≤ ε for all U1 ⊆ V1 and U2 ⊆ V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Informally this means that the graph f can be approximated with precision ε with the “low complexity” graph E(f, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If we consider the σ-algebras Bi generated by the partitions Pi and the σ-algebra B = B1 ∨ B2 generated by P1 × P2 then we have E(f|B), the so-called conditional expectation function of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover it is easy to see, using Cauchy-Schwarz, that estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10) ∥f − E(f|B1 ∨ B2)∥□(V1×V2) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' With this more probabilistic point of view the weak regularity lemma says that the function f can be approximated with precision ε by a low complexity function E(f|B1 � B2), corresponding to σ-algebras Bi on Vi generated by O(ε−2) sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This formulation is also referred to as a Koopman- von Neumann type decomposition, see Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will need a natural extension to k-regular hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' See [22, 8], and also [2] for extension to sparse hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Given an edge e′ ∈ Hd,k of k elements we define its boundary ∂e′ := {f′ ∈ Hd,k−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f′ ⊆ e′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For each f′ = e′\\{j} ∈ ∂e′ let B′ f be a σ-algebra on Vf′ := � j∈f′ Vj and ¯Bf′ := {U × Vj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' U ∈ Bf′} denote its pull-back over the space Ve′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The σ-algebra B = � f′∈∂e′ Bf′ is the smallest σ-algebra on ∂e′ containing ¯Bf′ for all f′ ∈ ∂e′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that the atoms of B are of the form A = � f′∈∂e′ Af′ where Af′ is an atom of ¯Bf′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We say that the complexity of a σ-algebra Bf′ is at most m, and write complex(Bf′) ≤ m, if it is generated by m sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (Weak hypergraph regularity lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 1 ≤ k ≤ d and fe : Vπ(e) → [−1, 1] be a given function for each e ∈ H2 d,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any ε > 0 there exists σ-algebras Bf′ on Vf′ for each f′ ∈ Hd,k−1 such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) complex(Bf′) = O(ε−2k+1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) ∥fe − E(fe| � f′∈∂π(e) Bf′)∥□(Vπ(e)) ≤ ε for all e ∈ H2 d,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The proof of Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 are presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We close this subsection by demon- strating how these lemmas can be combined to establish Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ε > 0, 2 ≤ k ≤ d and assume that the lemma holds for k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 that there exists σ-algebras Bf′ of complexity O(ε−2k+1) on Vf′ for each f′ ∈ Hd,k−1 for which (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) holds for all e ∈ H2 d,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For each e ∈ H2 d,k we let ¯fe := E(fe| � f′∈∂π(e) Bf′) and write fe = ¯fe + he.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and multi-linearity we have that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='13) Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = Nt( ¯fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) + O(ε) + O(q−1/2) and also by the Gowers-Cauchy-Schwarz inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14) M(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = M( ¯fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) + O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The conditional expectation functions ¯fe are linear combinations of the indicator functions 1Ae of the atoms Ae of the σ-algebras Be := � f′∈∂π(e) Bf′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the number of terms in this linear combination is at most 2Cε−2k+1 , with coefficients at most 1 in modulus, plugging these into the multi-linear expressions Nt( ¯fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) and M( ¯fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) one obtains a linear combination of expressions of the form Nt(1Ae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) and M(1Ae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) respectively with each Ae being an atoms of Be for all e ∈ H2 d,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The key observation is that these expressions are at level k − 1 instead of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed, 1Ae = � f′∈∂π(e) 1Aef′ where Aef′ = A′ ef′ × Vj, with A′ ef′ being an atom of Bf′ when f′ = π(e)\\{j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If e = (j1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , jl, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , jklk), WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 9 let pf′(e) := (j1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , jklk) ∈ H2 d,k−1, obtained from e by removing the jl-entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then we have 1Aef′ (xe) = 1A′ ef′(xp′ f(e)) since xjl ∈ Vj, and hence 1Ae(xe) = � f′∈∂π(e) 1A′ ef′ (xp′ f(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It therefore follows that Nt(1Ae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = Ex∈V 2 � e∈H2 d,k � f′∈∂π(e) 1A′ ef′(xpf′ (e)) d � j=1 σtj(xj2 − xj1) = Ex∈V 2 � f∈H2 d,k−1 � e∈H2 d,k, f′∈∂π(e) pf′ (e)=f 1A′ ef′(xpf′ (e)) � �� � =:gf d � j=1 σtj(xj2 − xj1) = Nt(gf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ H2 d,k−1) and similarly that M(1Ae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = M(gf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ H2 d,k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It then follows from the induction hypotheses that Nt(1Ae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = M(1Ae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) + O(ε1) + Oε1(q−1/2) for any ε1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If we choose ε1 := 2−C1 ε−2k+1 , with C1 ≫ 1 sufficiently large, then ε1 2Cε−2k+1 = O(ε) and it follows that Nt( ¯fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = M( ¯fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) + O(ε) + Oε(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This, together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='13) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14), establishes that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) hold for d = k as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We start by observing the following consequence of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6), namely that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15) ���Ex1,x2∈F2qf1(x1)f2(x2)σt(x2 − x1) ��� 2 ≤ Ex1,x2∈F2qf1(x1)f1(x2) + O(q−1/2) for any f1, f2 : F2 q → [−1, 1] and t ∈ F∗ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Now, fix an edge, say e0 = (11, 21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', k1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Partition the edges e ∈ H2 d,k into three groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' the first group consisting of edges e for which 1 /∈ π(e), the second where 11 ∈ e and write e = (11, e′) with e′ ∈ H2 d−1,k−1 and the third when 12 ∈ e, using the notation H2 d−1,k−1 := {(j2l2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , jklk)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Accordingly we can write (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = Ex∈V 2 � 1/∈π(e) fe(xe) � e′∈H2 d−1,k−1 f(11,e′)(x11, xe′) � e′∈H2 d−1,k−1 f(12,e′)(x12, xe′) d � j=1 σtj(xj2−xj1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If for given x ∈ V1 and x′ = (x21, x22, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd1, xd2) ∈ V 2 2 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × V 2 d we define g1(x, x′) := � e′∈H2 d−1,k−1 f(11,e′)(x, xe′) and g2(x, x′) := � e′∈H2 d−1,k−1 f(12,e′)(x, xe′) then we can write Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = Ex21,x22,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xd1,xd2 � 1/∈π(e) fe(xe) d � j=2 σtj(xj2 − xj1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17) × Ex11,x12 g1(x11, x′)g2(x12, x′) σt1(x12 − x11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15) we can estimate the inner sum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17) by the square root of Ex11,x12 g1(x11, x′)g1(x12, x′) + O(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 10 NEIL LYALL ´AKOS MAGYAR Thus by Cauchy-Schwarz, and the fact that fe : Vπ(e) → [−1, 1] for all e ∈ H2 d,k, we can conclude that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='18) Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k)2 ≤ Ex11,x12,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xd1,xd2 � e′∈H2 d−1,k−1 f(11,e′)(x11, xe′)f(11,e′)(x12, xe′) d � j=2 σtj(xj2 − xj2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The expression on the right hand side of the inequality above is similar to that in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) except for the following changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The functions fe for 1 /∈ e are eliminated i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' replaced by 1, as well as the factor σt1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The functions f(12,e′), are replaced by f(11,e′) for all e′ ∈ H2 d−1,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Repeating the same procedure for j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , k one eliminates all the factors σtj for 1 ≤ j ≤ k, moreover all the functions fe for edges e such that j /∈ π(e) for some 1 ≤ j ≤ k, which leaves only the edges e so that π(e) = (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , k), moreover for such edges the functions fe are eventually replaced by fe0 = f11,21,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The factors σtj(xj2 −xj1) are not changed for j > k however as the function fe0 does not depend on the variables xjl for j > k, averaging over these variables gives rise to a factor of 1 + O(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus one obtains the following final estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='19) Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k)2k ≤ Ex11,x12,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xk1,xk2 � π(e)=(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',k) fe0(xe) + O(q−1/2) = ∥fe0∥2k □(Vπ(e0)) + O(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This proves the lemma, as it is clear that the above procedure can be applied to any edge in place of e0 = (11, 21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', k1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For a function fe : Vπ(e) → [−1, 1] and a σ-algebra Bπ(e) on Vπ(e) define the energy of fe with respect to Bπ(e) as E(fe, Bπ(e)) := ∥E(fe|Bπ(e))∥2 2 = Ex∈Vπ(e) |E(fe|Bπ(e))(x)|2, and for a family of functions fe and σ-algebras Bπ(e), e ∈ H2 d,k its total energy as E(fe, Bπ(e);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) := � e∈H2 d,k E(fe, Bπ(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will show that if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) does not hold for a family of σ-algebras Bπ(e) = � f′∈∂π(e) Bf′ , then the σ-algebras Bf′ can be refined so that the total energy of the system increases by a quantity depending only on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the functions fe are bounded the total energy of the system is O(1), the energy increment process must stop in Oε(1) steps, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The idea of this procedure appears already in the proof of Szemer´edi’s regularity lemma [20], and have been used since in various places [7, 22, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Initially set Bf′ := {∅, Vf′} and hence Bπ(e) = {∅, Vπ(e)} to be the trivial σ-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Assume that in general (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) does not hold for a family of σ-algebras Bf′, with f′ ∈ Hd,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then there exists an edge e ∈ H2 d,k so that ∥ge∥□(Vπ(e)) ≥ ε, with ge := fe − E(fe|Bπ(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let e = (11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , k1) for simplicity of notation, hence π(e) = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then, with notation x′ = (x12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xk2), one has ε2k ≤ ∥ge∥2k □(Vπ(e)) = Ex11,x12,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xk1,xk2 � l1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',lk=1,2 ge(x1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xklk) ≤ Ex12,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xk2 ���Ex11,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xk1ge(x11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xk1) k � j=1 hj,x′(x11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xj−1 1, xj+1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xk1) ��� for some functions hj,x′ that are bounded by 1 in magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed if and edge e ̸= (11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , k1) then xe does not depend at least one of the variables xj1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus there must be an x′ for which the inner sum in the above expression is at least ε2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Fix such an x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Decomposing the functions hj,x′ into their positive and negative parts and then writing them as an average of indicator functions, one obtains that there sets Bj ⊆ Vπ(e)\\{j} such that ���Ex11,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xk1ge(x11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xk1) k � j=1 1Bj(x11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xj−1 1, xj+1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xk1) ��� ≥ 2−k ε2k WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 11 which can be written more succinctly, using the inner product notation, as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='20) ���⟨fe − E(fe|Bπ(e)), k � j=1 1Bj⟩ ��� ≥ 2−k ε2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For f′ = ∂π(e)\\{j} let B′ f′ be the σ-algebra generated by Bf′ and the set Bj and let B′ π(e) := � f′∈∂π(e) B′ f′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the functions 1Bj are measurable with respect to the σ-algebra B′ π(e) for all 1 ≤ j ≤ k, we have that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='21) ⟨fe − E(fe|B′ π(e)), k � j=1 1Bj⟩ = 0 and hence, by Cauchy-Schwarz, that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='22) ∥E(fe|B′ π(e)) − E(fe|Bπ(e))∥2 2 = ∥E(fe|B′ π(e))∥2 2 − ∥E(fe|Bπ(e))∥2 2 ≥ 2−2k ε2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that the first equality above follows from the fact that conditional expectation function E(f|B) is the orthogonal projection of f to the subspace of B-measurable functions in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This also implies that energy of a function is always increasing when the underlying σ-algebra is refined, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='22) tells us that the energy of fe is increased by at least ck ε2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For f′ /∈ ∂π(e) we set B′ f′ := Bf′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then the total energy of the family fe with respect to the system B′ π(e) = � f′∈∂π(e) B′ f′, e ∈ H2 d,k is also increased by at least ck ε2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is clear that the complexity of the σ-algebras Bf′ are increased by at most 1, hence, as explained above, the lemma follows by applying this energy increment process at most O(ε−2k+1) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The base case of an inductive strategy to establish Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 In this section we will ultimately establish the base case of our more general inductive argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We however start by giving a quick review of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 when d = 1 (which contains both Theorem B and Corollary B as stated in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1), namely the case of a single simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This was originally addressed in [1] and revisited in [12] and [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A Single Simplex in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q ⊆ Rn be a fixed cube and let l(Q) denotes its side length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆0 = {v1 = 0, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , vn} ⊆ Rn be a fixed non-degenerate simplex and define tkl := vk · vl for 2 ≤ k, l ≤ n where “ · ” is the dot product on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Given λ > 0, a simplex ∆ = {x1 = 0, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xn} ⊆ Rn is isometric to λ∆0 if and only if xk · xl = λ2tkl for all 2 ≤ k, l ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus the configuration space Sλ∆0 of isometric copies of λ∆0 is a non-singular real variety given by the above equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let σλ∆0 be natural normalized surface area measure on Sλ∆0, described in [1], [12], and [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is clear that the variable x1 can be replaced by any of the variables xi by redefining the constants tkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any family of functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn : Q → [−1, 1] and 0 < λ ≪ l(Q) we define the multi-linear expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1) N 1 λ∆0,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn) := x1∈Q ˆ x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xn f1(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' fn(xn) dσλ∆0(x2 − x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xn − x1) dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We note that all of our functions are 1-bounded and both integrals, in fact all integrals in this paper, are normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that we are using the normalized integral notation ffl A f := 1 |A| ´ A f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the normalized measure σλ∆0 is supported on Sλ∆0 we will not indicate the support of the variables (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xn) explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note also that if S ⊆ Q is a measurable set and N 1 λ∆0,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) > 0 then S must contain an isometric copy of λ∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The following proposition (with Q = [0, 1]n) is a quantitatively stronger version of Proposition B that appeared in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and hence immediately establishes Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 for d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < ε ≤ 1 there exists an integer J = O(ε−2 log ε−1) with the following property: Given any lacunary sequence l(Q) ≥ λ1 ≥ · · · ≥ λJ and S ⊆ Q, there is some 1 ≤ j < J such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) N 1 λ∆0,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) > � |S| |Q| �n − ε for all λ ∈ [λj+1, λj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 12 NEIL LYALL ´AKOS MAGYAR Our approach to establishing Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 is to compare the above expressions to simpler ones for which it is easy to obtain lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Given a scale 0 < λ ≪ l(Q) we define the multi-linear expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3) M1 λ,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn) := t∈Q x1,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xn∈t+Q(λ) f1(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' fn(xn) dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dxn dt where Q(λ) = [− λ 2 , λ 2 ]n and t + Q(λ) is the shift of the cube Q(λ) by the vector t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if S ⊆ Q is a set of measure |S| ≥ δ|Q| for some δ > 0, then for a given ε > 0, H¨older implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) M1 λ,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) = t∈Q � x∈t+Q(λ) 1S(x) dx �n dt ≥ � t∈Q x∈t+Q(λ) 1S(x) dx dt �n ≥ δn − O(ε), for all scales 0 < λ ≪ ε l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that for any ε > 0 we call a sequence L1 ≥ · · · ≥ LJ ε-admissible if Lj/Lj+1 ∈ N and Lj+1 ≪ ε2Lj for all 1 ≤ j < J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that given any lacunary sequence l(Q) ≥ λ1 ≥ · · · ≥ λJ′ with J′ ≫ (log ε−1) J, one can always finds an ε-admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJ such that for each 1 ≤ j < J the interval [Lj+1, Lj] contains at least two consecutive elements from the original lacunary sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In light of this observation, and the one above regarding a lower bound for M1 λ,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S), our proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 reduces to establishing the following “counting lemma”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer J1 = O(ε−2) such that for any ε-admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJ1 and S ⊆ Q there is some 1 ≤ j < J1 such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) N 1 λ∆0,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) = M1 λ,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) + O(ε) for all λ ∈ [Lj+1, Lj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There are two main ingredients in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, this will be typical to all of our arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The first ingredient is a result which establishes that the our multi-linear forms N 1 λ∆0,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn) are controlled by an appropriate norm which measures the uniformity of distribution of functions f : Q → [−1, 1] with respect to particular scales L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This is analogous to estimates in additive combinatorics [8] which are often referred to as generalized von-Neumann inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The result below was proved in [12] for Q = [0, 1]n, however a simple scaling of the variables xi transfers the result to an arbitrary cube Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 (A Generalized von-Neumann inequality [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ε > 0, 0 < λ ≪ l(Q), and 0 < L ≪ ε6λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any collections of functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn : Q → [−1, 1] we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) |N 1 λ∆0,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn)| ≤ min i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',n ∥fi∥U1 L(Q) + O(ε) where for any f : Q → [−1, 1] we define (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) ∥f∥2 U1 L(Q) := t∈Q ��� x∈t+Q(L) f(x) dx ��� 2 dt with t + Q(L) denoting the shift of the cube Q(L) = [− L 2 , L 2 ]n by the vector t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The corresponding inequality for the multilinear expression M1 λ,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn), namely the fact that M1 λ,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn) ≤ min i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',n ∥fi∥U1 L(Q) + O(ε) whenever 0 < L ≪ ε6λ follows easily from Cauchy-Schwarz together with the simple observation that ∥f∥U1 L(Q) ≤ ∥f∥U1 L′(Q) + O(ε) whenever L′ ≪ εL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The second key ingredient, proved in [13] and generalized in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 below, is a Koopman-von Neumann type decomposition of functions where the underlying σ-algebras are generated by cubes of a fixed length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' To recall it, let Q ⊆ Rn be a cube, L > 0 be scale that divides l(Q), Q(L) = [− L 2 , L 2 ]n, and GL,Q denote the collection of cubes t + Q(L) partitioning the cube Q and ΓL,Q denote the grids corresponding to the centers of the cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By a slightly abuse of notation we also write GL,Q for the σ-algebra generated by the WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 13 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that the conditional expectation function E(f|GL,Q) is constant and equal to ffl A f on each cube A ∈ GL,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (A Koopman-von Neumann type decomposition [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≤ 1 and Q ⊆ Rn be a cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer ¯J1 = O(ε−2) such that for any ε-admissible sequence l(Q) ≥ L1 ≥ · · · ≥ L ¯ J1 and function f : Q → [−1, 1] there is some 1 ≤ j < ¯J1 such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) ∥f − E(f|GLj,Q)∥U1 Lj+1 (Q) ≤ ε Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let GLj,Q be the grid obtained from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 for the functions f = 1S for some fixed ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ¯f := E(f|GLj,Q), then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) and multi-linearity, we have N 1 λ∆0,Q(f, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f) = N 1 λ∆0,Q( ¯f, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ¯f) + O(ε), and also M1 λ,Q(f, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f) = M1 λ,Q( ¯f, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ¯f) + O(ε) provided for ε−6Lj+1 ≪ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus in showing (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) one can replace the functions f with ¯f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If we make the additional assumption that λ ≪ εLj then it is easy to see, using the fact that the function ¯f is constant on the cubes Qt(Lj) ∈ GLj,Q, that N 1 λ∆0,Q( ¯f, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ¯f) = M1 λ,Q( ¯f, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ¯f) + O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the condition ε−6Lj+1 ≪ λ ≪ εLj can be replaced with Lj+1 ≪ λ ≪ Lj if one passes to a subsequence of scales, for example L′ j = L5j, this completes the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The base case of a general inductive strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In this section, as preparation to handle the case of products of simplices, we prove a parametric version of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, namely Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 below, which will serve as the base case for later inductive arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q = Q1 × · · · × Qd with Qi ⊆ Rni be cubes of equal side length l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let L be a scale dividing l(Q) and for each t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td) ∈ ΓL,Q let Qt(L) = t + Q(L) and Qti(L) = ti + Qi(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that Qt(L) = Qt1(L) × · · · × Qtd(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Here Q(L) = [− L 2 , L 2 ]n and Qi(L) = [− L 2 , L 2 ]ni for each 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆0 i = {vi 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , vi ni} ⊆ Rni be a non-degenerate simplex for each 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 (Parametric Counting Lemma on Rn for Simplices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≤ 1 and R ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer J1 = J1(ε, R) = O(R ε−4) such that for any ε-admissible sequence of scales L0 ≥ L1 ≥ · · · ≥ LJ1 with the property that L0 divides l(Q) and collection of functions f i,r k,t : Qti(L0) → [−1, 1] with 1 ≤ i ≤ d, 1 ≤ k ≤ ni, 1 ≤ r ≤ R and t ∈ ΓL0,Q there exists 1 ≤ j < J1 and a set Tε ⊆ ΓL0,Q of size |Tε| ≤ ε|ΓL0,Q| such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) N 1 λ∆0 i ,Qti (L0)(f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f i,r ni,t) = M1 λ,Qti (L0)(f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f i,r ni,t) + O(ε) for all λ ∈ [Lj+1, Lj] and t /∈ Tε uniformly in 1 ≤ i ≤ d and 1 ≤ r ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 will follow from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and the following generalization of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 in which we simultaneously consider a family of functions supported on the subcubes in a partition of an original cube Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 (A simultaneous Koopman-von Neumann type decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≤ 1, m ≥ 1, and Q ⊆ Rn be a cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer ¯J1 = O(mε−3) such that for any ε-admissible sequence L0 ≥ L1 ≥ · · · ≥ L ¯ J1 with the property that L0 divides l(Q), and collection of functions f1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fm,t : Qt(L0) → [−1, 1] defined for each t ∈ ΓL0,Q, there is some 1 ≤ j < ¯J1 and a set Tε ⊆ ΓL0,Q of size |Tε| ≤ ε|ΓL0,Q| such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10) ∥fi,t − E(fi,t|GLj,Qt(L0))∥U1 Lj+1(Qt(L0)) ≤ ε for all 1 ≤ i ≤ m and t /∈ Tε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 14 NEIL LYALL ´AKOS MAGYAR Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Fix 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For 1 ≤ k ≤ ni and t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td) ∈ ΓL0,Q , we will abuse notation and write f i,r k,t(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd) := f i,r k,t(xi) for (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd) ∈ Qt(L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If we apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 to the family of functions f i,r k,t on Qt(L0) for 1 ≤ i ≤ d, 1 ≤ k ≤ ni, and 1 ≤ r ≤ R, with m = (n1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' + nd)R, then this produces a grid GLj,Q for some 1 ≤ j ≤ ¯J1 = O(ε−3R), and a set Tε ⊆ ΓL0,Q of size |Tε| ≤ ε|ΓL0,Q|, such that ∥f i,r k,t − E(f i,r k,t|GLj,Q)∥U1 Lj+1 (Qt(L0)) ≤ ε uniformly for 1 ≤ i ≤ d, 1 ≤ k ≤ ni and 1 ≤ r ≤ R for t /∈ Tε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since f i,r k,t(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd) = f i,r k,t(xi) for (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd) ∈ Qt(L0) it is easy to see that ∥f i,r k,t − E(f i,r k,t|GLj,Q)∥U1 Lj+1 (Qt(L0)) = ∥f i,r k,t − E(f i,r k,t|GLj,Qi)∥U1 Lj+1(Qti (L0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ¯f i,r k,t := E(f i,r k,t|GLj,Qi) , then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, one has N 1 λ∆0 i ,Qti(L0)(f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f i,r ni,t) = N 1 λ∆0 i ,Qti (L0)( ¯f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ¯f i,r ni,t) + O(ε), and M1 λ,Qti (L0)(f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f i,r ni,t) = M1 λ,Qti (L0)( ¯f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ¯f i,r ni,t) + O(ε) for all t /∈ Tε provided ε−6Lj+1 ≪ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Finally, if we also have λ ≪ εLj then it is easy to see that N 1 λ∆0 i ,Qti (L0)( ¯f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ¯f i,r ni,t) = M1 λ,Qti (L0)( ¯f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ¯f i,r ni,t) + O(ε) as the functions ¯f i,r k,t are constant on cubes Qti(Lj) of GLj,Qi, which are of size Lj ≪ εL0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Passing first to a subsequence of scales, for example L′ j = L5j, the condition ε−6Lj+1 ≪ λ ≪ εLj can be replaced with Lj+1 ≪ λ ≪ Lj so this completes the proof of the Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ We conclude this section with a sketch of the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' These arguments are standard, see for example the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 given in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' First we make an observation about the U 1 L(Q)-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Suppose 0 < L′ ≪ ε2L with L′ dividing L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If s ∈ ΓL′,Q and t ∈ Qs(L′) then |t − s| = O(L′) and hence x∈Qt(L) g(x) dx = x∈Qs(L) g(x) dx + O(L′/L) for any function g : Q → [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover, since the cube Qs(L) is partitioned into the smaller cubes Qt(L′), we have by Cauchy-Schwarz ��� x∈Qs(L) g(x) dx ��� 2 ≤ Et∈ΓL′,Qs(L) ��� x∈Qt(L′) g(x) dx ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' From these observations it is easy to see that ∥g∥2 U1 L(Q) = t∈Q ��� x∈Qt(L) g(x) dx ��� 2 dt ≤ Et∈ΓL′,Q ��� x∈Qt(L′) g(x) dx ��� 2 + O(L′/L) and we note that the right side of the above expression is ∥E(g|GL′,Q)∥2 L2(Q) since the conditional expectation function E(g|GL′,Q) is constant and equal to ffl x∈Qt(L′) g(x) dx on the cubes Qt(L′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Suppose that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10) does not hold for some 1 ≤ i ≤ m for every t in some set Tε ⊆ ΓL0,Q of size |Tε| > ε |ΓL0,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If we apply the above observation to g := fi,t − E(fi,t|GLj,Qt(L0)), for every t ∈ Tε, we obtain by orthogonality that m � i=1 ∥E(fi,t|GLj+2,Qt(L0))∥2 L2(Qt(L0)) ≥ m � i=1 ∥E(fi,t|GLj,Qt(L0))∥2 L2(Qt(L0)) + cε2 for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 15 If we now define fi : Q → [−1, 1] such that fi|(Qt(L0)) = fi,t, for 1 ≤ i ≤ m, average over t ∈ ΓL0,Q, and use the fact ∥fi∥2 L2(Q) = Et∈ΓL0,Q∥fi,t∥2 L2(Qt(L0)), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) m � i=1 ∥E(fi|GLj+2,Q)∥2 L2(Q) ≥ m � i=1 ∥E(fi|GLj,Q)∥2 L2(Q) + cε3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is clear that the sums in the above expressions are bounded by m for all j ≥ 1, thus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) cannot hold for some 1 ≤ j ≤ ¯J1 for ¯J1 := C m ε−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This implies that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10) must hold for some 1 ≤ j ≤ ¯J1, for all 1 ≤ i ≤ m and all t /∈ Tε for a set Tε ⊆ ΓL0,Q of size |Tε| ≤ ε |ΓL0,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Product of two simplices in Rn Although not strictly necessary, we discuss in this section the special case d = 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This already gives an improvement of the main results of [12], but more importantly serves as a gentle preparation for the more complicated general case, presented in the Section 5, which involve both a plethora of different scales and the hypergraph bundle notation introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 with d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q = Q1 × Q2 with Q1 ⊆ Rn1 and Q2 ⊆ Rn2 be cubes of equal side length l(Q) and ∆0 = ∆0 1 × ∆0 2 with ∆0 1 = {v11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , v1n1} ⊆ Rn1 and ∆0 2 = {v11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , v2n2} ⊆ Rn2 two non-degenerate simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In order to “count” configurations of the form ∆ = ∆1 × ∆2 ⊆ Rn1+n2 with ∆1 and ∆2 isometric copies of λ∆0 1 and λ∆0 2 respectively for some 0 < λ ≪ l(Q) in a set S ⊆ Q we introduce the multi-linear expression N 2 λ∆0,Q({fkl}) := x11∈Q1 x21∈Q2 ˆ x12,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',x1n1 ˆ x22,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',x2n2 n1 � k=1 n2 � l=1 fkl(x1k, x2l) dσλ∆0 1(x12 − x11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , x1n1 − x11) dσλ∆0 2(x22 − x21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , x2n2 − x21) dx21 dx11 for any family of functions fkl : Q1 × Q2 → [−1, 1] with 1 ≤ k ≤ n1 and 1 ≤ l ≤ n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed, if fkl = 1S for all 1 ≤ k ≤ n1 and 1 ≤ l ≤ n2 then the above expression is 0 unless there exists a configuration ∆ ⊆ S of the form ∆1 × ∆2 with ∆1 and ∆2 isometric copies of λ∆0 1 and λ∆0 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The short argument presented in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 demonstrating how both Theorem B and Corollary B follow from Proposition B, and hence from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, applies equally well to each of our main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This reduces our main theorems to analogous quantitative results involving an arbitrary lacunary sequence of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In the case d = 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 this stronger quantitative result takes the following form: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < ε ≪ 1 there exists an integer J = O(exp(Cε−13)) with the following property: Given any lacunary sequence l(Q) ≥ λ1 ≥ · · · ≥ λJ and S ⊆ Q, there is some 1 ≤ j < J such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1) N 2 λ∆0,Q({1S}) > � |S| |Q| �n1n2 − ε for all λ ∈ [λj+1, λj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our approach to establishing Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 is again to compare the above expressions to simpler ones for which it is easy to obtain lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < λ ≪ l(Q) and family of functions fkl : Q1×Q2 → [−1, 1] with 1 ≤ k ≤ n1 and 1 ≤ l ≤ n2 we consider M2 λ,Q({fkl}) := t∈Q x1∈(t1+Q1(λ))n1 x2∈(t2+Q2(λ))n2 n1 � k=1 n2 � l=2 fkl(x1k, x2l) dx2 dx1 dt where t = (t1, t2) ∈ Q1 × Q2, xi = (xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xini) and Qi(λ) = [− λ 2 , λ 2 ]ni for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if S ⊆ Q is a set of measure |S| ≥ δ|Q| for some δ > 0, then careful applications of H¨older’s inequality give M2 λ,Q({1S}) ≥ t∈Q � (x1,x2)∈t+Q(λ) 1S(x1, x2) dx1dx2 �n1n2 dt ≥ δn1n2 − O(ε) for all scales 0 < λ ≪ ε l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 16 NEIL LYALL ´AKOS MAGYAR In light of the observation above, and the discussion preceding Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, we see that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, and hence Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 when d = 2, will follows as a consequence of the following Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer J2 = O(exp(Cε−12)) such that for any ε-admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJ2 and S ⊆ Q there is some 1 ≤ j < J2 such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) N 2 λ∆0,Q({1S}) = M2 λ,Q({1S}) + O(ε) for all λ ∈ [Lj+1, Lj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There are again two main ingredients in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The first establishes that the our multi-linear forms N 2 λ∆0,Q({fkl}) are controlled by an appropriate box-type norm attached to a scale L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q = Q1 × Q2 be a cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any scale 0 < L ≪ l(Q) and function f : Q → R we define its local box norm at scale L to be (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3) ∥f∥4 □L(Q1×Q2) := t∈Q ∥f∥4 □(t+Q(L)) dt where Q(L) = [− L 2 , L 2 ]n1+n2 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) ∥f∥4 □( � Q) := x11,x12∈ � Q1 x21,x22∈ � Q2 f(x11, x21)f(x12, x21)f(x11, x22)f(x12, x22) dx11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dx22 for any cube �Q ⊆ Q of the form �Q = �Q1 × �Q2 with �Qj ⊆ Qj for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 (A Generalized von-Neumann inequality [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ε > 0, 0 < λ ≪ l(Q), and 0 < L ≪ ε24λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any collections of functions fkl : Q1 × Q2 → [−1, 1] with 1 ≤ k ≤ n1 and 1 ≤ l ≤ n2 we have both (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) |N 2 λ∆0,Q({fkl})| ≤ min 1≤k≤n1, 1≤l≤n2 ∥fkl∥□L(Q1×Q2) + O(ε) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) |M2 λ,Q({fkl})| ≤ min 1≤k≤n1, 1≤l≤n2 ∥fkl∥□L(Q1×Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The result above was essentially proved in [12] for the multi-linear forms N 2 λ∆0,Q when Q = [0, 1]n1+n2, however a simple scaling argument transfers the result to an arbitrary cube Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For completeness we include its short proof in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The second and main ingredient is an analogue of a weak form of Szemer´edi’s regularity lemma due to Frieze and Kannan [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The more probabilistic formulation, we will use below, can be found for example in [21], [22], and [23], and is also sometimes referred to as a Koopman-von Neumann type decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any cube Q ⊆ Rn and scale L > 0 that divides l(Q) we will let Q(L) = [− L 2 , L 2 ]n and GL,Q denote the collection of cubes Qt(L) = t + Q(L) partitioning the cube Q and let ΓL,Q denote grid corresponding to the centers of these cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will say that a finite σ-algebra B on Q is of scale L if it contains GL,Q and for simplicity of notation will write Bt for B|Qt(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that if we have two σ-algebras B1 on a cube Q1 and B2 on Q2 then by B1 ∨ B2 we mean the σ-algebra on Q = Q1 × Q2 generated by the sets B1 × B2 with B1 ∈ B1 and B2 ∈ B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall also that we say the complexity of a σ-algebra B is at most m, and write complex(B) ≤ m, if it is generated by m sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (Weak regularity lemma in Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1 and Q = Q1 × Q2 with Q1 ⊆ Rn1 and Q2 ⊆ Rn2 be cubes of equal side length l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer ¯J2 = O(ε−12) such that for any ε4-admissible sequence l(Q) ≥ L1 ≥ · · · ≥ L ¯ J2 and function f : Q → [−1, 1] there is some 1 ≤ j ≤ ¯J2 and a σ-algebra B of scale Lj on Q such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) ∥f − E(f|B)∥□Lj+1(Q1×Q2) ≤ ε which has the additional local structure that for each t = (t1, t2) ∈ ΓLj,Q there exist σ-algebras B1,t on Qt1(Lj) and B2,t on Qt2(Lj) with complex(Bi,t) = O(j) for i = 1, 2 such that Bt = B1,t ∨ B2,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Comparing the above statement to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 for d = 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='e to the weak regularity lemma, note that the σ-algebra B of scale Lj has a direct product structure only locally, inside each cube Qt(Lj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover this product structure varies with t ∈ ΓLj,Q, however the “local complexity” remains uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 17 Assuming for now the validity of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 we prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will make crucial use of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3, namely our parametric counting lemma on Rn for simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1, ε1 := exp(−C1ε−12) for some C1 ≫ 1, and {Lj}j≥1 be an ε1- admissible sequence of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Set R = ε ε−1 1 and J1(ε1, R) be the parameter appearing in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3, noting that J1(ε1, R) = O(ε−5 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For L ∈ {Lj}j≥1 write index(L) = j if L = Lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We now choose a subsequence {L′ j} ⊆ {Lj} so that L′ 1 = L1 and index(L′ j+1) ≥ index(L′ j) + J1(ε1, R) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, with fkl = f := 1S for all 1 ≤ k ≤ n1 and 1 ≤ l ≤ n2, guarantees the existence of a σ-algebra B of scale L′ j on Q such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) ∥f − E(f|B)∥□L′ j+1(Q1×Q2) ≤ ε for some 1 ≤ j ≤ Cε−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover, we know that B has the additional local structure that for each t = (t1, t2) ∈ ΓL′ j,Q there exist σ-algebras B1,t on Qt1(L′ j) and B2,t on Qt2(L′ j) with complex(Bi,t) = O(ε−12) for i = 1, 2 such that Bt = B1,t ∨ B2,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus, if we let R1,t and R2,t denote the number of atoms in B1,t and B2,t respectively, then we can assume, by formally adding the empty set to these collections of atoms if necessary, that R1,t = R2,t = R′ := exp(Cε−12) for all t ∈ ΓL′ j,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If we let ¯f := E(f|B1 ∨ B2), then by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and multi-linearity we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) N 2 λ∆0,Q({f}) = N 2 λ∆0,Q({ ¯f}) + O(ε) and M2 λ,Q({f}) = M2 λ,Q({ ¯f}) + O(ε) provided for ε−24L′ j+1 ≪ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For a given t ∈ ΓQ,L′ j write ¯ft for the restriction of ¯f to the cube Qt(L′ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By localization, one then has (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10) N 2 λ∆0,Q({ ¯f}) = Et∈ΓL′ j ,Q N 2 λ∆0,Qt(L′ j)({ ¯ft}) + O(ε), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) M2 λ,Q({ ¯f}) = Et∈ΓL′ j,Q M2 λ,Qt(L′ j)({ ¯ft}) + O(ε) provided one also insists that λ ≪ ε L′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For given t ∈ ΓL′ j,Q, the functions ¯ft(x1, x2) are linear combinations of functions of the form 1Ar1 1,t(x1)1Ar2 2,t(x2), where {Ar1 1,t}1≤r1≤R′ and {Ar2 2,t}1≤r2≤R′ are the collections of the atoms of the σ-algebras B1,t and B2,t defined on the cubes Qt1(L′ j) and Qt2(L′ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus for each t ∈ ΓL′ j,Q one has ¯ft = R′ � r1=1 R′ � r2=1 αr,t1Ar1 1,t × 1Ar2 2,t where r = (r1, r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Plugging these linear expansions into the multi-linear expressions in above one obtains N 2 λ∆0,Qt(L′ j)({ ¯ft}) = � r={rkl}kl αr,t N 2 λ∆0,Qt(L′ j)({1A r1,kl 1,t × 1A r2,kl 2,t }) using the notations rkl = (r1,kl, r2,kl), αr,t = � kl αrkl,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Notice that the product n1 � k=1 n2 � l=1 1A r1,kl 1,t (x1k)1A r2,kl 2,t (x2l) is nonzero only if Ar1,kl 1,t = Ar1,k 1,t , that is if r1,kl = r1,k for all 1 ≤ l ≤ n2, as the atoms Ar 1,t are all disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Similarly, one has that r2,kl = r2,l for all 1 ≤ k ≤ n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus, in fact (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) N 2 λ∆0,Qt(L′ j)({ ¯ft}) = � r={rkl}kl αr,t N 2 λ∆0,Qt(L′ j)({1A r1,k 1,t × 1A r2,l 2,t }) and similarly (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='13) M2 λ,Qt(L′ j)({ ¯ft}) = � r={rkl}kl αr,t M2 λ,Qt(L′ j)({1A r1,k 1,t × 1A r2,l 2,t }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note, that indices r are running through the index set [1, R′]n1 × [1, R′]n2 of size at most R if C1 ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 18 NEIL LYALL ´AKOS MAGYAR The key observation is that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14) N 2 λ∆0,Qt(L′ j)(1A r1,k 1,t × 1A r2,l 2,t ) = N 1 λ∆0 1,Qt1 (L′ j)(1A r1,1 1,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1A r1,n1 1,t ) N 1 λ∆0 2,Qt2 (L′ j)(1A r2,1 2,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1A r2,n2 2,t ) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15) M2 λ,Qt(L′ j)(1A r1,k 1,t × 1A r2,l 2,t ) = M1 λ,Qt1 (L′ j)(1A r1,1 1,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1A r1,n1 1,t ) M1 λ,Qt2 (L′ j)(1A r2,1 2,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1A r2,n2 2,t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let r = {(r1,k, r2,l)}kl and g1,r k,t := 1A r1,k 1,t , g2,r l,t := 1A r2,l 2,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Writing j′ := index(L′ j) and J′ := index(L′ j+1), one may apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 for the families of functions g1,r k,t, g2,r l,t , where 1 ≤ k ≤ n1, 1 ≤ l ≤ n2 and r = (r1,k, r2,l)kl ∈ [1, R′]n1 × [1, R′]n2, with respect to the ε1-admissible sequence of scales Lj′+1 ≥ Lj′+2 ≥ · · · ≥ LJ′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This is possible as J′ − j′ = J1(ε1, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then there is a scale Lj with j′ ≤ j < J′ so that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) N 1 λ∆0 1,Qt1 (L′ j)(g1,r 1,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , g1,r n1t) = M1 λ,Qt1 (L′ j)(g1,r 1,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , g1,r n1,t) + O(ε1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17) N 1 λ∆0 2,Qt2 (L′ j)(g2,r 1,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , g2,r n2,t) = M1 λ,Qt2 (L′ j)(g2,r 1,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , g2,r n2,t) + O(ε1), for all λ ∈ [Lj+1, Lj] uniformly in r = {(r1,k, r2,l)}kl and t /∈ Tε1 ⊆ ΓL′ j,Q, for a set of size |Tε1| ≤ ε1|ΓL′ j,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='13), we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='18) N 2 λ∆0,Qt(L′ j)({ ¯ft}) = M2 λ,Qt(L′ j) ({ ¯ft}) + O(ε) for t /∈ Tε1, as |αr,t| ≤ 1 and Rε1 ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Finally, since |Tε1| ≤ ε1|ΓL′ j,Q|, by averaging in t ∈ ΓL′ j,Q, one has N 2 λ∆0,Q({ ¯f}) = M2 λ,Q ({ ¯f}) + O(ε) using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) and the Proposition follows by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) with an index 1 ≤ j < J2 = O(ε−12ε−5 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' First we note that if χL := L−n1[−L/2,L/2]n and ψL := χL ∗ χL, then ψL(x2 − x1) = ˆ t χL(x1 − t)χL(x2 − t) dt and hence for any function f : Q → [−1, 1], with Q ⊆ Rn being a cube of side length l(Q), one has ∥f∥2 U1 L(Q) = x1∈Q ˆ x2 f(x1)f(x2)ψL(x2 − x1) dx1dx2 + O(L/l(Q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Write x′ := (x21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , x2n2) and let gk,x′(x) := �n2 l=1 fkl(x, x2l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then one may write N 2 λ∆0,Q({fkl}) = x21∈Q2 ˆ x22,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',x2n2 N 1 λ∆0 1,Q1(g1,x′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , gn1,x′) dσλ∆0 2(x22 − x21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , x2n2 − x21) dx21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Using estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6), the above observation, and Cauchy-Schwarz one has |N 2 λ∆0,Q({fkl})|2 ≤ x11∈Q1 ˆ x12 ψL(x12 − x11) N 1 λ∆0 2,Q2(h1,x11,x12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , hn2,x11,x12) dx11dx12 + O(ε4) provided 0 < λ ≪ l(Q) and 0 < L ≪ ε24λ where hl,x11,x12(x) = f1l(x11, x)f1l(x12, x) for 1 ≤ l ≤ n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Applying the same procedure again ultimately gives |N 2 λ∆0,Q({fkl})|4 ≤ ∥f11∥4 □L(Q1×Q2) + O(ε4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The same estimate can of course be given for any function fkl in place of f11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This establishes (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) is established similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 19 Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For each t = (t1, t2) ∈ ΓL1,Q we will let B1,t(L1) := {∅, Qt1(L1)} and B2,t(L1) := {∅, Qt2(L1)}, in other words the trivial σ-algebras on Qt1(L1) and Qt2(L1) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) holds with B(L1) = GL1,Q, noting that Bt(L1) := B1,t(L1) ∨ B2,t(L1) in this case, then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We now assume that we have developed, for each t = (t1, t2) ∈ ΓLj,Q, σ-algebras B1,t(Lj) on Qt1(Lj) and B2,t(Lj) on Qt2(Lj) with complex(Bi,t(Lj)) ≤ j for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let B(Lj) be the σ-algebra such that Bt(Lj) = B1,t(Lj) ∨ B2,t(Lj) for all t ∈ ΓLj,Q and assume that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) does not hold, namely that ∥g∥□Lj+1(Q) ≥ ε where g := f − E(f|B(Lj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By the definition of the local box norm this means that t∈Q ∥g∥4 □(t+Q(Lj+1)) dt ≥ ε4 and hence, as Lj+2 ≪ ε4Lj+1, it is easy to see that Es∈ΓLj+2,Q ∥g∥4 □(s+Q(Lj+2)) ≥ ε4/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This implies that there is a set S ⊆ ΓLj+2,Q of size |S| ≥ (ε4/4)|ΓLj+2,Q| such that for all s = (s1, s2) ∈ S, one has that ∥g∥4 □(Qs(Lj+2)) ≥ ε4/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It therefore follows, as is well-known see for example [12] or [23], that there exist sets B1,s ⊆ Qs1(Lj+2) and B2,s ⊆ Qs2(Lj+2) such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='19) x1∈Qs1 (Lj+2) x2∈Qs2 (Lj+2) g(x1, x2) 1B1,s(x1)1B2,s(x2) dx1 dx2 ≥ ε4/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For a given s ∈ ΓLj+2,Q there is a unique t = t(s) such that Qs(Lj+2) ⊆ Qt(Lj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let B′ 1,s(Lj+2) := B1,t(Lj)|Qs1 (Lj+2) and B′ 2,s(Lj+2) := B2,t(Lj)|Qs2 (Lj+2) noting that complex(B′ i,s(Lj+2)) ≤ j for i = 1, 2, as the complexity of a σ-algebra does not increase when restricted to a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If, for i = 1, 2, we let Bi,s(Lj+2) denote the σ-algebra generated by B′ i,s(Lj+2) and the set Bi,s if s ∈ S and let Bi,s(Lj+2) := B′ i,s(Lj+2) otherwise, then clearly complex(Bi,s(Lj+2)) is at most j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We now define B(Lj+2) to the the sigma algebra of scale Lj+2 with the property that Bs(Lj+2) = B1,s(Lj+2) ∨ B2,s(Lj+2) for all s ∈ ΓLj+2,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Using the inner product notation ⟨f, g⟩Q = ffl Q f(x)g(x) dx we can rewrite (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='19) as ⟨f − E(f|B(Lj)) , 1B1,s × 1B2,s ⟩Qs(Lj+2) ≥ ε4/16 for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the function 1B1,s × 1B2,s is measurable with respect to B(Lj+2) one clearly has ⟨f − E(f|B(Lj+2)) , 1B1,s × 1B2,s⟩Qs(Lj+2) = 0 and hence ⟨E(f|B(Lj+2)) − E(f|B(Lj)) , 1B1,s × 1B2,s⟩Qt(Lj+2) ≥ ε4/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It then follows from Cauchy-Schwarz and orthogonality that ∥E(f|B(Lj+2))∥2 L2(Qs(Lj+2)) − ∥E(f|B1(Lj))∥2 L2(Qs(Lj+2)) ≥ ε8/256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since |S| ≥ (ε4/4)|ΓLj+2,Q| averaging over all s ∈ ΓLj+2,Q gives ∥E(f|B(Lj+2))∥2 L2(Q) ≥ ∥E(f|B(Lj))∥2 L2(Q) + ε12/210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Trivially both sides are at most 1 thus the process must stop at a step j = O(ε−12) where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) holds for a σ-algebra of “local complexity” at most j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This proves the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2: The general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' After these preparations we will now consider the general case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q = Q1×· · ·×Qd ⊆ Rn with Qi ⊆ Rni cubes of equal side length l(Q) and ∆0 = ∆0 1 ×· · ·×∆0 d with each ∆i ⊆ Rni a non-degenerate simplex of ni points for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will use a generalized version of the hypergraph terminology introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In particular, for a vertex set I = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', d} and set K = {il;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1 ≤ i ≤ d, 1 ≤ l ≤ ni} we will let π : K → I denote the 20 NEIL LYALL ´AKOS MAGYAR projection defined by π(il) := i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' As before we will let Hd,k := {e ⊆ I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' |e| = k} denote the complete k-regular hypergraph with vertex set I, and for the multi-index n = (n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , nd) define the hypergraph bundle Hn d,k := {e ⊆ K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' |e| = |π(e)| = k} noting that |π−1(i)| = ni for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In order to parameterize the vertices of direct products of simplices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' sets of the form ∆ = ∆1×· · ·×∆d with ∆i ⊆ Qi, we consider points x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd) with xi = (xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xini) ∈ Qni i for each i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Now for any 1 ≤ k ≤ d and any edge e′ ∈ Hd,k we will write Qe′ := � i∈e′ Qi, and for every x ∈ Qn1 1 × · · · × Qnd d and e ∈ Hn d,k we define xe := πe(x), where πe : Qn1 1 × · · · × Qnd d → Qπ(e) is the natural projection map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Writing ∆i = {xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xini} we have that ∆1 × · · · × ∆d = {xe : e ∈ Hn d,d} since every edge xe is of the form (x1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xdld).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We can therefore identify points x with configurations of the form ∆1 × · · · × ∆d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < λ ≪ l(Q) the measures dσλ∆0 i , introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, are supported on points (y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , yni) for which the simplex ∆i = {0, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , yni} is isometric to λ∆0 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For simplicity of notation we will write ˆ xi f(xi) dσλ i (xi) := xi1∈Qi ˆ xi2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xini f(xi) dσλ∆0 i (xi2 − xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xini − xi1) dxi1 Note that the support of the measure dσλ i is the set of points xi so that the simplex ∆i := {xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xini} is isometric to λ∆0 i and xi1 ∈ Qi, moreover the measure is normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus if S ⊆ Q is a set then the density of configurations ∆ in S of the form ∆ = ∆1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × ∆d with each ∆i ⊆ Qi an isometric copy of λ∆0 i is given by the expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1) N d λ∆0,Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) := ˆ x1 · · ˆ xd � e∈Hn d,d 1S(xe) dσλ 1 (x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dσλ d (xd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 reduces to establishing the following stronger quantitative result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < ε ≪ 1 there exists an integer Jd = Jd(ε) with the following property: Given any lacunary sequence l(Q) ≥ λ1 ≥ · · · ≥ λJd and S ⊆ Q, there is some 1 ≤ j < Jd such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) N d λ∆0,Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) > � |S| |Q| �n1··· nd − ε for all λ ∈ [λj+1, λj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Quantitative Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A careful analysis of our proof reveals that there is a choice of Jd(ε) which is less than Wd(log(C∆ε−3)), where Wk(m) is again the tower-exponential function defined by W1(m) = exp(m) and Wk+1(m) = exp(Wk(m)) for k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < λ ≪ l(Q) and set S ⊆ Q we define the expression: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3) Md λ,Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) := t∈Q Md t+Q(λ)(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) dt where Q(λ) = [− λ 2 , λ 2 ]n and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) Md � Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) := x1∈ � Qn1 1 · · xd∈ � Q nd d � e∈Hn d,d 1S(xe) dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dxd for any cube �Q ⊆ Q of the form �Q = �Q1 × · · · × �Qd with �Qi ⊆ Qi for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if S ⊆ Q is a set of measure |S| ≥ δ|Q| for some δ > 0, then careful applications of H¨older’s inequality give Md λ,Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) ≥ t∈Q � (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xd)∈t+Q(λ) 1S(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd) dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dxd �n1··· nd dt ≥ δn1··· nd − O(ε) for all scales 0 < λ ≪ ε l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In light of the discussion above, and that preceding Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, we see that Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, and hence Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 in general, will follows as a consequence of the following WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 21 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer Jd = Jd(ε) such that for any ε-admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJd and S ⊆ Q there is some 1 ≤ j < Jd such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) N d λ∆0,Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) = Md λ,Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) + O(ε) for all λ ∈ [Lj+1, Lj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The validity of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 will follow immediately from the d = k case of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Reduction of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 to a more general “local” counting lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any given 1 ≤ k ≤ d and collection of functions fe : Qπ(e) → [−1, 1] with e ∈ Hn d,k we define the following multi-linear expressions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) N d λ∆0,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) := ˆ x1 · · ˆ xd � e∈Hn d,k fe(xe) dσλ 1 (x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='dσλ d (xd) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) Md λ,Q(fe ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) := t∈Q Md t+Q(λ)(fe ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) dt where Q(λ) = [− λ 2 , λ 2 ]n and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) Md � Q(fe ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) := x1∈ � Qn1 1 · · xd∈ � Q nd d � e∈Hn d,k fe(xe) dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dxd for any cube �Q ⊆ Q of the form �Q = �Q1 × · · · × �Qd with �Qi ⊆ Qi for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our strategy to proving Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 is the same as illustrated in the finite field settings, that is we would like to compare averages Nλ∆0,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) to those of Md λ,Q(fe ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k), at certain scales λ ∈ [Lj+1, Lj], inductively for 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' However in the Euclidean case, an extra complication emerges due to the fact the (hypergraph) regularity lemma, the analogue of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, does not produce σ-algebras Bf, for f ∈ Hn d,k−1, on the cubes Qf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In a similar manner to the case for d = 2 discussed in the previous section, we will only obtain σ-algebras “local” on cubes Qtf(L0) at some scale L0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This will have the effect that the functions fe will be replaced by a family of functions fe,t, where t runs through a grid ΓL0,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' To be more precise, let L > 0 be a scale dividing the side-length l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For t ∈ ΓL,Q and e′ ∈ Hd,k we will use te′ to denote the projection of t onto Qe′ and Qte′ (L) := te′ + Qe′(L) to denote the projection of the cube Qt(L) centered at t onto Qe′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is then easy to see that for any ε > 0 we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) N d λ∆0,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Et∈ΓL,Q N d λ∆0,Qt(L)(fe,t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10) Md λ,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Et∈ΓL,Q Md λ,Qt(L)(fe,t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) provided 0 < λ ≪ εL where fe,t denotes the restriction of a function fe to the cube Qt(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' At this point the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 reduces to showing that the expressions in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) only differ by O(ε) at some scales λ ∈ [Lj+1, Lj], given an ε-admissible sequence L0 ≥ L1 ≥ · · · ≥ LJ, for any collection of bounded functions fe,t, e ∈ Hn d,k, t ∈ ΓL0,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed, our crucial result will be the following Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 (Local Counting Lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1 and M ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer Jk = Jk(ε, M) such that for any ε-admissible sequence of scales L0 ≥ L1 ≥ · · · ≥ LJk with the property that L0 divides l(Q), and collection of functions f m e,t : Qtπ(e)(L0) :→ [−1, 1] with e ∈ Hn d,k, 1 ≤ m ≤ M and t ∈ ΓL0,Q there exists 1 ≤ j < Jk and a set Tε ⊆ ΓL0,Q of size |Tε| ≤ ε|ΓL0,Q| such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) N d λ∆0,Qt(L0)(f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Mλ,Qt(L0)(f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) for all λ ∈ [Lj+1, Lj] and t /∈ Tε uniformly in e ∈ Hn d,k and 1 ≤ m ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 22 NEIL LYALL ´AKOS MAGYAR 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will prove Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 by induction on 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For k = 1 this is basically Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed, in this case for a given t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td) ∈ ΓL0,Q and edge e ∈ Hn d,1 = {il : 1 ≤ i ≤ d, 1 ≤ l ≤ ni} we have that f m e,t(xe) = f m il,t(xil) with xil ∈ Qti(L0) and hence both N d λ∆0,Qt(L0)(f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,1) = d � i=1 N 1 λ∆0 i ,Qti (L0)(f m i1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f m ini,t) Md λ,Qt(L0)(f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,1) = d � i=1 M1 λ,Qti (L0)(f m i1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f m ini,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 there exists an 1 ≤ j < J1 = O(Mε−4) and an exceptional set Tε ⊆ ΓL0,Q of size |Tε| ≤ ε|ΓL0,Q|, such that uniformly for t /∈ Tε and for 1 ≤ i ≤ d, one has N 1 λ∆0 i ,Qti(L0)(f m i1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f m ini,t) = M1 λ,Qti (L0)(f m i1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f m ini,t) + O(ε) hence N d λ∆0,Qt(L0)(f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,1) = Md λ,Qt(L0)(f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,1) + O(ε) as the all factors are trivially bounded by 1 in magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This implies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For the induction step we again need two main ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The first establishes that the our multi-linear forms N d λ∆0,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) are controlled by an appropriate box-type norm attached to a scale L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q = Q1 × · · · × Qd and 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any scale 0 < L ≪ l(Q) and function f : Qe′ → [−1, 1] with e′ ∈ Hd,k we define its local box norm at scale L by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) ∥f∥2k □L(Qe′ ) := s∈Qe′ ∥f∥2k □(s+Q(L)) ds where (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='13) ∥f∥2k □( � Q) := x11,x12∈ � Q1 · · xk1,xk2∈ � Qk � (ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',ℓk)∈{1,2}k f(x1ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xkℓk) dx11 dx12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dxk1 dxk2 for any cube �Q of the form �Q = �Q1 × · · · × �Qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 (Generalized von-Neumann inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ε > 0, 0 < λ ≪ l(Q) and 0 < L ≪ (ε2k)6λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 1 ≤ k ≤ d and collection of functions fe : Qπ(e) → [−1, 1] with e ∈ Hn d,k we have both (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14) |N d λ∆0,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k)| ≤ min e∈Hn d,k ∥fe∥□L(Qπ(e)) + O(ε) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15) |Md λ,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k)| ≤ min e∈Hn d,k ∥fe∥□L(Qπ(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The crucial ingredient is the following analogue of the weak hypergraph regularity lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (Parametric weak hypergraph regularity lemma for Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1, M ≥ 1, and 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists ¯Jk = O(Mε−2k+3) such that for any ε2k-admissible sequence L0 ≥ L1 ≥ · · · ≥ L ¯ Jk with the property that L0 divides l(Q) and collection of functions f m e,t : Qtπ(e)(L0) → [−1, 1] with e ∈ Hn d,k, 1 ≤ m ≤ M, and t ∈ ΓL0,Q there is some 1 ≤ j < ¯Jk and σ-algebras Be′,t of scale Lj on Qte′ (L0) for each t ∈ ΓL0,Q and e′ ∈ Hd,k such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) ∥f m e,t − E(f m e,t|Bπ(e),t)∥□Lj+1(Qtπ(e)(L0)) ≤ ε uniformly for all t /∈ Tε, e ∈ Hn d,k, and 1 ≤ m ≤ M, where Tε ⊆ ΓL0,Q with |Tε| ≤ ε|ΓL0,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 23 Moreover, the σ-algebras Be′,t have the additional local structure that the exist σ-algebras Be′,f′,s on Qsf′ (Lj) with complex(Be′,f′,s) = O(j) for each s ∈ ΓLj,Q, e′ ∈ Hd,k, and f′ ∈ ∂e′ such that if s ∈ Qt(L0), then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17) Be′,t �� Qse′ (Lj) = � f′∈∂e′ Be′,f′,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 is the parametric and simultaneous version of the extension of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7 to the product of d simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The difference is that in the general case one has to deal with a parametric family of functions f m e,t as t is running through a grid ΓL0,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The essential new content of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 is that one can develop σ-algebras Be′,t on the cubes Qt(L0) with respect to the family of functions f m e,t such that the local structure described above and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) hold simultaneously for almost all t ∈ ΓL0,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Assume the Proposition holds for k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ε > 0, ε1 := exp (−C1ε−2k+3) for some large constant C1 = C1(n, k, d) ≫ 1, and {Lj}j≥1 be an ε1-admissible sequence of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Set F(ε) := Jk−1(ε1, M) with M = ε ε−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For L ∈ {Lj}j≥1 we again write index(L) = j if L = Lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We now choose a subsequence {L′ j} ⊆ {Lj} so that L′ 0 = L0 and index(L′ j+1) ≥ index(L′ j) + F(ε) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 then guarantees the existence of σ-algebras Be′,t of scale L′ j on Qte′ (L0) for each t ∈ ΓL0,Q and e′ ∈ Hd,k, with the local structure described above, such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='18) ∥f m e,t − E(f m e,t|Bπ(e),t)∥□L′ j+1(Qtπ(e)(L0)) ≤ ε uniformly for all t /∈ T ′ ε, e ∈ Hn d,k, and 1 ≤ m ≤ M, for some 1 ≤ j < ¯Jk(ε, M) = O(Mε−2k+3), where T ′ ε ⊆ ΓL0,Q with |T ′ ε| ≤ ε|ΓL0,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ¯f m e,t := E(f m e,t|Bπ(e),t) for t ∈ ΓL0,Q and e ∈ Hn d,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If t /∈ T ′ ε, then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) we have both (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='19) N d λ∆0,Qt(L0)(f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = N d λ∆0,Qt(L0)( ¯f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='20) Md λ,Qt(L0)(f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Md λ,Qt(L0)( ¯f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' provided (ε−2k)6L′ j+1 ≪ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For given s ∈ ΓL′ j,Qt(L0) one may write ¯f m e,s for the restriction of ¯f m e,t on the cube Qs(L′ j) ⊆ Qt(L0), as s uniquely determines t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By localization, provided λ ≪ εL′ j, we then have both (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='21) N d λ∆0,Qt(L0)( ¯f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Es∈ΓL′ j,Qt(L0)N d λ∆0,Qs(L′ j)( ¯f m e,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='22) Md λ,Qt(L0)( ¯f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Es∈ΓL′ j,Qt(L0)Md λ,Qs(L′ j)( ¯f m e,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For a fixed cube Qs(L′ j) we have that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='23) ¯f m e,s = Re,s � re=1 αs,re,m 1Are π(e),s where {Are π(e),s}1≤r≤Re,s is the family of atoms of the σ-algebra Bπ(e),t restricted to the cube Qs(L′ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that |αs,re| ≤ 1 and |Re,s| = O(exp (Cε−2k+3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By adding the empty set to the collection of atoms one may assume |Re,s| = R := exp (Cε−2k+3) for all e ∈ Hn d,k and s ∈ ΓL′ j,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then, by multi-linearity, using the notations r = (re)e∈Hn d,k and αr,s = � e αs,re, one has both (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='24) N d λ∆0,Qs(L′ j)( ¯f m s,e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = � r αs,r,m N d λ∆0,Qs(L′ j)(1Are π(e),s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='25) Md λ,Qs(L′ j)( ¯f m s,e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = � r αs,r,m Md λ,Qs(L′ j)(1Are π(e),s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The key observation is that these expressions in the sum above are all at level k − 1 instead of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' To see this let e = (i1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , imlm, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , iklk) so e′ = π(e) = (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , im, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ik).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If f′ = e′\\{im} then recall that the 24 NEIL LYALL ´AKOS MAGYAR edge pf′(e) = (i1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , iklk) ∈ Hn d,k−1 is obtained from e by removing the imlm-entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus, for any atom Ae′,s of Bs,e′(L′ j) we have by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17), that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='26) 1Ae′,s(xe) = � f′∈∂e′ 1Ae′,f′,s,(xpf′ (e)) where Ae′,f′,s is an atom of the σ-algebra Be′,f′,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='27) � e∈Hn d,k 1Are π(e),s(xe) = � f∈Hn d,k−1 � e∈Hn d,k,f′∈∂π(e) pf′ (e)=f 1Are π(e),f′,s(xf) = � f∈Hn d,k−1 gr f,s (xf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It follows that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='28) N d λ∆0,Qs(L′ j)(1Are π(e),s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = N d λ∆0,Qs(L′ j) (gr f,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1) and hence that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='29) N d λ∆0,Qs(L′ j)( ¯f m e,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = � r αs,r,m N d λ∆0,Qs(L′ j) (gr f,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1) and similarly (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='30) Md λ,Qs(L′ j)( ¯f m e,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = � r αr,s,m Md λ,Qs(L′ j) (gr f,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that number of index vectors r = (re)e∈Hn d,k is RD with D := |Hn d,k| and hence RD ≤ M if C1 ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Writing j′ := index(L′ j) and J′ := index(L′ j+1) it then follows from our inductive hypothesis functions, applied with respect to the ε1-admissible sequence of scales Lj′+1 ≥ Lj′+2 ≥ · · · ≥ LJ′−1 which is possible as J′ − j′ ≫ Jk−1(ε1, RD), that there is a scale Lj with j′ ≤ j < J′ so that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='31) Nλ∆0,Qs(L′ j) (gr s,f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1) = Mλ,Qs(L′ j) (gr s,f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1) + O(ε1) for all λ ∈ [Lj+1, Lj] uniformly in r for s /∈ Sε1, where Sε1 ⊆ ΓL′ j,Q is a set of size |Sε1| ≤ ε1|ΓL′ j,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the cubes Qt(L0) form a partition of Q as t runs through the grid ΓL0,Q the relative density of the set Sε1 can substantially increase only of a few cubes Qt(L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed, it is easy to see that |T ′′ ε1| ≤ ε1/2 1 |ΓL0,Q| for the set T ′′ ε1 := {t ∈ ΓL0,Q : |Sε1 ∩ Qt(L0)| ≥ ε1/2 1 |ΓL′ j,Q ∩ Qt(L0)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We claim that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) holds for λ ∈ [Lj+1, Lj] uniformly in t /∈ Tε := T ′ ε ∪ T ′′ ε1, e ∈ Hn d,k, and 1 ≤ m ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed, from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='18), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='31) and the fact that |αs,r| ≤ 1, it follows N d λ∆0,Qs(L′ j) ( ¯fe,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Md λ,Qs(L′ j) ( ¯fe,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) for s /∈ Sε1 ∩ Qt(L0) since RDε1 ≪ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Finally, the fact that t /∈ T ′′ ε1 together with localization, namely (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='21) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='22), ensures that averaging over ΓL′ j,Qt(L0) gives N d λ∆0,Qt(L0) ( ¯fe,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Md λ,Qt(L0) ( ¯fe,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) + O(ε1/2 1 ) which in light of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='19), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='20), and the fact that ε1 ≪ ε2 complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The argument is similar to that of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Fix an edge, say e0 = (11, 12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', 1k), and partition the edges e ∈ Hn d,k in to as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let H0 be the set of those edges e for which 1 /∈ π(e), and for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , n1 let Hl denote the collection of edges of the form e = (1l, j2l2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , jklk), in other words e ∈ Hl if e = (1l, e′) for some edge e′ = (j2l2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , jklk) ∈ Hn d−1,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Accordingly write � e∈Hn d,k fe(xe) = � e∈H0 fe(xe) n1 � l=1 � e′∈Hn d−1,k−1 f1l,e′(x1l, xe′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 25 For x ∈ Q1 and x′ = (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd) with xi ∈ Qni i , define (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='32) gl(x, x′) := � e′∈Hn d−1,k−1 f1l,e′(x1l, xe′) Then one may write (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='33) N d λ∆0,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' xd � e∈H0 fe(xe) � x1 n1 � l=1 gl(x1l, x′) dσλ 1 (x1) � dσλ d (xd) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dσλ 2 (x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For the inner integrals we have, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6), the estimate � x1 n1 � l=1 gl(x1l, x′) dσλ 1 �2 ≤ ∥g1∥2 U1 L(Q) + O(ε2k) = y11 ˆ y12 g1(y11)g1(y12)ψ1 L(y12 − y11) dy11 dy12 + O(ε2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' provided 0 < L ≪ (ε2k)6λ, where as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 we use the notation ψi L(y2 − y1) = ˆ t χi L(y1 − t)χi L(y2 − t) dt with χi L := L−ni1[−L/2,L/2]ni for 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By Cauchy-Schwarz we then have ���N d λ∆0,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k ��� 2 ≤ ˆ y1 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' xd � e′∈Hn d−1,k−1 f11,e′(x11, xe′)f11,e′(x12, xe′) dσλ d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dσλ 2 dω1 L(y1) + O(ε2k) where dωi L(yi) = |Qi|−1ψi L(yi2 − yi1) dyi1 dyi2 with yi = (yi1, yi2) ∈ Q2 i for 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The expression we have obtained above is similar to the one in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) except for the following changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The variable x1 ∈ Qn1 1 is replaced by y1 ∈ Q2 1 and the measure dσλ 1 by dω1 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The functions f1l,e′ are replaced by f11,e′, for 1 ≤ l ≤ n1, while the functions fe for all e ∈ Hn d,k such that 1 /∈ π(e) are eliminated, that is replaced by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Repeating the same procedure for i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , k replaces all variables xi with variables yi as well as the measures dσλ i with dωi L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The procedure eliminates all functions fe when e is an edge such that i /∈ π(e) for some 1 ≤ i ≤ k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' for the remaining edges, when π(e) = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , k), it replaces the functions fe with fe0 = f11,21,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For k < i the variables xi and the measures dσλ i are not changed, however integrating in these variables will have no contribution as the measures are normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus one obtains the following final estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='34) ���Nλ∆0,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k ��� 2k ≤ 1 |Q1| ˆ y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1 |Qk| ˆ yk � e∈H2 k,k fe0(ye) k � i=1 ψi L(yi2 − yi1) dyi1 dyi2 + O(ε2k) noting that these integrals are not normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus, one may write the expression in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='34), using a change of variables yi1 := yi1 − ti, yi2 := yi2 − ti, as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='35) 1 |Q1| ˆ t1 y1∈t1+Q1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1 |Qk| ˆ tk yk∈tk+Qk � e∈H2 k,k fe0(ye) dy1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dyk dt = ∥fe0∥2k □L(Qπ(e0)) + O(ε2k) where the last equality follows from the facts that the function fe0 is supported on the cube Qπ(e0) and hence the integration in t is restricted to the cube Q + Q(L), giving rise an error of O(L/l(Q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14) follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='34) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='35) noting that the above procedure can be applied to any e ∈ Hn d,k in place of e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15) is established similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For j = 0 we set Be′,t(L0) := {Qt(L0), ∅} and Be′,f′,s(L0) := {Qsf′ (L0), ∅} for e′ ∈ Hd,k, f′ ∈ ∂e′, and t, s ∈ ΓL0,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will develop σ-algebras Be′,t(Lj) of scale Lj such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17) holds with complex(Be′,f′,s(Lj)) ≤ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We define the total energy of a family of functions f m e,t with respect to a family of σ-algebras Be′,t(Lj) as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='36) E(f m e,t|Be′,t(Lj)) := Et∈ΓL0,Q M � m=1 � e∈Hn d,k ∥E(f m e,t|Bπ(e),t(Lj))∥2 L2(Qtπ(e)(L0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 26 NEIL LYALL ´AKOS MAGYAR Since |f m e,t| ≤ 1 for all e, m, and t it follows that the total energy is bounded by M · |Hn d,k| = O(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our strategy will be to show that if (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) does not hold then there exist a family of σ-algebras Be′,t(Lj+2) such that the total energy of the family of functions f m e,t is increased by at least ckε2k+3 with respect to this new family of σ-algebras, and at the same time ensuring that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17) remains valid with complex(Be′,f′,s(Lj+2)) ≤ j + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This iterative process must stop at some j = O(M ε−2k+3) proving the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Assume that we have developed σ-algebras Be′,t(Lj) and Be′,f′,s(Lj) of scale Lj such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17) holds with complex(Be′,f′,s(Lj)) ≤ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) does not hold then |Tε| ≥ ε|ΓL0,Q| for the set Tε := {t ∈ ΓL0,Q : ∥f m e,t − E(f m e,t|Bπ(e),t(Lj))∥□Lj+1 (Qtπ(e) (L0)) ≥ ε for some e ∈ Hn d,k and 1 ≤ m ≤ M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Fix t ∈ Tε and let e ∈ Hn d,k and 1 ≤ m ≤ M be such that ∥f m e,t − E(f m e,t|Bπ(e),t(Lj))∥□Lj+1(Qtπ(e)(L0)) ≥ ε and write e′ := π(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Consider the partition of the cube Qte′ (L0) into small cubes Qse′ (Lj+2) where se′ ∈ ΓLj+2,Qe′ ∩Qte′ (L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By the localization properties of the □Lj+1(Q)-norm, and the fact that Lj+2 ≪ ε2kLj+1 we have that ∥f∥2k □Lj+1(Qte′ (L0)) ≤ Ese′ ∈ΓLj+2,Qte′ (L0) ∥f∥2k □(Qse′ (Lj+2)) + ε2k 2 for any function f : Qte′(L0) → [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus there exists a set Sε,e,t ⊆ ΓLj+2,Qte′ (L0) of size |Sε,e,t| ≥ ε2k 4 |ΓLj+2,Qte′ (L0)| such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='37) ∥f m e,t − E(f m e,t|Be′,t(Lj))∥2k □(Qse′ (Lj+2) ≥ ε2k 4 for all se′ ∈ Sε,e,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For a given cube Q and functions f, g : Q → R, define the normalized inner product of f and g as ⟨f, g⟩Q := Q f(x)g(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then by the well-known property of the □-norm, see for example [23] or the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, it follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='37) that there exits sets Bf′,se′ ,t ⊆ Qsf′ (Lj+2) for f′ ∈ ∂e′ such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='38) � f m e,t − E(f m e,t|Be′,t(Lj)) , � f′∈∂e′ 1Bf′,se′ ,t � Qse′ (Lj+2) ≥ ε2k 2k+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If s ∈ ΓLj+2,Q then there is a unique t = t(s) ∈ ΓL0,Q such that s ∈ Qt(L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If t ∈ Tε and se′ ∈ Sε,e,t then we define the σ-algebras Bf′,e′,s(Lj+2) on Qsf′ (Lj+2) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Write Bf′,e′,s = Bf′,se′ ,t where t = t(s) and let Bf′,e′,s(Lj+2) be the σ-algebra generated by the set Bf′,e′,s and the σ-algebra Bf′,e′,s′(Lj) restricted to Qsf′ (Lj+2) where s′ ∈ ΓLj,Q is the unique element so that s ∈ Qs′(Lj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that that the complexity of the σ-algebra Bf′,e′,s(Lj+2) is at most one larger then the complexity of the σ-algebra Bf′,e′,s′(Lj) as restricting a σ-algebra to a set does not increase its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If t = t(s) /∈ Tε or se′ /∈ Sε,e,t then let Bf′,e′,s(Lj+2) be simply the restriction of Bf′,e′,s′(Lj) to the cube Qsf′ (Lj+2), or equivalently define the sets Bf′,e′,s := Qsf′ (Lj+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Finally, let (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='39) Be′,s(Lj+2) := � f′∈∂e′ Bf′,e′,s(Lj+2) be the corresponding σ-algebra on the cube Qse′ (Lj+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the cubes Qse′ (Lj+2) partition the cube Qte′ (L0) as se′ runs through the grid ΓLj+2,Qe′ ∩ Qte′ (L0), these σ-algebras define a σ-algebra Be′,t(Lj+2) on Qte′(L0), such that its restriction to the cubes Qse′ (Lj+2) is equal to the σ-algebras Be′,s(Lj+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 27 Since the function � f′∈∂e′ 1Bf′,e′,s is measurable with respect to the σ-algebra Be′,t(Lj+2) restricted to the cube Qse′ (Lj+2) one clearly has (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='40) ⟨ f m e,t − E(f m e,t|Be′,t(Lj+2)), � f′∈∂e′ 1Bf′,e′,s ⟩Qse′ (Lj+2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' and hence, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='38), that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='41) ⟨ E(f m e,t|Be′,t(Lj+2)) − E(f m e,t|Be′,t(Lj)), � f′∈∂e′ 1Bf′,e′,s ⟩Qse′ (Lj+2) ≥ ε2k 2k+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It then follows from Cauchy-Schwarz and orthogonality, using the fact that the σ-algebra Be′,t(Lj+2)) is a refinement of Be′,t(Lj+2), that ∥E(f m e,t|Be′,t(Lj+2))−E(f m e,t|Be′,t(Lj))∥2 L2(Qse′ (Lj+2)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='42) = ∥E(f m e,t|Be′,t(Lj+2))∥2 L2(Qse′ (Lj+2)) − ∥E(f m e,t|Be′,t(Lj))∥2 L2(Qse′ (Lj+2)) ≥ � ε2k 2k+2 �2 for se′ ∈ Sε,e,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since |Sε,e,t| ≥ ε2k 4 |ΓLj+2,Qte′ (L0)| averaging over se′ ∈ ΓLj+2,Qte′ (L0) implies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='43) ∥E(f m e,t|Be′,t(Lj+2))∥2 L2(Qte′ (L0)) ≥ ∥E(f m e,t|Be′,t(Lj))∥2 L2(Qte′ (L0)) + ε2k+2 22k+6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' At this point we have shown that if t ∈ Tε then there exists an edge e ∈ Hn d,k, 1 ≤ m ≤ M, and σ-algebras Be′,t(Lj+2)) of scale Lj+2 on Qte′ (L0), with e′ = π(e), such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='43) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For all e′′ ∈ Hd,k with e′′ ̸= e′ let Bf′,e′′,s(Lj+2) be the restriction of the σ-algebra Bf′,e′′,s′(Lj) to the cube Qsf′ (Lj+2), where s′ is such that s ∈ Qs′(Lj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='39) this implies that Be′′,s(Lj+2) is also the restriction of Be′′,s′(Lj) to the cube Qse′′ (Lj+2), and hence the σ-algebra Be′′,t(Lj+2) is generated by the grid GLj+2,Qte′′ (L0) and the σ-algebra Be′′,t(Lj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We have therefore defined a family of the σ-algebras Be′,t(Lj+2) for e′ ∈ Hd,k, satisfying M � m=1 � e∈Hn d,k ∥E(f m e,t|Bπ(e),t(Lj+2))∥2 L2(Qtπ(e) (L0)) ≥ M � m=1 � e′∈Hn d,k ∥E(f m e,t|Bπ(e),t(Lj))∥2 L2(Qtπ(e)(L0)) + ε2k+2 22k+6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Using the fact that |Tε| ≥ ε|ΓL0,Q| and averaging over t ∈ ΓL0,Q it follows using the notations of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='36) that E(f m e,t|Be′,t(Lj+2)) ≥ E(f m e,t|Be′,t(Lj)) + ε2k+3 22k+6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' As the total energy E(f m e,t|Be′,t(Lj)) is bounded by O(M), the process must stop at a step j = O(M ε−2k+3) where (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) holds for a σ-algebra of “local complexity” at most j, completing the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The base case of an inductive strategy to establish Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4 In this section we will ultimately establish the base case of our more general inductive argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will however start by giving a (new) proof of Theorem B′, namely the case d = 1 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A Single Simplex in Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆0 = {v1 = 0, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , vn1} be a fixed non-degenerate simplex of n1 points in Zn with n = 2n1 + 3 and define tkl := vk · vl for 2 ≤ k, l ≤ n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall, see [17], that a simplex ∆ = {m1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1} ⊆ Zn is isometric to λ∆0 if and only if mk · ml = λ2tkl for all 2 ≤ k, l ≤ n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any positive integer q and λ ∈ q √ N we define Sλ∆0,q(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1) : Zn(n1−1) → {0, 1} be the function whose value is 1 if mk · ml = λ2tkl with both mk and ml in (qZ)n for all 2 ≤ k, l ≤ n1 and is equal to 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is a well-known fact in number theory, see [11] or [17], that for n ≥ 2n1 + 1 we have that � m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1 Sλ∆0,q(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1) = ρ(∆0) (λ/q)(n−n1)(n1−1)(1 + O(λ−τ)) 28 NEIL LYALL ´AKOS MAGYAR for some absolute constant τ > 0 and some constant ρ(∆0) > 0, the so-called singular series, which can be interpreted as the product of the densities of the solutions of the above system of equations among the p-adics and among the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus if we define σλ∆0,q := ρ(∆0)−1(λ/q)−(n−n1)(n1−1)Sλ∆0,q then σλ∆0,q is normalized in so much that � m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1 σλ∆0,q(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1) = 1 + O(λ−τ) for some absolute constant τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q ⊆ Zn be a fixed cube and let l(Q) denotes its side length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any family of functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn1 : Q → [−1, 1] and 0 < λ ≪ l(Q) we define the following two multi-linear expressions (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1) N 1 λ∆0,q,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn1) := Em1∈Q � m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1 f1(m1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' fn1(mn1) σλ∆0,q(m2 − m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1 − m1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) M1 λ,q,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn1) := Et∈Q Em1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1∈t+Q(q,λ) f1(m1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' fn1(mn1) where Q(q, λ) := [− λ 2 , λ 2 ]n ∩ (qZ)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if S ⊆ Q and N 1 λ∆0,q,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) > 0 then S must contain an isometric copy of λ∆0, while if |S| ≥ δ|Q| for some δ > 0 then as before H¨older implies that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3) M1 λ,q,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) ≥ δn − O(ε) for all scales λ ∈ q √ N with 0 < λ ≪ ε l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that for any 0 < ε ≪ 1 and positive integer q we call a sequence L1 ≥ · · · ≥ LJ (ε, q)-admissible if Lj/Lj+1 ∈ N and Lj+1 ≪ ε2Lj for all 1 ≤ j < J and LJ/q ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if λ1 ≥ · · · ≥ λJ′ ≥ 1 is any lacunary sequence in q √ N with J′ ≫ (log ε−1) J + log q, one can always finds an (ε, q)-admissible sequence of scales L1 ≥ · · · ≥ LJ with the property that for each 1 ≤ j < J the interval [Lj+1, Lj] contains at least two consecutive elements from the original lacunary sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In light of these observations we see that the following “counting lemma” ultimately establishes a quanti- tatively stronger version of Proposition B′ that appeared in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 and hence immediately establishes Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4 for d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1 and qj := q1(ε)j for j ≥ 1 with q1(ε) := lcm{1 ≤ q ≤ Cε−10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists J1 = O(ε−2) such that for any (ε, qJ1)-admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJ1 and S ⊆ Q there is some 1 ≤ j < J1 such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) N 1 λ∆0,qj,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) = M1 λ,qj,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) + O(ε) for all λ ∈ qj √ N with Lj+1 ≤ λ ≤ Lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' As in the continuous setting the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 has two main ingredients, namely Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In these lemmas, and for the remainder Sections 6 and 7, we will continue to use the notation q1(ε) := lcm{1 ≤ q ≤ Cε−10} for any given ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 (A Generalized von Neumann inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1, q, q′ ∈ N with qq1(ε)|q′, and λ ∈ q √ N with λ ≪ l(Q) and 1 ≪ L ≪ ε10λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any collection of functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn1 : Q → [−1, 1] we have (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) |N 1 λ∆0,q,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn1)| ≤ min 1≤i≤n1 ∥fi∥U1 q′,L(Q) + O(ε) where for any function f : Q → [−1, 1] we define (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) ∥f∥U1 q,L(Q) := � 1 |Q| � t∈Q |f ∗ χq,L(t)|2�1/2 WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 29 with χq,L denoting the normalized characteristic function of the cubes Q(q, L) := [− L 2 , L 2 ]n ∩ (qZ)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any cube Q ⊆ Zn of side length l(Q) and q, L ∈ N satisfying q ≪ L with L dividing l(Q), we shall now partition Q into cubic grids Qt(q, L) = t + ((qZ)n ∩ Q(L)), with Q(L) = [− L 2 , L 2 ]n as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' These grids form the atoms of a σ-algebra Gq,L,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if q|q′ and L′|L then Gq,L,Q ⊆ Gq′,L′,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (A Koopman-von Neumann type decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1 and qj := q1(ε)j for all j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer ¯J1 = O(ε−2) such that any (ε, q ¯ J1)- admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ L ¯ J1 and function f : Q → [−1, 1] there is some 1 ≤ j < ¯J1 such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) ∥f − E(f|Gqj,Lj,Q)∥U1 qj+1,Lj+1(Q) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The reduction of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 to these two lemmas is essentially identical to the analogous argument in the continuous setting as presented at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, we choose to omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will rely on some prior exponential sum estimates, specifically Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4 in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' First we deal with the case n1 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By the change of variables m1 := m1, mi := mi − m1 for 2 ≤ i ≤ n1, one may write N 1 λ∆0,q,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn1) := Em1∈QN � m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1 f1(m1)f2(m1 + m2) · · · fn1(m1 + mn1) σλ∆0,q(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We now write σλ∆0,q(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1) = σλ∆0′,q(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1−1) σ m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1−1 λ,q (mn1) where ∆0′ = {v1 = 0, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , vn1−1} and for each m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1−1 ∈ (qZ)n we are using σ m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1−1 λ,q (m) denote the (essentially) normalized indicator function of the subset of (qZ)n that contains m if and only if m · mk = λ2tkn1 for all 2 ≤ k ≤ n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Using the fact that |fi| ≤ 1, together with Cauchy-Schwarz and Plancherel, one can then easily see that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) |N 1 λ∆0,q,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn1)|2 ≤ |Q|−1 ˆ ξ∈Tn | �fn1(ξ)|2Hλ,q(ξ) dξ with Hλ,q(ξ) = � m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1 σλ∆0′,q(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1−1) | � σ m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1−1 λ,q (ξ)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It then follows by Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4 in [17], with δ = ε4 and after rescaling by q, that in addition to being non-negative and uniformly bounded in ξ we in fact have (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) Hλ,q(ξ) = O(ε) whenever ����qξ − l q1(ε) ���� ≥ q ε4λ, for all l ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We note that the expression Hλ,q(ξ) may be interpreted as the Fourier transform of the indicator function of the set of integer points on a certain variety, and estimate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) indicates that this concentrates near rational points of small denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is this crucial fact from number theory which makes results like Theorem B′ possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since �χq,L(ξ) = qn Ln � m∈[− L 2 , L 2 )n, q|m e−2πim·ξ it is easy to see that �χq,L(l/q) = 1 for all l ∈ Zn and that there exists some absolute constant C > 0 such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10) 0 ≤ 1 − �χq,L(ξ)2 ≤ C L |ξ − l/q| for all ξ ∈ Tn and l ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is then easy to see using our assumption that qq1(ε)|q′ that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) 0 ≤ Hλ,q(ξ)(1 − �χq′,L(ξ)2) ≤ Cε 30 NEIL LYALL ´AKOS MAGYAR for some constant C > 0 uniformly in ξ ∈ Tn provided L ≪ ε5λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Substituting inequality (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8), we obtain |N 1 λ∆0,q,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn1)|2 ≤ |Q|−1 �ˆ | ˆfn1(ξ)|2Hλ(ξ)�χq′,L(ξ)2 dξ + ˆ | ˆfn1(ξ)|2Hλ(ξ)(1 − �χq′,L(ξ)2) dξ � ≤ ∥fn1∥2 U1 q′,L(Q) + O(ε) provided L ≪ ε5λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This proves Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 for k ≥ 3, as it is clear that by re-indexing the above estimate holds for any of the functions fi in place of fn1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For n1 = 2 an easy modification of arguments in [14], specifically the proof of Lemma 3 therein, establishes that |N 1 λ∆0,q,Q(f1, f2)|2 ≤ ∥fi∥2 U1 q′,L(Q) + O(ε) for i = 1, 2 provided L ≪ ε5λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let q, L ∈ N such that L|N, q|L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The “modulo q” grids Qt(q, L) = t+Q(q, L) partition the cube Q with t running through the set Γq,L,Q = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', q}n + ΓL,Q, where ΓL,Q denote the centers of the “integer” grids t + Q(L) in an initial partition of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let q′, L′ be positive integers so that q|q′, L′|L and L′ ≪ ε2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If s ∈ Γq′,L′,Q and t ∈ Qs(q′, L′) then |t − s| = O(L′) and hence Ex∈Qt(q,L)g(x) = Ex∈Qs(q,L)g(x) + O(L′/L) for any function g : Q → [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover, since the cube Qs(q, L) is partitioned into the smaller cubes Qt(q′, L′), we have by Cauchy-Schwarz |Ex∈Qs(q,L) g(x)|2 ≤ Et∈Γq′,L′,Qs(q,L)|Ex∈Qt(q′,L′)g(x)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' From this it is easy to see that ∥g∥2 U1 q,L(Q) = Et∈Q|Ex∈Qt(q,L)g(x)|2 ≤ Et∈Γq′,L′,Q |Ex∈Qt(q′,L′)g(x)|2 + O(L′/L) and we note that the right side of the above expression is ∥E(g|Gq′,L′,Q)∥2 L2(Q) since the conditional expecta- tion function E(g|Gq′,L′,Q) is constant and equal to Ex∈Qt(q′,L′)g(x) on the cubes Qt(q′, L′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Now suppose (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) does not hold for some j ≥ 1, that is ∥f − E(f|Gqj,Lj,Q)∥2 U1 qj+1,Lj+1 (Q) ≥ ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since Lj+2 ≪ ε2Lj+1, Lj+2|Lj, and qj+1|qj+2 we can apply the above observations to g := f − E(f|Gqj,Lj,Q) and obtain, by orthogonality, that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) ∥E(f|Gqj+2,Lj+2,Q)∥2 L2(Q) ≥ ∥E(f|Gqj,Lj,Q)∥2 L2(Q) + cε2 for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the above expressions are clearly bounded by 1, the above procedure must stop in O(ε−2) steps at which (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) must hold for some 1 ≤ j ≤ ¯J1(ε) with ¯J1(ε) = O(ε−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The base case of our general inductive strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q = Q1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × Qd with Qi ⊆ Z2ni+3 be cubes of equal side length l(Q) and ∆0 i ⊆ Z2ni+3 be a non-degenerate simplex of ni points for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We note that for any q0 ∈ N and scale L0 dividing l(Q) if t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td) ∈ Γq0,L0,Q, then the corresponding grids Qt(q0, L0) in the partition of Q take the form Qt(q0, L0) = Qt1(q0, L0) × · · · × Qtd(q0, L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' As in the continuous setting we will ultimately need a parametric version of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, namely Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (Parametric Counting Lemma on Zn for Simplices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≤ 1 and R ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer J1 = J1(ε, R) = O(R ε−4) such that for any (ε, qJ1)-admissible sequence of scales L0 ≥ L1 ≥ · · · ≥ LJ1 with L0 dividing l(Q) and qj := q0q1(ε)j for 0 ≤ j ≤ J1 with q0 ∈ N, and collection of functions f i,r k,t : Qti(q0, L0) → [−1, 1] with 1 ≤ i ≤ d, 1 ≤ k ≤ ni, 1 ≤ r ≤ R and t ∈ Γq0,L0,Q there exists 1 ≤ j < J1 and a set Tε ⊆ Γq0,L0,Q of size |Tε| ≤ ε|Γq0,L0,Q| such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='13) N 1 λ∆0 i ,qj,Qti (q0,L0)(f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f i,r ni,t) = M1 λ,qj,Qti (q0,L0)(f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f i,r ni,t) + O(ε) WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 31 for all λ ∈ qj √ N with Lj+1 ≤ λ ≤ Lj and t /∈ Tε uniformly in 1 ≤ i ≤ d and 1 ≤ r ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This proposition follows, as the analogous result did in the continuous setting, from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and the follow parametric version of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 (A simultaneous Koopman-von Neumann type decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1, m ≥ 1, and Q ⊆ Zn be a cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer ¯J1 = O(mε−3) such that for any (ε, q ¯ J1)-admissible sequence L0 ≥ L1 ≥ · · · ≥ L ¯ J1 with L0 dividing l(Q) and qj := q0q1(ε)j for 0 ≤ j ≤ ¯J1 with q0 ∈ N, and collection of functions f1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' fm,t : Qt(q0, L0) → [−1, 1] defined for each t ∈ Γq0,L0,Q, there is some 1 ≤ j < ¯J1 and a set Tε ⊆ Γq0,L0,Q of size |Tε| ≤ ε|Γq0,L0,Q| such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14) ∥fi,t − E(fi,t|Gqj,Lj,Qt(q0,L0)∥U1 qj+1,Lj+1 (Qt(q0,L0)) ≤ ε for all 1 ≤ i ≤ m and t /∈ Tε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 above is of course the discrete analogue of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the proofs of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 are almost identical to the arguments presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 we choose to omit these details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4: The general case After the preparations in Section 6 we can proceed very similarly as in Section 5 to prove our main result in the discrete case, namely Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The main difference will be that given 0 < ε ≪ 1 and 1 ≤ k ≤ d, we construct a positive integer qk(ε) and assume that all our sequences of scales will be (ε, qk(ε))-admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The cubes Qt(L) will be naturally now be replaced by the grids Qt(q, L) of the form that already appear in Section 6 where we always assume q|L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆0 = ∆0 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × ∆0 d with each ∆0 i ⊆ Z2ni+3 a non-degenerate simplex of ni points for 1 ≤ i ≤ d and Q = Q1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × Qd ⊆ Zn with Qi ⊆ Z2ni+3 cubes of equal side length l(Q) (taken much larger than the diameter of ∆0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will use the same parameterizations in terms of hypergraph bundles Hn d,k and corresponding notations as in Section 5 to count the configurations ∆ = ∆1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × ∆d ⊆ Q with each ∆i ⊆ Qi an isometric copy of λ∆0 i for some λ ∈ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Given any positive integer q and λ ∈ q √ N we will make use of the notation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1) � xi f(xi) σi λ,q(xi) := Exi1∈Qi � xi2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xini f(xi) σλ∆0 i ,q(xi2 − xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xini − xi1) dxi1 with σλ∆0 i ,q as defined in the previous section and xi = (xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xini) ∈ Qni i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if S ⊆ Q then the density of configurations ∆ in S, of the form ∆ = ∆1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × ∆d with each ∆i ⊆ Qi an isometric copy of λ∆0 i for some λ ∈ q √ N is given by the expression (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) N d λ∆0,q,Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) := � x1 · · � xd � e∈Hn d,d 1S(xe) σ1 λ,q(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' σd λ,q(xd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' More generally, for any given 1 ≤ k ≤ d and a family of functions fe : Qπ(e) → [−1, 1] with e ∈ Hn d,k we define the multi-linear expression (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3) N d λ∆0,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) := � x1 · · � xd � e∈Hn d,k fe(xe) σ1 λ,q(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='σd λ,q(xd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' as well as (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) Md λ,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) := Et∈Q Md t+Q(q,L) (fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) where Q(q, L) = Q1(q, L) × · · · × Qd(q, L) with each Qi(q, L) = (qZ ∩ [− L 2 , L 2 ])2ni+3 and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) Md � Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) := Ex1∈ � Qn1 1 · · · Exd∈ � Q nd d � e∈Hn d,k fe(xe) for any cube �Q ⊆ Q of the form �Q = �Q1 × · · · × �Qd with �Qi ⊆ Qi for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 32 NEIL LYALL ´AKOS MAGYAR We note that it is easy to show, as in the continuous, that if S ⊆ Q with |S| ≥ δ|Q| for some δ > 0 then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) Md λ,q,Q(1S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) ≥ δn1··· nd − O(ε) for all scales λ ∈ q √ N with 0 < λ ≪ ε l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In light of this observation and the discussion preceding Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4 reduces, as it did in the continuous setting, to the following Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exist positive integers Jd = Jd(ε) and qd(ε) such that for any (ε, qd(ε)Jd)-admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJ1 and S ⊆ Q there is some 1 ≤ j < Jd such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) N d λ∆0,qj,Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) = Md λ,qj,Q(1S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) + O(ε), for all λ ∈ qj √ N with Lj+1 ≤ λ ≤ Lj with qj := qd(ε)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Quantitative Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A careful analysis of our proof reveals that there exist choices of Jd(ε) and qd(ε) which are less than Wd(log(C∆ε−3)) and Wd(C∆ε−13) respectively where Wk(m) is again the tower- exponential function defined by W1(m) = exp(m) and Wk+1(m) = exp(Wk(m)) for k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 follows along the same lines as the analogous result in the continuous setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' As before we will compare the averages N d λ∆0,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) to those of Md λ,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k), at certain scales q and λ ∈ q √ N with with Lj+1 ≤ λ ≤ Lj, inductively for 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' As the arguments closely follow those given in Section 5 we will be brief and emphasize mainly just the additional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Reduction of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 to a more general “local” counting lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any given 1 ≤ k ≤ d and a family of functions fe : Qπ(e) → [−1, 1] with e ∈ Hn d,k it is easy to see that for any ε > 0, scale L0 > 0 dividing the side-length l(Q), and q0|q we have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) N d λ∆0,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Et∈Γq0,L0,Q N d λ∆0,q,Qt(q0,L0)(fe,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) Md λ,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Et∈ΓL,Q Md λ,q,Qt(q0,L0)(fe,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) provided 0 < λ ≪ εL0 where fe,t denotes the restriction of a function fe to the cube Qt(q0, L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 reduces to showing that the expressions in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) only differ by O(ε) for all scales λ ∈ q √ N with Lj+1 ≤ λ ≤ Lj, given an (ε, q)-admissible sequence L0 ≥ L1 ≥ · · · ≥ LJ, for any collection of bounded functions fe,t, e ∈ Hn d,k, t ∈ Γq0,L0,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed, our crucial result will be the following Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (Local Counting Lemma in Zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1 and q0, M ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exist positive integers Jk = Jk(ε, M) and qk(ε) such that for any (ε, qJd)-admissible sequence of scales L0 ≥ L1 ≥ · · · ≥ LJ1 with L0 dividing l(Q) and qj := q0 qk(ε)j for j ≥ 1, and collection of functions f m e,t : Qtπ(e)(q0, L0) :→ [−1, 1] with e ∈ Hn d,k, 1 ≤ m ≤ M and t ∈ Γq0,L0,Q there exists 1 ≤ j < Jk and a set Tε ⊆ Γq0,L0,Q of size |Tε| ≤ ε|Γq0,L0,Q| such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10) N d λ∆0,qj,Qt(q0,L0)(fe,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Md λ,qj,Qt(q0,L0)(fe,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) for all λ ∈ qj √ N with Lj+1 ≤ λ ≤ Lj and t /∈ Tε uniformly in e ∈ Hn d,k and 1 ≤ m ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if k = d, L0 = l(Q), q0 = M = 1, then |Γq0,L0,Q| = 1, and moreover if fe,t = 1S for all e ∈ Hn d,k for a set S ⊆ Q, then Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 reduces to precisely Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In fact, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 is a parametric, multi-linear and simultaneous extension of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 which we need in the induction step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' when going from level k − 1 to level k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 33 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will prove Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 by induction on 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For k = 1 this is basically Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, exactly as it was in the base case of the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For the induction step we will again need two main ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The first establishes that the our multi- linear forms N d λ∆0,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) are controlled by a box-type norm attached to scales q′ and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q = Q1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × Qd with Qi ⊆ Z2ni+3 be cubes of equal side length l(Q) and 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any scale 0 < L ≪ l(Q) and function f : Qe′ → [−1, 1] with e′ ∈ Hd,k we define its local box norm at scales q′ and L by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) ∥f∥2k □q′,L(Qe′ ) := Es∈Qe′ ∥f∥2k □(Qs(q′,L)) where (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) ∥f∥2k □( � Q) := Ex11,x12∈ � Q1 · · · Exk1,xk2∈ � Qk � (ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',ℓk)∈{1,2}k f(x1ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xkℓk) for any cube �Q of the form �Q = �Q1 × · · · × �Qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We note that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) are special cases of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) with k = d, n = (2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 2), and fe = f for all e ∈ Hn d,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 (A Generalized von-Neumann inequality on Zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1, q, q′ ∈ N with qq1(ε)|q′, and λ ∈ q √ N with λ ≪ l(Q) and 1 ≪ L ≪ (ε2k)10λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any collection of functions fe : Qπ(e) → [−1, 1] with e ∈ Hn d,k we have both (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='13) |N d λ∆0,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k)| ≤ min e∈Hn d,k ∥fe∥□q′,L′ (Qπ(e)) + O(ε) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14) |Md λ,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k)| ≤ min e∈Hn d,k ∥fe∥□q′,L′ (Qπ(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The proof of inequalities (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='13) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14) follow exactly as in the continuous case, see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 in place of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The crucial ingredient is again a parametric weak hypergraph regularity lemma, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 adapted to the discrete settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The proof is essentially the same as in the continuous case, with exception that the □Lj-norms are replaced by □qj,Lj-norms where qj = q0qj is a given sequence of positive integers and L0 ≥ L1 ≥ · · · ≥ LJ is an (ε, qJ)-admissible sequence of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' To state it we say that a σ-algebra B on a cube Q is of scale (q, L) if it is refinement of the grid Gq,L,Q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' if its atoms partition each cube Qt(q, L) of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will always assume that q|L and L|l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall also that we say the complexity of a σ-algebra B is at most m, and write complex(B) ≤ m, if it is generated by m sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (Parametric weak hypergraph regularity lemma for Zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1, 1 ≤ k ≤ d, q0, q, L0, M ∈ N, and let qj := q0qj for j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists ¯Jk = O(Mε−2k+3) such that for any (ε2k, q ¯ Jk)-admissible sequence L0 ≥ L1 ≥ · · · ≥ L ¯ Jk with the property that L0 divides l(Q) and collection of functions f m e,t : Qtπ(e)(q0, L0) → [−1, 1] with e ∈ Hn d,k, 1 ≤ m ≤ M, and t ∈ Γq0,L0,Q there is some 1 ≤ j < ¯Jk and σ-algebras Be′,t of scale (qj, Lj) on Qte′ (q0, L0) for each t ∈ Γq0,L0,Q and e′ ∈ Hd,k such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15) ∥f m e,t − E(f m e,t|Bπ(e),t)∥□qj+1,Lj+1 (Qtπ(e) (L0)) ≤ ε uniformly for all t /∈ Tε, e ∈ Hn d,k, and 1 ≤ m ≤ M, where Tε ⊆ Γq0,L0,Q with |Tε| ≤ ε|Γq0,L0,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover, the σ-algebras Be′,t have the additional local structure that the exist σ-algebras Be′,f′,s on Qsf′ (qj, Lj) with complex(Be′,f′,s) = O(j) for each s ∈ Γqj,Lj,Q, e′ ∈ Hd,k, and f′ ∈ ∂e′ such that if s ∈ Qt(q0, L0), then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) Be′,t �� Qse′ (qj,Lj) = � f′∈∂e′ Be′,f′,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 34 NEIL LYALL ´AKOS MAGYAR The proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 follows exactly as the corresponding proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 in the continuous setting, so we will omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will however provide some details of how one deduces Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, from Lemmas 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The arguments are again very similar to those in the continuous setting, however one needs to make a careful choice of the integers qk(ε), appearing in the statement of the Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 2 ≤ k ≤ d and assume that the lemma holds for k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1 and ε1 := exp (−C1ε−2k+3) for some large constant C1 = C1(n, k, d) ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We then define qk(ε) := qk−1(ε1) recalling that q1(ε) := lcm{1 ≤ q ≤ Cε−10} and note that it is easy to see by induction that qk(ε)|qk(ε′) for 0 < ε′ ≤ ε and qk−1(ε)|qk(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We further define the function F(ε) := Jk−1(ε1, M) with M = ε ε−1 1 and recall that qj := q0 qk(ε)j for j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We now proceed exactly as in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 but with {Lj}j≥1 being a (ε1, q � J)-admissible sequence of scales, with �J ≫ F(ε) ¯Jk(ε, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We again choose a subsequence {L′ j} ⊆ {Lj} so that L′ 0 = L0 and index(L′ j+1) ≥ index(L′ j) + F(ε) + 2, but also now set q′ j = qj′, where j′ := index(L′ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 then guarantees the existence of σ-algebras Be′,t of scale (q′ j, L′ j) on Qte′ (q0, L0) for each t ∈ Γq0,L0,Q and e′ ∈ Hd,k, with the local structure described above, such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15) holds uniformly for all t /∈ T ′ ε, e ∈ Hn d,k, and 1 ≤ m ≤ M, for some 1 ≤ j < ¯Jk(ε, M) = O(Mε−2k+3), where T ′ ε ⊆ Γq0,L0,Q with |T ′ ε| ≤ ε|Γq0,L0,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Arguing as in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 we can conclude from this that for each j′ ≤ l < J′ we have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17) N d λ∆0,ql,Qs(q′ j,L′ j)(f m e,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = � r αs,r,m N d λ∆0,ql,Qs(q′ j,L′ j) (gr f,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1) + O(ε) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='18) Md λ,ql,Qs(q′ j,L′ j)(f m e,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = � r αr,s,m Md λ,ql,Qs(q′ j,L′ j) (gr f,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1) + O(ε) provided (ε−2k)10L′ j+1 ≪ λ with λ ∈ ql √ N, where each |αs,re| ≤ 1 and number of index vectors r = (re)e∈Hn d,k is RD with D := |Hn d,k| and hence RD ≤ M if C1 ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By induction, we apply Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 to the sequence of scales L′ j = Lj′ ≥ Lj′+1 ≥ · · · ≥ LJ′ = L′ j+1 with ε1 > 0 and for ql := q′ j qk(ε)l−j′ = qj′ qk−1(ε1)l−j′ where j′ ≤ l ≤ J′ with respect to the family of functions gr s,f : Qsf(q′ j, L′ j) → [−1, 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This is possible as J′ − j′ ≫ Jk−1(ε1, RD) and our sequence of scales is (ε1, qJ′)-admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus there exists an index j′ ≤ l < J′ such that for all λ ∈ ql √ N with Ll+1 ≤ λ ≤ Ll we have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='19) N d λ∆0,ql,Qs(q′ j,L′ j) (gr f,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1) = Md λ,ql,Qs(q′ j,L′ j) (gr f,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1) + O(ε1) uniformly in r for s /∈ Sε1, where Sε1 ⊆ Γq′ j,L′ j,Q is a set of size |Sε1| ≤ ε1|Γq′ j,L′ j,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The remainder of the proof follows as just as it did for Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Appendix: A short direct proof of Part (i) of Theorem B′ We conclude by providing a short direct proof of Part (i) of Theorem B′, namely the following Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 (Magyar [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < δ ≤ 1 and ∆ ⊆ Z2k+3 be a non-degenerate simplex of k points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If S ⊆ Z2k+3 has upper Banach density at least δ, then there exists an integer q0 = q0(δ) and λ0 = λ0(S, ∆) such that S contains an isometric copy of q0λ∆ for all λ ∈ √ N with λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any ε > 0 we define qε := lcm{1 ≤ q ≤ Cε−10} with C > 0 a (sufficiently) large absolute constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Following [14] we further define S ⊆ Zn to be ε-uniformly distributed (modulo qε) if its relative upper Banach density on any residue class modulo qε never exceeds (1 + ε2) times its density on Zn, namely if δ∗(S | s + (qεZ)d) ≤ (1 + ε2) δ∗(S) WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 35 for all s ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , qε}d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It turns out that this notion is closely related to the U 1 q,L(Q)-norm introduced in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that for any cube Q ⊆ Zn and function f : Q → [−1, 1] we define (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1) ∥f∥U1 q,L(Q) := � 1 |Q| � t∈Q |f ∗ χq,L(t)|2�1/2 with χq,L denoting the normalized characteristic function of the cubes Q(q, L) := [− L 2 , L 2 ]n ∩ (qZ)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that the U 1 q,L(Q)-norm measures the mean square oscillation of a function with respect to cubic grids of size L and gap q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The following observation from [14] (specifically Lemmas 1 and 2) is key to our short proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If S ⊆ Zn be ε-uniformly distributed with δ := δ∗(S) > 0, then there exists an integer L = L(S, ε) > 0 and cubes Q of arbitrarily large side length l(Q) with l(Q) ≫ ε−4L such that ∥1S − δ1Q∥U1 qε,L(Q) = O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆0 = {v1 = 0, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , vk} be a fixed non-degenerate simplex of k points in Zn with n = 2k + 3 and define tij := vi · vj for 2 ≤ i, j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We now define a function which counts isometric copies of λ∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall, see [17], that a simplex ∆ = {m1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mk} ⊆ Zn is isometric to λ∆0 if and only if mi · mj = λ2tij for all 2 ≤ i, j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any λ ∈ √ N we define Sλ∆0(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mk) : Zn(k−1) → {0, 1} be the function whose value is 1 if mi · mj = λ2tij for all 2 ≤ i, j ≤ k and is equal to 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is a well-known fact in number theory, see [11] or [17], that for n ≥ 2k + 1 we have that � m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mk Sλ∆0(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mk) = ρ(∆0) λ(n−k)(k−1)(1 + O(λ−τ)) for some absolute constant τ > 0 and constant ρ(∆0) > 0, the so-called singular series, which can be interpreted as the product of the densities of the solutions of the above system of equations among the p-adics and among the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus if we define σλ∆0 := ρ(∆0)−1λ−(n−k)(k−1)Sλ∆0 then σλ∆0 is normalized in so much that � m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mk σλ∆0(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mk) = 1 + O(λ−τ) for some absolute constant τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q ⊆ Zn be a fixed cube and let l(Q) denotes its side length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any family of functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fk : Q → [−1, 1] and 0 < λ ≪ l(Q) we define (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) N 1 λ∆0,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fk) := Em1∈Q � m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mk f1(m1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' fk(mk) σλ∆0(m2 − m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mk − m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is clear that if f1 = · · · = fk = 1S restricted to Q, then the above expression is a normalized count of the isometric copies of λ∆0 in S ∩ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 will follow from Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and the following special case (with q = 1) of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (A Generalized von Neumann inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If λ ∈ √ N with λ ≪ l(Q) and 1 ≪ L ≪ ε10λ then for any collection of functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fk : Q → [−1, 1] we have (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3) |N 1 λ∆0,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fk)| ≤ min 1≤j≤k ∥fj∥U1 qε,L(Q) + O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This compares with the purely number theoretic fact that the number of simplices ∆ = {v1 = 0, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , vk} ⊆ Zn isometric to λ∆0 is asymptotic to ρ(∆0) λ(n−k)(k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus, under the same conditions as in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, we have (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) N 1 λ∆0,Q(1Q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1Q) = 1 + O(λ−τ) + O(ε) 36 NEIL LYALL ´AKOS MAGYAR provided one also has λ ≪ εl(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ δk and S ⊆ Zn be a set of upper Banach density δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We assume first that S is ε-uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Select a scale L = L(ε, S) and a sufficiently large cube Q so that the conclusion of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For a given λ ∈ √ N with λ ≪ εl(Q) and L ≪ ε10λ write 1S = δ1Q + g and substitute this decomposition into the multi-linear expression N 1 λ∆0,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3)-(8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4), we have that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) N 1 λ∆0,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) ≥ δk − O(ε) and we can conclude that S must contain an isometric copy of λ∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If S is not ε-uniformly distributed, then its upper Banach density is increased to at least δ1 := (1 + ε2)δ when restricted to a residue class s+(qεZ)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Identify s+(qεZ)n with Zn and simultaneously the set S|s+(qεZ)n with a set S1 ⊆ Zn, via the map y → q−1 ε (y − s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if S1 is ε-uniformly distributed then it contains an isometric copy of λ∆0 for all sufficiently large λ ∈ √ N and hence S contains an isometric copy of qελ∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Repeating the above procedure one arrives to a set Sj = q−j ε (S − sj) ⊆ Zn for some sj ∈ Zn in j = O(log ε−1) steps which contains an isometric copy of λ∆0 for all sufficiently large λ ∈ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 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magyar@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='uga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} diff --git a/dtE0T4oBgHgl3EQfoQFr/content/tmp_files/2301.02523v1.pdf.txt b/dtE0T4oBgHgl3EQfoQFr/content/tmp_files/2301.02523v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8caea9406de10a8ea8a02af46f406a8512967bcf --- /dev/null +++ b/dtE0T4oBgHgl3EQfoQFr/content/tmp_files/2301.02523v1.pdf.txt @@ -0,0 +1,1043 @@ +1 + +Many-body Hybrid Excitons with strong molecular orientation +dependence +in +Organic-Inorganic +van +der +Waals +Heterostructures +Shaohua Fu,1,2,4 # Jianwei Ding3 #, Haifeng Lv,5 Shuangyan Liu,1 Kun Zhao1, Zhiying +Bai1, Dawei He,1 Rui Wang,6 Jimin Zhao,4 Xiaojun Wu,5 Dongsheng Tang,2 * Xiaohui +Qiu,3 * Yongsheng Wang1, Xiaoxian Zhang,1 * +1Key Laboratory of Luminescence and Optical Information, Ministry of Education, +Institute of Optoelectronic Technology, Beijing Jiaotong University, Beijing 100044, +China +2Synergetic Innovation Center for Quantum Effects an Application, Key Laboratory of +Low-dimensional Quantum Structures and Quantum Control of Ministry of Education, +School of Physics and Electronics, Hunan Normal University, Changsha 410081, China +3CAS Key Laboratory of Standardization and Measurement for Nanotechnology, CAS +Center for Excellence in Nanoscience, National Center for Nanoscience and +Technology, Beijing 100190, P. R. China. +4Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, +Chinese Academy of Sciences, Beijing 100190, China +5Hefei National Laboratory for Physical Sciences at the Microscale, CAS Key +Laboratory of Materials for Energy Conversion, Synergetic Innovation of Quantum +Information & Quantum Technology, School of Chemistry and Materials Sciences, and +CAS Center for Excellence in Nanoscience, University of Science and Technology of +China, Hefei, Anhui 230026, P.R. China + +2 + +6Beijing Information technology college, Beijing 100015, P. R. China +#These authors contributed equally +∗e-mail: dstang@hunnu.edu.cn; xhqiu@nanoctr.cn; +zhxiaoxian@bjtu.edu.cn + + + + + + + + + + + + + + + + +3 + +Abstract +The coherent many-body interaction at the organic-inorganic interface can give rise to +intriguing hybrid excitons that combine the advantages of the Wannier-Mott and +Frenkel excitons simultaneously. Unlike the 2D inorganic heterostructures that suffer +from moment mismatch, the hybrid excitons formed at the organic-inorganic interface +have a momentum-direct nature, which have yet to be explored. Here, we report hybrid +excitons at the copper phthalocyanine/molybdenum diselenide (CuPc/MoSe2) interface +with +strong +molecular +orientation +dependence +using +low-temperature +photoluminescence spectroscopy. The new emission peaks observed in the +CuPc/MoSe2 heterostructure indicate the formation of interfacial hybrid excitons. The +density functional theory (DFT) calculation confirms the strong hybridization between +the lowest unoccupied molecular orbital (LUMO) of CuPc and the conduction band +minimum (CBM) of MoSe2, suggesting that the hybrid excitons consist of electrons +extended in both layers and holes confined in individual layers. The temperature- +dependent measurements show that the hybrid excitons can gain the signatures of the +Frenkel excitons of CuPc and the Wannier-Mott excitons of MoSe2 simultaneously. The +out-of-plane molecular orientation is used to tailor the interfacial hybrid exciton states. +Our results reveal the hybrid excitons at the CuPc/MoSe2 interface with tunability by +molecular orientation, which suggests that the emerging organic-inorganic +heterostructure can be a promising platform for many-body exciton physics. + + + +4 + +Introduction +Hybrid excitons are many-body exciton states that originate from the hybridization +of electronic states at interfaces1,2, which have been realized in distinct systems, +including quantum dots coupled to a Fermi sea1,3, coupled quantum-dot molecules4,5, +and emerging van der Waals heterostructures2,6-8, displaying great potential in kondo +physics1,8, quantum optics4,7, and strongly correlated electronic physics2. Transition +metal dichalcogenides (TMDs) have become a promising building block for hybrid +excitons due to their strong light-matter interaction9,10 and rich exciton physics11-14, +such as valley polarized excitons15-17, long-lived interlayer excitons18 and moiré +excitons19-21. However, in TMD heterostructures, the momentum-mismatch problem +severely restricts the formation of prominent hybrid excitons, which is moment-direct +only at a small twist angle, requiring a precise control of the interlayer angle alignment +during fabrication2,6,7. +Unlike the momentum-mismatch issue encountered in inorganic heterostructures, +the hybrid excitons formed at the organic-inorganic heterostructures have a momentum- +direct nature22, which could simplify the fabrication process and maintain the novel +exciton physics at the same time. In addition, theoretical calculations have predicted +that the hybrid excitons at the organic-inorganic interfaces can gain the signature of the +Wannier-Mott excitons of inorganics and the Frenkel excitons of organics +simultaneously23,24. Nevertheless, the coupling at the organic-inorganic interfaces is +generally weak25, and an ordered structure of the organic materials is required to +achieve strong electronic coupling24. TMDs can be an ideal building brick for realizing + +5 + +hybrid excitons at the organic-inorganic interfaces, because they not only contain rich +exciton physics but can also serve as a suitable template for the growth of well-ordered +organic films through the van der Waals epitaxial method26-31. Moreover, the short- +range interactions, such as ultrafast charge transfer31-34 and interfacial spin orbital +coupling35, have been observed at organic-TMD interfaces, which suggest that the +coherent superposition of the electron wavefunctions and the formation of hybrid +excitons are possible at such interfaces. However, experimental evidence for hybrid +Wannier-Mott-Frenkel excitons at organic-TMD interfaces remains to be explored. + It is well-known that the interfacial hybridization strength depends sensitively on +the interlayer twist angle and stacking configuration in inorganic heterostructures6,7. +Similarly, the molecular orientation at the organic-inorganic interface can be used to +tune the interfacial distance, i.e., the hybridization strength, so the interfacial hybrid +exciton behavior could be effectively tailored, which is beneficial for designing the +organic-inorganic interface functionality. +In this article, we report the formation of hybrid excitons at the CuPc/MoSe2 +interface and their further modulation by molecular orientation. The new emission +peaks observed in photoluminescence spectroscopy and the further DFT calculations +confirm the emergence of new hybrid excitons. The temperature-dependent +measurements reveal that the hybrid excitons combine the signature of both Wannier- +Mott and Frenkel exciton species. The out-of-plane molecular orientation is also +applied to tailor the interfacial hybrid excitons. Our results suggest that the organic- +inorganic heterostructure is a promising platform to explore many-body exciton physics. + +6 + +Results + +Fig. 1 | Sample configuration and basic optical characterization. a Schematic illustration of the +different molecular orientations (face-on and edge-on) at CuPc/MoSe2 heterostructure interface. b +AFM topographic image of a typical CuPc/MoSe2 heterostructure on Si/SiO2 substrate. c PL spectra +of MoSe2 and CuPc/MoSe2 heterostructure at 298 K. d Time-resolved PL spectra of MoSe2 and +CuPc/MoSe2 heterostructure at 298 K. The solid curves are the fitted results. +Sample configuration and basic optical characterization +Figure 1a shows a schematic of the CuPc/MoSe2 heterostructure configuration. +We consider two different molecular orientations, i.e., face-on and edge-on, at the +CuPc/MoSe2 interface, which will sensitively influence the interfacial coupling strength. +The CuPc/MoSe2 heterostructure is prepared by directly evaporating CuPc molecules +on top of a monolayer MoSe2 surface in vacuum (see details in methods). A film +thickness of ~5 nm is determined by AFM (Fig. 1b and Supplementary Fig. 1c). The + +a +b +10 nm +edge-on +Cu +CuPc +N +face-on +CuPc +CuPclMoSe +C +H +Mo +Se +MoSe2 +600nm +0 +c +d +×20 +PL intensity (a.u.) +-MoSe, +Normalized PL +- CuPc/MoSe2 +0.1 +1.4 +1.5 +1.6 +1.7 +0 +1 +2 +Photon energy (eV) +Delay time (ns)7 + +optimal molecular orientation at interface of the as-grown sample is the face-on +orientation, which has been reported in similar systems34,36 and revealed by our +theoretical calculation. The edge-on orientation is introduced by using the CuPc single +crystal later. At the face-on orientation, the planar conjugated structure of the CuPc +molecule37 and the atomic flat surface of monolayer MoSe210 without dangling bonds +facilitate interfacial coupling between them (Fig. 1a). The photoluminescence (PL) +spectra of MoSe2 and CuPc/MoSe2 heterostructure acquired at room temperature are +shown in Fig. 1c and Supplementary Fig.1d. The MoSe2 exhibits a pronounced PL peak +located at ~1.58 eV from the A excitonic transition38. In contrast, a remarkable redshift +of ~20 meV in the PL peak energy and a strong quenching in the PL peak intensity are +observed in CuPc/MoSe2 heterostructure. The pure CuPc film shows no detectable PL +signal (Supplementary Fig. 2a) due to the weak absorption at approximately 514 nm34 +and its strong intersystem crossing39. Control experiments are further performed to +examine the underlying possibilities of the observed phenomena. By changing the +thickness of CuPc thin film, it is found that the PL quenching ratio remains almost +unchanged (Supplementary Fig. 2c), indicating that absorption of the CuPc film is not +the main reason and the phenomena may stem from interfacial interaction. We also +adopt another heterostructure configuration by dry transferring MoSe2 on top of the +CuPc film (Supplementary Fig.3) and observe similar phenomena, which indicates that +the dielectric environment change has negligible influence here40,41. Therefore, the +observed phenomena should originate from the intrinsic interfacial coupling between +CuPc and MoSe2. Time-resolved PL measurements are performed to compare the PL + +8 + +lifetimes of MoSe2 and CuPc/MoSe2 heterostructure (Fig. 1c). The PL decays of both +MoSe2 and CuPc/MoSe2 heterostructure can be well fitted with a biexponential function, +thus, two processes can be derived from both decay curves. For MoSe2, the fast decay +constituent has a lifetime of ~85 ps, which is consistent with the lifetime of the A +exciton42. The slow decay component has a lifetime of ~1495 ps and is likely to +originate from defect-bound excitons43,44. The PL decay of heterostructure exhibits +much shorter lifetime than the A exciton of MoSe2, which is not consistent with the +behavior of interfacial charge transfer exciton because it usually has a longer lifetime +due to the spatial indirect nature31,45. + +Fig. 2 | Hybrid excitons in CuPc/MoSe2 heterostructure. a PL spectra of the MoSe2 and +CuPc/MoSe2 heterostructure at 78 K. b The PL peak intensity of MoSe2 and CuPc/MoSe2 +heterostructure as a function of the excitation power. c PL spectra of the MoSe2 and CuPc/MoSe2 + +a +b +1.0 +x +hX2 +X +MoSe2 +105 +T +78 K +78 K +★ +hx +~p1.13 +CuPc/MoSe2 +intensity (a.u.) +hx, +PL +104 +hX +P1.13 +-P1.19 +103 +P1.26 +Normal +PLi +102 +P1.09 +hX? +T +hX4 +101 +~P1.26 +0.0 +1.50 +1.65 +1.80 +1.95 +1 +10 +100 +Photon energy (eV) +Excitation power(μW) +C 1.0 +d +hX2 +T +4 K +MoSe2 +10 +000 +Ix/lT +hX, +Peak intensity ratio +CuPc/MoSe2 +PL +Normalized I +Q00 +00.0000 +0.5 +Ihx1/lhx2 +X +hX3 +00 +hX4 +0.1 +0.0 +1.50 +1.65 +1.80 +1.95 +0 +50 +100 +150 +200 +Photon energy (eV) +Temperature (K)9 + +heterostructure at 4 K. d The PL peak intensity ratio of exciton versus trion in MoSe2 (IX/IT) and hx1 +versus hx2 in the CuPc/MoSe2 heterostructure (IhX1/IhX2) as a function of temperature. +Emergence of interfacial hybrid excitons +Low-temperature PL spectra under 514 nm excitation are obtained at 78 K to +further reveal the possible mechanism. We still observe no detectable PL signal in the +pure CuPc thin film (Supplementary Fig. 2b). As displayed in Fig. 2a, two emission +peaks located at ~1.648 eV and 1.618 eV are observed in the PL spectrum of MoSe2, +which can be ascribed to the emission from the A exciton (X) and trion (T) of MoSe238. +A striking contrast is observed in the PL spectrum of heterostructure, with four new +emission peaks located at ~1.630 eV, ~1.606 eV, ~1.727 eV and ~1.848 eV emerging, +which are labeled hX1, hX2, hX3, and hX4, respectively. The hX1 and hX2 show a clear +redshift compared with the A exciton of MoSe2, and the hX4 peak displays an obvious +redshift with respect to the B exciton of MoSe2 (Supplementary Fig. 4). The hX3 peak +is a totally new PL peak that are not observed in pure MoSe2 and CuPc films. Charge +transfer exciton31 or dark exciton46 of MoSe2 is also excluded because it has a much +higher energy (~79 meV) than the A exciton of MoSe2. In addition, the PL peaks of +heterostructure show clear broadening compared with those of MoSe2. We ascribe the +observed PL peak redshift and broadening to the signature of interfacial hybridization +as reported in similar MoSe2/WS2 heterostructure6. Power-dependent PL spectra are +further obtained to examine the origin of the new peaks in heterostructure (Fig. 2b and +Supplementary Fig. 5). It is obvious that the peak intensity is enhanced with increasing +power. The relationship between excitation power and PL intensity can be expressed +as47 𝐼 ∝ 𝑃𝛼, in which 𝐼 represents the PL intensity and 𝑃 represents the excitation + +10 + +power. The intensities of X and T peaks in MoSe2 show a linear relationship with +excitation power with a slope of ~1.13, which indicates recombination from excitons48. +Interestingly, all the new peaks in heterostructure also show the linear relationship with +similar slopes, which suggests similar exciton behavior with no biexciton49 or defect +effect50. Since we have observed new PL peaks with exciton behavior and the signature +of hybridization, it is possible that new hybrid excitons are formed in CuPc/MoSe2 +heterostructure due to interfacial hybridization. +We further perform PL measurements at 4 K to examine the influence of interfacial +hybridization. As illustrated in Fig. 2c, the PL spectrum of MoSe2 at 4 K is dominated +by trion rather than A exciton due to the enhanced trion localization, in great contrast +to that at 78 K42. On the contrary, the PL spectrum of heterostructure is still dominated +by hX1 and hX2, similar to that at 78 K. Figure 2d displays the evolution of X/T from 4 +K to 200 K, it is clear that the ratio of X/T has increased from 0.09 to 10 when the +temperature is increased from 4 K to 200 K due to the increased thermal perturbance to +trion formation. Nevertheless, the radio of hX1/hX2 shows a weak temperature +dependence from 4 K to 200 K, differing from the pure exciton and trion behavior in +MoSe2, which indicates that the interfacial hybridization effect has changed the exciton +behavior in heterostructure. In addition, the PL spectrum of heterostructure displays a +highly asymmetric line-shape with prominent low energy tail (Fig. 2a, c), which can be +ascribed to an energy shakeup process during the recombination of hybrid excitons 8. + + +11 + +Fig. 3 | Theoretical calculation of electronic structure in CuPc/MoSe2 heterostructure. a Top +-view and side-view for the optimized structure of CuPc/MoSe2 heterostructure. b Calculated band +structure of CuPc/MoSe2 heterostructure. c Conduction bands (CB), CB+1, CB+2 and CB+3 in the +energy range of 0.72 to 0.80 eV, which correspond to the blue square in b. d Projected charge +density for bands in b. ΔE (23 meV) is defined as the energy difference between CB+1 and CB+3, +which is mostly contributed by CuPc and MoSe2, respectively. e Schematic illustration of the +formation of interfacial hybrid excitons due to the hybridization between LUMO of CuPc and CBM +of MoSe2. +The above results indicate that the interfacial hybridization effect could lead to the +formation of hybrid excitons and change the exciton behavior at the CuPc/MoSe2 +interface. First-principles calculations are further performed to confirm this. The +heterostructure is built by adsorbing a CuPc molecule on a 5×3√3 supercell of MoSe2 +and the optimal molecular orientation is the face-on orientation (Fig. 3a). The calculated +electronic structure of CuPc/MoSe2 heterostructure with face-on orientation is shown + +a +Top-view +Side-view +e +CuPc +3.36A +hX +CuPc +h2o +MoSe2 +MoSe2 +b +1.0 +C 0.80 +CB +(eV) +CB+1 +0.5 +出 +AE +W-0.5 +CB+2 +CB+3 +CB +CB+1 +CB+2 +CB+3 +-1.0 +0.72 +x +Y12 + +in Fig. 3b. We could recognize two nearly flat bands near 0.5 and -0.5 eV, which +correspond to the singly occupied and unoccupied molecular orbitals (SOMO and +SUMO) of the CuPc molecule. Then, we concentrate on the bands in the energy range +of 0.72 to 0.80 eV (Fig. 3c), which are denoted as conduction band CB, CB+1, CB+2 +and CB+3. As shown in Fig. 3d, the projected charge density shows that CB and CB+1 +are mostly contributed by CuPc, and CB+3 is mostly contributed by MoSe2. Notably, +CB+2 is contributed both by CuPc and MoSe2, which could be regarded as the +hybridization between the LUMO of CuPc and the conduction band of MoSe2. The +energy difference between CB+3 and CB+1 in the same spin channel is approximately +23 meV, leading to strong hybridization between CuPc and MoSe2 at the face-on +orientation, which can explain the observed new hybrid excitons at CuPc/MoSe2 +interface. The calculation also reveals that the formed hybrid excitons consist of +electrons extended in both layers and holes confined in individual layers (Fig. 3e), +which can be used to achieve novel quantum control at organic-inorganic interfaces7. + + + +13 + + +Fig. 4 | Temperature dependence of the hybrid excitons. a Two-dimensional PL spectrum of +CuPc/MoSe2 heterostructure as a function of temperature. b PL spectra of CuPc/MoSe2 +heterostructure in the energy range of 1.4 - 1.95 eV at the temperatures of 78 K, 98 K, 138 K, 178 +K, and 208 K, respectively. c PL spectra of CuPc/MoSe2 heterostructure in the energy range of 1.68- +1.95 eV at the temperatures of 78 K, 98 K, 138 K, 178 K, and 208 K, respectively. d The peak +energy of hX1 (black), hX2 (red), and hX3 (purple) as a function of temperature. +Temperature-dependent behavior of interfacial hybrid excitons +The temperature-dependent behavior of the observed hybrid excitons is carefully +examined from 78 K to 298 K. For CuPc /MoSe2 heterostructure (Fig. 4a), we clearly +observed a remarkable increase in the whole PL intensity when cooling from room +temperature (298 K) to low temperature (78 K), which can be explained by the +suppression of nonradiative recombination51. When the temperature is higher than 178 +K, hX2 becomes undistinguishable and hX1 dominates the PL spectra (Fig. 4b). For +MoSe2, the trion peak disappears at 98 K and the A exciton becomes dominant + +a +78 +b +hX +C +6 +hX, +hX4 +hX3 +2 +208 K +hX3 +208 K +max +98 +hX4 +1 +3 +hX3hX4 +118 +0 +0 +hX +148 +178K +178 K +2 +4 +hX4 +1 +188 +hX3 hX4 +2 +(×103) +0 +0 +228 +138K +hx +138 K +min +intensity +intensity +4 +278 +2 +hX +1.4 +1.6 +1.8 +2.0 +hX3 hX4 +2 +Photon energy (eV) +0 +d 1.75 +μ20 +xyaxy +PL +0 +98K +hX. +98 K +14 +1.70 +10 +hX +7 +4 +(eV) +hX3 +hX +hX4 +hX +0 +hX2/ +0 +40 +hX, +78 K +78 K +18 +1.60 +... +20 +hX3 +hX3 hX4 +9 +0 +1.55 +0 +100 +150 +200 +250 +300 +1.4 +1.6 +1.8 +1.71 +1.80 +1.89 +Temperature (K) +Photon energy (eV) +Photonenergy (eV)14 + +(Supplementary Fig. 6). To our surprise, the peak energy of the hybrid excitons shows +different temperature dependence (Fig. 4a, d). We first focus on the hX1, hX2, and hX4 +peaks, of which the peak energy shows obvious redshift with increasing temperature +(Fig. 4b, d) due to the increased electron-phonon interactions51, similar to the +temperature-dependent behavior of exciton and trion in MoSe2 (Supplementary Fig. 6). +By fitting the peak energy with the standard semiconductor bandgap model52:𝐸𝑔(0) = +𝐸𝑔(𝑇) − 𝑆ℏ𝜔 [𝑐𝑜𝑡ℎ ( +ℏ𝜔 +2𝑘𝐵𝑇) −1] , where 𝐸𝑔 represents the bandgap, ℏ𝜔 represents +the phonon energy, 𝑆 represents the electron-phonon coupling strength, and 𝑇 +represents the temperature, we obtain a similar phonon energy for the CuPc/MoSe2 +heterostructure and MoSe2 (Supplementary Fig. 7a, and Supplementary Table 1), +suggesting that these hybrid excitons are also influenced by the phonons of MoSe2. +The hX3 peak located at ~1.72 eV is more unique among the four hybrid excitons. +The peak energy shows a rather weak temperature dependence (Fig 4c, d), which is +totally different from the other three peaks. Such weak temperature-dependent behavior +of excitons has been observed in organic molecules, in which the effects of thermal +expansion and exciton-phonon coupling almost cancel out53, indicating that the hX3 +peak displays the signature of Frenkel excitons in CuPc. However, this peak cannot be +simply assigned to the emission of the CuPc molecules since no PL signals of CuPc +film were observed at 78 K (Supplementary Fig. 2b). Furthermore, we can even observe +this peak in the heterostructure region when we dry transferred monolayer MoSe2 on +top of the CuPc thin film immediately without any further treatment (Supplementary +Fig. 8), which can exclude the influence of the CuPc film morphology. This also + +15 + +indicates that the coupling at the CuPc/MoSe2 interface is very robust and the hybrid +excitons can be formed immediately once they are in contact without any further +treatment. On the other hand, it also combines the character of the Wannier-Mott +exciton in MoSe2. For example, the peak intensity of hX3 shows similar temperature- +dependent behavior with the A exciton of MoSe2 (Supplementary Fig. 7b), which +suggests that it gains a large oscillator strength from MoSe2 and shows a detectable PL +signal compared with the pure CuPc film. Therefore, the peak energy of hX3 shows the +signature of the Frenkel excitons in the organic CuPc film and its emission properties +display the character of Wannier-Mott excitons in the inorganic MoSe2 monolayer, +which unambiguously reveal the formation of hybrid Frenkel-Wannier-Mott excitons +at the CuPc/MoSe2 interface. + + +a +b +hX, +100°C +?200°C +X +100°℃ +200°C +anneal +hX2 +MOSe +CL +UPC +MoSe,/cuPc +(crvstal +crystal) +Normalized PL +Edge-on +MoSe,/CuPccrystal region +C +hX3 +hX4 +hx +mixed Face-on +hX3 +3mixedFace-on +and Edge-on +andEdge-on +hX +hX +hx. +hx4 +MoSe, region +Face-on +Face-on +hX3 +hX4 +1.5 +1.6 +1.7 +1.8 +1.91.5 +1.6 +1.7 +1.8 +1.9 +1.7 +1.8 +1.9 +Photon energy (ev) +Photon energy (ev) +Photonenergy (eV) +d +e +HS(film) +Face-on +('n'e) +CuPc +hybridization +MoSe, +Raman intensity +? +h +Hybrid Exciton +HybridExciton +CuPc +200°℃ +Mixed +Face-on +(h) +Intralayer Exciton +Interlayer +HS(crystal)Edge-on +MoSe,region +100℃ +@h +200240 +1380 +1610 +100°C +200°C +Ramanshift(cm-1) +Annealingtemperature16 + +Fig. 5 | Molecular orientation-dependent hybrid excitons. a The PL spectra in the MoSe2 region +and MoSe2/CuPc crystal region for the same MoSe2/CuPc crystal heterostructure after annealing at +100°C and 200°C. After annealing at 200°C, the CuPc crystal partially decomposes and the CuPc +molecules can migrate on MoSe2, which lead to the formation hybrid excitons in both regions. b +AFM topographic image of the MoSe2/CuPc crystal heterostructure after annealing at 100℃ and +200℃. c Enlarged view of the image in the gray dotted box in a. d Raman spectra of the MoSe2 +region in the MoSe2/CuPc crystal heterostructure sample after annealing at 100°C and 200°C. e +Schematic illustration of the relationship between molecular orientation (face-on, edge-on) of CuPc +and interlayer hybridization. +Tailoring the hybrid exciton using molecular orientation +The molecular orientation is introduced as a new degree of freedom to modulate the +interfacial hybridization strength, which will further tailor the interfacial hybrid +excitons. In general, the CuPc molecule tends to adopt a face-on orientation on the +MoSe2 surface that allows efficient interfacial hybridization, as revealed by our +theoretical calculation. In contrast, the edge-on orientation will experience insufficient +interfacial hybridization due to the larger interfacial distance. To demonstrate that the +molecular orientation can be used to tailor the interfacial hybrid exciton states, we +carefully prepared MoSe2/CuPc film heterostructure and MoSe2/CuPc crystal +heterostructure simultaneously by the dry transfer method. Since the CuPc molecule +stacks randomly in the CuPc film, it easily adopts a face-on orientation on the MoSe2 +surface. However, because the CuPc molecule shows a herringbone stacking in the +crystal54, it can only adopt an edge-on orientation on MoSe2 surface before crystal +decomposition. Because the monolayer MoSe2 partially covers the CuPc crystal, we +could compare the measurements from the MoSe2 region and MoSe2/CuPc crystal + +17 + +region in the same sample (Supplementary Fig. 9a). After annealing simultaneously at +100°C, the PL spectra of MoSe2/CuPc film heterostructure and MoSe2/CuPc crystal +heterostructure display great contrast as expected. The PL of crystal heterostructure +shows similar spectral features and slight quenching compared with monolayer MoSe2 +(Fig. 5a and Supplementary Fig. 9b), indicating a weak interfacial hybridization +strength. However, the PL of film heterostructure presents obvious quenching and the +formation of hybrid excitons (Supplementary Fig. 9b, c). The influence of the +morphology of CuPc can be excluded since the surface of CuPc crystal is flatter than +that of the CuPc film (Supplementary Fig. 10). Therefore, the above results suggest that +the molecular orientation can be used to tune the interlayer hybridization strength and +further tailor the interfacial hybrid excitons. +To confirm this deduction, the MoSe2/CuPc crystal heterostructure is further +annealed at 200oC to decompose the CuPc crystal. When the CuPc crystal decomposes, +the CuPc molecules can easily adopt a face-on orientation on the MoSe2 surface, thus, +the interfacial hybrid excitons should also be observed. After annealing at 200°C, the +PL spectra display obvious quenching in both the MoSe2/CuPc crystal region and +MoSe2 region (Supplementary Fig. 9d), and clearly shows the formation of hybrid +excitons (Fig. 5a, c), which indicates that the molecular orientation is changed from +edge-on to face-on after CuPc crystal decomposition. The decomposition of the CuPc +crystal is confirmed by AFM, as shown in Fig. 5b. It is obvious that the CuPc crystal +partially decomposes after annealing at 200°C, as evidenced by the change in the AFM +height profile. Note that we can even observe hybrid excitons in the MoSe2 region + +18 + +because the CuPc molecules can migrate on MoSe2 surface after the decomposition of +CuPc crystal. The Raman spectra further confirms this, as the Raman peaks of both +MoSe2 and CuPc molecules appear in the MoSe2 region after annealing at 200°C (Fig. +5d). These results unambiguously show that we can successfully tailor the interfacial +hybrid excitons by changing the molecular orientation (Fig. 5e). The theoretical +calculation also supports our results. As shown in the electronic structure of +heterostructure at the edge-on orientation (Supplementary Fig. 11), we observe no +obvious interfacial hybridization, which coincides with the observed phenomena in +MoSe2/CuPc crystal heterostructure. +Discussion + We have demonstrated the formation of interfacial hybrid excitons in CuPc/MoSe2 +heterostructure due to the hybridization between CuPc and MoSe2. The observed +phenomenon is unusual as the coupling at the organic-inorganic interface is generally +weak24. The first principles calculations rationalize the results as the LUMO of CuPc +strongly hybridized with the CBM of MoSe2, which leads to the emergence of new +eigenstates. The new hybrid excitons consist of electrons delocalized in both layers and +holes confined in individual layer, enabling simultaneous large optical and electrical +dipoles7. The temperature-dependent behavior suggests that the hybrid excitons +simultaneously gain the signature of the Wannier-Mott excitons in MoSe2 and the +Frenkel excitons in CuPc. The excellent agreement between the theoretical and +experimental results not only validates the observed strong coupling phenomenon, but +also provides a basis for manipulating hybrid excitons at the organic-inorganic interface. + +19 + +For instance, the large electrical dipole in the out-of-plane direction can be used to +achieve novel electrical control of the hybrid excitons55,56. Our result is also of great +importance for realizing tunable interlayer hybridization strength by changing the +molecular orientation, which can be used to tailor the exciton states at the organic- +inorganic interface. In conclusion, we report the formation of interfacial hybrid excitons +with strong molecular orientation dependence that originate from the hybridization +between CuPc and MoSe2, which is meaningful for many-body exciton physics at the +organic-inorganic interface. + +Methods +Sample preparation +(1) Construction of the CuPc film /MoSe2 heterostructure +Monolayer MoSe2 was mechanically exfoliated on a SiO2/Si substrate from bulk +crystals and further annealed in vacuum at 200 °C to remove surface contaminants. The +thickness of MoSe2 was confirmed by optical contrast, atomic force microscopy (AFM), +and Raman measurements (Supplementary Fig. 1). To construct the CuPc (film)/MoSe2 +heterostructure, CuPc thin film was directly deposited on top of monolayer MoSe2 using +thermal evaporation in vacuum (home-built evaporator). The heating current was +maintained at 5 amperes and the average evaporation speed was 0.25 nm/min. +(2) Construction of the MoSe2/CuPc film heterostructure +The CuPc film was firstly thermally evaporated on a SiO2/Si substrate using the same +conditions as (1). Then, monolayer MoSe2 was mechanically exfoliated on the PDMS + +20 + +substrate from bulk crystals, and further transferred on top of the CuPc film using the +dry transfer method. +(3) Construction of the MoSe2/CuPc crystal heterostructure +The single crystals of CuPc were grown by the physical vapor deposition (PVT) method +in a quartz tube with a hot zone temperature of 400°C. To construct the MoSe2/CuPc +(crystal) heterostructure, monolayer MoSe2 was mechanically exfoliated on a PDMS +substrate, and further transferred on top of a CuPc single crystal using the dry transfer +method. +Low temperature PL Measurements. +The measurements at 78 K were conducted in a temperature-controlled cryostat +(THMS600, Linkam) with a diffraction-limited excitation beam diameter of 1µm.The +signal was collected using a 50X long-working distance objective and detected on a +commercial Renishaw inVia spectrometer. The excitation power was selected to be +below 200 µW to avoid heating damage to the sample. The measurements at 4 K were +conducted in a temperature-controlled cryostat (Montana Instruments) with an +excitation beam diameter of 1µm. The signal was collected using a 100X objective and +detected on a commercial Ocean Optics spectrometer. +Theoretical calculation. First-principles calculations were carried out based on the +density functional theory (DFT) framework by utilizing the Vienna Ab initio Simulation +Package (VASP) 5.4.4 package57,58. Pseudopotentials were used to describe the +electron-ion interactions within the PAW approach and generalized gradient +approximations (GGA) of Perdew-Burke-Ernzerhof (PBE) were adopted for the + +21 + +exchange-correlation potential59-61. To better describe the interlayer van der Waals +(vdW) interactions, we adopt optB88-vdW corrections for the optimization of +structures62. The electron wave functions are expanded on a plane-wave basis set with +an energy cutoff of 520 eV. The atomic coordinates of all structures were allowed to +relax until the forces acting on the ions were less than 0.01 eV Å-1. The convergence +criterion for the electronic self-consistent cycle is fixed at 1×10-5 eV. The integrations +in the reduced Brillouin zone are performed on a 3×3×1 Monkhorst-Pack special k- +points for optimization and self-consistent calculations63,64. A vacuum slab above 15 Å +was used in all calculations to avoid interlayer interactions. The CuPc/MoSe2 +heterostructure is modeled by adsorbing one CuPc molecule on a 5×3√3 supercell of +MoSe2, which can be written as Mo30Se60C32N8H16Cu. The lattice parameters of the +CuPc/MoSe2 heterostructure were calculated to be a = 16.62 Å, b = 17.27 Å, and +α=β=γ=90°. The interlayer distance between CuPc and MoSe2 substrate is +approximately 3.36 Å.× + +Data availability +The data that support the findings of this study are available from the corresponding +authors upon reasonable request. + +References +1. +Kleemans, N.A.J.M., et al. Many-body exciton states in self-assembled quantum dots coupled +to a Fermi sea. Nat. Phys. 6, 534-538 (2010). +2. +Shimazaki, Y., Schwartz, I., Watanabe, K., Taniguchi, T., Kroner, M. & Imamoglu, A. 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B 40, 3616-3621 (1989). + +25 + + +Acknowledgements +This work was supported by the National Nature Science Foundation of China (Grant +Nos. 11974088, 12074116, 21790353, 61875236, 61975007), the National Key +Research and Development Program of China (Grant Nos.2016YFA0202302, +2017YFA0205000, 2021YFA1400201), the Strategic Priority Research Program of +CAS (Grant No. XDB30000000), the CAS Project for Young Scientists in Basic +Research (Grant No. YSBR-059). +Author contributions +X.Z., X.Q. and D.T. conceived the idea; S.F. and J.D. prepared the samples and +conducted all the optical measurements and the corresponding data analysis; X.W. and +H.L. performed the DFT calculations; This manuscript was prepared primarily by X.Z., +S.F. and J.D., and all authors contributed to discussing and commenting on the paper. +Competing interests +The authors declare no competing interests +Additional information +Correspondence and requests for materials should be addressed to Xiaoxian Zhang. + 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='6 Jimin Zhao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='4 Xiaojun Wu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='5 Dongsheng Tang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='2 * Xiaohui Qiu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='3 * Yongsheng Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Xiaoxian Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='1 * 1Key Laboratory of Luminescence and Optical Information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Ministry of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Institute of Optoelectronic Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Beijing Jiaotong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Beijing 100044,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' China 2Synergetic Innovation Center for Quantum Effects an Application,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Key Laboratory of Low-dimensional Quantum Structures and Quantum Control of Ministry of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' School of Physics and Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Hunan Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Changsha 410081,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' China 3CAS Key Laboratory of Standardization and Measurement for Nanotechnology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' CAS Center for Excellence in Nanoscience,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' National Center for Nanoscience and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 4Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 5Hefei National Laboratory for Physical Sciences at the Microscale, CAS Key Laboratory of Materials for Energy Conversion, Synergetic Innovation of Quantum Information & Quantum Technology, School of Chemistry and Materials Sciences, and CAS Center for Excellence in Nanoscience, University of Science and Technology of China, Hefei, Anhui 230026, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' China 2 6Beijing Information technology college, Beijing 100015, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' China #These authors contributed equally ∗e-mail: dstang@hunnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' xhqiu@nanoctr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' zhxiaoxian@bjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='cn 3 Abstract The coherent many-body interaction at the organic-inorganic interface can give rise to intriguing hybrid excitons that combine the advantages of the Wannier-Mott and Frenkel excitons simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Unlike the 2D inorganic heterostructures that suffer from moment mismatch, the hybrid excitons formed at the organic-inorganic interface have a momentum-direct nature, which have yet to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Here, we report hybrid excitons at the copper phthalocyanine/molybdenum diselenide (CuPc/MoSe2) interface with strong molecular orientation dependence using low-temperature photoluminescence spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The new emission peaks observed in the CuPc/MoSe2 heterostructure indicate the formation of interfacial hybrid excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The density functional theory (DFT) calculation confirms the strong hybridization between the lowest unoccupied molecular orbital (LUMO) of CuPc and the conduction band minimum (CBM) of MoSe2, suggesting that the hybrid excitons consist of electrons extended in both layers and holes confined in individual layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The temperature- dependent measurements show that the hybrid excitons can gain the signatures of the Frenkel excitons of CuPc and the Wannier-Mott excitons of MoSe2 simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The out-of-plane molecular orientation is used to tailor the interfacial hybrid exciton states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Our results reveal the hybrid excitons at the CuPc/MoSe2 interface with tunability by molecular orientation, which suggests that the emerging organic-inorganic heterostructure can be a promising platform for many-body exciton physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 4 Introduction Hybrid excitons are many-body exciton states that originate from the hybridization of electronic states at interfaces1,2, which have been realized in distinct systems, including quantum dots coupled to a Fermi sea1,3, coupled quantum-dot molecules4,5, and emerging van der Waals heterostructures2,6-8, displaying great potential in kondo physics1,8, quantum optics4,7, and strongly correlated electronic physics2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Transition metal dichalcogenides (TMDs) have become a promising building block for hybrid excitons due to their strong light-matter interaction9,10 and rich exciton physics11-14, such as valley polarized excitons15-17, long-lived interlayer excitons18 and moiré excitons19-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' However, in TMD heterostructures, the momentum-mismatch problem severely restricts the formation of prominent hybrid excitons, which is moment-direct only at a small twist angle, requiring a precise control of the interlayer angle alignment during fabrication2,6,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Unlike the momentum-mismatch issue encountered in inorganic heterostructures, the hybrid excitons formed at the organic-inorganic heterostructures have a momentum- direct nature22, which could simplify the fabrication process and maintain the novel exciton physics at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' In addition, theoretical calculations have predicted that the hybrid excitons at the organic-inorganic interfaces can gain the signature of the Wannier-Mott excitons of inorganics and the Frenkel excitons of organics simultaneously23,24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Nevertheless, the coupling at the organic-inorganic interfaces is generally weak25, and an ordered structure of the organic materials is required to achieve strong electronic coupling24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' TMDs can be an ideal building brick for realizing 5 hybrid excitons at the organic-inorganic interfaces, because they not only contain rich exciton physics but can also serve as a suitable template for the growth of well-ordered organic films through the van der Waals epitaxial method26-31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Moreover, the short- range interactions, such as ultrafast charge transfer31-34 and interfacial spin orbital coupling35, have been observed at organic-TMD interfaces, which suggest that the coherent superposition of the electron wavefunctions and the formation of hybrid excitons are possible at such interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' However, experimental evidence for hybrid Wannier-Mott-Frenkel excitons at organic-TMD interfaces remains to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' It is well-known that the interfacial hybridization strength depends sensitively on the interlayer twist angle and stacking configuration in inorganic heterostructures6,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Similarly, the molecular orientation at the organic-inorganic interface can be used to tune the interfacial distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=', the hybridization strength, so the interfacial hybrid exciton behavior could be effectively tailored, which is beneficial for designing the organic-inorganic interface functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' In this article, we report the formation of hybrid excitons at the CuPc/MoSe2 interface and their further modulation by molecular orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The new emission peaks observed in photoluminescence spectroscopy and the further DFT calculations confirm the emergence of new hybrid excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The temperature-dependent measurements reveal that the hybrid excitons combine the signature of both Wannier- Mott and Frenkel exciton species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The out-of-plane molecular orientation is also applied to tailor the interfacial hybrid excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Our results suggest that the organic- inorganic heterostructure is a promising platform to explore many-body exciton physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 6 Results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 1 | Sample configuration and basic optical characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' a Schematic illustration of the different molecular orientations (face-on and edge-on) at CuPc/MoSe2 heterostructure interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' b AFM topographic image of a typical CuPc/MoSe2 heterostructure on Si/SiO2 substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' c PL spectra of MoSe2 and CuPc/MoSe2 heterostructure at 298 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' d Time-resolved PL spectra of MoSe2 and CuPc/MoSe2 heterostructure at 298 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The solid curves are the fitted results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Sample configuration and basic optical characterization Figure 1a shows a schematic of the CuPc/MoSe2 heterostructure configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' We consider two different molecular orientations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=', face-on and edge-on, at the CuPc/MoSe2 interface, which will sensitively influence the interfacial coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The CuPc/MoSe2 heterostructure is prepared by directly evaporating CuPc molecules on top of a monolayer MoSe2 surface in vacuum (see details in methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' A film thickness of ~5 nm is determined by AFM (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 1b and Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The a b 10 nm edge-on Cu CuPc N face-on CuPc CuPclMoSe C H Mo Se MoSe2 600nm 0 c d ×20 PL intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=') MoSe, Normalized PL CuPc/MoSe2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='7 0 1 2 Photon energy (eV) Delay time (ns)7 optimal molecular orientation at interface of the as-grown sample is the face-on orientation, which has been reported in similar systems34,36 and revealed by our theoretical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The edge-on orientation is introduced by using the CuPc single crystal later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' At the face-on orientation, the planar conjugated structure of the CuPc molecule37 and the atomic flat surface of monolayer MoSe210 without dangling bonds facilitate interfacial coupling between them (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The photoluminescence (PL) spectra of MoSe2 and CuPc/MoSe2 heterostructure acquired at room temperature are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 1c and Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The MoSe2 exhibits a pronounced PL peak located at ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='58 eV from the A excitonic transition38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' In contrast, a remarkable redshift of ~20 meV in the PL peak energy and a strong quenching in the PL peak intensity are observed in CuPc/MoSe2 heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The pure CuPc film shows no detectable PL signal (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 2a) due to the weak absorption at approximately 514 nm34 and its strong intersystem crossing39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Control experiments are further performed to examine the underlying possibilities of the observed phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' By changing the thickness of CuPc thin film, it is found that the PL quenching ratio remains almost unchanged (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 2c), indicating that absorption of the CuPc film is not the main reason and the phenomena may stem from interfacial interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' We also adopt another heterostructure configuration by dry transferring MoSe2 on top of the CuPc film (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='3) and observe similar phenomena, which indicates that the dielectric environment change has negligible influence here40,41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Therefore, the observed phenomena should originate from the intrinsic interfacial coupling between CuPc and MoSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Time-resolved PL measurements are performed to compare the PL 8 lifetimes of MoSe2 and CuPc/MoSe2 heterostructure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The PL decays of both MoSe2 and CuPc/MoSe2 heterostructure can be well fitted with a biexponential function, thus, two processes can be derived from both decay curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' For MoSe2, the fast decay constituent has a lifetime of ~85 ps, which is consistent with the lifetime of the A exciton42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The slow decay component has a lifetime of ~1495 ps and is likely to originate from defect-bound excitons43,44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The PL decay of heterostructure exhibits much shorter lifetime than the A exciton of MoSe2, which is not consistent with the behavior of interfacial charge transfer exciton because it usually has a longer lifetime due to the spatial indirect nature31,45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 2 | Hybrid excitons in CuPc/MoSe2 heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' a PL spectra of the MoSe2 and CuPc/MoSe2 heterostructure at 78 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' b The PL peak intensity of MoSe2 and CuPc/MoSe2 heterostructure as a function of the excitation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' c PL spectra of the MoSe2 and CuPc/MoSe2 a b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='0 x hX2 X MoSe2 105 T 78 K 78 K ★ hx ~p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='13 CuPc/MoSe2 intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=') hx, PL 104 hX P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='13 P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='19 103 P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='26 Normal PLi 102 P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='09 hX?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' T hX4 101 ~P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='95 1 10 100 Photon energy (eV) Excitation power(μW) C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='0 d hX2 T 4 K MoSe2 10 000 Ix/lT hX, Peak intensity ratio CuPc/MoSe2 PL Normalized I Q00 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='5 Ihx1/lhx2 X hX3 00 hX4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='95 0 50 100 150 200 Photon energy (eV) Temperature (K)9 heterostructure at 4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' d The PL peak intensity ratio of exciton versus trion in MoSe2 (IX/IT) and hx1 versus hx2 in the CuPc/MoSe2 heterostructure (IhX1/IhX2) as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Emergence of interfacial hybrid excitons Low-temperature PL spectra under 514 nm excitation are obtained at 78 K to further reveal the possible mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' We still observe no detectable PL signal in the pure CuPc thin film (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' As displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 2a, two emission peaks located at ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='648 eV and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='618 eV are observed in the PL spectrum of MoSe2, which can be ascribed to the emission from the A exciton (X) and trion (T) of MoSe238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' A striking contrast is observed in the PL spectrum of heterostructure, with four new emission peaks located at ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='630 eV, ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='606 eV, ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='727 eV and ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='848 eV emerging, which are labeled hX1, hX2, hX3, and hX4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The hX1 and hX2 show a clear redshift compared with the A exciton of MoSe2, and the hX4 peak displays an obvious redshift with respect to the B exciton of MoSe2 (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The hX3 peak is a totally new PL peak that are not observed in pure MoSe2 and CuPc films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Charge transfer exciton31 or dark exciton46 of MoSe2 is also excluded because it has a much higher energy (~79 meV) than the A exciton of MoSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' In addition, the PL peaks of heterostructure show clear broadening compared with those of MoSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' We ascribe the observed PL peak redshift and broadening to the signature of interfacial hybridization as reported in similar MoSe2/WS2 heterostructure6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Power-dependent PL spectra are further obtained to examine the origin of the new peaks in heterostructure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 2b and Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' It is obvious that the peak intensity is enhanced with increasing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The relationship between excitation power and PL intensity can be expressed as47 𝐼 ∝ 𝑃𝛼, in which 𝐼 represents the PL intensity and 𝑃 represents the excitation 10 power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The intensities of X and T peaks in MoSe2 show a linear relationship with excitation power with a slope of ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='13, which indicates recombination from excitons48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Interestingly, all the new peaks in heterostructure also show the linear relationship with similar slopes, which suggests similar exciton behavior with no biexciton49 or defect effect50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Since we have observed new PL peaks with exciton behavior and the signature of hybridization, it is possible that new hybrid excitons are formed in CuPc/MoSe2 heterostructure due to interfacial hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' We further perform PL measurements at 4 K to examine the influence of interfacial hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 2c, the PL spectrum of MoSe2 at 4 K is dominated by trion rather than A exciton due to the enhanced trion localization, in great contrast to that at 78 K42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' On the contrary, the PL spectrum of heterostructure is still dominated by hX1 and hX2, similar to that at 78 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Figure 2d displays the evolution of X/T from 4 K to 200 K, it is clear that the ratio of X/T has increased from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='09 to 10 when the temperature is increased from 4 K to 200 K due to the increased thermal perturbance to trion formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Nevertheless, the radio of hX1/hX2 shows a weak temperature dependence from 4 K to 200 K, differing from the pure exciton and trion behavior in MoSe2, which indicates that the interfacial hybridization effect has changed the exciton behavior in heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' In addition, the PL spectrum of heterostructure displays a highly asymmetric line-shape with prominent low energy tail (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 2a, c), which can be ascribed to an energy shakeup process during the recombination of hybrid excitons 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 3 | Theoretical calculation of electronic structure in CuPc/MoSe2 heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' a Top view and side-view for the optimized structure of CuPc/MoSe2 heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' b Calculated band structure of CuPc/MoSe2 heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' c Conduction bands (CB), CB+1, CB+2 and CB+3 in the energy range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='72 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='80 eV, which correspond to the blue square in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' d Projected charge density for bands in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' ΔE (23 meV) is defined as the energy difference between CB+1 and CB+3, which is mostly contributed by CuPc and MoSe2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' e Schematic illustration of the formation of interfacial hybrid excitons due to the hybridization between LUMO of CuPc and CBM of MoSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The above results indicate that the interfacial hybridization effect could lead to the formation of hybrid excitons and change the exciton behavior at the CuPc/MoSe2 interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' First-principles calculations are further performed to confirm this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The heterostructure is built by adsorbing a CuPc molecule on a 5×3√3 supercell of MoSe2 and the optimal molecular orientation is the face-on orientation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The calculated electronic structure of CuPc/MoSe2 heterostructure with face-on orientation is shown a Top-view Side-view e CuPc 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='36A hX CuPc h2o MoSe2 MoSe2 b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='0 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='80 CB (eV) CB+1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='5 出 AE W-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='5 CB+2 CB+3 CB CB+1 CB+2 CB+3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='72 x Y12 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' We could recognize two nearly flat bands near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='5 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='5 eV, which correspond to the singly occupied and unoccupied molecular orbitals (SOMO and SUMO) of the CuPc molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Then, we concentrate on the bands in the energy range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='72 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='80 eV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 3c), which are denoted as conduction band CB, CB+1, CB+2 and CB+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 3d, the projected charge density shows that CB and CB+1 are mostly contributed by CuPc, and CB+3 is mostly contributed by MoSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Notably, CB+2 is contributed both by CuPc and MoSe2, which could be regarded as the hybridization between the LUMO of CuPc and the conduction band of MoSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The energy difference between CB+3 and CB+1 in the same spin channel is approximately 23 meV, leading to strong hybridization between CuPc and MoSe2 at the face-on orientation, which can explain the observed new hybrid excitons at CuPc/MoSe2 interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The calculation also reveals that the formed hybrid excitons consist of electrons extended in both layers and holes confined in individual layers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 3e), which can be used to achieve novel quantum control at organic-inorganic interfaces7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 4 | Temperature dependence of the hybrid excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' a Two-dimensional PL spectrum of CuPc/MoSe2 heterostructure as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' b PL spectra of CuPc/MoSe2 heterostructure in the energy range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='4 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='95 eV at the temperatures of 78 K, 98 K, 138 K, 178 K, and 208 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' c PL spectra of CuPc/MoSe2 heterostructure in the energy range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='68- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='95 eV at the temperatures of 78 K, 98 K, 138 K, 178 K, and 208 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' d The peak energy of hX1 (black), hX2 (red), and hX3 (purple) as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Temperature-dependent behavior of interfacial hybrid excitons The temperature-dependent behavior of the observed hybrid excitons is carefully examined from 78 K to 298 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' For CuPc /MoSe2 heterostructure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 4a), we clearly observed a remarkable increase in the whole PL intensity when cooling from room temperature (298 K) to low temperature (78 K), which can be explained by the suppression of nonradiative recombination51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' When the temperature is higher than 178 K, hX2 becomes undistinguishable and hX1 dominates the PL spectra (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' For MoSe2, the trion peak disappears at 98 K and the A exciton becomes dominant a 78 b hX C 6 hX, hX4 hX3 2 208 K hX3 208 K max 98 hX4 1 3 hX3hX4 118 0 0 hX 148 178K 178 K 2 4 hX4 1 188 hX3 hX4 2 (×103) 0 0 228 138K hx 138 K min intensity intensity 4 278 2 hX 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='0 hX3 hX4 2 Photon energy (eV) 0 d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='75 μ20 xyaxy PL 0 98K hX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 98 K 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='70 10 hX 7 4 (eV) hX3 hX hX4 hX 0 hX2/ 0 40 hX, 78 K 78 K 18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='60 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 20 hX3 hX3 hX4 9 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='55 0 100 150 200 250 300 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='89 Temperature (K) Photon energy (eV) Photonenergy (eV)14 (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' To our surprise, the peak energy of the hybrid excitons shows different temperature dependence (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 4a, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' We first focus on the hX1, hX2, and hX4 peaks, of which the peak energy shows obvious redshift with increasing temperature (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 4b, d) due to the increased electron-phonon interactions51, similar to the temperature-dependent behavior of exciton and trion in MoSe2 (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' By fitting the peak energy with the standard semiconductor bandgap model52:𝐸𝑔(0) = 𝐸𝑔(𝑇) − 𝑆ℏ𝜔 [𝑐𝑜𝑡ℎ ( ℏ𝜔 2𝑘𝐵𝑇) −1] , where 𝐸𝑔 represents the bandgap, ℏ𝜔 represents the phonon energy, 𝑆 represents the electron-phonon coupling strength, and 𝑇 represents the temperature, we obtain a similar phonon energy for the CuPc/MoSe2 heterostructure and MoSe2 (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 7a, and Supplementary Table 1), suggesting that these hybrid excitons are also influenced by the phonons of MoSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The hX3 peak located at ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='72 eV is more unique among the four hybrid excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The peak energy shows a rather weak temperature dependence (Fig 4c, d), which is totally different from the other three peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Such weak temperature-dependent behavior of excitons has been observed in organic molecules, in which the effects of thermal expansion and exciton-phonon coupling almost cancel out53, indicating that the hX3 peak displays the signature of Frenkel excitons in CuPc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' However, this peak cannot be simply assigned to the emission of the CuPc molecules since no PL signals of CuPc film were observed at 78 K (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Furthermore, we can even observe this peak in the heterostructure region when we dry transferred monolayer MoSe2 on top of the CuPc thin film immediately without any further treatment (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 8), which can exclude the influence of the CuPc film morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' This also 15 indicates that the coupling at the CuPc/MoSe2 interface is very robust and the hybrid excitons can be formed immediately once they are in contact without any further treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' On the other hand, it also combines the character of the Wannier-Mott exciton in MoSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' For example, the peak intensity of hX3 shows similar temperature- dependent behavior with the A exciton of MoSe2 (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 7b), which suggests that it gains a large oscillator strength from MoSe2 and shows a detectable PL signal compared with the pure CuPc film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Therefore, the peak energy of hX3 shows the signature of the Frenkel excitons in the organic CuPc film and its emission properties display the character of Wannier-Mott excitons in the inorganic MoSe2 monolayer, which unambiguously reveal the formation of hybrid Frenkel-Wannier-Mott excitons at the CuPc/MoSe2 interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' a b hX, 100°C ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='200°C X 100°℃ 200°C anneal hX2 MOSe CL UPC MoSe,/cuPc (crvstal crystal) Normalized PL Edge-on MoSe,/CuPccrystal region C hX3 hX4 hx mixed Face-on hX3 3mixedFace-on and Edge-on andEdge-on hX hX hx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' hx4 MoSe, region Face-on Face-on hX3 hX4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content="9 Photon energy (ev) Photon energy (ev) Photonenergy (eV) d e HS(film) Face-on ('n'e) CuPc hybridization MoSe, Raman intensity ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' h Hybrid Exciton HybridExciton CuPc 200°℃ Mixed Face-on (h) Intralayer Exciton Interlayer HS(crystal)Edge-on MoSe,region 100℃ @h 200240 1380 1610 100°C 200°C Ramanshift(cm-1) Annealingtemperature16 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 5 | Molecular orientation-dependent hybrid excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' a The PL spectra in the MoSe2 region and MoSe2/CuPc crystal region for the same MoSe2/CuPc crystal heterostructure after annealing at 100°C and 200°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' After annealing at 200°C, the CuPc crystal partially decomposes and the CuPc molecules can migrate on MoSe2, which lead to the formation hybrid excitons in both regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' b AFM topographic image of the MoSe2/CuPc crystal heterostructure after annealing at 100℃ and 200℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' c Enlarged view of the image in the gray dotted box in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' d Raman spectra of the MoSe2 region in the MoSe2/CuPc crystal heterostructure sample after annealing at 100°C and 200°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' e Schematic illustration of the relationship between molecular orientation (face-on, edge-on) of CuPc and interlayer hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Tailoring the hybrid exciton using molecular orientation The molecular orientation is introduced as a new degree of freedom to modulate the interfacial hybridization strength, which will further tailor the interfacial hybrid excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' In general, the CuPc molecule tends to adopt a face-on orientation on the MoSe2 surface that allows efficient interfacial hybridization, as revealed by our theoretical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' In contrast, the edge-on orientation will experience insufficient interfacial hybridization due to the larger interfacial distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' To demonstrate that the molecular orientation can be used to tailor the interfacial hybrid exciton states, we carefully prepared MoSe2/CuPc film heterostructure and MoSe2/CuPc crystal heterostructure simultaneously by the dry transfer method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Since the CuPc molecule stacks randomly in the CuPc film, it easily adopts a face-on orientation on the MoSe2 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' However, because the CuPc molecule shows a herringbone stacking in the crystal54, it can only adopt an edge-on orientation on MoSe2 surface before crystal decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Because the monolayer MoSe2 partially covers the CuPc crystal, we could compare the measurements from the MoSe2 region and MoSe2/CuPc crystal 17 region in the same sample (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 9a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' After annealing simultaneously at 100°C, the PL spectra of MoSe2/CuPc film heterostructure and MoSe2/CuPc crystal heterostructure display great contrast as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The PL of crystal heterostructure shows similar spectral features and slight quenching compared with monolayer MoSe2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 5a and Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 9b), indicating a weak interfacial hybridization strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' However, the PL of film heterostructure presents obvious quenching and the formation of hybrid excitons (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 9b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The influence of the morphology of CuPc can be excluded since the surface of CuPc crystal is flatter than that of the CuPc film (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Therefore, the above results suggest that the molecular orientation can be used to tune the interlayer hybridization strength and further tailor the interfacial hybrid excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' To confirm this deduction, the MoSe2/CuPc crystal heterostructure is further annealed at 200oC to decompose the CuPc crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' When the CuPc crystal decomposes, the CuPc molecules can easily adopt a face-on orientation on the MoSe2 surface, thus, the interfacial hybrid excitons should also be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' After annealing at 200°C, the PL spectra display obvious quenching in both the MoSe2/CuPc crystal region and MoSe2 region (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 9d), and clearly shows the formation of hybrid excitons (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 5a, c), which indicates that the molecular orientation is changed from edge-on to face-on after CuPc crystal decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The decomposition of the CuPc crystal is confirmed by AFM, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' It is obvious that the CuPc crystal partially decomposes after annealing at 200°C, as evidenced by the change in the AFM height profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Note that we can even observe hybrid excitons in the MoSe2 region 18 because the CuPc molecules can migrate on MoSe2 surface after the decomposition of CuPc crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The Raman spectra further confirms this, as the Raman peaks of both MoSe2 and CuPc molecules appear in the MoSe2 region after annealing at 200°C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' These results unambiguously show that we can successfully tailor the interfacial hybrid excitons by changing the molecular orientation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 5e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The theoretical calculation also supports our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' As shown in the electronic structure of heterostructure at the edge-on orientation (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 11), we observe no obvious interfacial hybridization, which coincides with the observed phenomena in MoSe2/CuPc crystal heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Discussion We have demonstrated the formation of interfacial hybrid excitons in CuPc/MoSe2 heterostructure due to the hybridization between CuPc and MoSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The observed phenomenon is unusual as the coupling at the organic-inorganic interface is generally weak24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The first principles calculations rationalize the results as the LUMO of CuPc strongly hybridized with the CBM of MoSe2, which leads to the emergence of new eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The new hybrid excitons consist of electrons delocalized in both layers and holes confined in individual layer, enabling simultaneous large optical and electrical dipoles7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The temperature-dependent behavior suggests that the hybrid excitons simultaneously gain the signature of the Wannier-Mott excitons in MoSe2 and the Frenkel excitons in CuPc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The excellent agreement between the theoretical and experimental results not only validates the observed strong coupling phenomenon, but also provides a basis for manipulating hybrid excitons at the organic-inorganic interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 19 For instance, the large electrical dipole in the out-of-plane direction can be used to achieve novel electrical control of the hybrid excitons55,56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Our result is also of great importance for realizing tunable interlayer hybridization strength by changing the molecular orientation, which can be used to tailor the exciton states at the organic- inorganic interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' In conclusion, we report the formation of interfacial hybrid excitons with strong molecular orientation dependence that originate from the hybridization between CuPc and MoSe2, which is meaningful for many-body exciton physics at the organic-inorganic interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Methods Sample preparation (1) Construction of the CuPc film /MoSe2 heterostructure Monolayer MoSe2 was mechanically exfoliated on a SiO2/Si substrate from bulk crystals and further annealed in vacuum at 200 °C to remove surface contaminants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The thickness of MoSe2 was confirmed by optical contrast, atomic force microscopy (AFM), and Raman measurements (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' To construct the CuPc (film)/MoSe2 heterostructure, CuPc thin film was directly deposited on top of monolayer MoSe2 using thermal evaporation in vacuum (home-built evaporator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The heating current was maintained at 5 amperes and the average evaporation speed was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='25 nm/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' (2) Construction of the MoSe2/CuPc film heterostructure The CuPc film was firstly thermally evaporated on a SiO2/Si substrate using the same conditions as (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Then, monolayer MoSe2 was mechanically exfoliated on the PDMS 20 substrate from bulk crystals, and further transferred on top of the CuPc film using the dry transfer method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' (3) Construction of the MoSe2/CuPc crystal heterostructure The single crystals of CuPc were grown by the physical vapor deposition (PVT) method in a quartz tube with a hot zone temperature of 400°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' To construct the MoSe2/CuPc (crystal) heterostructure, monolayer MoSe2 was mechanically exfoliated on a PDMS substrate, and further transferred on top of a CuPc single crystal using the dry transfer method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Low temperature PL Measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The measurements at 78 K were conducted in a temperature-controlled cryostat (THMS600, Linkam) with a diffraction-limited excitation beam diameter of 1µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='The signal was collected using a 50X long-working distance objective and detected on a commercial Renishaw inVia spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The excitation power was selected to be below 200 µW to avoid heating damage to the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The measurements at 4 K were conducted in a temperature-controlled cryostat (Montana Instruments) with an excitation beam diameter of 1µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The signal was collected using a 100X objective and detected on a commercial Ocean Optics spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Theoretical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' First-principles calculations were carried out based on the density functional theory (DFT) framework by utilizing the Vienna Ab initio Simulation Package (VASP) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='4 package57,58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Pseudopotentials were used to describe the electron-ion interactions within the PAW approach and generalized gradient approximations (GGA) of Perdew-Burke-Ernzerhof (PBE) were adopted for the 21 exchange-correlation potential59-61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' To better describe the interlayer van der Waals (vdW) interactions, we adopt optB88-vdW corrections for the optimization of structures62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The electron wave functions are expanded on a plane-wave basis set with an energy cutoff of 520 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The atomic coordinates of all structures were allowed to relax until the forces acting on the ions were less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='01 eV Å-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The convergence criterion for the electronic self-consistent cycle is fixed at 1×10-5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The integrations in the reduced Brillouin zone are performed on a 3×3×1 Monkhorst-Pack special k- points for optimization and self-consistent calculations63,64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' A vacuum slab above 15 Å was used in all calculations to avoid interlayer interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The CuPc/MoSe2 heterostructure is modeled by adsorbing one CuPc molecule on a 5×3√3 supercell of MoSe2, which can be written as Mo30Se60C32N8H16Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The lattice parameters of the CuPc/MoSe2 heterostructure were calculated to be a = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='62 Å, b = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='27 Å, and α=β=γ=90°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' The interlayer distance between CuPc and MoSe2 substrate is approximately 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='36 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='× Data availability The data that support the findings of this study are available from the corresponding authors upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Kleemans, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='J.' metadata={'source': 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CAS (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' XDB30000000), the CAS Project for Young Scientists in Basic Research (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' YSBR-059).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Author contributions X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=', X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' conceived the idea;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' prepared the samples and conducted all the optical measurements and the corresponding data analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' performed the DFT calculations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' This manuscript was prepared primarily by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=', and all authors contributed to discussing and commenting on the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} +page_content=' Competing interests The authors declare no competing interests Additional information Correspondence and requests for materials should be addressed to Xiaoxian Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfoQFr/content/2301.02523v1.pdf'} diff --git a/dtE1T4oBgHgl3EQfyAVc/content/tmp_files/2301.03428v1.pdf.txt b/dtE1T4oBgHgl3EQfyAVc/content/tmp_files/2301.03428v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b725c6d62ecfafe9a603969241bffa6a06aa8395 --- /dev/null +++ b/dtE1T4oBgHgl3EQfyAVc/content/tmp_files/2301.03428v1.pdf.txt @@ -0,0 +1,760 @@ +Regularized Optimal Mass Transport with Nonlinear Diffusion +Kaiming Xu, Xinan Chen, Helene Benveniste, Allen Tannenbaum ∗†‡§ +January 10, 2023 +Abstract +In this paper, we combine nonlinear diffusion with the regularized optimal mass +transport (rOMT) model. As we will demonstrate, this new approach provides further +insights into certain applications of fluid flow analysis in the brain. From the point +of view of image processing, the anisotropic diffusion method, based on Perona-Malik, +explicitly considers edge information. Applied to rOMT analysis of glymphatic trans- +port based on dynamic contrast-enhanced magnetic resonance imaging data, this new +framework appears to capture a larger advection-dominant volume. +1 +Introduction +The theory of optimal mass transport(OMT) was first proposed by Gaspard Monge in 1781 +and has since evolved into a unique scientific field which has had significant impact on +research in many disciplines [22, 23]. Mass transport theory has been applied to diverse +fields including physics, biology, economics and engineering. OMT defines a distance called +the Wasserstein distance, and thus creates a natural geometry on the space of probability +distributions. +Our study is based on a fluid dynamics reformulation of OMT [1] which +allows us to calculate the flow fields between two density distributions. +Regularized optimal mass transport (rOMT), an extension of fluid dynamics reformulation +of OMT, is a tool to study temporal flow fields as a physically inspired model of optical flow. +It has the ability to capture the flow dynamics, handle noise and simulate diffusion [3, 5, 9]. +rOMT utilizes an advection-diffusion equation as its flow-driven partial different equation +and is endpoint free. A source term may be added to rOMT in which case the total mass +preservation condition can be circumvented. This line of research will be pursued in other +work. +Anisotropic diffusion, a major tool for image segmentation, edge detection and image de- +noising, was first proposed by Perona and Malik [17]. Notably, instead of using a constant +diffusion coefficient, Perona and Malik considered a nonnegative function (conductivity +∗K. Xu is with the Department of Applied Mathematics & Statistics, Stony Brook University, NY; email: +kaiming.xu@stonybrook.edu +†X. Chen is with the Department of Medical Physics, Memorial Sloan Kettering Cancer Center, NY +‡H. Benveniste is with the Department of Anesthesiology, Yale School of Medicine, CT +§A. Tannenbaum is with the Departments of Computer Science and Applied Mathematics & Statistics, +Stony Brook University, NY; email: allen.tannenbaum@stonybrook.edu +1 +arXiv:2301.03428v1 [physics.flu-dyn] 3 Dec 2022 + +coefficient) of the magnitude of the local density gradient; see equation (8). The authors +suggested two possible conductivity coefficients (see (9) and (10)), wherein the diffusion will +be very small near the edges, i.e. reflecting the fact that near edges images tend to have +very large intensity gradients. In this work, we show that anisotropic diffusion enhances +the interpretation of glymphatic dynamic contrast-enhanced magnetic resonance imaging +(DCE-MRI) flow data and may be used in conjunction with the constant diffusion coefficient +approach [3]. The anisotropic diffusion equation may be derived via the steepest descend +method for solving an energy minimization problem [25]. +The glymphatic system is involved in transporting waste products from the brain to the +meningeal lymphatic system which connects to the cervical lymph nodes [14]. The function- +ing of the glymphatic and lymphatic systems decrease with age and have been implicated in +the pathophysiology of a wide range of neurodegenerative diseases including cerebral amy- +loid angiopathy [3, 24] and Alzheimer’s disease [4, 10, 13, 16]. We study glymphatic trans- +port using a temporal series of DCE-MRI data acquired from the rodent brain [6, 11, 12]. +Since the data are acquired at discrete time points, our work is motivated by the need to +find a dynamic physically based model of the transport. Several different versions of OMT +[18] and rOMT [3, 5, 9] have been used to model the glymphatic flow. +In the present work, we propose a new version of rOMT. Specifically, we replace the lin- +ear diffusion in rOMT [3, 5, 9] with the Perona-Malik based anisotropic diffusion. Here, +we argue that this gives us enhanced flexibility to study image-based flows inherent to +glymphatic transport. Notably, many diffusion processes in fluids are better captured by +nonlinear models, e.g., axisymmetric surface diffusion [2] and thin fluid films [7, 8]. We +utilize Lagrangian coordinates for visualizing the glymphatic transport pathlines. Several +properties of solute particle movement are computed along the pathlines such as speed and +the P´eclet number. Here we compare various parameters of the anisotropic diffusion coef- +ficient, and observe the impact of different values on several data metrics including P´eclet +plots which can map diffusion dominated versus advection dominated regions of the brain. +We briefly summarize the contents of the present paper. In Section 2, we review the theory +of OMT, rOMT and nonlinear diffusion. Section 3 introduces the algorithm and numerical +methods we employ for our current work. In Section 4, we explicate the application of the +model to glymphatic DCE-MRI data and analyze the experimental results and we conclude +our paper in Section 5. +2 +Model +2.1 +OMT +In this section, we introduce OMT and its fluid dynamics formulation. All the technical +details as well as a complete set of references may be found in [22, 23]. +The original +formulation of OMT was given by Gaspard Monge and may be expressed as +inf +T { +� +Ω +c(x, T(x))ρ0(x)dx | T#ρ0 = ρ1}, +(1) +where c(x, y) is the cost function of moving the unit mass from x to y, ρ0 and ρ1 are two +probability distributions in the domain Ω ⊆ Rd, T is the transport map, and T# is the +2 + +push-forward of T. This formulation assumes that ρ0 and ρ1 have the same total mass, i.e. +� +Ω ρ0(x)dx = +� +Ω ρ1(x)dx and then seeks for the optimal transport map T to minimize the +total cost, the integral in equation (1), subject to the push-forward constraint. +Later, Leonid Kantorovich formulated a relaxed version of OMT as follows: +inf +π∈Π(ρ0,ρ1) +� +Ω×Ω +c(x, y)π(dx, dy), +(2) +where Π(ρ0, ρ1) denotes the set of all couplings (joint distributions) between the marginals +ρ0 and ρ1. From here on, the cost function c will be taken as the square of the Euclidean +distance c(x, y) = ∥x − y∥2. +Benemou and Brenier [1] proved that for c(x, y) = ∥x − y∥2, the specific infimum of Monge- +Kantorovich formulation is equal to the result in following fluid dynamics formulation for +density/probability distributions with compact support: +inf +ρ,v +� 1 +0 +� +Ω +ρ(t, x)|v(t, x)|2dxdt, +(3) +∂ρ +∂t + ∇ · (ρv) = 0, +(4) +ρ(0, x) = ρ0(x), +ρ(1, x) = ρ1(x), +(5) +where ρ : [0, 1]×Ω → R≥0 is the family of density/probability distributions defining geodesic +path from ρ0 to ρ1, and v : [0, 1] × Ω → Rd is the velocity vector field. +2.2 +rOMT +The regularized OMT model (rOMT) [5, 9] adds two assumptions: 1. the image data we +use are noisy observations and thus we do not want to make the final density we calculate +coincide with the MR images; and 2. the flow is driven by an advection-diffusion equation. +Based on these two assumptions, the rOMT formulation may be written as: +inf +ρ,v +� 1 +0 +� +Ω +ρ(t, x)|v(t, x)|2dxdt + β +� +Ω +(ρ(1, x) − ρ1(x))2dx, +(6) +∂ρ +∂t + ∇ · (ρv) = ∇ · (σ0∇ρ), +(7) +ρ(0, x) = ρ0(x). +In this formulation, the final marginal condition is removed and a penalty of the error +between final density and ground truth is added in the objective function (6), where β is +the penalty parameter. Equation (7) is an advection-diffusion equation with a constant σ0 +denoting the diffusion coefficient. +2.3 +Nonlinear diffusion +Instead of using linear diffusion in which σ0 is a constant, nonlinear diffusion seems to have +certain advantages that we will now describe. Perona and Malik proposed an anisotropic +3 + +diffusion [17], which is a useful tool for image segmentation, edge detection and image +denoising. The anisotropic diffusion equation is +∂ρ +∂t = ∇ · (σ(|∇ρ|)∇ρ), +(8) +where σ(·) is a nonnegative strictly decreasing function. If we consider a 3D problem, then +|∇ρ| = +� +ρ2x + ρ2y + ρ2z. The proper diffusion should be large in smooth homogeneous areas +and become smaller near edges, the places where |∇ρ| is large. +Perona and Malik [17] +suggested two versions of the diffusion (conductivity) coefficient: +σ(x) = σ0 +1 +1 + ( x +K )2 , +(9) +σ(x) = σ0e−( x +K )2. +(10) +Both are 0 when x approaches ∞ and attend upper bound σ0 while x = 0. K is a constant +and controls the sensitivity to edges and can be tuned for different applications. +Following [25], we may derive the anisotropic diffusion equation (8) via the steepest descent +from an energy minimization problem. More precisely, considering the following minimiza- +tion problem: +min +� +Ω +f(|∇ρ|)dΩ, +(11) +then the steepest descend equation may be computed to be +∂ρ +∂t = ∇ · (f′(|∇ρ| ∇ρ +|∇ρ|)). +(12) +Obviously, (12) is identical to (8) if +f′(x) = xσ(x). +(13) +For example, the corresponding f function of σ function (9) is +f(x) = σ0K2 +2 +ln[1 + ( x +K )2] +(14) +2.4 +rOMT with nonlinear diffusion +In this section, we present our new rOMT formulation. We replace the diffusion in (7) by +anisotropic diffusion in (8) and obtain the following formulation: +inf +ρ,v +� 1 +0 +� +Ω +ρ(t, x)|v(t, x)|2dxdt + β +� +Ω +(ρ(1, x) − ρ1(x))2dx, +∂ρ +∂t + ∇ · (ρv) = ∇ · (σ(|∇ρ|)∇ρ), +(15) +ρ(0, x) = ρ0(x). +One may employ various versions of the σ function and in this work, we choose the function +given in (9). Note that, there are two parameters σ0 and K which may be tuned based on +the data we use. +4 + +Equation (15) may be written in conservation form as +∂ρ +∂t + ∇ · (ρ(v − σ(|∇ρ|)∇ log ρ)) = 0, +and after defining an augmented velocity +vaug = v − σ(|∇ρ|)∇ log ρ, +we derive a simple conservation form of equation (15) +∂ρ +∂t + ∇ · (ρvaug) = 0. +The Lagrangian representation X = X(x, t) of the optimal trajectory for this rOMT with +nonlinear diffusion model is given by +X(x, 0) = x, +∂X(x, t) +∂t += vaug +opt (X(x, t), t), +(16) +where +vaug +opt = vopt − σ(|∇ρopt|)∇ log ρopt, +(17) +and vopt and ρopt denote the optimal solution of the rOMT with nonlinear diffusion model. +In Section 4, we exhibit the pathlines in Figure 2 and Figure 3 derived from the Lagrangian +coordinates (16). +3 +Numerical scheme +In this section, we focus on the numerical solution of the nonlinear diffusive rOMT model. +The pipeline that comes from [5, 9] is based on the Gauss-Newton method: +1. Give initial guess of v at each time and spatial point. +2. Use v, ρ0 and the advection-diffusion equation (15) to calculate ρ at each subsequent +time step. +3. Calculate the objective function (6), which we will denote with Γ(v) as the discrete +form. +4. Calculate the gradient g(v) and the Hessian matrix H(v) of Γ(v) with respect to v. +5. Solve the descent direction s by solving H(v)s = −g(v). +6. Do line search to find l and update v by setting v = v + ls. +7. Repeat step 2-6 until the results attain the final condition. +Space is discretized into a cell-center grid of size nx × ny × nz with a total number of N +cells, each with width ∆x, height ∆y and depth ∆z. Time is divided into m intervals of +length ∆t with m + 1 time steps. Moreover, the superscript 0 corresponds to initial time +t = 0, M corresponds to final time t = 1 and dt × m = 1. We use ρ = [(ρ0)T , . . . , (ρm)T ]T +and v = [(v1)T , . . . , (vm)T ]T to represent temporal density and velocity, respectively. Note +that the velocity vi describes the velocity field from (i − 1)th time step to ith time step. +5 + +3.1 +Advection-diffusion equation +Here we describe the numerical scheme for equation (15). +The discrete form of equation (15) between time tn and tn+1 is +ρn+1 − ρn +∆t ++ A(ρ, v) = D(ρ), +(18) +where A and D are discretizations of advective and diffusive terms, respectively. We will +describe these in greater detail below. Following the work of Steklova and Haber [21], we +split equation (18) into two parts, +ρadv − ρn +∆t ++ A(ρ, v) = 0, +(19) +ρn+1 − ρadv +∆t += D(ρ), +(20) +where ρadv is an auxiliary variable. Simply by adding (19) and (20), we obtain the equation +(18). +So far we have not chosen the time step of ρ in the advective part A(ρ, v) and +diffusive part D(ρ). We use a standard forward scheme, i.e. ρ = ρn in our implementation. +Summarizing up to this point, to solve for the next time step density ρn+1, we first calculate +ρadv by solving equation (19) and use ρadv and ρn to calculate ρn+1 following equation +(20). +For the advective part A(ρ, v), we utilize a particle-in-cell method which is also how Steklova +and Haber[21] dealt with their advective part to solve equation (19): +ρadv = S(v)ρ. +(21) +S(v) is the averaging matrix with respect to v. +The basic idea of particle-in-cell method is moving density the ρi in the cell center to +the target ρnew +i +according to its velocity vi and using its nearest neighbor cell centers to +interpolate. +The numerical techniques of solving equation (20) are based on hyperbolic conservation +laws and the theory of viscosity solutions [15, 19, 20], and we explicitly write D in the next +section. +3.2 +Anisotropic diffusion +From now on, we explore in 3-dimension (d = 3), following [19, 25] and discretize the +anisotropic diffusion as follows: +D(ρi,j,k) = ∆x +−{σ[ +� +(∆x ++ρi,j,k)2 + m2(∆y ++ρi,j,k, ∆y +−ρi,j,k) + m2(∆z ++ρi,j,k, ∆z +−ρi,j,k)]∆x ++ρi,j,k} ++ ∆y +−{σ[ +� +(∆y ++ρi,j,k)2 + m2(∆x ++ρi,j,k, ∆x +−ρi,j,k) + m2(∆z ++ρi,j,k, ∆z +−ρi,j,k)]∆y ++ρi,j,k} ++ ∆z +−{σ[ +� +(∆z ++ρi,j,k)2 + m2(∆x ++ρi,j,k, ∆x +−ρi,j,k) + m2(∆y ++ρi,j,k, ∆y +−ρi,j,k)]∆z ++ρi,j,k}. +(22) +6 + +Here, +∆x +−ai,j,k = ai,j,k − ai−1,j,k +∆x +, +∆x ++ai,j,k = ai+1,j,k − ai,j,k +∆x +, +∆y +−ai,j,k = ai,j,k − ai,j−1,k +∆y +, +∆y ++ai,j,k = ai,j+1,k − ai,j,k +∆y +, +∆z +−ai,j,k = ai,j,k − ai,j,k−1 +∆z +, +∆z ++ai,j,k = ai,j,k+1 − ai,j,k +∆z +, +m(a, b) = median(a, b, 0). +We note that the solution of equation (18) may be written recursively: +ρn+1 = S(vn)ρn + ∆tD(ρn). +(23) +3.3 +Objective function Γ(v) +A straightforward way [5, 9] to discretize the objective function Γ(v) in (6) is +hd ∗ ∆t ∗ ρT (Im ⊗ [IN|IN|IN])(v ⊙ v) + β|ρm − ρT |2. +(24) +Here hd = ∆x∗∆y∗∆z, ρ, v are column vectors, ⊗ is Kronecker product and ⊙ is Hadamard +product. +3.4 +Gradient, hessian and sensitivity +In order to apply the Gauss-Newton minimization procedure such as described in Steklova +and Haber [21], we need expressions for the gradient g(v) and the Hessian H(v). Taking +the gradient of (24) with respect to v, we find +g(v) = ∂Γ(v) +∂v += hd ∗ ∆t ∗ [2ρT Mdiag(v) + (M(v ⊙ v))T J] + β(ρm − ρ1)T ∂ρm +∂v , +(25) +where M = Im ⊗ [IN|IN|IN], matrix J = (Jk +j ). Here Jk +j = ∂ρk +∂vj , k = 1, . . . , m and j = +0, . . . , m − 1. +The Hessian matrix is +H(v) = ∂g +∂v = hd∗∆t∗[2ρT ∇(Mdiag(v))+2∇(ρ)Mdiag(v)+M(v⊙v)∇J +∇[M(v⊙v)]J] ++ β[(∂ρm +∂v )T (∂ρm +∂v ) + (ρm − ρ1)∂2ρm +∂v2 ]. +(26) +Numerically we approximate the Hessian by +H(v) = 2hd ∗ ∆t ∗ ρT ∇(Mdiag(v)) + β(∂ρm +∂v )T (∂ρm +∂v ) += 2hd ∗ ∆t ∗ diag(ρT M) + β(∂ρm +∂v )T (∂ρm +∂v ). +(27) +In the formulae for the gradient (25) and Hessian (27), we still need to know the sensitivity +of the density ρ with respect to the velocity v. We recall equation (23) +ρn+1 = S(vn)ρn + ∆tD(ρn). +7 + +From that, the sensitivity can be calculated as below: +∂ρk +∂vj = +� +� +� +S(vk−1) ∂ρk−1 +∂vj ++ ∆tD′(ρk−1) ∂ρk−1 +∂vj +k ≥ j + 2 +∂ +∂vj (S(vj)ρj) +k = j + 1 +0 +k ≤ j +(28) +4 +Experimental results +In this section, we test our proposed methodology on 3D DCE-MRI data derived from +[3]. In this dataset, rats were anesthetized, and a Gd-tagged tracer was injected into the +cerebrospinal fluid (CSF). The rat underwent dynamic 3D MRI scanning every 5 minutes +to collect a total 29 3D brain images with a voxel size of 100×106×100. Post-processing of +the DCE-MRI data included head motion correction, intensity normalization, and voxel-by- +voxel conversion to percentage of baseline signal. In our experiment, we chose a 12-month- +old wild type rat for demonstrating the results. +The new algorithm was run for data covering a 100-minute time period (60 minutes to +160 minutes) which includes 23 frames, and we used every other image as inputs to reduce +runtime, leaving 12 frames for the numerical experiment. +We use In, n = 1, . . . , 12 to +represent these frames. To derive the interpolations, we applied our model between each of +two consecutive frames, i.e. Ik and Ik+1. To ensure continuity, (except for the first step), +the initial density originates from the previous step. For example, if we are considering +the problem between I2 and I3, and we will use the final density I′ +2 calculated between I1 +and I2 as the new initial density here and apply our model between I′ +2 and I3. One of the +metrics that can measure the model accuracy is the error between the final density I′ +k and +the ground truth Ik at each step. +Here we are using σ function (9) with σ0 = 0.002. The choice of σ0 follows [3]. We tested +rOMT on the 3D DCE-MRI data set with σ0 = 0.00002, 0.0002, 0.002, 0.02, 0.2. The speed +maps in Figure 4 show a stable trend between σ0 = 0.00002 and σ0 = 0.002 and among +these three σ0 (0.00002,0.0002 and 0.002), 0.002 has the minimal interpolation error (see +Figure 5). +We computed pathlines based on Lagrangian coordinates (16). We compared different K’s +and the results are shown in Figures 1-3. Figure 1 shows the relative error +e = |I′ − I|2 +|I|2 +on each frame with different K’s. The x-axis represents the indices of frames and the y- +axis is the relative error. From Figure 1, we observe that rOMT with anisotropic diffusion +has similar accuracy as the original rOMT model. Figure 2 compares the P´eclet number +along pathlines in the right lateral view plane for different K’s. Further, Figure 3 shows +the ventral surface of the brain. Red color represents larger P´eclet numbers (advection +dominant) and blue represents smaller P´eclet numbers (diffusion dominant). As shown in +Figure 2 and Figure 3, a smaller K value results in more advection dominated transport +in ‘surface’ areas of the brain which corresponds to the CSF compartment. When we set +K = ∞, then clearly σ(x) = σ0, since +lim +K→∞ σ0 +1 +1 + ( x +K )2 = σ0. +8 + +Figure 1: Relative interpolation error plot for different parameter K’s. +Original means +constant diffusion coefficient, i.e. K = ∞. +9 + +relativeinterpolationerror +0.25 +0.2 +0.15 +original +0.1 +K=10 +K=100 +K=1000 +K=10000 +K=100000 +0.05 +K=1000000 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11Figure 2: Pathlines endowed with P´eclet Number shown in the lateral view plane. Parameter +K = 10, 100, 1000, 10000, 100000, ∞. The maximal limit of color bar is 300. When K is +small, the advective (red) pathline dominates in CSF rich areas. +10 + +K=10 +Pseudocolor +Var: PathPoint +300.0 +225.0 +150.0 +75.00 +0.000 +Max: 2.210e+13 +Min: 0.000K=100 +Pseudocolor +Var: PathPoint +300.0 +225.0 +150.0 +75.00 +0.000 +Max: 6.551e+13 +Min: 0.000K= 1000 +Pseudocolor +Var: PathPoint +300.0 +225.0 +150.0 +75.00 +0.000 +Max: 7.892e+13 +Min: 0.000K=10000 +Pseudocolor +Var: PathPoint +300.0 +225.0 +150.0 +75.00 +0.000 +Max:1.703e+14 +Min: 0.000K=100000 +Pseudocolor +Var: PathPoint +300.0 +225.0 +150.0 +75.00 +0.000 +Max:1.770e+14 +Min: 0.000K=Infinity +Pseudocolor +Var: PathPoint +300.0 +225.0 +150.0 +75.00 +0.000 +Max:1.839e+13 +Min: 0.000Figure 3: P´eclet number endowed pathlines shown in ventral view plane. Parameter K = +10, 100, 1000, 10000, 100000, ∞. The maximal limit of the color bar is 300. +11 + +K=1000 +Pseudocolor +Var: PathPoint +300.0 +225.0 +150.0 +75.00 +0.000 +Max:7.892e+13 +Min: 0.000K= 10000 +Pseudocolor +Var: PathPoint +300.0 +225.0 +150.0 +75.00 +0.000 +Max: 1.703e+1 +Min: 0.000K=100000 +Pseudocolor +Var: PathPoint +300.0 +225.0 +150.0 +75.00 +0.000 +Max:1.770e+14 +Min: 0.000K=Infinity +Pseudocolor +Var: PathPoint +300.0 +225.0 +150.0 +75.00 +0.000 +Max:1.839e+13 +Min: 0.000K=10 +Pseudocolor +Var: PathPoint +300.0 +225.0 +150.0 +75.00 +0.000 +Max: 2.210e+13 +Min: 0.000K=100 +Pseudocolor +Var: PathPoint +300.0 +225.0 +150.0 +75.00 +0.000 +Max: 6.551e+13 +Min: 0.000Figure 4: Speed map for different σ0’s. The maximal limit of the color bar is 0.6. The first +three speed maps exhibit a stable trend. The last two speed maps with higher values of +diffusion dramatically (and erroneously) increase speed suggesting that σ0 is too large. +Figure 5: Mean speed (blue line) and interpolation error (orange line) of different σ0’s. The +interpolation error is the relative error between interpolated frames and data image of the +last frame. The interpolation error reflects the closeness between interpolations from rOMT +and the data image. Lower interpolation error means more accurate the rOMT is fitting +the real data. This figure shows larger σ0 has better interpolation error but when σ0 goes +to 0.2, the mean speed accelerates dramatically, which is unrealistic given previous data of +the expected magnitude of solute transport in brain tissue. +12 + +。=0.00020.=0.002.=0.020.6 +0。=0.2 +0.5 +0.4 +0.3 +0.2 +0.10.25 +0.2 +meanspeed +interpolation error +0.15 +0.1 +0.05 +e +0.00002 +0.0002 +0.002 +0.02 +0.2 +do5 +Discussion +In this paper, we proposed a novel extension of the rOMT model. Specifically, we replaced +the linear diffusion term in the advection-diffusion equation by a nonlinear diffusion term +based on the Perona-Malik anisotropic diffusion approach. The updated model was tested +on glymphatic DCE-MRI data comparing different parameter K’s in the conductivity co- +efficient (σ) function and we observed that smaller K yields increased number of advective +pathlines in CSF rich areas. More uniform advective solutes flow in the CSF compartment +including at the level of the basal cisterns, ambient cistern and subarachnoid space above +the cerebellum may be more biologically realistic. +This paper only applied the model on glymphatic DCE-MRI data, but it can be generally +applied to other types of biological imaging data. +In the future, we plan to apply our +approach to tumor vasculature imagery also derived from DCE-MRI, since the mass (tracer) +is injected and may leak, we also plan to explore an unbalanced version of rOMT with +nonlinear diffusion. +Acknowledgments +This research was funded in part by AFOSR grant FA9550-20-1-0029, NIH grant R01- +AG048769, a grant from Breast Cancer Research Foundation BCRF-17-193, Army Research +Office grant W911NF2210292, and a grant from the Cure Alzheimer’s Foundation. +References +[1] Jean-David Benamou and Yann Brenier. A computational fluid mechanics solution to +the monge-kantorovich mass transfer problem. 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IEEE Transactions on Image Processing, +5(11):1539–1553, 1996. +15 + diff --git a/dtE1T4oBgHgl3EQfyAVc/content/tmp_files/load_file.txt b/dtE1T4oBgHgl3EQfyAVc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9dabb499cccbfb4f81c116d6977572e2a42aba24 --- /dev/null +++ b/dtE1T4oBgHgl3EQfyAVc/content/tmp_files/load_file.txt @@ -0,0 +1,431 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf,len=430 +page_content='Regularized Optimal Mass Transport with Nonlinear Diffusion Kaiming Xu, Xinan Chen, Helene Benveniste, Allen Tannenbaum ∗†‡§ January 10, 2023 Abstract In this paper, we combine nonlinear diffusion with the regularized optimal mass transport (rOMT) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' As we will demonstrate, this new approach provides further insights into certain applications of fluid flow analysis in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' From the point of view of image processing, the anisotropic diffusion method, based on Perona-Malik, explicitly considers edge information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Applied to rOMT analysis of glymphatic trans- port based on dynamic contrast-enhanced magnetic resonance imaging data, this new framework appears to capture a larger advection-dominant volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 1 Introduction The theory of optimal mass transport(OMT) was first proposed by Gaspard Monge in 1781 and has since evolved into a unique scientific field which has had significant impact on research in many disciplines [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Mass transport theory has been applied to diverse fields including physics, biology, economics and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' OMT defines a distance called the Wasserstein distance, and thus creates a natural geometry on the space of probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Our study is based on a fluid dynamics reformulation of OMT [1] which allows us to calculate the flow fields between two density distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Regularized optimal mass transport (rOMT), an extension of fluid dynamics reformulation of OMT, is a tool to study temporal flow fields as a physically inspired model of optical flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' It has the ability to capture the flow dynamics, handle noise and simulate diffusion [3, 5, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' rOMT utilizes an advection-diffusion equation as its flow-driven partial different equation and is endpoint free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' A source term may be added to rOMT in which case the total mass preservation condition can be circumvented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' This line of research will be pursued in other work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Anisotropic diffusion, a major tool for image segmentation, edge detection and image de- noising, was first proposed by Perona and Malik [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Notably, instead of using a constant diffusion coefficient, Perona and Malik considered a nonnegative function (conductivity ∗K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Xu is with the Department of Applied Mathematics & Statistics, Stony Brook University, NY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' email: kaiming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='xu@stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='edu †X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Chen is with the Department of Medical Physics, Memorial Sloan Kettering Cancer Center, NY ‡H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Benveniste is with the Department of Anesthesiology, Yale School of Medicine, CT §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Tannenbaum is with the Departments of Computer Science and Applied Mathematics & Statistics, Stony Brook University, NY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' email: allen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='tannenbaum@stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='03428v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='flu-dyn] 3 Dec 2022 coefficient) of the magnitude of the local density gradient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' see equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The authors suggested two possible conductivity coefficients (see (9) and (10)), wherein the diffusion will be very small near the edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' reflecting the fact that near edges images tend to have very large intensity gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' In this work, we show that anisotropic diffusion enhances the interpretation of glymphatic dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) flow data and may be used in conjunction with the constant diffusion coefficient approach [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The anisotropic diffusion equation may be derived via the steepest descend method for solving an energy minimization problem [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The glymphatic system is involved in transporting waste products from the brain to the meningeal lymphatic system which connects to the cervical lymph nodes [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The function- ing of the glymphatic and lymphatic systems decrease with age and have been implicated in the pathophysiology of a wide range of neurodegenerative diseases including cerebral amy- loid angiopathy [3, 24] and Alzheimer’s disease [4, 10, 13, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' We study glymphatic trans- port using a temporal series of DCE-MRI data acquired from the rodent brain [6, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Since the data are acquired at discrete time points, our work is motivated by the need to find a dynamic physically based model of the transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Several different versions of OMT [18] and rOMT [3, 5, 9] have been used to model the glymphatic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' In the present work, we propose a new version of rOMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Specifically, we replace the lin- ear diffusion in rOMT [3, 5, 9] with the Perona-Malik based anisotropic diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Here, we argue that this gives us enhanced flexibility to study image-based flows inherent to glymphatic transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Notably, many diffusion processes in fluids are better captured by nonlinear models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=', axisymmetric surface diffusion [2] and thin fluid films [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' We utilize Lagrangian coordinates for visualizing the glymphatic transport pathlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Several properties of solute particle movement are computed along the pathlines such as speed and the P´eclet number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Here we compare various parameters of the anisotropic diffusion coef- ficient, and observe the impact of different values on several data metrics including P´eclet plots which can map diffusion dominated versus advection dominated regions of the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' We briefly summarize the contents of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' In Section 2, we review the theory of OMT, rOMT and nonlinear diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Section 3 introduces the algorithm and numerical methods we employ for our current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' In Section 4, we explicate the application of the model to glymphatic DCE-MRI data and analyze the experimental results and we conclude our paper in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 2 Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='1 OMT In this section, we introduce OMT and its fluid dynamics formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' All the technical details as well as a complete set of references may be found in [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The original formulation of OMT was given by Gaspard Monge and may be expressed as inf T { � Ω c(x, T(x))ρ0(x)dx | T#ρ0 = ρ1}, (1) where c(x, y) is the cost function of moving the unit mass from x to y, ρ0 and ρ1 are two probability distributions in the domain Ω ⊆ Rd, T is the transport map, and T# is the 2 push-forward of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' This formulation assumes that ρ0 and ρ1 have the same total mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' � Ω ρ0(x)dx = � Ω ρ1(x)dx and then seeks for the optimal transport map T to minimize the total cost, the integral in equation (1), subject to the push-forward constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Later, Leonid Kantorovich formulated a relaxed version of OMT as follows: inf π∈Π(ρ0,ρ1) � Ω×Ω c(x, y)π(dx, dy), (2) where Π(ρ0, ρ1) denotes the set of all couplings (joint distributions) between the marginals ρ0 and ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' From here on, the cost function c will be taken as the square of the Euclidean distance c(x, y) = ∥x − y∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Benemou and Brenier [1] proved that for c(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' y) = ∥x − y∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' the specific infimum of Monge- Kantorovich formulation is equal to the result in following fluid dynamics formulation for density/probability distributions with compact support: inf ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='v � 1 0 � Ω ρ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' x)|v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' x)|2dxdt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' (3) ∂ρ ∂t + ∇ · (ρv) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' (4) ρ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' x) = ρ0(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' ρ(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' x) = ρ1(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' (5) where ρ : [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 1]×Ω → R≥0 is the family of density/probability distributions defining geodesic path from ρ0 to ρ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' and v : [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 1] × Ω → Rd is the velocity vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='2 rOMT The regularized OMT model (rOMT) [5, 9] adds two assumptions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' the image data we use are noisy observations and thus we do not want to make the final density we calculate coincide with the MR images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' the flow is driven by an advection-diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Based on these two assumptions, the rOMT formulation may be written as: inf ρ,v � 1 0 � Ω ρ(t, x)|v(t, x)|2dxdt + β � Ω (ρ(1, x) − ρ1(x))2dx, (6) ∂ρ ∂t + ∇ · (ρv) = ∇ · (σ0∇ρ), (7) ρ(0, x) = ρ0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' In this formulation, the final marginal condition is removed and a penalty of the error between final density and ground truth is added in the objective function (6), where β is the penalty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Equation (7) is an advection-diffusion equation with a constant σ0 denoting the diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='3 Nonlinear diffusion Instead of using linear diffusion in which σ0 is a constant, nonlinear diffusion seems to have certain advantages that we will now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Perona and Malik proposed an anisotropic 3 diffusion [17], which is a useful tool for image segmentation, edge detection and image denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The anisotropic diffusion equation is ∂ρ ∂t = ∇ · (σ(|∇ρ|)∇ρ), (8) where σ(·) is a nonnegative strictly decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' If we consider a 3D problem, then |∇ρ| = � ρ2x + ρ2y + ρ2z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The proper diffusion should be large in smooth homogeneous areas and become smaller near edges, the places where |∇ρ| is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Perona and Malik [17] suggested two versions of the diffusion (conductivity) coefficient: σ(x) = σ0 1 1 + ( x K )2 , (9) σ(x) = σ0e−( x K )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' (10) Both are 0 when x approaches ∞ and attend upper bound σ0 while x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' K is a constant and controls the sensitivity to edges and can be tuned for different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Following [25], we may derive the anisotropic diffusion equation (8) via the steepest descent from an energy minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' More precisely, considering the following minimiza- tion problem: min � Ω f(|∇ρ|)dΩ, (11) then the steepest descend equation may be computed to be ∂ρ ∂t = ∇ · (f′(|∇ρ| ∇ρ |∇ρ|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' (12) Obviously, (12) is identical to (8) if f′(x) = xσ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' (13) For example, the corresponding f function of σ function (9) is f(x) = σ0K2 2 ln[1 + ( x K )2] (14) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='4 rOMT with nonlinear diffusion In this section, we present our new rOMT formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' We replace the diffusion in (7) by anisotropic diffusion in (8) and obtain the following formulation: inf ρ,v � 1 0 � Ω ρ(t, x)|v(t, x)|2dxdt + β � Ω (ρ(1, x) − ρ1(x))2dx, ∂ρ ∂t + ∇ · (ρv) = ∇ · (σ(|∇ρ|)∇ρ), (15) ρ(0, x) = ρ0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' One may employ various versions of the σ function and in this work, we choose the function given in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Note that, there are two parameters σ0 and K which may be tuned based on the data we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 4 Equation (15) may be written in conservation form as ∂ρ ∂t + ∇ · (ρ(v − σ(|∇ρ|)∇ log ρ)) = 0, and after defining an augmented velocity vaug = v − σ(|∇ρ|)∇ log ρ, we derive a simple conservation form of equation (15) ∂ρ ∂t + ∇ · (ρvaug) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The Lagrangian representation X = X(x, t) of the optimal trajectory for this rOMT with nonlinear diffusion model is given by X(x, 0) = x, ∂X(x, t) ∂t = vaug opt (X(x, t), t), (16) where vaug opt = vopt − σ(|∇ρopt|)∇ log ρopt, (17) and vopt and ρopt denote the optimal solution of the rOMT with nonlinear diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' In Section 4, we exhibit the pathlines in Figure 2 and Figure 3 derived from the Lagrangian coordinates (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 3 Numerical scheme In this section, we focus on the numerical solution of the nonlinear diffusive rOMT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The pipeline that comes from [5, 9] is based on the Gauss-Newton method: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Give initial guess of v at each time and spatial point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Use v, ρ0 and the advection-diffusion equation (15) to calculate ρ at each subsequent time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Calculate the objective function (6), which we will denote with Γ(v) as the discrete form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Calculate the gradient g(v) and the Hessian matrix H(v) of Γ(v) with respect to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Solve the descent direction s by solving H(v)s = −g(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Do line search to find l and update v by setting v = v + ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Repeat step 2-6 until the results attain the final condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Space is discretized into a cell-center grid of size nx × ny × nz with a total number of N cells, each with width ∆x, height ∆y and depth ∆z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Time is divided into m intervals of length ∆t with m + 1 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Moreover, the superscript 0 corresponds to initial time t = 0, M corresponds to final time t = 1 and dt × m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' We use ρ = [(ρ0)T , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' , (ρm)T ]T and v = [(v1)T , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' , (vm)T ]T to represent temporal density and velocity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Note that the velocity vi describes the velocity field from (i − 1)th time step to ith time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='1 Advection-diffusion equation Here we describe the numerical scheme for equation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The discrete form of equation (15) between time tn and tn+1 is ρn+1 − ρn ∆t + A(ρ, v) = D(ρ), (18) where A and D are discretizations of advective and diffusive terms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' We will describe these in greater detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Following the work of Steklova and Haber [21], we split equation (18) into two parts, ρadv − ρn ∆t + A(ρ, v) = 0, (19) ρn+1 − ρadv ∆t = D(ρ), (20) where ρadv is an auxiliary variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Simply by adding (19) and (20), we obtain the equation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' So far we have not chosen the time step of ρ in the advective part A(ρ, v) and diffusive part D(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' We use a standard forward scheme, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' ρ = ρn in our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Summarizing up to this point, to solve for the next time step density ρn+1, we first calculate ρadv by solving equation (19) and use ρadv and ρn to calculate ρn+1 following equation (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' For the advective part A(ρ, v), we utilize a particle-in-cell method which is also how Steklova and Haber[21] dealt with their advective part to solve equation (19): ρadv = S(v)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' (21) S(v) is the averaging matrix with respect to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The basic idea of particle-in-cell method is moving density the ρi in the cell center to the target ρnew i according to its velocity vi and using its nearest neighbor cell centers to interpolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The numerical techniques of solving equation (20) are based on hyperbolic conservation laws and the theory of viscosity solutions [15, 19, 20], and we explicitly write D in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='2 Anisotropic diffusion From now on, we explore in 3-dimension (d = 3), following [19, 25] and discretize the anisotropic diffusion as follows: D(ρi,j,k) = ∆x −{σ[ � (∆x +ρi,j,k)2 + m2(∆y +ρi,j,k, ∆y −ρi,j,k) + m2(∆z +ρi,j,k, ∆z −ρi,j,k)]∆x +ρi,j,k} + ∆y −{σ[ � (∆y +ρi,j,k)2 + m2(∆x +ρi,j,k, ∆x −ρi,j,k) + m2(∆z +ρi,j,k, ∆z −ρi,j,k)]∆y +ρi,j,k} + ∆z −{σ[ � (∆z +ρi,j,k)2 + m2(∆x +ρi,j,k, ∆x −ρi,j,k) + m2(∆y +ρi,j,k, ∆y −ρi,j,k)]∆z +ρi,j,k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' (22) 6 Here, ∆x −ai,j,k = ai,j,k − ai−1,j,k ∆x , ∆x +ai,j,k = ai+1,j,k − ai,j,k ∆x , ∆y −ai,j,k = ai,j,k − ai,j−1,k ∆y , ∆y +ai,j,k = ai,j+1,k − ai,j,k ∆y , ∆z −ai,j,k = ai,j,k − ai,j,k−1 ∆z , ∆z +ai,j,k = ai,j,k+1 − ai,j,k ∆z , m(a, b) = median(a, b, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' We note that the solution of equation (18) may be written recursively: ρn+1 = S(vn)ρn + ∆tD(ρn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' (23) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='3 Objective function Γ(v) A straightforward way [5, 9] to discretize the objective function Γ(v) in (6) is hd ∗ ∆t ∗ ρT (Im ⊗ [IN|IN|IN])(v ⊙ v) + β|ρm − ρT |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' (24) Here hd = ∆x∗∆y∗∆z, ρ, v are column vectors, ⊗ is Kronecker product and ⊙ is Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='4 Gradient, hessian and sensitivity In order to apply the Gauss-Newton minimization procedure such as described in Steklova and Haber [21], we need expressions for the gradient g(v) and the Hessian H(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Taking the gradient of (24) with respect to v, we find g(v) = ∂Γ(v) ∂v = hd ∗ ∆t ∗ [2ρT Mdiag(v) + (M(v ⊙ v))T J] + β(ρm − ρ1)T ∂ρm ∂v , (25) where M = Im ⊗ [IN|IN|IN], matrix J = (Jk j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Here Jk j = ∂ρk ∂vj , k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' , m and j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' , m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The Hessian matrix is H(v) = ∂g ∂v = hd∗∆t∗[2ρT ∇(Mdiag(v))+2∇(ρ)Mdiag(v)+M(v⊙v)∇J +∇[M(v⊙v)]J] + β[(∂ρm ∂v )T (∂ρm ∂v ) + (ρm − ρ1)∂2ρm ∂v2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' (26) Numerically we approximate the Hessian by H(v) = 2hd ∗ ∆t ∗ ρT ∇(Mdiag(v)) + β(∂ρm ∂v )T (∂ρm ∂v ) = 2hd ∗ ∆t ∗ diag(ρT M) + β(∂ρm ∂v )T (∂ρm ∂v ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' (27) In the formulae for the gradient (25) and Hessian (27), we still need to know the sensitivity of the density ρ with respect to the velocity v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' We recall equation (23) ρn+1 = S(vn)ρn + ∆tD(ρn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 7 From that, the sensitivity can be calculated as below: ∂ρk ∂vj = � � � S(vk−1) ∂ρk−1 ∂vj + ∆tD′(ρk−1) ∂ρk−1 ∂vj k ≥ j + 2 ∂ ∂vj (S(vj)ρj) k = j + 1 0 k ≤ j (28) 4 Experimental results In this section, we test our proposed methodology on 3D DCE-MRI data derived from [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' In this dataset, rats were anesthetized, and a Gd-tagged tracer was injected into the cerebrospinal fluid (CSF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The rat underwent dynamic 3D MRI scanning every 5 minutes to collect a total 29 3D brain images with a voxel size of 100×106×100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Post-processing of the DCE-MRI data included head motion correction, intensity normalization, and voxel-by- voxel conversion to percentage of baseline signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' In our experiment, we chose a 12-month- old wild type rat for demonstrating the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The new algorithm was run for data covering a 100-minute time period (60 minutes to 160 minutes) which includes 23 frames, and we used every other image as inputs to reduce runtime, leaving 12 frames for the numerical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' We use In, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' , 12 to represent these frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' To derive the interpolations, we applied our model between each of two consecutive frames, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Ik and Ik+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' To ensure continuity, (except for the first step), the initial density originates from the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' For example, if we are considering the problem between I2 and I3, and we will use the final density I′ 2 calculated between I1 and I2 as the new initial density here and apply our model between I′ 2 and I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' One of the metrics that can measure the model accuracy is the error between the final density I′ k and the ground truth Ik at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Here we are using σ function (9) with σ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The choice of σ0 follows [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' We tested rOMT on the 3D DCE-MRI data set with σ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='00002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='0002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The speed maps in Figure 4 show a stable trend between σ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='00002 and σ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='002 and among these three σ0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='00002,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='0002 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='002), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='002 has the minimal interpolation error (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' We computed pathlines based on Lagrangian coordinates (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' We compared different K’s and the results are shown in Figures 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Figure 1 shows the relative error e = |I′ − I|2 |I|2 on each frame with different K’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The x-axis represents the indices of frames and the y- axis is the relative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' From Figure 1, we observe that rOMT with anisotropic diffusion has similar accuracy as the original rOMT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Figure 2 compares the P´eclet number along pathlines in the right lateral view plane for different K’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Further, Figure 3 shows the ventral surface of the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Red color represents larger P´eclet numbers (advection dominant) and blue represents smaller P´eclet numbers (diffusion dominant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' As shown in Figure 2 and Figure 3, a smaller K value results in more advection dominated transport in ‘surface’ areas of the brain which corresponds to the CSF compartment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' When we set K = ∞, then clearly σ(x) = σ0, since lim K→∞ σ0 1 1 + ( x K )2 = σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 8 Figure 1: Relative interpolation error plot for different parameter K’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Original means constant diffusion coefficient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' K = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 9 relativeinterpolationerror 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='15 original 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='1 K=10 K=100 K=1000 K=10000 K=100000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='05 K=1000000 0 1 2 3 4 5 6 7 8 9 10 11Figure 2: Pathlines endowed with P´eclet Number shown in the lateral view plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Parameter K = 10, 100, 1000, 10000, 100000, ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The maximal limit of color bar is 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' When K is small, the advective (red) pathline dominates in CSF rich areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 10 K=10 Pseudocolor Var: PathPoint 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='0 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} 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PathPoint 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='0 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='0 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='000 Max:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='770e+14 Min: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='000K=Infinity Pseudocolor Var: PathPoint 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='0 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='0 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='000 Max:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='839e+13 Min: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='000Figure 3: P´eclet number endowed pathlines shown in ventral view plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Parameter K = 10, 100, 1000, 10000, 100000, ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The maximal limit of the color bar is 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 11 K=1000 Pseudocolor Var: PathPoint 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='0 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='0 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Pseudocolor Var: PathPoint 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='0 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='0 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='000 Max: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='551e+13 Min: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='000Figure 4: Speed map for different σ0’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The maximal limit of the color bar is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The first three speed maps exhibit a stable trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The last two speed maps with higher values of diffusion dramatically (and erroneously) increase speed suggesting that σ0 is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Figure 5: Mean speed (blue line) and interpolation error (orange line) of different σ0’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The interpolation error is the relative error between interpolated frames and data image of the last frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The interpolation error reflects the closeness between interpolations from rOMT and the data image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Lower interpolation error means more accurate the rOMT is fitting the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' This figure shows larger σ0 has better interpolation error but when σ0 goes to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='2, the mean speed accelerates dramatically, which is unrealistic given previous data of the expected magnitude of solute transport in brain tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 12 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='00020.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='2 meanspeed interpolation error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='05 e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='00002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content='2 do5 Discussion In this paper, we proposed a novel extension of the rOMT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Specifically, we replaced the linear diffusion term in the advection-diffusion equation by a nonlinear diffusion term based on the Perona-Malik anisotropic diffusion approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' The updated model was tested on glymphatic DCE-MRI data comparing different parameter K’s in the conductivity co- efficient (σ) function and we observed that smaller K yields increased number of advective pathlines in CSF rich areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' More uniform advective solutes flow in the CSF compartment including at the level of the basal cisterns, ambient cistern and subarachnoid space above the cerebellum may be more biologically realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' This paper only applied the model on glymphatic DCE-MRI data, but it can be generally applied to other types of biological imaging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' In the future, we plan to apply our approach to tumor vasculature imagery also derived from DCE-MRI, since the mass (tracer) is injected and may leak, we also plan to explore an unbalanced version of rOMT with nonlinear diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Acknowledgments This research was funded in part by AFOSR grant FA9550-20-1-0029, NIH grant R01- AG048769, a grant from Breast Cancer Research Foundation BCRF-17-193, Army Research Office grant W911NF2210292, and a grant from the Cure Alzheimer’s Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' References [1] Jean-David Benamou and Yann Brenier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' A computational fluid mechanics solution to the monge-kantorovich mass transfer problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Numerische Mathematik, 84(3):375–393, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' [2] Andrew J 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' European Journal of Neurology, 29(10):2895–2904, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' [25] Yu-Li You, Wenyuan Xu, Allen Tannenbaum, and Mostafa Kaveh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' Behavioral analysis of anisotropic diffusion in image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' IEEE Transactions on Image Processing, 5(11):1539–1553, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} +page_content=' 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'} diff --git a/e9E2T4oBgHgl3EQfGgbf/content/tmp_files/2301.03659v1.pdf.txt b/e9E2T4oBgHgl3EQfGgbf/content/tmp_files/2301.03659v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..04aedee36638fc42d29100a21096910a711d7f7c --- /dev/null +++ b/e9E2T4oBgHgl3EQfGgbf/content/tmp_files/2301.03659v1.pdf.txt @@ -0,0 +1,1307 @@ +1 + +Multifunctional Fiber-based Optoacoustic Emitter for Non-genetic +Bidirectional Neural Communication +Author Information +Nan Zheng1, Ying Jiang2, Shan Jiang3, Jongwoon Kim3, Yueming Li4, Ji-Xin Cheng2, 6, Xiaoting Jia3 * +and Chen Yang5, 6 * +Affiliations +1 Division of Materials Science and Engineering, Boston University, Boston, MA, USA +2 Department of Biomedical Engineering, Boston University, Boston, MA, USA +3 Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA, +USA +4 Department of Mechanical Engineering, Boston University, Boston, MA, USA +5 Department of Chemistry, Boston University, Boston, MA, USA +6 Department of Electrical and Computer Engineering, Boston University, Boston, MA, USA +Contributions +C.Y. conceived the project. N.Z. and S.J. performed fabrication and characterization of materials. N.Z. +and Y.J. performed the stimulation and recording experiments in vitro and in vivo. N.Z. and Y.L. +performed the in vivo biocompatibility evaluations. X.J. provided guidance on the multifunctional fiber +system. J.X.C. provided guidance on the design of fiber optoacoustic emitter. J.K. provided guidance on +optimization of recording and data analysis. The manuscript was written through contributions of all +authors. All authors have given approval to the final version of the manuscript. +Corresponding author +Correspondence to: Chen Yang (cheyang@bu.edu) and Xiaoting Jia (xjia@vt.edu) + + + +2 + +Abstract +A bidirectional brain interface with both “write” and “read” functions can be an important tool for +fundamental studies and potential clinical treatments for neurological diseases. Here we report a +miniaturized multifunctional fiber based optoacoustic emitter (mFOE) that first integrates simultaneous +non-genetic optoacoustic stimulation for “write” and electrophysiology recording of neural circuits for +“read”. The non-genetic feature addresses the challenges of the viral transfection required by optogenetics +in primates and human. The orthogonality between optoacoustic waves and electrical field provides a +solution to avoid the interference between electrical stimulation and recording. We first validated the non- +genetic stimulation function of the mFOE in rat cultured neurons using calcium imaging. In vivo +application of mFOE for successful simultaneous optoacoustic stimulation and electrical recording of +brain activities was confirmed in mouse hippocampus in both acute and chronical applications up to 1 +month. Minimal brain tissue damage has been confirmed after these applications. The capability of non- +genetic neural stimulation and recording enabled by mFOE opens up new possibilities for the +investigation of neural circuits and brings new insights into the study of ultrasound neurostimulation. +Introduction +Bidirectional communication with dynamic local circuits inside the brain of individual behaving animals +or humans has been an invaluable approach for fundamental studies of neural circuits and for effective +clinical treatment of neurological diseases, like epilepsy, Parkinson’ s disease, and depression1, 2. +Additionally, bidirectional neural interface paves the way for the closed-loop control, as it could enable +more sophisticated, real-time control over neural dynamics3, behaviors4 and achieve effective therapeutic +effect in neurological disease5, 6. To achieve real time assessment of the stimulated outcome, neural +interfaces with ability to simultaneously manipulate and directly monitor the neural activities are +preferred. Among the technologies developed in past decades, electrical stimulation and +electrophysiology recording have been widely used and forms the basis of current implantable devices, +which has been applied to clinical applications7. For example, to restore both the motor and sensory + +3 + +modalities, electric stimulation of the cortical surface is often associated with electrophysiology +recording8, 9, like electrocorticography (ECoG). Also, the bidirectional electrical stimulation has +demonstrated promising treatment effect in neurological diseases, such as epilepsy. The responsive focal +cortical stimulation (RNS), leveraging ECoG recording as the trigger to provide stimulation, showed a +statistically significantly greater reduction in seizure frequency and the benefits increased over time in a +two-year study10, 11. However, electrical stimulation has a limited spatial resolution due to current spread. +It also interferes with the electrical signals used for recording, leading to “contamination” in +electrophysiology recording2, 12. Although researchers are improving its performance through +technologies such as current steering13, novel electrode design14, and artifacts cancellation15, considering +the intrinsic physical properties of brain tissue16, the current spread, root cause of above-mentioned issues +is hard to be fully eliminated. Therefore, electrical stimulation for the bidirectional communication of +brain may not be the ideal candidate. +Being orthogonal with electrical recording, optical stimuli not only avoids the interference but +also enables a high spatial resolution. To take this advantage, early efforts developed so-called +optoelectrodes by simply assembling the optical fibers for optogenetics stimulation with the electrodes, +such as Utah arrays17-19, Michigan probes20, 21 and microwires22. Semiconductor fabrication techniques +and multiple material processing methods have recently been applied to improve the integration of those +bidirectional devices. New processing techniques not only make the device more compact but also +strengthen its functionality and biocompatibility. For example, monolithically integrated micro-light- +emitting-diodes (µLEDs) were used to reduce the complexity of light-guide structures and significantly +boosted the number of stimulation sites and stimulation resolution 23, 24. Alternatively, a high-throughput +thermal drawing method has been used to integrate the function components, for example, electrodes, +microfluidic channels, and optical waveguides, to the flexible multifunctional polymer fiber 25, 26. +Through this approach, the flexible fiber probes showed low bending-stiffness and enabled +multifunctionalities, including optogenetics, electrical recording and drug delivery 27-29. Since + +4 + +optogenetics relies on the expression of light-sensitive opsins in neurons through gene modification26, it is +challenging to apply optogenetics to non-human primates and human effectively and safely30. +Recently, our team showed non-genetic optoacoustic neural stimulation with a high spatial +resolution up to single neuron level31, 32. In an optoacoustic process, the pulsed light is illuminated on an +absorber, causing transient heating and thermal expansion, and generating broadband acoustic pulses at +ultrasonic frequencies33, 34. As a light mediated neural modulation method, optoacoustic is an ideal +candidate to work with electrical recording for bidirectional neural communication. Compared with +existing technologies, it exhibited the advantages as a light mediated method, including a high spatial +resolution and minimal crosstalk noise with electrical recording. Importantly, the non-genetic +optoacoustic neurostimulation alleviates the challenges and safety concern in optogenetics since no viral +transfection is required. +Here, we developed a multifunctional fiber-based optoacoustic emitter (mFOE) as a miniaturized +bidirectional brain interface performing simultaneously non-genetic neural stimulation and electrical +recording of the neural activities. Through a thermal drawing process,25, 35 fabrication of mFOE integrated +an optical waveguide and multiple electrodes within a single fiber with a total diameter of 300 µm, +compatible to the typical size of silica fibers used in optogenetic studies. An optoacoustic coating was +selectively deposited to the tip of the core optical waveguide in the mFOE through a controlled micro- +injection process. Upon nanosecond pulse laser delivered to the photoacoustic coating, the mFOE +generates a peak-to-peak pressure greater than 1 MPa, confirmed by the hydrophone measurement, which +is sufficient for successful neural stimulation in vitro and in vivo. By calcium imaging, the optoacoustic +stimulation function of the mFOE was validated in Oregon green-loaded rat primary neurons. +Importantly, we demonstrated the reliable functions of the chronic implanted mFOE for simultaneously +stimulating and recording neurons in mouse hippocampus. Chronic recording also demonstrated that the +embedded electrodes could provide long-term neural monitoring with a single-unit resolution. The +histological evaluation of the brain tissue response confirmed that our flexible mFOE established a stable + +5 + +and biocompatible multifunctional neural interface. mFOE is the first device integrated both optoacoustic +stimulation with electrical recording for bidirectional neural communication. With the bidirectional +capabilities and excellent biocompatibility, it offers a non-generic tools probing brain circuits, alternative +to the optoelectrode devices, with improved feasibility in non-human primates and human. It also opens +up potentials for closed-loop neural stimulation and brain machine interface. +Results +Design, fabrication and characterization of mFOE +Towards bidirectional neural communication, we have designed the mFOE to utilize the optoacoustic +stimulation as “writing” and electrophysiological recording as “reading” of the neural interface (Fig. 1a). +Previously, fiber based optoacoustic emitters have been developed as a miniature invasive ultrasound +transducer for the biomedical applications, such as intravascular imaging and interventional cardiology36, +37. Recently, our work showed that fiber based optoacoustic emitters can also be applied to neural +stimulation in vitro and in vivo, with single neuron resolution and dual site capability32, 38. In these +studies, typically commercial silica fibers were used, together with optoacoustic coating. However, the +silica fiber, with Young’s modulus of ~70 GPa, is mismatched with mechanical properties of native +neural tissue (kilo- to mega pascals)2 and not easy to integrate with miniaturized electrodes for recording. +In this study, we took advantage of the fiber fabrication method developed by Anikeeva and Yoel25, and +utilized the polymer multifunctional fiber design as the base for the mFOE to delivering nanosecond laser +to the optoacoustic coating and to record electrical signals. Specifically, a multifunctional fiber with a +core optical waveguide and miniaturized electrodes was fabricated using the thermal drawing process +(TDP) as previously reported27 (Fig. 1b). The waveguide is made of polycarbonate core (PC, refractive +index nPC = 1.586, diameter = 150 µm) and polyvinylidene difluoride cladding (PVDF, refractive index +nPVDF = 1.426, thickness = 50 µm) as the core and the shell, respectively (Fig. 1c). BiSn alloy is used in +surrounding electrodes with diameters of 35 µm because of its conductivity and compatibility with TDP + +6 + +(Fig. 1c).This multifunctional fiber showed broadband transmission across the visible range to near +infrared region and sub-megaohm impedance when it has been prepared into two centimetres long27, 39. +To integrate the optoacoustic converter to the multifunctional fiber, the optoacoustic coating, +composed of light absorbers and thermal expansion matrix, is needed to be selectively coated on the core +waveguide distal end while keeping the surrounding electrodes exposed and conducting. Compared to +previously reported FOE fabrication, here we took several innovative steps. First, a pressure-driven pico- +litter injector was used to precisely deposit the optoacoustic materials to the core waveguide distal end. +The coating area was controlled through varying the injection volume (0.1 – 0.5 nL), which is controlled +by the regulated pressure (2-4 psi) over a set period of time (1-2 s, Supplementary Fig. S1) as described in +equation (1), +������������ = ������������ ∙ ������������������������������������������������������������������������ +3 +∙ ������������ ∙ ������������ + + + + + +(1) +where ������������ is the injection volume, ������������ is a constant attributed to the unit conversion factors, effects of liquid +viscosity and the taper angle of micropipette, ������������������������������������������������������������������������ is the inner diameter of the pico-litter injector, ������������ is +the pressure, and ������������ is the deposition time. Two 3D translational stages with stereo microscopes were used +to precisely control the deposition localization. Second, instead of using carbon nanotubes (CNT), we +used carbon black (CB) embedded polydimethylsiloxane (PDMS) as the composite optoacoustic material. +CB exhibited similar wideband light absorption40, assuring the sufficient photoacoustic conversion for +neural stimulation. Importantly, due to its relative low viscosity 41, 42, CB/PDMS composite shows much +higher injectability compared with CNT/PDMS, therefore more comparable to the pico-liter deposition +process. Through these steps, we successfully coated 10-20 µm thick 10% w/w CB/PDMS composite +onto the 150 µm diameter core waveguide distal end while electrodes were still exposed as shown in Fig. +1e. Collectively mFOE with the photoacoustic emitter and multiple electrodes has been successfully +fabricated. +To characterize the optoacoustic performance of mFOE, a Q-switched 1030 nm pulsed +nanosecond laser was applied with pulse energies of 16.6 µJ, 27.3 µJ and 41.8 µJ, respectively. The + +7 + +generated acoustic waves were measured by a 40 µm needle hydrophone placed at about 100 µm away +from the fiber tip. Representative pulse acoustic pulse with a width of approximately 0.08 µs was +generated by a single laser pulse as shown in Fig. 1f. Higher input laser pulse energy led to larger acoustic +pressure. A peak-to-peak pressure of 1.0, 1.6 and 2.3 MPa were measured with the pulse energy of 16.6, +27.3 and 41.8 µJ, respectively. The frequency spectrum shows the broadband characteristic of typical +optoacoustic waves34, and the peak frequencies are around 12.5 MHz (Fig. 1g). Based on previous work, +we expected that such pressure and frequency is capable to successfully stimulate neurons in vitro and in +vivo. We also calculated the mechanical index (MI), a commonly used matrix, to evaluate the probability +of mechanical damage due to ultrasound generated. The MI of acoustic waves generated by 2.3 MPa is +0.198, lower than 1.9, the safety threshold suggested by the Food and Drug Administration (FDA) safety +guidelines. + +8 + +Figure. 1 Design, fabrication and characterization of mFOE +a. Schematic of mFOE for bidirectional communication with neurons. Input laser pulse (red) is used to +generate optoacoustic waves (black) by the converter and the neural activities are recorded by embedded +electrodes as the output electrical signal (blue). b. Illustration of the thermal drawing +process. c. Components of the multifunctional fiber, including a PC/PVDF waveguide, BiSn alloy +electrodes and PC sacrifice layer. d. The selective deposition process for integrating the optoacoustic +converter to the core wave guide in the multifunctional fiber. A pressure-driven micro-injector is used to +control the volume of CS/PDMS deposited. 3D translation stages and microscope are used to control the +deposition location. Zoom-in: The micro pipette was aligned to the center of the fiber under the microscope. +e. Top view microscope image of the mFOE. Scale bar: 100 µm. f. Representative acoustic waveforms +under different laser pulse energy recorded by a needle hydrophone. g. Frequency spectrum of acoustic +waveforms shown in f. +mFOE stimulation of cultured primary neurons + +a +ptoacousticwave +Pulsed laser +Electrical signal +q +C +d +Injector on 3D stage +Wave guide +Micro pipette +(PC/PVDF) +Heat +Fiber +Sacrificelayer(PC) +Vdrawing speed +Electrode (BiSn alloy) +Fiber holder +on 3D stage +e +Optoacoustic +g +0.4 +41.8 μJ +41.8 μJ +converter +1.5 +27.3 μJ +27.3 μJ +Pressure (MPa) +16.6 μJ +0.3 +16.6μJ +(a.u +Magnitude +0.5 +0.2 +0 +0.1 +0.5 +Electrode +0 +0 +0.1 +0.2 +0.3 +0.4 +0 +20 +40 +60 +80 +100 +Time (μs) +Frequency (MHz)9 + +To investigate mFOE can directly trigger the neuronal activity, we examined the response of cultured +primary neurons under mFOE stimulation. Because of the presence of calcium channels in neuronal +membrane and their activation during the depolarization, calcium imaging has been widely used to +monitor the neuronal activities43, 44. Here, we cultured and loaded the rat cortical neurons (days in vitro +10-14) with a calcium indicator, Oregon Green™ 488 BAPTA-1 dextran (OGD-1)45 , and performed the +calcium imaging with an inverted wide-field fluorescence microscope (Supplementary Fig. S2). To +perform the optoacoustic stimulation, mFOE was placed approximately 50 µm above the in-focus target +neurons (Fig. 2a) by a micromanipulator under the microscope. 1030 nm 3 ns pulsed laser with a +repetition rate of 1.7 kHz was delivered to the mFOE through an optical fiber. The energy of laser pulse +was 41.8 µJ, corresponding to a peak-to-peak pressure of 2.3 MPa generated. Lower energy was tested +but did not induce calcium transient. The stimulation duration determined by each laser burst was 100 ms, +corresponding to 170 pulses (Supplementary Fig. S3). By applying 5 bursts of laser pulses with interval +of 1s, we investigated the reproducibility of the stimulation. +Using calcium imaging, we monitored the activities of all neurons in the field of view and divided +them into two groups: groups within the converter area (Fig. 2b) and outside the converter area (Fig. 2c). +For neurons within the converter area, i.e. the 100 µm from the center of the mFOE, Fig. 2b shows that 8 +of 10 neurons showed successful and repeatable calcium transient (ΔF/F > 1%, the baseline standard +deviation) corresponding to each stimulation. Calcium transients are also repeatable for each burst applied +over the 1 s period, indicating the evoked neuronal activities and confirming the reliability and biosafety +of mFOE stimulation. For neurons outside the converter area, only 2 of 10 neurons responded. This result +also suggested the mFOE with the 150 µm center waveguide with photoacoustic coating provided a +spatial precision of ~200 µm for stimulation in vitro. This observation is consistent with that fiber based +optoacoustic converters generate a confined ultrasound fields with sizes comparable with the radius of +converter31. +Next, to investigate the threshold of mFOE stimulation, we varied the stimulation duration from 5 +ms, 50 ms, 100 ms to 200 ms on neurons in different cultures (N = 15) under the same laser pulse energy + +10 + +of 41.8 µJ and the same repetition rate of 1.7 kHz. mFOE stimulation with duration of 5 ms did not +evoked any observable fluorescence change (n.s., p > 0.05) (Fig. 2g). Only when the duration was 50 ms +or longer, the mFOE successfully produced neural activation (ΔF/F > 1%, p < 0.01) as shown in Fig. 2d-f, +and Fig. 2h. Longer pulse durations leads to larger peak fluorescence changes, from 2.9 ± 1.1%, 6.0 ± +2.8% to 7.8 ± 1.3% corresponding to 50 ms, 100 ms and 200 ms, respectively. For the longest stimulation +duration of 200 ms tested, no obvious change on morphology or elevation of baseline fluorescence +intensity was detected in neurons after multiple stimulations (Supplementary Fig. 4), indicating the safety +of stimulation. +Laser only control experiment was also performed. Laser light with same pulse energy of 41.8 µJ +and duration (200 ms, 100 ms and 50 ms) was delivered to OGD-1 loaded neurons through +multifunctional fiber without optoacoustic coating. None of neuron culture showed detectable calcium +response, distinct from the observed in mFOE stimulated neurons (Supplementary Fig. 5). +To evaluate the photothermal effect of the mFOE stimulation and its potential impact on neurons, +we also characterized the thermal profile of the mFOE in PBS during the acoustic generation. +Temperature was measured by an ultrafast thermal sensor with a sampling rate of 2000 Hz placed in +contact with mFOE optoacoustic coating under the microscope. The laser conditions were consistent with +neural stimulation test, i.e., the pulse energy was maintained at 41.8 µJ and the burst duration was varied +from 50 ms, 100 ms to 200 ms. The temperature increase on the mFOE surface was found to be 1.23 ± +0.09 °C, 1.07 ± 0.08 °C, 0.96 ± 0.08 °C for 200, 100, 50 ms laser durations, respectively (Supplementary +Fig. 6). Such temperature increase is far below the previously reported threshold of thermal-induced +neural stimulation (ΔT > 5 °C)46, 47. Taken together, we conclude that activation of neurons was due to the +mFOE optoacoustic stimulation. + +11 + + +Figure 2. Calcium transients induced by mFOE in cultured primary neurons. +a. Calcium image of primary cultured neurons loaded with OGD-1. Twenty neurons within (orange) and +outside (blue) the optoacoustic converter area are circled and labelled. Scale bar: 100 µm. Solid circle: +area outside the converter area; dashed line circle: area within the optoacoustic converter area. b-c. +Calcium traces of neurons undergone repeated mFOE stimulations with a laser pulse train duration of 100 +ms (red dots). Each pulse train was repeated 5 times. Colors and numbers of the traces are corresponding +to the neurons labelled in a. d-g. Average calcium traces of neurons triggered by mFOE stimulation with +durations of 200 ms (d), 100 ms (e), 50 ms (f) and 5 ms (g), respectively. Shaded area: the standard +deviation (SD). N=15 h. Average maximum ΔF/F of neurons stimulated by mFOE. N = 15. (n.s.: non- +significant, p > 0.05; *: p < 0.05; **: p < 0.01; ***: p < 0.001, One-Way ANOVA and Tukey’s mean +comparison test) +In vivo simultaneous optoacoustic stimulation and electrophysiological recording +Since the animal experiment is a significant part of the study in neuroscience and neurological diseases, +we further investigated the performance of mFOE in the wild type C57BL/6J mice. In vivo optoacoustic +stimulation was performed by delivering pulsed laser to the implanted mFOE, and the optoacoustically +stimulated neuronal activities were recorded through electrodes in the mFOE (Fig. 3a). Experimentally, +we implanted the mFOE into the hippocampus of mice (N =5). The chronically implanted mFOE allows + +a +b +10 +hhyeh10 +C +9 +9 +9 +8 +8 +Me +40 +8 +6 +6 +5 +4 +3 +2 +2 +10% +AA +5s' +d +e ++ +g +h +0.12 +200 ms +100 ms +50 ms +5 ms +0.1 +0.1 +0.1 +0.1 +0.1 +T +0.08 +F 0.05 +0.05 +0.05 +0.05 +F +0.06 +A +F +0.04 +n.s. +0.02 +0 +A +0 +0 +0 +0 +0 +1 +2 +3 +0 +1 +2 +3 +0 +1 +2 +3 +0 +1 +2 +3 +-0.02 +Time (s) +Time (s) +Time (s) +Time (s) +200ms100ms50ms5ms12 + +mice to move freely after surgery (Fig 3b). During stimulation and recording tests, the mFOE was +coupled with the laser source and electrophysiological recording headstage through the standard ferrule +and pin connector, respectively. The stimulation and recording were conducted in the mice under +continuous anesthesia induced and maintained by isoflurane. Based on the threshold of optoacoustic +stimulation obtained in in vitro studies, 50 ms bursts of laser pulses with a pulse energy of 41.8 µJ were +delivered to the mFOE at 1Hz during the 5 second treatment period. The simultaneous +electrophysiological recording by mFOE electrodes was bandpass filtered to examine the local field +potential (LFP, 0.5-300 Hz). Simultaneous optoacoustic stimulation and electrophysiological recording +were performed at multiple time points, including 3 days, 7 days, 2 weeks and 1 month (Fig. 3c-f). Three +out of five mice tested showed successful simultaneous stimulation and recording functions for testing +periods of 3 days to one month. +The evoked brain activities corresponding to the optoacoustic stimulation were confirmed by +monitoring the LFP response. LFP response at two weeks after implantation was detected with latency of +7.19 ± 2.29 ms (N = 15, from three mice). The amplitude of LFP response varied at four time points. The +largest and smallest responses occurred at 2 weeks and 1 month, respectively. A possible reason for this +observation may be the brain tissue injury and healing after the implantation. These results collectively +demonstrate the reliability of the optoacoustic stimulation and recording functions of the implanted +mFOE in the animals. +To eliminate the possibility that LFP response was induced by electrical noise or laser artifacts, +we also conducted two sham control experiments. In the light only control group, we implanted a +multifunctional fiber without optoacoustic coating to the mouse hippocampus and delivered the laser light +with the same condition. The LFP recorded didn’t correlate to the laser pulse train, indicating the +spontaneous brain activities were recorded and light only did not invoke the LFP response +(Supplementary Fig. 7a). In the dead brain control group, we tested the optoacoustic stimulation through +mFOE implanted to the euthanized mouse and did not observe the corresponding LFP response + +13 + +(Supplementary Fig. 7b). These results collectively confirm the signals we detected from mFOE +stimulation were not artifacts. +We further evaluated the recording performance of implanted mFOE. To evaluate the ability of +mFOE for single unit recording, the electrophysiological signals recorded were bandpass filtered for spike +activity (0.5-3 kHz, Fig. 3g). Through a principal-component analysis (PCA) based spike sorting +algorithm, two spike clusters can be isolated from an endogenous neural recording (Fig. 3j). The cluster +quality was assessed by two common measures48, Lratio and isolation distance. Lratio is 0.0017 and isolation +distance has the value of 99.37. The first averaged spike shape (Fig. 3h) showed a narrower and larger +depolarization than that of the second spike shape (Fig. 3i). The different spike waveform and the cluster +analysis suggested that the action potentials were recorded from at least two different groups of neurons49, +50. Thus, the successfully spike sorted neural activities from CA3 confirmed the ability of mFOE +electrodes for the single-unit recording. +To examine the sensitivity of LFP recording, at one month after implantation we altered the +anesthesia level via adjusting the induced isoflurane concentration during the recording to see if the +characteristic anesthesia dosage-dependent changes can be observed (Fig. 3k). Initially, a low level of +anesthesia was maintained at 0.5% v/v isoflurane, and recorded LFP showed that spontaneous brain +activities occurred continuously (i in Fig. 3h. and Fig. 3l). Then a higher-level anesthesia (3% v/v +isoflurane) was applied for 3 minutes. After increased the isoflurane level, some spontaneous brain +activities were suppressed and a hyperexcitable brain state was induced, where the voltage alternation +(bursts) and isoelectric quiescence (suppression) appeared quasiperiodically27, 51 (ii in Fig. 3h and Fig. +3m). With maintaining 3% v/v isoflurane, a deep anesthesia state was induced in the animal. At the same +time, both respiration rate and responsiveness to toe pinch decreased due to the higher anesthetic level. + +Less voltage alternation occurred and for the most of time the LFP signal was a flat line +(suppression, iii in Fig. 3h and Fig. 3n). Compared with initial stage, γ band LFP activity in 30-100 Hz +was decreased due to the higher concentration of isoflurane as shown in the power spectrum52 (Fig. 3n). +Later, when the concentration of isoflurane was reduced to 0.5% v/v again, the LFP activity returned to a + +14 + +similar level as measured in the initial stage. Taken together, this isoflurane dosage-dependent +characteristic confirmed the accuracy of LFP recording by mFOE. + +Figure. 3 Simultaneous optoacoustic stimulation and electrophysiological recording by implanted +mFOE in mouse hippocampus. +a. Illustration of the mFOE enabled bidirectional neural communication using laser signal as input and +electrical signal as readout. b. mFOE was implanted into hippocampus of a wild type C57BL/6J mouse. +c-f. Simultaneous optoacoustic stimulation and electrophysiological recording performed at 3 days (c), 7 +days (d), two weeks (e) and one month (f) after implantation. Blue dots the laser pulse trains. For each +laser train: 50 ms burst of pulses, pulse energy of 41.8 µJ, laser repetition rate 1.7 kHz. g. Part of the +filtered spontaneous activity containing two separable units recorded by mFOE electrode at one month +after implantation. h-i. Spike shapes of two separable units in g. j. Principal-components analysis (PCA) +of the two units. k. Local field potential (LFP) recorded by mFOE one month after implantation with an +alternating anaesthesia level (0.5-3% v/v isoflurane). l-n. different LFP responses induced by varying the +concentration of isoflurane: l corresponds to the initial stage (0.5% of isoflurane level); m corresponds to +the burst/suppression transition stage (after increasing the isoflurane level to 3%); n corresponds to the +suppression stage (the isoflurane level was maintained at 3% and took effect). + +a +c +d +Optical +Electrical +0.5 +3 days +0.5 +7 days +input +readout +(mV) +0.0 +Voltage ( +0.0 +-0.5 +-1.0- +-1.0- +0 +2 +4 +6 +10 +0 +4 +6 +8 +10 +Time (s) +Time (s) +b +e + (mV) +0.5 +2 weeks +0.5- +1 month +(mV) +Voltage ( +0.0 +Voltage +0.0 +-0.5 +-0.5 +-1.0 +-1.0 +0 +2 +4 +6 +8 +10 +0 +2 +4 +6 +8 +10 +Time (s) +Time (s) +g +h +100 +40 +40 +20 +20 +Yoltage (μV) +(Λ) +50 +0 +0 +-20 +PC2 +40 +40 +-50 +60 +-60 +0 +0.5 +1 +1.5 +2 +0 +2 s +0.5 +1 +1.5 +2 +-100 +Time (ms) +Time(ms) +-200 +-150 +-100 +-50 +0 +50 +100 +PC1 +k +3 % +i Initial stage +m +ii Burst/suppression +n +ii Suppression +0.5 % +M +2 s +N +(ZH) +100 +100 +Frequency +(dB) +Frequency +(p) +Frequency +-50 +50 +-50 +50 +Power +50 +100 +-100 +50 +-100 +150 +ii +ili +2 +4 +68101214 +2 +468101214 +2 +68101214 +50 s +Time (s) +Time (s) +Time (s)15 + +Foreign body response comparison between mFOE and standard optical fiber using +immunohistochemistry +Foreign body response is a critical property of implantable neural interface to assure their usage in a safe +and chronic way, since the physical insertion into brain tissue commonly initiates a progressive +inflammatory tissue response53. To evaluate the biocompatibility of mFOE, we compared the foreign +body response of mouse brain to mFOE with the similar size standard silica optical fibers (diameter = 300 +µm), which is widely used in optogenetic technologies54, 55. The immunohistochemistry analysis of +surrounding brain tissue was performed from mice (N = 3) implanted with the mFOE and a conventional +silica fiber 3 days and 1 month after implantation (Fig. 4a). The damage to surrounding neurons from +implant was assessed through evaluating neuronal density using the neuronal nuclei (NeuN) markers (Fig. +4b). Number of neurons was calculated by counting the NeuN-positive cells per field of view (650 × 650 +µm). The presence of ionized calcium-binding adaptor molecule 1 (Iba1, Fig. 4c) and glial fibrillary +acidic protein (GFAP, Fig. 4d) were used as the markers for activated microglia and astrocytic response, +respectively. +Compared with the silica fiber, mFOE induced significantly less microglial response (p < 0.01, +Fig. 4c, f) and astrocyte reactivity (p < 0.001, Fig. 4d, g), but no significant difference was observed on +the neuronal density (Fig. 4b, e) 3 days after implantation. A decrease in foreign body response, +specifically, higher neuronal density and lower microglia and astrocytic response (Fig. 4e-g), was +observed from 3 days to 1 month after implantation of both mFOE and silica fiber and no significant +difference was observed between mFOE and silica fiber 1 month after implantation. Taken together, the +immunohistochemistry analysis confirmed that mFOE yielded less foreign body response in the short +period, i.e., 3 days, after implantation and showed similar biocompatibilty with silica fiber at longer +implantation time, i.e., 1 month. + +16 + + +Figure. 4 Foreign body response comparison of mFOE and silica fiber using +immunohistochemistry. +a-d. Immunohistochemistry images of mouse brains implanted with mFOE and silica fiber one month +after implantation (N = 3). Scale bar: 100 µm. Brain slices were labelled with the neuron-specific protein +(NeuN, cyan), ionized calcium-binding adaptor molecule 1 (Iba1, red) and glial fibrillary acidic protein +(GFAP, green). e. Number of neurons in the field of view, calculated by counting the NeuN-positive cells +for mFOE and silica fiber at 3 days and 1 mon after implantation. f. Microglial reactivity, assessed by +counting the Iba-1 labelled area, for mFOE and silica fiber at 3 days and 1 mon after implantation. g. +Astrocyte reactivity, assessed by counting the GFAP labelled area, for mFOE and silica fiber at 3 days +and 1 mon after implantation. For each experimental group, two to four brain slices were used from each + +a +mFOE +Silica Fiber +800 +e +n.s. +700 +neurons +Composite +600 +500 +Number +n.s. +400 +300 +mFOE +b +Silica fiber +200 +Day 3 +Day30 +TimePoint +NeuN +** +mFOE +3×104 + Silica fiber +n.s. +2 ×10 +C +10 +6×103 +Iba1 +Day 3 +Day30 +TimePoint +g +3×10 +mFOE +Silica fiber +d +2 × 10 +GFAP area (μm" +n.s. +GFAP +10 +6 ×10° +Day 3 +Day30 +TimePoint17 + +mouse (N= 3). (n.s.: non-significant, p > 0.05; *: p < 0.05; **: p < 0.01; ***: p < 0.001, One-Way +ANOVA and Tukey’s mean comparison test) + +Discussion +In this study, we designed and developed a miniaturized fiber-based device, i.e. mFOE, for +bidirectional neural communication. mFOE performs the “write” function, i.e. non-genetic optoacoustic +stimulation and the “read” function, i.e. simultaneous electrophysiological recording. The broadband +acoustic wave with a broadband ultrasound pulse with pulse width about 0.1 µs and a center frequency at +12.5 MHz and a peak pressure of 2.3 MPa with pulse numbers >85 generated by mFOE successfully +stimulate neurons with a spatial resolution of approximately 200 µm in primary rat cortical neuron +culture. By implanting mFOE into mouse hippocampus, we demonstrated its ability for simultaneous +optoacoustic stimulation and electrophysiological recording and superior biocompatibility as a chronic +bidirectional neural interface. Reliable stimulation and LFP recording have been achieved up to one +month post implantation. Recording quality has been demonstrated by single unit recording. +For the first time, combining this pico-liter deposition and thermal fiber pulling, we successfully +integrated an optoacoustic converter to the polymer multifunctional fiber. Different from the conventional +dip-coating method36, 56, the selective deposition through micro-injection allows the easy fabrication of +optoacoustic emitter in a volume and position-controlled way. Through the selective deposition, the +dimension of optoacoustic emitter is no longer limited by the tip sizes of optical fibers. Our choice of +CB/PDMS composite as the optoacoustic material is also essential as it is comparable with this deposition +process with a fine volume control at pico liter level. Besides the application in neural interface, such +design and fabrication method can also be applied to optical ultrasound probes used in imaging37, 57, for +example, in the tip engineering and the integration to photonics crystal fibers. +We introduced the optoacoustic stimulation as a new strategy for “writing” in the bidirectional +neural interface. Compared with previous optoelectrode devices based on optogenetics24, 25, 27 and + +18 + +photothermal58, 59, the non-genetic optoacoustic stimulation enabled by mFOE reduces the barrier of +transgenic techniques for applications in primate and potentially human, and avoids the thermal toxicity. +At the same time, it offers the spatial precision benefit from the confined ultrasound field. It is orthogonal +to electrical recording, therefore minimizing crosstalk with electrical recording. As an emerging +neuromodulation method, the mechanism of optoacoustic stimulation is still not fully understood but +more studies indicated that mechanosensitive ion channels are responsible for the activation of neurons60, +61. +Bidirectional brain interfaces are important research tools to understand brain circuits, potential +treatments for neurological disease and bridges to brain computer interface for broad applications. New +features of mFOE compared to the previous fiber based interface, such as non-genetic and non-electrical +stimulation are critical to advance these applications. For example, closed-loop neuromodulation has been +demonstrated to be superior to the conventional open-loop system, as it can achieve more responsive and +real-time control over neural dynamics. In neurological diseases treatment, combining the detection and +in situ intervention improves the treatment effectiveness and safety. Because of its bidirectional +capabilities, mFOE has the potential to be used as a new brain interface with closed-loop capability. +Using epilepsy as an example, by implanting the mFOE into seizure foci, the continuous LFP recording +can guide the localized optoacoustic stimulation and intervene can be triggered at the early stage before +seizure progresses into a generalized seizure. The unique orthogonal non-electrical optoacoustic +stimulation and electrical recording prevents “contamination” of the recording signals, potentially +offering a more effective closed-loop strategy. +In comparison of the optoelectrodes fabricated through semiconductor fabrication process, the +recording and stimulation sites of the current mFOE design is fixed at the core waveguide and the number +of channels is limited because of the nature of multifunctional fiber. Some post processing methods have +been proposed to tackle this challenge, like the laser micromachining technique27. In addition, it is +possible to further engineer the fiber to offer multiple and selective stimulation sites62. With the further + +19 + +development of multifunctional fiber strategy, we believe the bandwidth of mFOE would be improved +and open more opportunities in the research of neuroscience and neurological diseases. + +Methods +Multifunctional fiber fabrication and optoacoustic emitter integration +Multifunctional fibers were fabricated from a preform fiber and then drawn into thin fibers through TDP +in a customized furnace. For the preform fiber, PVDF film (Mcmaster) and PC film (laminated plastics) +were rolled onto a PC rod (Mcmaster) and followed by a consolidation process in vacuum at 200 °C. +Next, four rectangular grooves (2 mm × 2 mm) were machined on the solid PC layer and inserted with the +BiSn (Indium Corporate) electrodes. Then, another PVDF layer was rolled over the rod to form an +insulation layer for the electrodes and followed by an additional PC as the sacrifice layer for the +convenience of TDP. The detailed fabrication process was discussed in the previous paper27. +A composite of 10% carbon black (diameter < 500 nm, Sigma Aldrich) and 90% +polydimethylsiloxane (PDMS, Sylgard 184, Dow Corning Corporation, USA) were used as the +optoacoustic material. The mixture was sonicated for 1 hour followed by degassing in vacuum for 30 +minutes. The mixture was then filled in the glass micropipette (Inner diameter = 30 µm, TIP30TW1, +World Precision Instruments, USA) connected to the pico-liter injector (PLI-100A, Warner Instruments, +USA). Under the microscope, the glass micropipette was aligned with the core waveguide of +multifunctional fiber and the mixture was deposited to the surface of the core waveguide by controlling +the injection pressure and time. The deposited fiber was then cured vertically at room temperature for 2 +days. +Before use, mFOE was further prepared for the optical coupling and electrodes connection. For +the optical coupling, a ceramic ferrule (Thorlabs, USA) was added and affixed to the end of the fiber by +the 5-min epoxy (Devcon, ITW Performance Polymers, USA). Then the end surface was polished by + +20 + +optical polishing papers to reduce roughness from 30 µm to 1 µm. For the connection to electrodes +embedded in the multifunctional fiber, the electrodes were exposed manually along the side wall of the +fiber by using a blade and silver paint (SPI Supplies, USA). Then copper wires were wrapped around the +fiber at each exposure locations along the fiber and the silver paint were applied for the fixation and lower +resistance. The copper wires connected to fiber electrodes were soldered to the pin connector while a +stainless-steel wire was also soldered as the ground wire for later extracellular recording. In addition, the +5-min epoxy (Devcon, ITW Performance Polymers, USA) was applied to the connection interface for +strengthening affixation and better electrical insulation. +Optoacoustic wave characterization +To generate the optoacoustic signal, a compact Q-switched diode-pumped solid-state laser (1030 nm, 3 +ns, 100 μJ, repetition rate of 1.7 kHz, RPMC Lasers Inc., USA) was used as the excitation laser source. +The laser was first connected to an optical fiber through a 200 µm fiber coupling module and then +connected to the mFOE with a SubMiniature version A (SMA) connector. The pulse energy was adjusted +through a fiber optic attenuator (varied gap SMA Connector, Thorlabs, Inc., USA). The acoustic signal +was measured through a homebuilt system including a needle hydrophone (ID. 40 µm; OD, 300 µm) with +a frequency range of 1–30 MHz (NH0040, Precision Acoustics Inc., Dorchester, UK), an amplifier and an +oscilloscope. The mFOE tip and hydrophone tip were both immersed in degassed water. The pressure +values were calculated based on the calibration factor provided by the hydrophone manufacturer. The +frequency data was obtained through a fast Fourier transform (FFT) calculation using the OriginPro 2019. +Embryonic neuron culture +All experimental procedures complied with all relevant guidelines and ethical regulations for animal +testing and research established and approved by Institutional Animal Care and Use Committee (IACUC) +of Boston University (PROTO201800534). Primary cortical neurons were isolated from embryonic day +15 (E15) Sprague−Dawley rat embryos of either sex (Charles River Laboratories, MA, USA). Cortices + +21 + +were isolated and digested in TrypLE Express (ThermoFisher Scientific, USA). Then the neurons were +plated on poly-D-lysine (50 μgmL−1, ThermoFisher Scientific, USA)-coated glass bottom dish (P35G- +1.5-14-C, MatTek Corporation, USA). Neurons were first cultured with a seeding medium composed of +90% Dulbecco’s modified Eagle medium (ThermoFisher Scientific, USA) and 10% fetal bovine serum +(ThermoFisher Scientific, USA) and 1% GlutaMAX (ThermoFisher Scientific, USA), which was then +replaced 24 h later by a growth medium composed of Neurobasal Media (ThermoFisher Scientific, USA) +supplemented with 1× B27 (ThermoFisher Scientific, USA), 1× N2 (ThermoFisher Scientific, USA), and +1× GlutaMAX (ThermoFisher Scientific, USA). Half of the medium was replaced with fresh growth +medium every 3 or 4 days. Cells cultured in vitro for 10−14 days were used for Oregon Green labelling +and PA stimulation experiments. +In vitro neurostimulation and calcium imaging +Oregon Green™ 488 BAPTA-1 dextran (OGD-1) (ThermoFisher Scientific, USA) was dissolved in 20% +Pluronic F-127 in dimethyl sulfoxide (DMSO) at a concentration of 1 mM as stock solution. Before +imaging, neurons were incubated with 2 µM OGD-1 for 30 min, followed by incubation with normal +medium for 30 min. Q-switched 1030 nm nanosecond laser was used to generate light and delivered to +mFOE. The pulse energy was adjusted through a fiber optic attenuator (varied gap SMA Connector, +Thorlabs, Inc., USA). Notably, 1030 nm is far from the excitation peak of Oregon Green (494 nm) and +pass band of emission filter (500-540 nm), therefore assuring no effect from direct excitation of OGD by +any light leak from the fiber. A 3D translational stage was used to position the mFOE approaching the +target neurons. +Calcium fluorescence imaging was performed on a lab-built wide-field fluorescence microscope +based on an Olympus IX71 microscope frame with a 20× air objective (UPLSAPO20X, 0.75NA, +Olympus, USA), illuminated by a 470 nm LED (M470L2, Thorlabs, USA), an emission filter (FBH520- +40, Thorlabs, USA), an excitation filter (MF469-35, Thorlabs) and a dichroic mirror (DMLP505R, +Thorlabs, USA). Image sequences were acquired with a scientific CMOS camera (Zyla 5.5, Andor, + +22 + +Oxfords Instruments, UK) at 20 frames per second. The fluorescence intensities, data analysis, and +exponential curve fitting were analyzed using ImageJ (Fiji) and MATLAB 2022. +Implantation surgery procedure +All surgery procedures complied with all relevant guidelines and ethical regulations for animal testing and +research established and approved by Institutional Animal Care and Use Committee (IACUC) of Boston +University (PROTO201800534). Eight to ten weeks old male wildtype C57BL/6-E mice (Charles River +Laboratories, US) were received and allowed to acclimate for at least 3 days before enrolling them in +experiments. All mice in experiments had access to food and water ad libitum and were kept in the BU +animal facility maintained for 12-h light/dark cycle. During the implantation surgery, mice were +anesthetized by isoflurane (5% for induction, 1-3.5% during the procedure) and positioned on a +stereotaxic apparatus (51500D, Stoelting Co., USA). After hair removal, a small incision was made by +sterile surgery scalpel at the target region and then a small craniotomy was made by using a dental drill. +Assembled mFOE was inserted into mice hippocampus (−2.0 mm AP, 1.5 mm ML, 2 mm DV) using the +manipulator with respect to the Mouse Brain Atlas. The ground stainless steel wire was soldered to a +miniaturized screw (J.I. Morris) on the skull. Finally, the whole exposed skull area was fully covered by a +layer of Metabond (C&B METABOND, Parkell, USA) and dental cement (51458, Stoelting Co., USA). +Buprenorphine SR was used to provide long effective analgesia after the surgery. +In vivo electrophysiology recording and optoacoustic stimulation +Extracellular recording was performed through an electrophysiology system (Molecular Devices, LLC, +USA). mFOE electrodes were connected to the amplifier (Multiclamp 700B, Molecular Devices, LLC, +USA) through the pin connector and headstages after the animals recovered from surgeries. The amplified +analog signal was then converted and recorded by the digitizer (Digidata 1550, Molecular Devices, LLC, +USA). + +23 + +Q-switched 1030 nm nanosecond laser was used to generate light and delivered to mFOE. During the +extracellular electrophysiological recording, the preset trigger signal was generated by the digitizer and +used to trigger the Q-switch laser for optoacoustic stimulation. The pulse energy was adjusted through a +fiber optic attenuator (varied gap SMA Connector, Thorlabs, Inc., USA). +Data analysis was performed with Matlab and OriginPro and custom scripts were used to analyse the local +field potential and spike sorting. The raw extracellular recordings were first band filtered for local field +potential results (LFP, 0.5 – 300 Hz) and spike results (300 – 5000 Hz). A custom Matlab script was used +to create spectrograms to visually support the analysis of the LFPs in both the time domain and the +frequency domain. The spike sorting algorithm was implemented through several steps: first, individual +spike signals with length of 3 ms were picked up from the full recording through a standard amplitude +threshold method; then the dimensionality of each spike signal was reduced via the principal component +analysis (PCA) and unsupervised learning algorithms (K-means clustering) was used to separate out the +clusters. +Foreign body response assessment via immunohistochemistry +To compare the tissue response, animals were implanted with a silica optical fiber (diameter = 300 µm, +FT300EMT, Thorlabs, Inc, USA) and mFOE for 3 days or 4 weeks. Then at target timepoints, animals +were euthanized and transcardially perfused with phosphate-buffered saline (PBS, ThermoFisher +Scientific, USA) followed by 4% paraformaldehyde (PFA, ThermoFisher Scientific, USA) in PBS. The +fiber probes were carefully extracted before the extraction and then the brains were kept in 4% PFA +solution for one day at 4 °C. Brains were sectioned in the horizontal plane at 75 µm on a vibrating blade +vibratome. Free-floating brain slices were washed in PBS and blocked for 1 hour at room temperature in a +blocking solution consisting of 0.3% Triton X-100 (vol/vol) and 2.5% goat serum (vol/vol) in PBS. After +blocking, brain slices were incubated with the primary antibodies in the PBS solution with 2.5% goat +serum (vol/vol) for 24 hours at 4 °C. Primary antibodies used included rat anti-GFAP (Abcam Cat. # + +24 + +ab279291, 1:500), chicken anti-NeuN (Millipore Cat. # ABN91, 1:500), and rabbit anti-Iba1 (Abcam Cat. +# ab178846, 1:500). Following primary incubation, slices were washed three times with PBS for 10 min +at room temperature. The brain slices were then incubated with secondary antibodies in the PBS solution +with 2.5% goat serum (vol/vol) for 2 hours at room temperature. Secondary antibodies used included goat +anti-rat Alexa Fluor 488 (Abcam Cat. # ab150157, 1:1000), goat anti-rabbit Alexa Fluor 568 (Abcam Cat. +# ab175471, 1:1000) and goat anti-chicken Alexa Fluor 647 (Abcam Cat. # ab150171, 1:1000). Slices +were then washed three times with PBS for 10 min at room temperature. Before imaging, slices were +stained with DAPI solution (1 µg/ml, Millipore, USA) for 15 minutes at room temperature. All +fluorescent images were acquired with a laser scanning confocal microscope (Olympus FV3000) with an +air 20× objective with a numerical aperture NA = 0.75 unless otherwise noted. Neuron density was then +calculated within the normalized area by counting NeuN labeled cell bodies using the cell counter plugin +(ImageJ). Area analysis of Iba1 and GFAP labeled cells was performed by creating binary layers of the +fluorescence images using the threshold function and quantified using the measurement tool (ImageJ). +Statistical information +Data shown are mean ± standard deviation. For the comparison on peak fluorescence change of in vitro +optoacoustic stimulation, one-way ANOVA and Tukey’s mean comparison test were conducted by using +OriginLab. 15 stimulation events were compared for each condition. For the comparison of foreign body +response between silica fiber and mFOE, N > 8 brain slices from 3 animals were analysed using one-way +ANOVA and Tukey’s mean comparison test. The p values were determined as n.s.: nonsignificant, p > +0.05; *: p < 0.05; **: p < 0.01; ***: p < 0.001. Statistic analysis were conducted using OriginPro. +Data Availability +The raw data that support the findings of this study are available from the corresponding author upon +request. +Code Availability + +25 + +The MATLAB scripts for analysis are available from the corresponding author upon request. +Acknowledgements +This work was supported by National Institute of Health Brain Initiative R01 NS109794 to J-XC and CY. +Research reported in this publication was supported by the Boston University Micro and Nano Imaging +Facility and the Office of the Director, National Institutes of Health of the National Institutes of Health +under award Number S10OD024993 +References +1. +Wilbrecht, L. & Shohamy, D. Neural Circuits can Bridge Systems and Cognitive Neuroscience. +Front Hum Neurosci 3, 81 (2010). +2. +Chen, R., Canales, A. & Anikeeva, P. Neural Recording and Modulation Technologies. Nat Rev +Mater 2 (2017). +3. +Sohal, V.S., Zhang, F., Yizhar, O. & Deisseroth, K. 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Neurophotonics 9, 032207 (2022). +62. +Pisanello, F. et al. Multipoint-emitting optical fibers for spatially addressable in vivo +optogenetics. Neuron 82, 1245-1254 (2014). + +Supplementary Information + +Supplementary Figure 1. Microscope images of deposited carbon black and PDMS composite. +The coverage area was controlled through tuning the injection pressure and time. Injection time was +varied between 1 second and 2 seconds, and the pressure was varied from 2 psi, 3 psi and 4 psi. Scale bar: +50 µm. + + +2 psi +3 psi +4 psi +2 s1030 nm +Optical fiber +pulsedlaser +Function +generator +mFOE +Neurons cultured +Micromanipulator +loadedwithOGD-1 +Objective +Lens +Lens +470nm +DM +LED +Lens +CMOS +Mirror +camera29 + +Supplementary Figure 2. Schematic of in vitro mFOE stimulation and calcium imaging set up. +Stimulation: 1030 nm pulsed laser is triggered by a function generator and delivered to the mFOE through +an optical fiber. Calcium imaging: Oregon green is excited by 470 nm LED and the fluorescence signal is +detected through a CMOS camera. + +Supplementary Figure 3. Illustration of the laser pulse train for 5 bursts with 100 ms duration at +1Hz. + + +Repetition rate: 1.7 kHz +Pulse numbers: 170 +3 ns pulse width +100 msPre +Post30 + +Supplementary Fig. 4 Calcium imaging of neurons before and after mFOE stimulation. Scale bar: +100 µm. + +Supplementary Fig. S5 Average calcium traces of laser only control groups. The laser duration was +same with three conditions tested in mFOE stimulation (200 ms, 100 ms and 50 ms). Laser light with +pulse energy of 41.8 µJ was triggered at the time point labelled by the red bar. Shaded areas: standard +deviation. (N=3) + + + +200 ms +100 ms +50 ms +0.1 +0.1 +0.1 + 0.05 +F 0.05 +F 0.05 +△F/ +△F/ +△F/ +0 +1 +2 +3 +0 +1 +2 +3 +0 +1 +2 +3 +Time (s) +Time (s) +Time (s)1.5 +1.5 +1.5 +50 ms +100 ms +200 ms +1 +1 +1 +(0。) . +(0。) +(0。) +0.5 +0.5 +0.5 +△T +△T +△T +0 +0 +0 +-0.5 +-0.5 +-0.5 +0 +2 +4 +0 +2 +4 +0 +2 +4 +Time (s) +Time (s) +Time (s)31 + +Supplementary Fig. S6 Temperature change of the optoacoustic emitter integrated on mFOE. The +pulse energy was maintained at 41.8 µJ and the burst duration was varied from 50 ms (blue), 100 ms +(yellow) to 200 ms (orange). Laser was trigger at 2.5 second as labelled by the red bar. + + +Supplementary Fig. S7 LFP recording of sham control stimulation experiments. +a. Electrophysiological recording under light only stimulations delivered through a bare multifunctional +fiber without optoacoustic emitter. b. Simultaneous optoacoustic stimulation and electrophysiological +recording of an euthanized mouse. 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+page_content=' performed fabrication and characterization of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' performed the stimulation and recording experiments in vitro and in vivo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' performed the in vivo biocompatibility evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' provided guidance on the multifunctional fiber system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' provided guidance on the design of fiber optoacoustic emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' provided guidance on optimization of recording and data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The manuscript was written through contributions of all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' All authors have given approval to the final version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Corresponding author Correspondence to: Chen Yang (cheyang@bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='edu) and Xiaoting Jia (xjia@vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='edu) 2 Abstract A bidirectional brain interface with both “write” and “read” functions can be an important tool for fundamental studies and potential clinical treatments for neurological diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Here we report a miniaturized multifunctional fiber based optoacoustic emitter (mFOE) that first integrates simultaneous non-genetic optoacoustic stimulation for “write” and electrophysiology recording of neural circuits for “read”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The non-genetic feature addresses the challenges of the viral transfection required by optogenetics in primates and human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The orthogonality between optoacoustic waves and electrical field provides a solution to avoid the interference between electrical stimulation and recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' We first validated the non- genetic stimulation function of the mFOE in rat cultured neurons using calcium imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' In vivo application of mFOE for successful simultaneous optoacoustic stimulation and electrical recording of brain activities was confirmed in mouse hippocampus in both acute and chronical applications up to 1 month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Minimal brain tissue damage has been confirmed after these applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The capability of non- genetic neural stimulation and recording enabled by mFOE opens up new possibilities for the investigation of neural circuits and brings new insights into the study of ultrasound neurostimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Introduction Bidirectional communication with dynamic local circuits inside the brain of individual behaving animals or humans has been an invaluable approach for fundamental studies of neural circuits and for effective clinical treatment of neurological diseases, like epilepsy, Parkinson’ s disease, and depression1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Additionally, bidirectional neural interface paves the way for the closed-loop control, as it could enable more sophisticated, real-time control over neural dynamics3, behaviors4 and achieve effective therapeutic effect in neurological disease5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' To achieve real time assessment of the stimulated outcome, neural interfaces with ability to simultaneously manipulate and directly monitor the neural activities are preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Among the technologies developed in past decades, electrical stimulation and electrophysiology recording have been widely used and forms the basis of current implantable devices, which has been applied to clinical applications7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' For example, to restore both the motor and sensory 3 modalities, electric stimulation of the cortical surface is often associated with electrophysiology recording8, 9, like electrocorticography (ECoG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Also, the bidirectional electrical stimulation has demonstrated promising treatment effect in neurological diseases, such as epilepsy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The responsive focal cortical stimulation (RNS), leveraging ECoG recording as the trigger to provide stimulation, showed a statistically significantly greater reduction in seizure frequency and the benefits increased over time in a two-year study10, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' However, electrical stimulation has a limited spatial resolution due to current spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' It also interferes with the electrical signals used for recording, leading to “contamination” in electrophysiology recording2, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Although researchers are improving its performance through technologies such as current steering13, novel electrode design14, and artifacts cancellation15, considering the intrinsic physical properties of brain tissue16, the current spread, root cause of above-mentioned issues is hard to be fully eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Therefore, electrical stimulation for the bidirectional communication of brain may not be the ideal candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Being orthogonal with electrical recording, optical stimuli not only avoids the interference but also enables a high spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' To take this advantage, early efforts developed so-called optoelectrodes by simply assembling the optical fibers for optogenetics stimulation with the electrodes, such as Utah arrays17-19, Michigan probes20, 21 and microwires22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Semiconductor fabrication techniques and multiple material processing methods have recently been applied to improve the integration of those bidirectional devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' New processing techniques not only make the device more compact but also strengthen its functionality and biocompatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' For example, monolithically integrated micro-light- emitting-diodes (µLEDs) were used to reduce the complexity of light-guide structures and significantly boosted the number of stimulation sites and stimulation resolution 23, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Alternatively, a high-throughput thermal drawing method has been used to integrate the function components, for example, electrodes, microfluidic channels, and optical waveguides, to the flexible multifunctional polymer fiber 25, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Through this approach, the flexible fiber probes showed low bending-stiffness and enabled multifunctionalities, including optogenetics, electrical recording and drug delivery 27-29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Since 4 optogenetics relies on the expression of light-sensitive opsins in neurons through gene modification26, it is challenging to apply optogenetics to non-human primates and human effectively and safely30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Recently, our team showed non-genetic optoacoustic neural stimulation with a high spatial resolution up to single neuron level31, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' In an optoacoustic process, the pulsed light is illuminated on an absorber, causing transient heating and thermal expansion, and generating broadband acoustic pulses at ultrasonic frequencies33, 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' As a light mediated neural modulation method, optoacoustic is an ideal candidate to work with electrical recording for bidirectional neural communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Compared with existing technologies, it exhibited the advantages as a light mediated method, including a high spatial resolution and minimal crosstalk noise with electrical recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Importantly, the non-genetic optoacoustic neurostimulation alleviates the challenges and safety concern in optogenetics since no viral transfection is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Here, we developed a multifunctional fiber-based optoacoustic emitter (mFOE) as a miniaturized bidirectional brain interface performing simultaneously non-genetic neural stimulation and electrical recording of the neural activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Through a thermal drawing process,25, 35 fabrication of mFOE integrated an optical waveguide and multiple electrodes within a single fiber with a total diameter of 300 µm, compatible to the typical size of silica fibers used in optogenetic studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' An optoacoustic coating was selectively deposited to the tip of the core optical waveguide in the mFOE through a controlled micro- injection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Upon nanosecond pulse laser delivered to the photoacoustic coating, the mFOE generates a peak-to-peak pressure greater than 1 MPa, confirmed by the hydrophone measurement, which is sufficient for successful neural stimulation in vitro and in vivo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' By calcium imaging, the optoacoustic stimulation function of the mFOE was validated in Oregon green-loaded rat primary neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Importantly, we demonstrated the reliable functions of the chronic implanted mFOE for simultaneously stimulating and recording neurons in mouse hippocampus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Chronic recording also demonstrated that the embedded electrodes could provide long-term neural monitoring with a single-unit resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The histological evaluation of the brain tissue response confirmed that our flexible mFOE established a stable 5 and biocompatible multifunctional neural interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' mFOE is the first device integrated both optoacoustic stimulation with electrical recording for bidirectional neural communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' With the bidirectional capabilities and excellent biocompatibility, it offers a non-generic tools probing brain circuits, alternative to the optoelectrode devices, with improved feasibility in non-human primates and human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' It also opens up potentials for closed-loop neural stimulation and brain machine interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Results Design, fabrication and characterization of mFOE Towards bidirectional neural communication, we have designed the mFOE to utilize the optoacoustic stimulation as “writing” and electrophysiological recording as “reading” of the neural interface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Previously, fiber based optoacoustic emitters have been developed as a miniature invasive ultrasound transducer for the biomedical applications, such as intravascular imaging and interventional cardiology36, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Recently, our work showed that fiber based optoacoustic emitters can also be applied to neural stimulation in vitro and in vivo, with single neuron resolution and dual site capability32, 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' In these studies, typically commercial silica fibers were used, together with optoacoustic coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' However, the silica fiber, with Young’s modulus of ~70 GPa, is mismatched with mechanical properties of native neural tissue (kilo- to mega pascals)2 and not easy to integrate with miniaturized electrodes for recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' In this study, we took advantage of the fiber fabrication method developed by Anikeeva and Yoel25, and utilized the polymer multifunctional fiber design as the base for the mFOE to delivering nanosecond laser to the optoacoustic coating and to record electrical signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Specifically, a multifunctional fiber with a core optical waveguide and miniaturized electrodes was fabricated using the thermal drawing process (TDP) as previously reported27 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The waveguide is made of polycarbonate core (PC, refractive index nPC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='586, diameter = 150 µm) and polyvinylidene difluoride cladding (PVDF, refractive index nPVDF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='426, thickness = 50 µm) as the core and the shell, respectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' BiSn alloy is used in surrounding electrodes with diameters of 35 µm because of its conductivity and compatibility with TDP 6 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='This multifunctional fiber showed broadband transmission across the visible range to near infrared region and sub-megaohm impedance when it has been prepared into two centimetres long27, 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' To integrate the optoacoustic converter to the multifunctional fiber, the optoacoustic coating, composed of light absorbers and thermal expansion matrix, is needed to be selectively coated on the core waveguide distal end while keeping the surrounding electrodes exposed and conducting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Compared to previously reported FOE fabrication, here we took several innovative steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' First, a pressure-driven pico- litter injector was used to precisely deposit the optoacoustic materials to the core waveguide distal end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The coating area was controlled through varying the injection volume (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='1 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 nL), which is controlled by the regulated pressure (2-4 psi) over a set period of time (1-2 s, Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' S1) as described in equation (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' ������������ = ������������ ∙ ������������������������������������������������������������������������ 3 ������������ ∙ ������������ (1) where ������������ is the injection volume,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' ������������ is a constant attributed to the unit conversion factors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' effects of liquid viscosity and the taper angle of micropipette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' ������������������������������������������������������������������������ is the inner diameter of the pico-litter injector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' ������������ is the pressure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' and ������������ is the deposition time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Two 3D translational stages with stereo microscopes were used to precisely control the deposition localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Second, instead of using carbon nanotubes (CNT), we used carbon black (CB) embedded polydimethylsiloxane (PDMS) as the composite optoacoustic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' CB exhibited similar wideband light absorption40, assuring the sufficient photoacoustic conversion for neural stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Importantly, due to its relative low viscosity 41, 42, CB/PDMS composite shows much higher injectability compared with CNT/PDMS, therefore more comparable to the pico-liter deposition process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Through these steps, we successfully coated 10-20 µm thick 10% w/w CB/PDMS composite onto the 150 µm diameter core waveguide distal end while electrodes were still exposed as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 1e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Collectively mFOE with the photoacoustic emitter and multiple electrodes has been successfully fabricated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' To characterize the optoacoustic performance of mFOE, a Q-switched 1030 nm pulsed nanosecond laser was applied with pulse energies of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='6 µJ, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='3 µJ and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='8 µJ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The 7 generated acoustic waves were measured by a 40 µm needle hydrophone placed at about 100 µm away from the fiber tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Representative pulse acoustic pulse with a width of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='08 µs was generated by a single laser pulse as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 1f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Higher input laser pulse energy led to larger acoustic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' A peak-to-peak pressure of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='3 MPa were measured with the pulse energy of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='6, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='3 and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='8 µJ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The frequency spectrum shows the broadband characteristic of typical optoacoustic waves34, and the peak frequencies are around 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 MHz (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 1g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Based on previous work, we expected that such pressure and frequency is capable to successfully stimulate neurons in vitro and in vivo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' We also calculated the mechanical index (MI), a commonly used matrix, to evaluate the probability of mechanical damage due to ultrasound generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The MI of acoustic waves generated by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='3 MPa is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='198, lower than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='9, the safety threshold suggested by the Food and Drug Administration (FDA) safety guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 8 Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 1 Design, fabrication and characterization of mFOE a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Schematic of mFOE for bidirectional communication with neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Input laser pulse (red) is used to generate optoacoustic waves (black) by the converter and the neural activities are recorded by embedded electrodes as the output electrical signal (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Illustration of the thermal drawing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Components of the multifunctional fiber, including a PC/PVDF waveguide, BiSn alloy electrodes and PC sacrifice layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The selective deposition process for integrating the optoacoustic converter to the core wave guide in the multifunctional fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' A pressure-driven micro-injector is used to control the volume of CS/PDMS deposited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 3D translation stages and microscope are used to control the deposition location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Zoom-in: The micro pipette was aligned to the center of the fiber under the microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Top view microscope image of the mFOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Scale bar: 100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Representative acoustic waveforms under different laser pulse energy recorded by a needle hydrophone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Frequency spectrum of acoustic waveforms shown in f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' mFOE stimulation of cultured primary neurons a ptoacousticwave Pulsed laser Electrical signal q C d Injector on 3D stage Wave guide Micro pipette (PC/PVDF) Heat Fiber Sacrificelayer(PC) Vdrawing speed Electrode (BiSn alloy) Fiber holder on 3D stage e Optoacoustic g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='4 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='8 μJ 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='8 μJ converter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='3 μJ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='3 μJ Pressure (MPa) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='6 μJ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='6μJ (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='u Magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 Electrode 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='4 0 20 40 60 80 100 Time (μs) Frequency (MHz)9 To investigate mFOE can directly trigger the neuronal activity, we examined the response of cultured primary neurons under mFOE stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Because of the presence of calcium channels in neuronal membrane and their activation during the depolarization, calcium imaging has been widely used to monitor the neuronal activities43, 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Here, we cultured and loaded the rat cortical neurons (days in vitro 10-14) with a calcium indicator, Oregon Green™ 488 BAPTA-1 dextran (OGD-1)45 , and performed the calcium imaging with an inverted wide-field fluorescence microscope (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' To perform the optoacoustic stimulation, mFOE was placed approximately 50 µm above the in-focus target neurons (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 2a) by a micromanipulator under the microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 1030 nm 3 ns pulsed laser with a repetition rate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='7 kHz was delivered to the mFOE through an optical fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The energy of laser pulse was 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='8 µJ, corresponding to a peak-to-peak pressure of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='3 MPa generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Lower energy was tested but did not induce calcium transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The stimulation duration determined by each laser burst was 100 ms, corresponding to 170 pulses (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' By applying 5 bursts of laser pulses with interval of 1s, we investigated the reproducibility of the stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Using calcium imaging, we monitored the activities of all neurons in the field of view and divided them into two groups: groups within the converter area (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 2b) and outside the converter area (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' For neurons within the converter area, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' the 100 µm from the center of the mFOE, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 2b shows that 8 of 10 neurons showed successful and repeatable calcium transient (ΔF/F > 1%, the baseline standard deviation) corresponding to each stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Calcium transients are also repeatable for each burst applied over the 1 s period, indicating the evoked neuronal activities and confirming the reliability and biosafety of mFOE stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' For neurons outside the converter area, only 2 of 10 neurons responded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' This result also suggested the mFOE with the 150 µm center waveguide with photoacoustic coating provided a spatial precision of ~200 µm for stimulation in vitro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' This observation is consistent with that fiber based optoacoustic converters generate a confined ultrasound fields with sizes comparable with the radius of converter31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Next, to investigate the threshold of mFOE stimulation, we varied the stimulation duration from 5 ms, 50 ms, 100 ms to 200 ms on neurons in different cultures (N = 15) under the same laser pulse energy 10 of 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='8 µJ and the same repetition rate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='7 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' mFOE stimulation with duration of 5 ms did not evoked any observable fluorescence change (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=', p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='05) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 2g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Only when the duration was 50 ms or longer, the mFOE successfully produced neural activation (ΔF/F > 1%, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='01) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 2d-f, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Longer pulse durations leads to larger peak fluorescence changes, from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='1%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='8% to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='3% corresponding to 50 ms, 100 ms and 200 ms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' For the longest stimulation duration of 200 ms tested, no obvious change on morphology or elevation of baseline fluorescence intensity was detected in neurons after multiple stimulations (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 4), indicating the safety of stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Laser only control experiment was also performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Laser light with same pulse energy of 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='8 µJ and duration (200 ms, 100 ms and 50 ms) was delivered to OGD-1 loaded neurons through multifunctional fiber without optoacoustic coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' None of neuron culture showed detectable calcium response, distinct from the observed in mFOE stimulated neurons (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' To evaluate the photothermal effect of the mFOE stimulation and its potential impact on neurons, we also characterized the thermal profile of the mFOE in PBS during the acoustic generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Temperature was measured by an ultrafast thermal sensor with a sampling rate of 2000 Hz placed in contact with mFOE optoacoustic coating under the microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The laser conditions were consistent with neural stimulation test, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=', the pulse energy was maintained at 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='8 µJ and the burst duration was varied from 50 ms, 100 ms to 200 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The temperature increase on the mFOE surface was found to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='09 °C, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='08 °C, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='08 °C for 200, 100, 50 ms laser durations, respectively (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Such temperature increase is far below the previously reported threshold of thermal-induced neural stimulation (ΔT > 5 °C)46, 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Taken together, we conclude that activation of neurons was due to the mFOE optoacoustic stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 11 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Calcium transients induced by mFOE in cultured primary neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Calcium image of primary cultured neurons loaded with OGD-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Twenty neurons within (orange) and outside (blue) the optoacoustic converter area are circled and labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Scale bar: 100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Solid circle: area outside the converter area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' dashed line circle: area within the optoacoustic converter area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' b-c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Calcium traces of neurons undergone repeated mFOE stimulations with a laser pulse train duration of 100 ms (red dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Each pulse train was repeated 5 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Colors and numbers of the traces are corresponding to the neurons labelled in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' d-g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Average calcium traces of neurons triggered by mFOE stimulation with durations of 200 ms (d), 100 ms (e), 50 ms (f) and 5 ms (g), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Shaded area: the standard deviation (SD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' N=15 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Average maximum ΔF/F of neurons stimulated by mFOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' N = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' : non- significant, p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' *: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' **: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' ***: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='001, One-Way ANOVA and Tukey’s mean comparison test) In vivo simultaneous optoacoustic stimulation and electrophysiological recording Since the animal experiment is a significant part of the study in neuroscience and neurological diseases, we further investigated the performance of mFOE in the wild type C57BL/6J mice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' In vivo optoacoustic stimulation was performed by delivering pulsed laser to the implanted mFOE, and the optoacoustically stimulated neuronal activities were recorded through electrodes in the mFOE (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Experimentally, we implanted the mFOE into the hippocampus of mice (N =5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=" The chronically implanted mFOE allows a b 10 hhyeh10 C 9 9 9 8 8 Me 40 8 6 6 5 4 3 2 2 10% AA 5s' d e + g h 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='12 200 ms 100 ms 50 ms 5 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='1 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='08 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='05 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='06 A F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='04 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='02 0 A 0 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='02 Time (s) Time (s) Time (s) Time (s) 200ms100ms50ms5ms12 mice to move freely after surgery (Fig 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' During stimulation and recording tests, the mFOE was coupled with the laser source and electrophysiological recording headstage through the standard ferrule and pin connector, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The stimulation and recording were conducted in the mice under continuous anesthesia induced and maintained by isoflurane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Based on the threshold of optoacoustic stimulation obtained in in vitro studies, 50 ms bursts of laser pulses with a pulse energy of 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='8 µJ were delivered to the mFOE at 1Hz during the 5 second treatment period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The simultaneous electrophysiological recording by mFOE electrodes was bandpass filtered to examine the local field potential (LFP, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5-300 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Simultaneous optoacoustic stimulation and electrophysiological recording were performed at multiple time points, including 3 days, 7 days, 2 weeks and 1 month (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 3c-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Three out of five mice tested showed successful simultaneous stimulation and recording functions for testing periods of 3 days to one month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The evoked brain activities corresponding to the optoacoustic stimulation were confirmed by monitoring the LFP response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' LFP response at two weeks after implantation was detected with latency of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='19 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='29 ms (N = 15, from three mice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The amplitude of LFP response varied at four time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The largest and smallest responses occurred at 2 weeks and 1 month, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' A possible reason for this observation may be the brain tissue injury and healing after the implantation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' These results collectively demonstrate the reliability of the optoacoustic stimulation and recording functions of the implanted mFOE in the animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' To eliminate the possibility that LFP response was induced by electrical noise or laser artifacts, we also conducted two sham control experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' In the light only control group, we implanted a multifunctional fiber without optoacoustic coating to the mouse hippocampus and delivered the laser light with the same condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The LFP recorded didn’t correlate to the laser pulse train, indicating the spontaneous brain activities were recorded and light only did not invoke the LFP response (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 7a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' In the dead brain control group, we tested the optoacoustic stimulation through mFOE implanted to the euthanized mouse and did not observe the corresponding LFP response 13 (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 7b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' These results collectively confirm the signals we detected from mFOE stimulation were not artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' We further evaluated the recording performance of implanted mFOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' To evaluate the ability of mFOE for single unit recording, the electrophysiological signals recorded were bandpass filtered for spike activity (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5-3 kHz, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 3g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Through a principal-component analysis (PCA) based spike sorting algorithm, two spike clusters can be isolated from an endogenous neural recording (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 3j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The cluster quality was assessed by two common measures48, Lratio and isolation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Lratio is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='0017 and isolation distance has the value of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The first averaged spike shape (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 3h) showed a narrower and larger depolarization than that of the second spike shape (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 3i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The different spike waveform and the cluster analysis suggested that the action potentials were recorded from at least two different groups of neurons49, 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Thus, the successfully spike sorted neural activities from CA3 confirmed the ability of mFOE electrodes for the single-unit recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' To examine the sensitivity of LFP recording, at one month after implantation we altered the anesthesia level via adjusting the induced isoflurane concentration during the recording to see if the characteristic anesthesia dosage-dependent changes can be observed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 3k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Initially, a low level of anesthesia was maintained at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5% v/v isoflurane, and recorded LFP showed that spontaneous brain activities occurred continuously (i in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 3h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 3l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Then a higher-level anesthesia (3% v/v isoflurane) was applied for 3 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' After increased the isoflurane level, some spontaneous brain activities were suppressed and a hyperexcitable brain state was induced, where the voltage alternation (bursts) and isoelectric quiescence (suppression) appeared quasiperiodically27, 51 (ii in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 3h and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 3m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' With maintaining 3% v/v isoflurane, a deep anesthesia state was induced in the animal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' At the same time, both respiration rate and responsiveness to toe pinch decreased due to the higher anesthetic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Less voltage alternation occurred and for the most of time the LFP signal was a flat line (suppression, iii in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 3h and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 3n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Compared with initial stage, γ band LFP activity in 30-100 Hz was decreased due to the higher concentration of isoflurane as shown in the power spectrum52 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 3n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Later, when the concentration of isoflurane was reduced to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5% v/v again, the LFP activity returned to a 14 similar level as measured in the initial stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Taken together, this isoflurane dosage-dependent characteristic confirmed the accuracy of LFP recording by mFOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 3 Simultaneous optoacoustic stimulation and electrophysiological recording by implanted mFOE in mouse hippocampus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Illustration of the mFOE enabled bidirectional neural communication using laser signal as input and electrical signal as readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' mFOE was implanted into hippocampus of a wild type C57BL/6J mouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' c-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Simultaneous optoacoustic stimulation and electrophysiological recording performed at 3 days (c), 7 days (d), two weeks (e) and one month (f) after implantation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Blue dots the laser pulse trains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' For each laser train: 50 ms burst of pulses, pulse energy of 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='8 µJ, laser repetition rate 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='7 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Part of the filtered spontaneous activity containing two separable units recorded by mFOE electrode at one month after implantation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' h-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Spike shapes of two separable units in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Principal-components analysis (PCA) of the two units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Local field potential (LFP) recorded by mFOE one month after implantation with an alternating anaesthesia level (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5-3% v/v isoflurane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' l-n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' different LFP responses induced by varying the concentration of isoflurane: l corresponds to the initial stage (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5% of isoflurane level);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' m corresponds to the burst/suppression transition stage (after increasing the isoflurane level to 3%);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' n corresponds to the suppression stage (the isoflurane level was maintained at 3% and took effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' a c d Optical Electrical 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 3 days 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 7 days input readout (mV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='0 Voltage ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='0- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='0- 0 2 4 6 10 0 4 6 8 10 Time (s) Time (s) b e (mV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 2 weeks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5- 1 month (mV) Voltage ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='0 Voltage 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='0 0 2 4 6 8 10 0 2 4 6 8 10 Time (s) Time (s) g h 100 40 40 20 20 Yoltage (μV) (Λ) 50 0 0 20 PC2 40 40 50 60 60 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 2 0 2 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 2 100 Time (ms) Time(ms) 200 150 100 50 0 50 100 PC1 k 3 % i Initial stage m ii Burst/suppression n ii Suppression 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 % M 2 s N (ZH) 100 100 Frequency (dB) Frequency (p) Frequency 50 50 50 50 Power 50 100 100 50 100 150 ii ili 2 4 68101214 2 468101214 2 68101214 50 s Time (s) Time (s) Time (s)15 Foreign body response comparison between mFOE and standard optical fiber using immunohistochemistry Foreign body response is a critical property of implantable neural interface to assure their usage in a safe and chronic way,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' since the physical insertion into brain tissue commonly initiates a progressive inflammatory tissue response53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' To evaluate the biocompatibility of mFOE, we compared the foreign body response of mouse brain to mFOE with the similar size standard silica optical fibers (diameter = 300 µm), which is widely used in optogenetic technologies54, 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The immunohistochemistry analysis of surrounding brain tissue was performed from mice (N = 3) implanted with the mFOE and a conventional silica fiber 3 days and 1 month after implantation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The damage to surrounding neurons from implant was assessed through evaluating neuronal density using the neuronal nuclei (NeuN) markers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Number of neurons was calculated by counting the NeuN-positive cells per field of view (650 × 650 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The presence of ionized calcium-binding adaptor molecule 1 (Iba1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 4c) and glial fibrillary acidic protein (GFAP, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 4d) were used as the markers for activated microglia and astrocytic response, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Compared with the silica fiber, mFOE induced significantly less microglial response (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='01, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 4c, f) and astrocyte reactivity (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='001, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 4d, g), but no significant difference was observed on the neuronal density (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 4b, e) 3 days after implantation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' A decrease in foreign body response, specifically, higher neuronal density and lower microglia and astrocytic response (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 4e-g), was observed from 3 days to 1 month after implantation of both mFOE and silica fiber and no significant difference was observed between mFOE and silica fiber 1 month after implantation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Taken together, the immunohistochemistry analysis confirmed that mFOE yielded less foreign body response in the short period, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=', 3 days, after implantation and showed similar biocompatibilty with silica fiber at longer implantation time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=', 1 month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 16 Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 4 Foreign body response comparison of mFOE and silica fiber using immunohistochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' a-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Immunohistochemistry images of mouse brains implanted with mFOE and silica fiber one month after implantation (N = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Scale bar: 100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Brain slices were labelled with the neuron-specific protein (NeuN, cyan), ionized calcium-binding adaptor molecule 1 (Iba1, red) and glial fibrillary acidic protein (GFAP, green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Number of neurons in the field of view, calculated by counting the NeuN-positive cells for mFOE and silica fiber at 3 days and 1 mon after implantation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Microglial reactivity, assessed by counting the Iba-1 labelled area, for mFOE and silica fiber at 3 days and 1 mon after implantation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Astrocyte reactivity, assessed by counting the GFAP labelled area, for mFOE and silica fiber at 3 days and 1 mon after implantation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' For each experimental group, two to four brain slices were used from each a mFOE Silica Fiber 800 e n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 700 neurons Composite 600 500 Number n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 400 300 mFOE b Silica fiber 200 Day 3 Day30 TimePoint NeuN ** mFOE 3×104 Silica fiber n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 2 ×10 C 10 6×103 Iba1 Day 3 Day30 TimePoint g 3×10 mFOE Silica fiber d 2 × 10 GFAP area (μm" n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' GFAP 10 6 ×10° Day 3 Day30 TimePoint17 mouse (N= 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' : non-significant, p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' *: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' **: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' ***: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='001, One-Way ANOVA and Tukey’s mean comparison test) Discussion In this study, we designed and developed a miniaturized fiber-based device, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' mFOE, for bidirectional neural communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' mFOE performs the “write” function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' non-genetic optoacoustic stimulation and the “read” function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' simultaneous electrophysiological recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The broadband acoustic wave with a broadband ultrasound pulse with pulse width about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='1 µs and a center frequency at 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 MHz and a peak pressure of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='3 MPa with pulse numbers >85 generated by mFOE successfully stimulate neurons with a spatial resolution of approximately 200 µm in primary rat cortical neuron culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' By implanting mFOE into mouse hippocampus, we demonstrated its ability for simultaneous optoacoustic stimulation and electrophysiological recording and superior biocompatibility as a chronic bidirectional neural interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Reliable stimulation and LFP recording have been achieved up to one month post implantation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Recording quality has been demonstrated by single unit recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' For the first time, combining this pico-liter deposition and thermal fiber pulling, we successfully integrated an optoacoustic converter to the polymer multifunctional fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Different from the conventional dip-coating method36, 56, the selective deposition through micro-injection allows the easy fabrication of optoacoustic emitter in a volume and position-controlled way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Through the selective deposition, the dimension of optoacoustic emitter is no longer limited by the tip sizes of optical fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Our choice of CB/PDMS composite as the optoacoustic material is also essential as it is comparable with this deposition process with a fine volume control at pico liter level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Besides the application in neural interface, such design and fabrication method can also be applied to optical ultrasound probes used in imaging37, 57, for example, in the tip engineering and the integration to photonics crystal fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' We introduced the optoacoustic stimulation as a new strategy for “writing” in the bidirectional neural interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Compared with previous optoelectrode devices based on optogenetics24, 25, 27 and 18 photothermal58, 59, the non-genetic optoacoustic stimulation enabled by mFOE reduces the barrier of transgenic techniques for applications in primate and potentially human, and avoids the thermal toxicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' At the same time, it offers the spatial precision benefit from the confined ultrasound field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' It is orthogonal to electrical recording, therefore minimizing crosstalk with electrical recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' As an emerging neuromodulation method, the mechanism of optoacoustic stimulation is still not fully understood but more studies indicated that mechanosensitive ion channels are responsible for the activation of neurons60, 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Bidirectional brain interfaces are important research tools to understand brain circuits, potential treatments for neurological disease and bridges to brain computer interface for broad applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' New features of mFOE compared to the previous fiber based interface, such as non-genetic and non-electrical stimulation are critical to advance these applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' For example, closed-loop neuromodulation has been demonstrated to be superior to the conventional open-loop system, as it can achieve more responsive and real-time control over neural dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' In neurological diseases treatment, combining the detection and in situ intervention improves the treatment effectiveness and safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Because of its bidirectional capabilities, mFOE has the potential to be used as a new brain interface with closed-loop capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Using epilepsy as an example, by implanting the mFOE into seizure foci, the continuous LFP recording can guide the localized optoacoustic stimulation and intervene can be triggered at the early stage before seizure progresses into a generalized seizure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The unique orthogonal non-electrical optoacoustic stimulation and electrical recording prevents “contamination” of the recording signals, potentially offering a more effective closed-loop strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' In comparison of the optoelectrodes fabricated through semiconductor fabrication process, the recording and stimulation sites of the current mFOE design is fixed at the core waveguide and the number of channels is limited because of the nature of multifunctional fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Some post processing methods have been proposed to tackle this challenge, like the laser micromachining technique27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' In addition, it is possible to further engineer the fiber to offer multiple and selective stimulation sites62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' With the further 19 development of multifunctional fiber strategy, we believe the bandwidth of mFOE would be improved and open more opportunities in the research of neuroscience and neurological diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Methods Multifunctional fiber fabrication and optoacoustic emitter integration Multifunctional fibers were fabricated from a preform fiber and then drawn into thin fibers through TDP in a customized furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' For the preform fiber, PVDF film (Mcmaster) and PC film (laminated plastics) were rolled onto a PC rod (Mcmaster) and followed by a consolidation process in vacuum at 200 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Next, four rectangular grooves (2 mm × 2 mm) were machined on the solid PC layer and inserted with the BiSn (Indium Corporate) electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Then, another PVDF layer was rolled over the rod to form an insulation layer for the electrodes and followed by an additional PC as the sacrifice layer for the convenience of TDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The detailed fabrication process was discussed in the previous paper27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' A composite of 10% carbon black (diameter < 500 nm, Sigma Aldrich) and 90% polydimethylsiloxane (PDMS, Sylgard 184, Dow Corning Corporation, USA) were used as the optoacoustic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The mixture was sonicated for 1 hour followed by degassing in vacuum for 30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The mixture was then filled in the glass micropipette (Inner diameter = 30 µm, TIP30TW1, World Precision Instruments, USA) connected to the pico-liter injector (PLI-100A, Warner Instruments, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Under the microscope, the glass micropipette was aligned with the core waveguide of multifunctional fiber and the mixture was deposited to the surface of the core waveguide by controlling the injection pressure and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The deposited fiber was then cured vertically at room temperature for 2 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Before use, mFOE was further prepared for the optical coupling and electrodes connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' For the optical coupling, a ceramic ferrule (Thorlabs, USA) was added and affixed to the end of the fiber by the 5-min epoxy (Devcon, ITW Performance Polymers, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Then the end surface was polished by 20 optical polishing papers to reduce roughness from 30 µm to 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' For the connection to electrodes embedded in the multifunctional fiber, the electrodes were exposed manually along the side wall of the fiber by using a blade and silver paint (SPI Supplies, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Then copper wires were wrapped around the fiber at each exposure locations along the fiber and the silver paint were applied for the fixation and lower resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The copper wires connected to fiber electrodes were soldered to the pin connector while a stainless-steel wire was also soldered as the ground wire for later extracellular recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' In addition, the 5-min epoxy (Devcon, ITW Performance Polymers, USA) was applied to the connection interface for strengthening affixation and better electrical insulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Optoacoustic wave characterization To generate the optoacoustic signal, a compact Q-switched diode-pumped solid-state laser (1030 nm, 3 ns, 100 μJ, repetition rate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='7 kHz, RPMC Lasers Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=', USA) was used as the excitation laser source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The laser was first connected to an optical fiber through a 200 µm fiber coupling module and then connected to the mFOE with a SubMiniature version A (SMA) connector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The pulse energy was adjusted through a fiber optic attenuator (varied gap SMA Connector, Thorlabs, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=', USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The acoustic signal was measured through a homebuilt system including a needle hydrophone (ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 40 µm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' OD, 300 µm) with a frequency range of 1–30 MHz (NH0040, Precision Acoustics Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=', Dorchester, UK), an amplifier and an oscilloscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The mFOE tip and hydrophone tip were both immersed in degassed water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The pressure values were calculated based on the calibration factor provided by the hydrophone manufacturer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The frequency data was obtained through a fast Fourier transform (FFT) calculation using the OriginPro 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Embryonic neuron culture All experimental procedures complied with all relevant guidelines and ethical regulations for animal testing and research established and approved by Institutional Animal Care and Use Committee (IACUC) of Boston University (PROTO201800534).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Primary cortical neurons were isolated from embryonic day 15 (E15) Sprague−Dawley rat embryos of either sex (Charles River Laboratories, MA, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Cortices 21 were isolated and digested in TrypLE Express (ThermoFisher Scientific, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Then the neurons were plated on poly-D-lysine (50 μgmL−1, ThermoFisher Scientific, USA)-coated glass bottom dish (P35G- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5-14-C, MatTek Corporation, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Neurons were first cultured with a seeding medium composed of 90% Dulbecco’s modified Eagle medium (ThermoFisher Scientific, USA) and 10% fetal bovine serum (ThermoFisher Scientific, USA) and 1% GlutaMAX (ThermoFisher Scientific, USA), which was then replaced 24 h later by a growth medium composed of Neurobasal Media (ThermoFisher Scientific, USA) supplemented with 1× B27 (ThermoFisher Scientific, USA), 1× N2 (ThermoFisher Scientific, USA), and 1× GlutaMAX (ThermoFisher Scientific, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Half of the medium was replaced with fresh growth medium every 3 or 4 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Cells cultured in vitro for 10−14 days were used for Oregon Green labelling and PA stimulation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' In vitro neurostimulation and calcium imaging Oregon Green™ 488 BAPTA-1 dextran (OGD-1) (ThermoFisher Scientific, USA) was dissolved in 20% Pluronic F-127 in dimethyl sulfoxide (DMSO) at a concentration of 1 mM as stock solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Before imaging, neurons were incubated with 2 µM OGD-1 for 30 min, followed by incubation with normal medium for 30 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Q-switched 1030 nm nanosecond laser was used to generate light and delivered to mFOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The pulse energy was adjusted through a fiber optic attenuator (varied gap SMA Connector, Thorlabs, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=', USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Notably, 1030 nm is far from the excitation peak of Oregon Green (494 nm) and pass band of emission filter (500-540 nm), therefore assuring no effect from direct excitation of OGD by any light leak from the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' A 3D translational stage was used to position the mFOE approaching the target neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Calcium fluorescence imaging was performed on a lab-built wide-field fluorescence microscope based on an Olympus IX71 microscope frame with a 20× air objective (UPLSAPO20X, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='75NA, Olympus, USA), illuminated by a 470 nm LED (M470L2, Thorlabs, USA), an emission filter (FBH520- 40, Thorlabs, USA), an excitation filter (MF469-35, Thorlabs) and a dichroic mirror (DMLP505R, Thorlabs, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Image sequences were acquired with a scientific CMOS camera (Zyla 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5, Andor, 22 Oxfords Instruments, UK) at 20 frames per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The fluorescence intensities, data analysis, and exponential curve fitting were analyzed using ImageJ (Fiji) and MATLAB 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Implantation surgery procedure All surgery procedures complied with all relevant guidelines and ethical regulations for animal testing and research established and approved by Institutional Animal Care and Use Committee (IACUC) of Boston University (PROTO201800534).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Eight to ten weeks old male wildtype C57BL/6-E mice (Charles River Laboratories, US) were received and allowed to acclimate for at least 3 days before enrolling them in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' All mice in experiments had access to food and water ad libitum and were kept in the BU animal facility maintained for 12-h light/dark cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' During the implantation surgery, mice were anesthetized by isoflurane (5% for induction, 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5% during the procedure) and positioned on a stereotaxic apparatus (51500D, Stoelting Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=', USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' After hair removal, a small incision was made by sterile surgery scalpel at the target region and then a small craniotomy was made by using a dental drill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Assembled mFOE was inserted into mice hippocampus (−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='0 mm AP, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 mm ML, 2 mm DV) using the manipulator with respect to the Mouse Brain Atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The ground stainless steel wire was soldered to a miniaturized screw (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Morris) on the skull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Finally, the whole exposed skull area was fully covered by a layer of Metabond (C&B METABOND, Parkell, USA) and dental cement (51458, Stoelting Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=', USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Buprenorphine SR was used to provide long effective analgesia after the surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' In vivo electrophysiology recording and optoacoustic stimulation Extracellular recording was performed through an electrophysiology system (Molecular Devices, LLC, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' mFOE electrodes were connected to the amplifier (Multiclamp 700B, Molecular Devices, LLC, USA) through the pin connector and headstages after the animals recovered from surgeries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The amplified analog signal was then converted and recorded by the digitizer (Digidata 1550, Molecular Devices, LLC, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 23 Q-switched 1030 nm nanosecond laser was used to generate light and delivered to mFOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' During the extracellular electrophysiological recording, the preset trigger signal was generated by the digitizer and used to trigger the Q-switch laser for optoacoustic stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The pulse energy was adjusted through a fiber optic attenuator (varied gap SMA Connector, Thorlabs, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=', USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Data analysis was performed with Matlab and OriginPro and custom scripts were used to analyse the local field potential and spike sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The raw extracellular recordings were first band filtered for local field potential results (LFP, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 – 300 Hz) and spike results (300 – 5000 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' A custom Matlab script was used to create spectrograms to visually support the analysis of the LFPs in both the time domain and the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The spike sorting algorithm was implemented through several steps: first, individual spike signals with length of 3 ms were picked up from the full recording through a standard amplitude threshold method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' then the dimensionality of each spike signal was reduced via the principal component analysis (PCA) and unsupervised learning algorithms (K-means clustering) was used to separate out the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Foreign body response assessment via immunohistochemistry To compare the tissue response, animals were implanted with a silica optical fiber (diameter = 300 µm, FT300EMT, Thorlabs, Inc, USA) and mFOE for 3 days or 4 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Then at target timepoints, animals were euthanized and transcardially perfused with phosphate-buffered saline (PBS, ThermoFisher Scientific, USA) followed by 4% paraformaldehyde (PFA, ThermoFisher Scientific, USA) in PBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The fiber probes were carefully extracted before the extraction and then the brains were kept in 4% PFA solution for one day at 4 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Brains were sectioned in the horizontal plane at 75 µm on a vibrating blade vibratome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Free-floating brain slices were washed in PBS and blocked for 1 hour at room temperature in a blocking solution consisting of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='3% Triton X-100 (vol/vol) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5% goat serum (vol/vol) in PBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' After blocking, brain slices were incubated with the primary antibodies in the PBS solution with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5% goat serum (vol/vol) for 24 hours at 4 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Primary antibodies used included rat anti-GFAP (Abcam Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' # 24 ab279291, 1:500), chicken anti-NeuN (Millipore Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' # ABN91, 1:500), and rabbit anti-Iba1 (Abcam Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' # ab178846, 1:500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Following primary incubation, slices were washed three times with PBS for 10 min at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The brain slices were then incubated with secondary antibodies in the PBS solution with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5% goat serum (vol/vol) for 2 hours at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Secondary antibodies used included goat anti-rat Alexa Fluor 488 (Abcam Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' # ab150157, 1:1000), goat anti-rabbit Alexa Fluor 568 (Abcam Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' # ab175471, 1:1000) and goat anti-chicken Alexa Fluor 647 (Abcam Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' # ab150171, 1:1000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Slices were then washed three times with PBS for 10 min at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Before imaging, slices were stained with DAPI solution (1 µg/ml, Millipore, USA) for 15 minutes at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' All fluorescent images were acquired with a laser scanning confocal microscope (Olympus FV3000) with an air 20× objective with a numerical aperture NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='75 unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Neuron density was then calculated within the normalized area by counting NeuN labeled cell bodies using the cell counter plugin (ImageJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Area analysis of Iba1 and GFAP labeled cells was performed by creating binary layers of the fluorescence images using the threshold function and quantified using the measurement tool (ImageJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Statistical information Data shown are mean ± standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' For the comparison on peak fluorescence change of in vitro optoacoustic stimulation, one-way ANOVA and Tukey’s mean comparison test were conducted by using OriginLab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 15 stimulation events were compared for each condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' For the comparison of foreign body response between silica fiber and mFOE, N > 8 brain slices from 3 animals were analysed using one-way ANOVA and Tukey’s mean comparison test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The p values were determined as n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' : nonsignificant, p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' *: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' **: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' ***: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Statistic analysis were conducted using OriginPro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Data Availability The raw data that support the findings of this study are available from the corresponding author upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Code Availability 25 The MATLAB scripts for analysis are available from the corresponding author upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Acknowledgements This work was 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' & Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' High-precision neural stimulation through optoacoustic emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Neurophotonics 9, 032207 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Pisanello, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Multipoint-emitting optical fibers for spatially addressable in vivo optogenetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Neuron 82, 1245-1254 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Supplementary Information Supplementary Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Microscope images of deposited carbon black and PDMS composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The coverage area was controlled through tuning the injection pressure and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Injection time was varied between 1 second and 2 seconds, and the pressure was varied from 2 psi, 3 psi and 4 psi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Scale bar: 50 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 2 psi 3 psi 4 psi 2 s1030 nm Optical fiber pulsedlaser Function generator mFOE Neurons cultured Micromanipulator loadedwithOGD-1 Objective Lens Lens 470nm DM LED Lens CMOS Mirror camera29 Supplementary Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Schematic of in vitro mFOE stimulation and calcium imaging set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Stimulation: 1030 nm pulsed laser is triggered by a function generator and delivered to the mFOE through an optical fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Calcium imaging: Oregon green is excited by 470 nm LED and the fluorescence signal is detected through a CMOS camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Supplementary Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Illustration of the laser pulse train for 5 bursts with 100 ms duration at 1Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Repetition rate: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='7 kHz Pulse numbers: 170 3 ns pulse width 100 msPre Post30 Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' 4 Calcium imaging of neurons before and after mFOE stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Scale bar: 100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' S5 Average calcium traces of laser only control groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The laser duration was same with three conditions tested in mFOE stimulation (200 ms, 100 ms and 50 ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Laser light with pulse energy of 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='8 µJ was triggered at the time point labelled by the red bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Shaded areas: standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' (N=3) 200 ms 100 ms 50 ms 0.' metadata={'source': 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+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 50 ms 100 ms 200 ms 1 1 1 (0。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=') .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' (0。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=') (0。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 △T △T △T 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 0 2 4 0 2 4 0 2 4 Time (s) Time (s) Time (s)31 Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' S6 Temperature change of the optoacoustic emitter integrated on mFOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' The pulse energy was maintained at 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='8 µJ and the burst duration was varied from 50 ms (blue), 100 ms (yellow) to 200 ms (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Laser was trigger at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 second as labelled by the red bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' S7 LFP recording of sham control stimulation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Electrophysiological recording under light only stimulations delivered through a bare multifunctional fiber without optoacoustic emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Simultaneous optoacoustic stimulation and electrophysiological recording of an euthanized mouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' Same laser condition was used: pulse energy of 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='8 µJ, 50 ms burst of pulses, 1 Hz, blue dots indicate the laser onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content=' b a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='25 Voltage (mV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} +page_content='0 Voltage (mV) 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E2T4oBgHgl3EQfGgbf/content/2301.03659v1.pdf'} diff --git a/edFST4oBgHgl3EQfFzg4/content/tmp_files/2301.13719v1.pdf.txt b/edFST4oBgHgl3EQfFzg4/content/tmp_files/2301.13719v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..862cf823b456bc43808eb0ec155964d7cfd91b2d --- /dev/null +++ b/edFST4oBgHgl3EQfFzg4/content/tmp_files/2301.13719v1.pdf.txt @@ -0,0 +1,815 @@ +arXiv:2301.13719v1 [math.GT] 31 Jan 2023 +FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS +CRISTINA COSTOYA, VICENTE MU˜NOZ, AND ANTONIO VIRUEL +Abstract. In this paper we solve in the positive the question of whether any finite set +of integers A, containing the zero, is the mapping degree set between two oriented closed +connected manifolds of the same dimension. We extend this question to the rational +setting, where an affirmative answer is also given. +1. Introduction +In this paper, we settle in the positive various questions which have been raised about +D(M, N), the set of mapping degrees between two oriented closed connected manifolds +M and N of the same dimension: +D(M, N) = {d ∈ Z | ∃f : M → N, deg(f) = d}. +In [14, Problem 1.1], the authors discuss the problem of finding, for any set A ⊂ +Z containing the zero, two oriented closed connected manifolds M and N of the same +dimension such that A = D(M, N). Note that 0 ∈ A is a necessary condition as the +constant map M → N is of degree zero. +A quick argument shows that when A is an infinite set, this problem is solved in the +negative [14, Theorem 1.3]: there are uncountably many infinite subsets A ⊂ Z containing +the zero, compared to the countably many mapping degree sets D(M, N) that exist for +pairs of oriented closed connected manifolds of the same dimension. Hence not every +infinite set, containing the zero, is realizable as the mapping degree set of manifolds. +Thus, one might ask: +Question 1.1 ([14, Problem 1.4]). Let A be a finite set of integers containing the zero. +Is A = D(M, N) for some oriented closed manifolds M, N of the same dimension? +Remark 1.2. It is important to notice that if {0} ⊊ A = D(M, N), A finite, for some +manifolds M and N, then D(M, M) and D(N, N) must both be contained in {0, 1, −1}. +Otherwise, if there exists g : M → M with | deg(g)| > 1, then for any non-zero degree +f : M → N (which exists by assumption), the subset {deg(f ◦ gm) | m ∈ N} of D(M, N) +is unbounded. This leads to a contradiction as A = D(M, N) is finite. The same follows +for D(N, N). +2020 Mathematics Subject Classification. 55M25, 57N65, 55P62, 55R10. +Key words and phrases. Mapping degree sets, inflexible manifold, fiber bundle, unstable Adams +operation. +The first author was partially supported by MINECO (Spain) grants PID2020-115155GB-I00 and +TED2021-131201B-I00. The second author was partially supported MINECO (Spain) grant PID2020- +118452GB-I00. The third author was partially supported by MINECO (Spain) grant PID2020-118753GB- +I00, and by PAIDI 2020 (Andalusia) grant PROYEXCEL-00827. +1 + +2 +C. COSTOYA, V. MU˜NOZ, AND A. VIRUEL +An oriented closed manifold M satisfying D(M, M) ⊆ {0, 1, −1} is called an inflexible +manifold [7, Definition 1.4]. +This condition is equivalent to asking that D(M, M) is +bounded: since it is a multiplicative semi-group, if there exists any ℓ ∈ D(M, M) with +|ℓ| > 1, then D(M, M) is unbounded. Simply connected inflexible manifolds are rare +objects that have appeared quite recently in literature using rational homotopy theory +and surgery theory (see [5] for an account on the simply connected inflexible manifolds +that are known at present). Not surprisingly, and in lights of Remark 1.2, part of our key +constructions will use rational homotopy methods. +The main result in this work answers Question 1.1 positively: +Theorem A. Let A be a finite set of integers containing the zero. Then, A = D(M, N) +for some oriented closed connected 3-manifolds M, N. +The proof of this theorem will be carried out at the end of Section 3. Appealing to [14, +Example 1.5], we point out that the 3-dimension of the manifolds is the lowest possible. +A second problem related to Question 1.1 is also treated in this paper. More precisely, +let the rational mapping degree set between oriented closed connected n-manifolds M, N +be the following set +DQ(M, N) = {d ∈ Q | ∃f : (M(0), [M]Q) → (N(0), [N]Q), deg(f) = d}, +where [M] ∈ Hn(M; Z) denotes the cohomological fundamental class of M, [M]Q ∈ +Hn(M; Q) denotes the rational cohomological fundamental class of M, and M(0) the +rationalization of M. Then, we raise the following question, which can be thought of +as a rational version of [14, Problem 1.4]: +Question 1.3. Let A be a finite set of rational numbers containing the zero. Is A = +DQ(M, N) for some oriented closed connected manifolds M, N of the same dimension? +In Section 4 we solve this problem in the positive by proving: +Theorem B. Let A be a finite set of rational numbers containing the zero. Then A = +DQ(M, N) for some oriented closed manifolds M, N. Moreover, given any integer k ≥ 1, +the manifolds M, N above can be chosen (30k + 17)-connected. +The proofs of Theorem A and Theorem B consist of mainly two steps: +• Arithmetical decomposition of finite sets: +In Section 2 we show how to decompose +the candidate A to be realized as the mapping degree set of manifolds, as an in- +tersection of sums over specifically designed sequences of integers SBi, i = 0, . . . , n +(see Definition 2.1). Each of those sums gradually approaches A (Proposition 2.2, +Corollary 2.3). +• Spherical fibrations: In Sections 3 and 4, we use certain inflexible manifolds (resp. +inflexible Sullivan algebras) as the basis of spherical fibrations where the total +spaces are also inflexible manifolds (resp. inflexible Sullivan algebras). Relations +between connected sums and mapping degree sets (see Propositions 3.3 and 4.3) +allow one to consider iterated connected sums of the total spaces, in a first stage +to realize the sums SBi above mentioned, and in a second stage to realize the +candidate A. + +FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS +3 +Looking at the connectivity, while manifolds from Theorem B are simply connected +(indeed, they are as highly connected as desired) the ones from Theorem A have non- +trivial fundamental group. In Section 5 we will use unstable Adams operations to prove +the following results that guarantee that manifolds realizing finite sets of integers can be +chosen simply connected: +Theorem C. Suppose that there exists an oriented closed k-connected 2m-manifold Σ, +m > 1, verifying that Σ(0) is inflexible and πj(Σ(0)) = 0 for j ≥ 2m − 1. Then any finite +set of integers A containing the zero can be realized as A = D(M, N) for some oriented +closed k-connected (4m − 1)-manifolds M, N. +Examples of simply connected manifolds fulfilling the hypotheses of Theorem C can be +found in [1, Example 3.8] and [7, Theorem 6.8]. Hence, the following holds: +Corollary D. Any finite set of integers A containing the zero can be realized as A = +D(M, N) for some oriented closed simply connected manifolds M, N. +2. Some arithmetic combinatorics +In this section we show that every finite set A ⊂ Z (resp. ⊂ Q) containing the zero can +be expressed as the intersection of sums over certain sequences of integers, that gradually +approach A. The sequences have an additional property (see Proposition 2.2) that will +be crucial to prove Theorem C in Section 5 below. +Following the notation in [7, Section II.1], [14, Section 3], given A, B ⊂ Z (resp. ⊂ Q) +we write A + B := {a + b : a ∈ A, b ∈ B} ⊂ Z (resp. ⊂ Q). +Definition 2.1. Let B be a finite sequence of not necessarily pairwise distinct non-zero +integers (resp. rational numbers). We write +SB := +� +b∈B +{0, b} ⊂ Z (resp. ⊂ Q), +and we refer to it as the sum over the sequence B. +Proposition 2.2. Let d1, . . . , dn be pairwise distinct non-zero integers. For any positive +integer m ≥ 1, there exist finite sequences B(i), i = 0, . . . , n, of not necessarily pairwise +distinct non-zero integers, such that +{0, d1, . . . , dn} = +n� +i=0 +SB(i). +Moreover, every element in B(i) can be written as a power ±km for some positive integer +k coprime to m!. +Proof. Fix m ≥ 1. Since the construction of B(i), i = 0, . . . , n, depends on the sign of +the pairwise distinct di ∈ Z, i = 1, . . . , n, we write them as an ordered sequence +{−ar < . . . < −a1 < 0 < e1 < . . . < es} +where n = r + s. We assume a0 = 0 = e0. + +4 +C. COSTOYA, V. MU˜NOZ, AND A. VIRUEL +In the first place, let B(0) be the sequence consisting of ar copies of −1 = −1m and es +copies of 1 = 1m. Thus +{−ar < . . . < es} ⊂ SB(0) = [−ar, es] ∩ Z. +In the second place, for j = 1, . . . , s, choose a positive kj ∈ Z coprime with m! such that +km +j > max{es, ej + ar}. Then, let B(j) be the sequence consisting of km +j − ej copies of +−1 = −1m, ej−1 copies of 1 = 1m, and one copy of km +j . Hence, +{−ar < . . . < es} ⊂ SB(j) = +� +[−(km +j − ej), ej−1] ∪ [ej, km +j + ej−1] +� +∩ Z. +Finally, for j = s + 1, . . . , n, choose a positive kj ∈ Z coprime with m! such that km +j > +max{ar, aj−s + es}. Then, let B(j) be the sequence consisting of km +j − aj−s copies of +1 = 1m, aj−s−1 copies of −1 = −1m, and one copy of −km +j . Hence, +{−ar < . . . < es} ⊂ SB(j) = +� +[−km +j − aj−s−1, −aj−s] ∪ [−aj−s−1, km +j − aj−s−1] +� +∩ Z. +All of the above implies that +{0, d1, . . . , dn} = {−ar < . . . < es} = +n� +i=0 +SB(i). +□ +For A ⊂ Q and λ ∈ Q we write +λA := {λa : a ∈ A}. +Notice that if B(i) is a finite sequence of not necessarily pairwise distinct non-zero rational +numbers, i = 0, . . . , n, for any λ ∈ Q, we have that +λ +� n� +i=0 +SB(i) +� += +n� +i=0 +SλB(i). +Therefore, the following is a direct consequence of Proposition 2.2: +Corollary 2.3. Let d1, d2, . . . , dn be pairwise distinct non-zero rational numbers. Then, +there exist finite sequences B(i), i = 0, . . . , n, of not necessarily pairwise distinct non-zero +rational numbers, such that +{0, d1, . . . , dn} = +n� +i=0 +SB(i). +3. Circle bundles over inflexible 2-manifolds: mapping degree set +This section is devoted to prove Theorem A. As explained in the introduction (see +Remark 1.2) if we want to realize a finite set of integer strictly containing the zero as a +mapping degree set D(M, N), then both M and N need to be inflexible manifolds. We +are going to consider circle bundles over certain inflexible 2-manifolds, with prescribed +Euler class, whose total space is again an inflexible 3-manifold. These 3-manifolds will be +used as building blocks to construct, be means of iterated connected sums, the manifolds +M and N. +We first collect from literature a couple of results that are needed: + +FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS +5 +Lemma 3.1 ([5, Lemma 7.8], [14, Lemma 3.5]). Let M1, M2 and N be oriented closed +connected n-manifolds. Then +D(M1, N) + D(M2, N) ⊆ D(M1#M2, N) +Moreover, if πn−1(N) = 0, then +D(M1, N) + D(M2, N) = D(M1#M2, N). +We reformulate [14, Lemma 4.3] as follows: +Lemma 3.2. Let M and N1, N2 be oriented closed n-manifolds. Then +D(M, N1#N2) ⊆ D(M, N1) ∩ D(M, N2). +Using the previous two lemmas, we prove the following result: +Proposition 3.3. Let M1, M2 and N1, N2 be oriented closed n-manifolds verifying that +πn−1(Nj) = 0, j = 1, 2, and D(Mi, Nj) = {0}, for i ̸= j. Then +D(M1#M2, N1#N2) = D(M1, N1) ∩ D(M2, N2). +Proof. By combining Lemma 3.2 and Lemma 3.1, it follows directly that: +D(M1#M2, N1#N2) ⊆ D(M1#M2, N1) ∩ D(M1#M2, N2) = D(M1, N1) ∩ D(M2, N2). +Conversely, let fi : Mi → Ni, i = 1, 2, be maps both of the same degree d. Without +loss of generality we may assume that fi is cellular, i = 1, 2. +Therefore it induces a +commutative diagram of cofibration sequences +M[n−1] +i +Mi +Sn +N[n−1] +i +Ni +Sn +fi +˜fi +where X[n−1] stands for the (n−1)-skeleton of X, and ˜fi is a pointed map of degree d (the +base points in Sn are the class represented by M[n−1] +i +and N[n−1] +i +). Hence, there exists a +pointed homotopy deforming ˜fi to a pointed map ˜gi such that ˜gi stabilizes the equator +Sn−1 ⊂ Sn and ˜gi|Sn−1 = g for some fixed g : Sn−1 → Sn−1 of degree d. Then, the pointed +homotopy deforming ˜fi can be lifted to Mi and defines gi : Mi → Ni that induces a maps +between disks gi : DMi → DNi such that gi|∂DMi = g, i = 1, 2. +Finally, gluing M1#M2 along ∂DMi, i = 1, 2, and N1#N2 along ∂DNi, i = 1, 2, give +rise to a well defined a map +g1#g2: M1#M2 → N1#N2 +whose degree is precisely d by construction. Therefore +D(M1, N1) ∩ D(M2, N2) ⊆ D(M1#M2, N1#N2) +and we conclude the proof. +□ +A rational version of Proposition 3.3 will be required in order to prove Theorem B. This +will be done in Section 4. Although we will not give the details, previous results (Lemma +3.1 and Lemma 3.2) can be easily generalized to finite iterated connected sums. Hence, +following along the lines of the proof in Proposition 3.3 we obtain: + +6 +C. COSTOYA, V. MU˜NOZ, AND A. VIRUEL +Corollary 3.4. Let Mi, Ni, i = 1, . . . , r, be oriented closed connected n-manifolds such +that πn−1(Ni) = 0, i = 1, . . . , r, and D(Mi, Nj) = {0}, for i ̸= j. Then +D(M1# · · · #Mr, N1# · · · #Nr) = +r� +i=1 +D(Mi, Ni). +We now have all the ingredients to prove our main theorem. +Proof of Theorem A. Let A = {0, d1, . . . , dn} be a finite set of pairwise distinct integers. +We need to show that A is realized by two oriented closed 3-manifolds M, N in the sense +that A = D(M, N). +For this purpose, we consider an oriented closed hyperbolic surface of genus g > 1, Σg. +Then, for any i ∈ Z, let Ki be the total space in the circle bundle +S1 → Ki → Σg +with Euler number e(Ki) = i. Observe that Ki, i ∈ Z, is an aspherical 3-manifold. The +mapping degree set between these 3-manifolds is fully described in [14, Lemma 3.4]: +D(Ki, Kj) = +� +{0, j/i}, +if i|j, +{0}, +if i̸ |j. +(1) +According to Proposition 2.2, for any positive integer m > 0 that we fix, there exist +finite sequences, B(i), i = 0, . . . , n, of not necessarily pairwise distinct non-zero integers, +satisfying that +A = +n� +i=0 +SB(i). +Now, we choose particular pairwise distinct primes q0, q1, . . . , qn fulfilling the condition +qi > max{|b| : b ∈ B(i)}, i = 0, . . . , n, +and we denote +αi = qi +� +b∈B(i) +b, i = 0, . . . , n. +Then, we construct the following “intermediate” manifolds (that will serve us to realize +each of the sums SB(i)), for i = 0, . . . , n: +Mi = +# +b∈B(i) +Kαi/b +Ni = Kαi. +Because Kαi are aspherical 3-manifolds, for i = 0, . . . , n, we have that π2(Kαi) = 0, +and conditions to apply Lemma 3.1 hold. Therefore: +D(Mi, Nj) = D( # +b∈B(i) +Kαi/b, Kαj) = +� +b∈B(i) +D(Kαi/b, Kαj). +Using (1), we then get that, for i = 0, . . . , n, +D(Mi, Ni) = SB(i) , and +D(Mi, Nj) = {0}, for i ̸= j. + +FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS +7 +Finally, we consider the following iterated connected sums: +M = M0#M1# . . . #Mn, +N = N0#N1# . . . #Nn, +for which all the conditions to apply Corollary 3.4 plainly hold. Hence, +D(M, N) = +n� +i=0 +SB(i) = A, +and the proof of Theorem A is complete. +□ +Remark 3.5. We end this section by pointing out that all the 3-manifolds involved in the +previous theorem are inflexible (see also Remark 1.2). It is clear, by (1), that Ki, i ∈ Z, +are inflexible. Now, proceeding along the lines of Theorem A, we apply repeatedly Lemma +3.2 and Lemma 3.1 to get the inflexibility property. On the one hand, we obtain that +D(Mi, Mj) = {0} for i ̸= j, and on the other hand +D(M, M) ⊆ +n� +i=0 +D(Mi, Mi). +Also, by Lemma 3.2, +D(Mi, Mi) = D(Mi, +# +b∈B(i) +Kαi/b) ⊂ +� +b∈B(i) +D(Mi, Kαi/b) +and using Lemma 3.1, +D(Mi, Kαi/b) = D( # +b′∈B(i) +Kαi/b′, Kαi/b) = +� +b′∈B(i) +D(Kαi/b′, Kαi/b). +Now, by Equation (1), D(Kαi/b′, Kαi/b) is either {0} or {0, b′/b} whenever b|b′. Hence, +D(Mi, Kαi/b) is bounded, and so D(Mi, Mi) and D(M, M) are bounded. Hence Mi and +M are inflexible manifolds, i = 1, . . . , n. The same arguments work for N so we conclude. +4. Spherical fibrations over inflexible Sullivan models: +rational mapping degree set +In this section we prove Theorem B, which can be thought of as the rational version +of Theorem A. Rational homotoy theory provides an equivalence of categories between +the category of simply connected rational spaces and the category of certain differential +graded algebras, the so-called Sullivan minimal models. We refer to [8] for basics facts in +Rational Homotopy Theory. +More concretely, if V is a graded rational vector space, we write ΛV for the free com- +mutative graded algebra on V . A Sullivan model (ΛV, ∂) is a commutative differential +graded algebra (cdga for short) which is free as commutative graded algebra on a simply +connected graded vector space V of finite dimension in each degree. It is minimal if in +addition ∂(W) ⊂ Λ≥2W. +Now, if M is an oriented closed simply connected manifold, then the cohomology of +the associated minimal model AM coincides with the rational cohomology of M. +In + +8 +C. COSTOYA, V. MU˜NOZ, AND A. VIRUEL +particular AM has a cohomological fundamental class [AM] ∈ H∗(AM) ∼= H∗(M; Q) +which is isomorphic to the rational cohomological fundamental class [M]Q of M. +Ellipticity for a Sullivan minimal model (ΛV, ∂) means that both V and H∗(ΛV ) are +finite-dimensional. +Hence, the cohomology is a Poincar´e duality algebra [9] and one +can easily compute the degree of its fundamental cohomological class [8, Theorem 32.6]. +In particular one can introduce the notion of mapping degree between elliptic Sullivan +minimal models and also translate the notion of inflexibility: +Let (ΛV, ∂) be an elliptic Sullivan minimal model. Let µ ∈ (ΛV )n be a representative of +its cohomological fundamental class. Then (ΛV, ∂) is inflexible if for every cdga-morphism +ϕ: (ΛV, ∂) → (ΛV, ∂) +we have deg(ϕ) = 0, ±1, where H([µ]) = deg(ϕ)[µ]. +4.1. Rational mapping degree set and connected sums. The following results es- +tablish, under certain restrictions, the relationship between rational mapping degree sets +and connected sums of manifolds: +Lemma 4.1 ([7, Lemma II.2]). Let M1, M2 and N be oriented closed n-manifolds with +πn−1(N(0)) = 0. Then +DQ(M1#M2, N) = DQ(M1, N) + DQ(M2, N). +Proof. Under the same assumptions, in [7, Lemma II.2] is asserted that the following +holds: +D(M1#M2, N) ⊆ DQ(M1, N) + DQ(M2, N). +However, a stronger result is demonstrated in the proof. Namely, +DQ(M1#M2, N) ⊆ DQ(M1, N) + DQ(M2, N). +Hence, it suffices to prove the other inclusion. +To that end, one can apply the same +arguments as in [5, Lemma 7.8]: let q(0) : (M1)(0)#(M2)(0) → (M1)(0) ∨ (M2)(0) denote the +rationalization of the pinching map. Then for any given maps fi : (Mi)(0) → N(0), the +composition +(f1 ∨ f2) ◦ q: (M1)(0)#(M2)(0) → N(0) +has degree deg(f1) + deg(f2) and the result follows. +□ +A precise definition of connected sum in the world of cdga’s: +Definition 4.2. Let Ai, i = 1, 2, be connected cdgas and let ai ∈ Ai, i = 1, 2, be elements +of the same degree. The connected sum of the pairs (Ai, [ai]), i = 1, 2, is the dga +(A1, [a1])#(A2, [a2]) +def +:= (A1 ⊕Q A2)/I , +where A1 ⊕Q A2 +def +:= (A1 ⊕ A2)/Q{(1, −1)}, and I ⊂ A1 ⊕Q A2 is the differential ideal +generated by a1 − a2. + +FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS +9 +Connected sums of cdgas provide rational models for connected sums of oriented mani- +folds. Indeed, for Mi, i = 1, 2 oriented closed simply connected n-manifold, with Sullivan +minimal model AMi, let mi be a representative of the cohomological fundamental class of +AMi, for i = 1, 2. By [5, Theorem 7.12] +(AM1, [m1])#(AM 2, [m2]) +(2) +is a rational model of M1#M2. +We use (2) above to prove the rational version of Proposition 3.3: +Proposition 4.3. Let M1, M2 and N1, N2 be oriented closed simply connected n-manifolds +such that πn−1(Nj) ⊗ Q = 0, j = 1, 2, and DQ(Mi, Nj) = {0}, i ̸= j. Then +DQ(M1#M2, N1#N2) = DQ(M1, N1) ∩ DQ(M2, N2). +Proof. According to Lemma 4.1 and the rational version of Lemma 3.2 (which can be +proved following the same arguments as in [14, Lemma 4.3]), we get that +DQ(M1#M2, N1#N2) ⊂ DQ(M1, N1) ∩ DQ(M2, N2). +Conversely, let (AMi, [mi]) and (ANi, [ni]) be Sullivan minimal models of (Mi, [Mi]) and +(Ni, [Ni]) respectively, i = 1, 2. For +d ∈ DQ(M1, N1) ∩ DQ(M2, N2) +there exists fi : ANi → AMi with fi(ni) = d · mi + αi and where αi is a coboundary, +i = 1, 2. +Because (AM1, [m1 + α1])#(AM2, [m2 + α2]) and (AN1, [n1])#(AN2, [n2]) are +Sullivan minimal models for M1#M2 and N1#N2 respectively, then f1 and f2 give rise to +a well defined cdga-morphism +f1#f2 : (AN1, [n1])#(AN2, [n2]) → (AM1, [m1 + α1])#(AM2, [m2 + α2]) +defined by +(f1#f2)(x) = +� +f1(x), +if x ∈ AN1, +f2(x), +if x ∈ AN2 +and whose degree is deg(f1#f2) = d. +□ +Remark 4.4. The previous result can be generalized to an arbitrary finite iterated con- +nected sum, as in Corollary 3.4. Namely, if Mi, Ni, i = 1, . . . , r, are oriented closed simply +connected n-manifolds such that πn−1(Nj) = 0, j = 1, . . . , r, and DQ(Mi, Nj) = {0}, for +i ̸= j, then +DQ(M1# · · · #Mr, N1# · · · #Nr) = +r� +i=1 +DQ(Mi, Ni). +4.2. Inflexible Sullivan minimal models of inflexible manifolds. Following the +same strategy as in Section 3, we consider spherical fibrations over certain elliptic and +inflexible Sullivan minimal models (Definition 4.5), whose total spaces are the Sullivan +minimal models of inflexible manifolds (see Lemma 4.7). These manifolds will be the +building blocks to construct, by means of iterated connected sums, manifolds that realize +finite sets of rational numbers. + +10 +C. COSTOYA, V. MU˜NOZ, AND A. VIRUEL +Definition 4.5. Let (A, ∂) be an elliptic, inflexible Sullivan minimal model of formal +dimension 2m, m ≥ 1, such that πj(A) = 0 for j ≥ 2m−1. Fix µ ∈ A a representative of +its cohomological fundamental class. Then for any non-zero q ∈ Q, define the following +Sullivan minimal model +(Kq(A), ∂) := (A ⊗ Λ(y2m−1), ∂) +that extends the differential of A by ∂(y2m−1) = qµ. +Remark 4.6. Notice that (Kq(A), ∂) is the total space in the rational S2m−1-fiber sequence: +(Λ(y2m−1), 0) ←− (Kq(A), ∂) ←− (A, ∂), +whose Euler class is q[µ]. +Lemma 4.7. Let (A, ∂) be an elliptic, inflexible Sullivan minimal model of formal di- +mension 2m, m ≥ 1, such that πj(A) = 0 for j ≥ 2m − 1. Fix µ ∈ A a representative +of the fundamental class of A, and let x ∈ A such that ∂(x) = µ2. Then for any non- +zero q ∈ Q, +� +Kq(A), [y2m−1µ − qx] +� +is the Sullivan minimal model of an oriented closed +inflexible (4m − 1)-manifold MKq, with the same connectivity as (A, ∂). +Proof. According to [6, Proposition 3.1], (Kq(A), ∂) is an elliptic Sullivan model of formal +dimension 4m−1 where y2m−1µ−qx is a representative of its cohomological fundamental +class. By [6, Lemma 3.2], (Kq(A), ∂) is an inflexible algebra because (A, ∂) is so. Now, +since its formal dimension is 4m − 1 ≡ 3 mod 4, the obstruction theory of Sullivan [15, +Theorem (13.2)] and Barge [2, Th´eor`eme 1] guarantees that (Kq(A), [y2m−1µ − qx]) is the +Sullivan minimal model of an oriented closed simply-connected manifold MKq. Finally, +by [4, Proposition A.1], MKq and (A, ∂) have the same connectivity. +□ +We compute the rational mapping degree set between the manifolds appearing in the +previous lemma: +Lemma 4.8. For any non-zero q ∈ Q, let MKq be the oriented closed manifold from +Lemma 4.7 whose Sullivan minimal model is (Kq(A), ∂) from Definition 4.5. Then +DQ(MKp, MKq) = {0, q/p}. +Proof. We follow the ideas in [6, Lemma 3.2]. +Let f : (Kq(A, ∂) → (Kp(A), ∂) be a +morphism of non-trivial degree d ∈ Q, that is, +f(y2m−1µ − qx) = d(y2m−1µ − px) + α +(3) +where α is a coboundary. By a degree argument, f induces a non-trivial degree morphism +f|A : (A, ∂) → (A, ∂). On the one hand f(µ) = �dµ + β1 and f(x) = �d 2x + β2 where β1, β2 +are coboundaries, and �d ∈ {−1, 1}. On the other hand, f(y2m−1) = ay2m−1 + γ where +a ∈ Q and γ is a coboundary. +Because f(∂y2m−1) = ∂f(y2m−1), we get that ap = q �d and β1 = 0. Hence a = �d (q/p) +and +f(y2m−1µ − qx) = +� +(�d q/p y2m−1 + γ +� +(�dµ) − q(�d 2x + β2) += (�d 2q/p)(y2m−1µ − px) − qβ2 += (q/p)(y2m−1µ − px) − qβ2 +(recall �d ∈ {−1, 1}). +By comparing this equation to (3), we obtain that d = q/p and the proof is complete. +□ + +FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS +11 +We illustrate the existence of elliptic, inflexible Sullivan minimal models satisfying the +conditions from Definition 4.5 and Lemma 4.7: +Definition 4.9. Let Γ be a connected finite simple graph with more that one vertex, i.e., +|V (Γ)| > 1. Given an integer k ≥ 1, let (Ak(Γ), ∂) be the (30k +17)-connected elliptic and +inflexible Sullivan algebra constructed in [4, Definition 2.1], whose formal dimension is +2m = 540k2+984k +396+|V (Γ)|(360k2+436k +132) and πj +� +Ak(Γ) +� += 0 for j ≥ 2m−1. +Fix µ ∈ Ak(Γ) a representative of the cohomological fundamental class. Then for any +nonz-zero q ∈ Q, define the following Sullivan minimal model +(Kq(Γ, k), ∂) := (Ak(Γ) ⊗ Λ(y2m−1), ∂) +that extends the differential of Ak(Γ) by ∂(y2m−1) = qµ. +Remark 4.10. Because conditions from Lemma 4.7 hold, (Kq(Γ, k), ∂) is a Sullivan model +of an oriented closed (30k + 17)-connected inflexible (4m − 1)-manifold MKq(Γ,k), where +2m = 540k2 + 984k + 396 + |V (Γ)|(360k2 + 436k + 132). +Lemma 4.11. Let Γ1 and Γ2 be connected finite simple graphs with |V (Γ1)| = |V (Γ2)| > 1. +Given a positive integer k ≥ 1, and a non-zero pi ∈ Q, i = 1, 2, consider the manifold +MKpi(Γi,k), i = 1, 2 as in Remark 4.10. Then +DQ(MKp1(Γ1,k), MKp2(Γ2,k)) = +� +{0, p2/p1}, +if +Γ1 ∼= Γ2, +{0}, +otherwise. +Proof. Let (Kpi(Γi, k), ∂) = (Ak(Γi) ⊗ Λ(yi), ∂), introduced in Definition 4.9, be the Sul- +livan model of the manifold MKpi(Γi,k), where ∂(yi) = piµi for µi a representative of the +cohomological fundamental class of Ak(Γi), i = 1, 2. Recall from Lemma 4.7 that for +xi ∈ Ak(Γi) satisfying ∂(xi) = µ2 +i , the element yiµi − pixi is a representative of the +cohomological fundamental class of (Kpi(Γi, k), ∂), i = 1, 2. +With these constructions in mind, we follow the ideas from [6, Lemma 3.2]. Consider +a morphism of non-trivial degree d ∈ Q: +f : (Kp2(Γ2, k), ∂) → (Kp1(Γ1, k), ∂). +Then f(y2µ2 − p2x2) = d(y1µ1 − p1x1) + α with α a coboundary. By a degree argument, +the morphism f induces a non-trivial degree morphism +f|Ak(Γ2): (Ak(Γ2), ∂) → (Ak(Γ1), ∂). +Focusing specifically on this former morphism, the arguments in [4, Lemma 2.12] (see +also [6, Remark 2.8]), show that it is induced by a graph full monomorphism σ: Γ1 → Γ2. +Now, since |V (Γ1)| = |V (Γ2)|, σ is indeed an isomorphism of graphs, and f(µ2) = µ1 + β1 +and f(x2) = x1 + β2 with β1, β2 coboundaries, by [4, Lemma 2.12]. +Finally, by another degree reasoning argument, one obtains that f(y2) = ay1 + γ where +a is a non-zero rational number, and γ is a coboundary. We conclude as in the proof of +Lemma 4.8. +□ + +12 +C. COSTOYA, V. MU˜NOZ, AND A. VIRUEL +4.3. Proof of Theorem B. Let A = {0, d1, . . . , dn} where d1, d2, . . . , dn are pairwise +different non-zero rational numbers. Fix an integer k ≥ 1. According to Corollary 2.3, +there exist finite sequences of not necessarily pairwise distinct non-zero rational numbers +B(i), i = 0, . . . , n, such that +A = +n� +i=0 +SB(i). +Choose Γ0, Γ1, . . . , Γn, pairwise non-isomorphic connected finite simple graphs, such +that |V (Γi)| = |V (Γj)| > 1 for every i, j = 0, . . . , n. According to Remark 4.10, we define +the (30k + 17)-connected manifolds +Mi = +# +b∈B(i) +MKb−1(Γi,k) +Ni = MK1(Γi,k), +for i = 0, . . . , n. By Lemmas 4.11 and the rational version of Lemma 3.2 (which can be +proved following the same arguments as in [14, Lemma 4.3]), we have that +DQ(Mi, Ni) = SB(i) , and +DQ(Mi, Nj) = {0}, for i ̸= j. +Finally, define +M = M0#M1# . . . #Mn, +N = N0#N1# . . . #Nn. +and use Proposition 4.3 (see also Remark 4.4) to get +DQ(M, N) = +N� +i=0 +SB(i) = A. +5. From unstable Adams operations to mapping degree sets +We recall the basics on unstable Adams operations following Jackowski-McCLure- +Oliver’s work [10, 11]. Given a compact connected Lie group G, a self-map f : BG → BG +is called an unstable Adams operation of degree r ≥ 0, if H2i(f; Q) is the multiplication +by ri for each i > 0 [10, p. 183]. For a given simple Lie group G with Weyl group WG, +an unstable Adams operation of degree r > 0 exists if and only if (r, |WG|) = 1, and +moreover, this operation is unique [10, Theorem 2]. In particular, when G = SO(2m − 1) +or G = SO(2m), m > 1, unstable Adams operations of degree r > 0 exist if r and m! +are coprime numbers. In what follows, we denote by ϕr the unstable Adams operation +of degree r > 0 on BSO(2m − 1) and BSO(2m). Notice that since they are unique, then +ϕs ◦ ϕr = ϕrs. +Henceforward, (Σ, [Σ]) is a fixed oriented closed connected 2m-manifold whose ratio- +nalization (Σ(0), [Σ]Q) is inflexible and πj(Σ(0)) = 0 for j ≥ 2m − 1. Let (AΣ, ∂) be a +Sullivan minimal model of Σ. Denote by π: Σ → S2m the map obtained by collapsing the +(2m − 1)-skeleton of Σ. + +FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS +13 +Lemma 5.1. Let X2m ∈ H2m(BSO(2m); Z) be the Euler class of the spherical fiber se- +quence +S2m−1 → BSO(2m − 1) → BSO(2m), +thus X2m is a torsion free integral cohomology class [3, Theorem 1.5, Equation (2.1)], and +S2m is thought of as an oriented closed manifold. There exists ι: S2m → BSO(2m), a tor- +sion free element in π2m(BSO(2m)), and a non-zero integer κ ∈ Z such that H∗(ι; Z)(X2m) = +κ[S2m]. +Proof. Recall that π2m(BSO(2m)) ∼= π2m−1(SO(2m)). By [13, Corollary IV.6.14] (see also +[12, p. 161]), π2m−1(SO(2m)) contains a copy of Z inducing the p-local (thus rational) +splitting SO(2m) ≃(p) SO(2m − 1) × S2m−1 [13, Corollary IV.6.21]. Let ι be a generator +of such a copy of Z in π2m(BSO(2m)). +By construction, H∗(ι; Q) is non-trivial on the Euler class of the rational fiber sequence +S2m−1 +(0) +→ BSO(2m − 1)(0) → BSO(2m)(0), +which is just X2m ⊗Q 1. Therefore, H∗(ι; Z)(X2m) = κ[S2m] for some non-zero κ ∈ Z. +□ +Definition 5.2. Given any integers r > 0, m > 1, with r coprime to m!, we define: +(1) The oriented (4m − 1)-manifold Erm as the total space in the principal spherical +SO(2m)-fiber bundle +S2m−1 +S2m−1 +Erm +BSO(2m − 1) +Σ +BSO(2m), +⌟ +φr +(4) +where φr = ϕr ◦ ι ◦ π. +(2) The oriented (4m−1)–manifold E−rm obtained by reversing the original orientation +on the manifold Erm above introduced. +Remark 5.3. The Euler class of the spherical fiber bundle over Σ given in diagram (4) is +κrm[Σ] by construction. +Recall from the beginning of this section that (Σ, [Σ]) is a fixed oriented closed connected +2m-manifold where (AΣ, ∂) is its Sullivan minimal model. +Lemma 5.4. Let Erm be the manifold introduced in Definition 5.2. A Sullivan mini- +mal model of Erm is Kκrm(AΣ) as given in Definition 4.5. Therefore Erm is rationally +equivalent to MKκrm, the manifold given in Lemma 4.7. +Proof. As it was pointed out in Remark 4.6, Kκrm(AΣ) is a Sullivan minimal model for +the total space in a rational S2m−1-fiber sequence whose Euler class is κrm[Σ]Q. It coin- +cides with the Euler class of the rationalization of the spherical SO(2m)-fiber bundle in +diagram (4). Therefore Kκrm(AΣ) is a Sullivan minimal model for Erm. +□ + +14 +C. COSTOYA, V. MU˜NOZ, AND A. VIRUEL +Lemma 5.5. Let i, j, m be positive integers, m > 1, such that (i, m!) = (j, m!) = 1, and +let Erm, r = i, j, be the (4m − 1)-manifold introduced in Definition 5.2. Then +D(Eim, Ejm) = +� +{0, (j/i)m}, +if i|j, +{0}, +if i̸ |j. +Proof. By Lemma 5.4, the manifolds Erm and MKκrm are rationally equivalent, for every +0 < r ∈ Z. Therefore: +D(Eim, Ejm) ⊂ DQ(Eim, Ejm) ∩ Z = DQ(MKκim, MKκjm) ∩ Z += {0, (j/i)m} ∩ Z (by Lemma 4.8) += +� +{0, (j/i)m}, +if i|j, +{0}, +if i̸ |j. +The proof will be completed if we construct a map f : Eim → Ejm of degree (j/i)m +when i|j. To this end, let us suppose that j = di, d ∈ Z, and recall that unstable Adams +operations satisfy that ϕj = ϕd ◦ ϕi. +Therefore, by construction (see Definition 5.2) +φj = ϕd ◦ φi. +Let f : Eim → Ejm be the map obtained by the universal property of +pullbacks in the following commutative diagram: +Eim +BSO(2m − 1) +Ejm +BSO(2m − 1) +Σ +BSO(2m) +BSO(2m) +f +ϕd +⌟ +φi +ϕd +(5) +Diagram (5) gives rise to a commutative diagram of spherical fiber sequences +S2m−1 +S2m−1 +Eim +Ejm +Σ +Σ, +�f +f +(6) +whose associated Serre spectral sequences (Sss) can be compared via the edge morphisms +given by naturality: the Sss associated to the left (resp. right) side of diagram (6) is fully +determined by the differential +d2m([S2m−1]) = κim[Σ] (resp. d2m([S2m−1]) = κjm[Σ]), +and since by naturality +d2m +� +H∗( �f)([S2m−1]) +� += H∗(IdΣ) +� +d2m([S2m−1]) +� +we obtain that deg( �f) = (j/i)m. + +FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS +15 +Now, the cohomological fundamental class [Eim] (resp. [Ejm]) is represented by the +class [S2m−1]⊗[Σ] in the E2m−1,2m +∞ +-term of the Sss associated to the left (resp. right) fiber +sequence in diagram (6). Hence by naturality +H∗(f)([Ejm]) = H∗( �f)([S2m−1]) ⊗ H∗(IdΣ)([Σ])] += +� +(j/i)m[S2m−1] +� +⊗ [Σ] += (j/i)m[Eim] +and therefore deg(f) = (j/i)m. +□ +Remark 5.6. Notice that manifolds Erm and E−rm differ in just the orientation. Hence, +for any other oriented closed connected (4m − 1)-manifold N, the mapping set degree is +D(E−rm, N) = −D(Erm, N) and D(N, E−rm) = −D(N, Erm). +Proof of Theorem C. Let Σ be an oriented closed k-connected 2m-manifold verifying +that Σ(0) is inflexible and πj(Σ(0)) = 0 for j ≥ 2m − 1. Let A = {0, d1, . . . , dn} where +d1, d2, . . . , dn are pairwise different non-zero integers. +According to Proposition 2.2, there exist finite sequences B(i), i = 0, . . . , n, of not +necessarily pairwise distinct non-zero integers, such that every element in B(i) can be +written as ±rm for 0 < r ∈ Z with (r, m!) = 1, and +A = +n� +i=0 +SB(i). +Choose pairwise distinct prime numbers q0, q1, . . . , qn, in such a way that +qj > max{|b| : b ∈ B(i), i = 0, . . . , n} +and (qj, m!) = 1, for j = 0, . . . , n. Let αi = qm +i +� +b∈B(i) +b, for every i = 0, . . . , n. Notice +that αi and αi/b, b ∈ B(i), are integers that can be written up to a sign as rm for some +positive integer r such that (r, m!) = 1 +Following the notation in Definition 5.2, we define the following (4m − 1)-manifolds +Mi = +# +b∈B(i) +Eαi/b +Ni = Eαi +for i = 0, . . . , n. According to Lemma 5.5 and Lemma 3.1, we deduce that +D(Mi, Ni) = SB(i) , and +D(Mi, Nj) = {0}, for i ̸= j. +Finally, we construct +M = M0#M1# . . . #Mn, +N = N0#N1# . . . #Nn, +and, according to Corollary 3.4, we obtain that +D(M, N) = +N +� +i=0 +SB(i) = A. + +16 +C. COSTOYA, V. MU˜NOZ, AND A. VIRUEL +References +[1] M. Amann. Degrees of self-maps of simply connected manifolds. Int. Math. Res. Not., 2015(18):8545– +8589, 2015. +[2] J. Barge. Structures diff´erentiables sur les types d’homotopie rationnelle simplement connexes. Ann. +Sci. ´Ec. Norm. Sup´er. (4), 9:469–501, 1976. +[3] E. H. j. Brown. The cohomology of BSOnand BOnwith integer coefficients. Proc. Am. Math. Soc., +85:283–288, 1982. +[4] C. Costoya, D. M´endez, and A. Viruel. Homotopically rigid Sullivan algebras and their applications. +In An alpine bouquet of algebraic topology, volume 708 of Contemp. Math., pages 103–121. Amer. +Math. Soc., Providence, RI, 2018. +[5] C. Costoya, V. Mu˜noz, and A. Viruel. On Strongly Inflexible Manifolds. International Mathematics +Research Notices, 03 2022. rnac064. +[6] C. Costoya and A. Viruel. Every finite group is the group of self-homotopy equivalences of an elliptic +space. Acta Math., 213(1):49–62, 2014. +[7] D. Crowley and C. L¨oh. Functorial seminorms on singular homology and (in)flexible manifolds. +Algebr. Geom. Topol., 15(3):1453–1499, 2015. +[8] Y. F´elix, S. Halperin, and J.-C. Thomas. Rational homotopy theory, volume 205 of Springer-Verlag. +Springer-Verlag, New York, 2001. +[9] S. Halperin. Finiteness in the minimal models of sullivan. Trans. Amer. Math. Soc., 230:173–199, +1977. +[10] S. Jackowski, J. E. McClure, and B. Oliver. Homotopy classification of self-maps of BG via G-actions. +I. Ann. Math. (2), 135(1):183–226, 1992. +[11] S. Jackowski, J. E. McClure, and B. Oliver. Homotopy classification of self-maps of BG via G-actions. +II. Ann. Math. (2), 135(2):227–270, 1992. +[12] M. A. Kervaire. Some non-stable homotopy groups of Lie groups. Ill. J. Math., 4:161–169, 1960. +[13] M. Mimura and H. Toda. Topology of Lie groups, I and II. Transl. from the Jap. by Mamoru Mimura +and Hirosi Toda, volume 91 of Transl. Math. Monogr. Providence, RI: American Mathematical +Society, 1991. +[14] C. Neofytidis, S. Wang, and Z. Wang. Realising sets of integers as mapping degree sets. Bull. Lond. +Math. Soc. (to appear), 2022. +[15] D. Sullivan. Infinitesimal computations in topology. Publ. Math., Inst. Hautes ´Etud. Sci., 47:269–331, +1977. +CITIC, Departamento de Computaci´on, Universidade da Coru˜na, 15071-A Coru˜na, Spain. +Email address: cristina.costoya@udc.es +Departamento de ´Algebra, Geometr´ıa y Topolog´ıa, Universidad Complutense de Madrid, +Plaza de las Ciencias, 3, 28040-Madrid, Spain +Email address: vicente.munoz@ucm.es +Departamento de ´Algebra, Geometr´ıa y Topolog´ıa, Universidad de M´alaga, Campus +de Teatinos, s/n, 29071-M´alaga, Spain +Email address: viruel@uma.es + diff --git a/edFST4oBgHgl3EQfFzg4/content/tmp_files/load_file.txt b/edFST4oBgHgl3EQfFzg4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ba050b33dc963d74c2b10be6841aee3f6421e70d --- /dev/null +++ b/edFST4oBgHgl3EQfFzg4/content/tmp_files/load_file.txt @@ -0,0 +1,780 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf,len=779 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='13719v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='GT] 31 Jan 2023 FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS CRISTINA COSTOYA, VICENTE MU˜NOZ, AND ANTONIO VIRUEL Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' In this paper we solve in the positive the question of whether any finite set of integers A, containing the zero, is the mapping degree set between two oriented closed connected manifolds of the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' We extend this question to the rational setting, where an affirmative answer is also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Introduction In this paper, we settle in the positive various questions which have been raised about D(M, N), the set of mapping degrees between two oriented closed connected manifolds M and N of the same dimension: D(M, N) = {d ∈ Z | ∃f : M → N, deg(f) = d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' In [14, Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1], the authors discuss the problem of finding, for any set A ⊂ Z containing the zero, two oriented closed connected manifolds M and N of the same dimension such that A = D(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Note that 0 ∈ A is a necessary condition as the constant map M → N is of degree zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' A quick argument shows that when A is an infinite set, this problem is solved in the negative [14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3]: there are uncountably many infinite subsets A ⊂ Z containing the zero, compared to the countably many mapping degree sets D(M, N) that exist for pairs of oriented closed connected manifolds of the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Hence not every infinite set, containing the zero, is realizable as the mapping degree set of manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Thus, one might ask: Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1 ([14, Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let A be a finite set of integers containing the zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Is A = D(M, N) for some oriented closed manifolds M, N of the same dimension?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' It is important to notice that if {0} ⊊ A = D(M, N), A finite, for some manifolds M and N, then D(M, M) and D(N, N) must both be contained in {0, 1, −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Otherwise, if there exists g : M → M with | deg(g)| > 1, then for any non-zero degree f : M → N (which exists by assumption), the subset {deg(f ◦ gm) | m ∈ N} of D(M, N) is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' This leads to a contradiction as A = D(M, N) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' The same follows for D(N, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' 55M25, 57N65, 55P62, 55R10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Mapping degree sets, inflexible manifold, fiber bundle, unstable Adams operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' The first author was partially supported by MINECO (Spain) grants PID2020-115155GB-I00 and TED2021-131201B-I00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' The second author was partially supported MINECO (Spain) grant PID2020- 118452GB-I00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' The third author was partially supported by MINECO (Spain) grant PID2020-118753GB- I00, and by PAIDI 2020 (Andalusia) grant PROYEXCEL-00827.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' 1 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' COSTOYA, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' MU˜NOZ, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' VIRUEL An oriented closed manifold M satisfying D(M, M) ⊆ {0, 1, −1} is called an inflexible manifold [7, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' This condition is equivalent to asking that D(M, M) is bounded: since it is a multiplicative semi-group, if there exists any ℓ ∈ D(M, M) with |ℓ| > 1, then D(M, M) is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Simply connected inflexible manifolds are rare objects that have appeared quite recently in literature using rational homotopy theory and surgery theory (see [5] for an account on the simply connected inflexible manifolds that are known at present).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Not surprisingly, and in lights of Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2, part of our key constructions will use rational homotopy methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' The main result in this work answers Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1 positively: Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let A be a finite set of integers containing the zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then, A = D(M, N) for some oriented closed connected 3-manifolds M, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' The proof of this theorem will be carried out at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Appealing to [14, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='5], we point out that the 3-dimension of the manifolds is the lowest possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' A second problem related to Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1 is also treated in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' More precisely, let the rational mapping degree set between oriented closed connected n-manifolds M, N be the following set DQ(M, N) = {d ∈ Q | ∃f : (M(0), [M]Q) → (N(0), [N]Q), deg(f) = d}, where [M] ∈ Hn(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Z) denotes the cohomological fundamental class of M, [M]Q ∈ Hn(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Q) denotes the rational cohomological fundamental class of M, and M(0) the rationalization of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then, we raise the following question, which can be thought of as a rational version of [14, Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='4]: Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let A be a finite set of rational numbers containing the zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Is A = DQ(M, N) for some oriented closed connected manifolds M, N of the same dimension?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' In Section 4 we solve this problem in the positive by proving: Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let A be a finite set of rational numbers containing the zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then A = DQ(M, N) for some oriented closed manifolds M, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Moreover, given any integer k ≥ 1, the manifolds M, N above can be chosen (30k + 17)-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' The proofs of Theorem A and Theorem B consist of mainly two steps: Arithmetical decomposition of finite sets: In Section 2 we show how to decompose the candidate A to be realized as the mapping degree set of manifolds, as an in- tersection of sums over specifically designed sequences of integers SBi, i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Each of those sums gradually approaches A (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Spherical fibrations: In Sections 3 and 4, we use certain inflexible manifolds (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' inflexible Sullivan algebras) as the basis of spherical fibrations where the total spaces are also inflexible manifolds (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' inflexible Sullivan algebras).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Relations between connected sums and mapping degree sets (see Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3) allow one to consider iterated connected sums of the total spaces, in a first stage to realize the sums SBi above mentioned, and in a second stage to realize the candidate A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS 3 Looking at the connectivity, while manifolds from Theorem B are simply connected (indeed, they are as highly connected as desired) the ones from Theorem A have non- trivial fundamental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' In Section 5 we will use unstable Adams operations to prove the following results that guarantee that manifolds realizing finite sets of integers can be chosen simply connected: Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Suppose that there exists an oriented closed k-connected 2m-manifold Σ, m > 1, verifying that Σ(0) is inflexible and πj(Σ(0)) = 0 for j ≥ 2m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then any finite set of integers A containing the zero can be realized as A = D(M, N) for some oriented closed k-connected (4m − 1)-manifolds M, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Examples of simply connected manifolds fulfilling the hypotheses of Theorem C can be found in [1, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='8] and [7, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Hence, the following holds: Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Any finite set of integers A containing the zero can be realized as A = D(M, N) for some oriented closed simply connected manifolds M, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Some arithmetic combinatorics In this section we show that every finite set A ⊂ Z (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' ⊂ Q) containing the zero can be expressed as the intersection of sums over certain sequences of integers, that gradually approach A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' The sequences have an additional property (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2) that will be crucial to prove Theorem C in Section 5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Following the notation in [7, Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1], [14, Section 3], given A, B ⊂ Z (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' ⊂ Q) we write A + B := {a + b : a ∈ A, b ∈ B} ⊂ Z (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' ⊂ Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let B be a finite sequence of not necessarily pairwise distinct non-zero integers (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' rational numbers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' We write SB := � b∈B {0, b} ⊂ Z (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' ⊂ Q), and we refer to it as the sum over the sequence B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , dn be pairwise distinct non-zero integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' For any positive integer m ≥ 1, there exist finite sequences B(i), i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n, of not necessarily pairwise distinct non-zero integers, such that {0, d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , dn} = n� i=0 SB(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Moreover, every element in B(i) can be written as a power ±km for some positive integer k coprime to m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='. Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Fix m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Since the construction of B(i), i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n, depends on the sign of the pairwise distinct di ∈ Z, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n, we write them as an ordered sequence {−ar < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' < −a1 < 0 < e1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' < es} where n = r + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' We assume a0 = 0 = e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' COSTOYA, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' MU˜NOZ, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' VIRUEL In the first place, let B(0) be the sequence consisting of ar copies of −1 = −1m and es copies of 1 = 1m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Thus {−ar < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' < es} ⊂ SB(0) = [−ar, es] ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' In the second place, for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , s, choose a positive kj ∈ Z coprime with m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' such that km j > max{es, ej + ar}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then, let B(j) be the sequence consisting of km j − ej copies of −1 = −1m, ej−1 copies of 1 = 1m, and one copy of km j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Hence, {−ar < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' < es} ⊂ SB(j) = � [−(km j − ej), ej−1] ∪ [ej, km j + ej−1] � ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Finally, for j = s + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n, choose a positive kj ∈ Z coprime with m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' such that km j > max{ar, aj−s + es}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then, let B(j) be the sequence consisting of km j − aj−s copies of 1 = 1m, aj−s−1 copies of −1 = −1m, and one copy of −km j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Hence, {−ar < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' < es} ⊂ SB(j) = � [−km j − aj−s−1, −aj−s] ∪ [−aj−s−1, km j − aj−s−1] � ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' All of the above implies that {0, d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , dn} = {−ar < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' < es} = n� i=0 SB(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' □ For A ⊂ Q and λ ∈ Q we write λA := {λa : a ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Notice that if B(i) is a finite sequence of not necessarily pairwise distinct non-zero rational numbers, i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n, for any λ ∈ Q, we have that λ � n� i=0 SB(i) � = n� i=0 SλB(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Therefore, the following is a direct consequence of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , dn be pairwise distinct non-zero rational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then, there exist finite sequences B(i), i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n, of not necessarily pairwise distinct non-zero rational numbers, such that {0, d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , dn} = n� i=0 SB(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Circle bundles over inflexible 2-manifolds: mapping degree set This section is devoted to prove Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' As explained in the introduction (see Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2) if we want to realize a finite set of integer strictly containing the zero as a mapping degree set D(M, N), then both M and N need to be inflexible manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' We are going to consider circle bundles over certain inflexible 2-manifolds, with prescribed Euler class, whose total space is again an inflexible 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' These 3-manifolds will be used as building blocks to construct, be means of iterated connected sums, the manifolds M and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' We first collect from literature a couple of results that are needed: FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS 5 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1 ([5, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='8], [14, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let M1, M2 and N be oriented closed connected n-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then D(M1, N) + D(M2, N) ⊆ D(M1#M2, N) Moreover, if πn−1(N) = 0, then D(M1, N) + D(M2, N) = D(M1#M2, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' We reformulate [14, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3] as follows: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let M and N1, N2 be oriented closed n-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then D(M, N1#N2) ⊆ D(M, N1) ∩ D(M, N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Using the previous two lemmas, we prove the following result: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let M1, M2 and N1, N2 be oriented closed n-manifolds verifying that πn−1(Nj) = 0, j = 1, 2, and D(Mi, Nj) = {0}, for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then D(M1#M2, N1#N2) = D(M1, N1) ∩ D(M2, N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' By combining Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1, it follows directly that: D(M1#M2, N1#N2) ⊆ D(M1#M2, N1) ∩ D(M1#M2, N2) = D(M1, N1) ∩ D(M2, N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Conversely, let fi : Mi → Ni, i = 1, 2, be maps both of the same degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Without loss of generality we may assume that fi is cellular, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Therefore it induces a commutative diagram of cofibration sequences M[n−1] i Mi Sn N[n−1] i Ni Sn fi ˜fi where X[n−1] stands for the (n−1)-skeleton of X, and ˜fi is a pointed map of degree d (the base points in Sn are the class represented by M[n−1] i and N[n−1] i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Hence, there exists a pointed homotopy deforming ˜fi to a pointed map ˜gi such that ˜gi stabilizes the equator Sn−1 ⊂ Sn and ˜gi|Sn−1 = g for some fixed g : Sn−1 → Sn−1 of degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then, the pointed homotopy deforming ˜fi can be lifted to Mi and defines gi : Mi → Ni that induces a maps between disks gi : DMi → DNi such that gi|∂DMi = g, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Finally, gluing M1#M2 along ∂DMi, i = 1, 2, and N1#N2 along ∂DNi, i = 1, 2, give rise to a well defined a map g1#g2: M1#M2 → N1#N2 whose degree is precisely d by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Therefore D(M1, N1) ∩ D(M2, N2) ⊆ D(M1#M2, N1#N2) and we conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' □ A rational version of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3 will be required in order to prove Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' This will be done in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Although we will not give the details, previous results (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2) can be easily generalized to finite iterated connected sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Hence, following along the lines of the proof in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3 we obtain: 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' COSTOYA, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' MU˜NOZ, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' VIRUEL Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let Mi, Ni, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , r, be oriented closed connected n-manifolds such that πn−1(Ni) = 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , r, and D(Mi, Nj) = {0}, for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then D(M1# · · · #Mr, N1# · · · #Nr) = r� i=1 D(Mi, Ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' We now have all the ingredients to prove our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let A = {0, d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , dn} be a finite set of pairwise distinct integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' We need to show that A is realized by two oriented closed 3-manifolds M, N in the sense that A = D(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' For this purpose, we consider an oriented closed hyperbolic surface of genus g > 1, Σg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then, for any i ∈ Z, let Ki be the total space in the circle bundle S1 → Ki → Σg with Euler number e(Ki) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Observe that Ki, i ∈ Z, is an aspherical 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' The mapping degree set between these 3-manifolds is fully described in [14, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='4]: D(Ki, Kj) = � {0, j/i}, if i|j, {0}, if i̸ |j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' (1) According to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2, for any positive integer m > 0 that we fix, there exist finite sequences, B(i), i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n, of not necessarily pairwise distinct non-zero integers, satisfying that A = n� i=0 SB(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Now, we choose particular pairwise distinct primes q0, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , qn fulfilling the condition qi > max{|b| : b ∈ B(i)}, i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n, and we denote αi = qi � b∈B(i) b, i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then, we construct the following “intermediate” manifolds (that will serve us to realize each of the sums SB(i)), for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n: Mi = # b∈B(i) Kαi/b Ni = Kαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Because Kαi are aspherical 3-manifolds, for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n, we have that π2(Kαi) = 0, and conditions to apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Therefore: D(Mi, Nj) = D( # b∈B(i) Kαi/b, Kαj) = � b∈B(i) D(Kαi/b, Kαj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Using (1), we then get that, for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n, D(Mi, Ni) = SB(i) , and D(Mi, Nj) = {0}, for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS 7 Finally, we consider the following iterated connected sums: M = M0#M1# .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' #Mn, N = N0#N1# .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' #Nn, for which all the conditions to apply Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='4 plainly hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Hence, D(M, N) = n� i=0 SB(i) = A, and the proof of Theorem A is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' We end this section by pointing out that all the 3-manifolds involved in the previous theorem are inflexible (see also Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' It is clear, by (1), that Ki, i ∈ Z, are inflexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Now, proceeding along the lines of Theorem A, we apply repeatedly Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1 to get the inflexibility property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' On the one hand, we obtain that D(Mi, Mj) = {0} for i ̸= j, and on the other hand D(M, M) ⊆ n� i=0 D(Mi, Mi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Also, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2, D(Mi, Mi) = D(Mi, # b∈B(i) Kαi/b) ⊂ � b∈B(i) D(Mi, Kαi/b) and using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1, D(Mi, Kαi/b) = D( # b′∈B(i) Kαi/b′, Kαi/b) = � b′∈B(i) D(Kαi/b′, Kαi/b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Now, by Equation (1), D(Kαi/b′, Kαi/b) is either {0} or {0, b′/b} whenever b|b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Hence, D(Mi, Kαi/b) is bounded, and so D(Mi, Mi) and D(M, M) are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Hence Mi and M are inflexible manifolds, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' The same arguments work for N so we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Spherical fibrations over inflexible Sullivan models: rational mapping degree set In this section we prove Theorem B, which can be thought of as the rational version of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Rational homotoy theory provides an equivalence of categories between the category of simply connected rational spaces and the category of certain differential graded algebras, the so-called Sullivan minimal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' We refer to [8] for basics facts in Rational Homotopy Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' More concretely, if V is a graded rational vector space, we write ΛV for the free com- mutative graded algebra on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' A Sullivan model (ΛV, ∂) is a commutative differential graded algebra (cdga for short) which is free as commutative graded algebra on a simply connected graded vector space V of finite dimension in each degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' It is minimal if in addition ∂(W) ⊂ Λ≥2W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Now, if M is an oriented closed simply connected manifold, then the cohomology of the associated minimal model AM coincides with the rational cohomology of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' In 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' COSTOYA, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' MU˜NOZ, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' VIRUEL particular AM has a cohomological fundamental class [AM] ∈ H∗(AM) ∼= H∗(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Q) which is isomorphic to the rational cohomological fundamental class [M]Q of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Ellipticity for a Sullivan minimal model (ΛV, ∂) means that both V and H∗(ΛV ) are finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Hence, the cohomology is a Poincar´e duality algebra [9] and one can easily compute the degree of its fundamental cohomological class [8, Theorem 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' In particular one can introduce the notion of mapping degree between elliptic Sullivan minimal models and also translate the notion of inflexibility: Let (ΛV, ∂) be an elliptic Sullivan minimal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let µ ∈ (ΛV )n be a representative of its cohomological fundamental class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then (ΛV, ∂) is inflexible if for every cdga-morphism ϕ: (ΛV, ∂) → (ΛV, ∂) we have deg(ϕ) = 0, ±1, where H([µ]) = deg(ϕ)[µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Rational mapping degree set and connected sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' The following results es- tablish, under certain restrictions, the relationship between rational mapping degree sets and connected sums of manifolds: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1 ([7, Lemma II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let M1, M2 and N be oriented closed n-manifolds with πn−1(N(0)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then DQ(M1#M2, N) = DQ(M1, N) + DQ(M2, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Under the same assumptions, in [7, Lemma II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2] is asserted that the following holds: D(M1#M2, N) ⊆ DQ(M1, N) + DQ(M2, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' However, a stronger result is demonstrated in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Namely, DQ(M1#M2, N) ⊆ DQ(M1, N) + DQ(M2, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Hence, it suffices to prove the other inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' To that end, one can apply the same arguments as in [5, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='8]: let q(0) : (M1)(0)#(M2)(0) → (M1)(0) ∨ (M2)(0) denote the rationalization of the pinching map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then for any given maps fi : (Mi)(0) → N(0), the composition (f1 ∨ f2) ◦ q: (M1)(0)#(M2)(0) → N(0) has degree deg(f1) + deg(f2) and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' □ A precise definition of connected sum in the world of cdga’s: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let Ai, i = 1, 2, be connected cdgas and let ai ∈ Ai, i = 1, 2, be elements of the same degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' The connected sum of the pairs (Ai, [ai]), i = 1, 2, is the dga (A1, [a1])#(A2, [a2]) def := (A1 ⊕Q A2)/I , where A1 ⊕Q A2 def := (A1 ⊕ A2)/Q{(1, −1)}, and I ⊂ A1 ⊕Q A2 is the differential ideal generated by a1 − a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS 9 Connected sums of cdgas provide rational models for connected sums of oriented mani- folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Indeed, for Mi, i = 1, 2 oriented closed simply connected n-manifold, with Sullivan minimal model AMi, let mi be a representative of the cohomological fundamental class of AMi, for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' By [5, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='12] (AM1, [m1])#(AM 2, [m2]) (2) is a rational model of M1#M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' We use (2) above to prove the rational version of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let M1, M2 and N1, N2 be oriented closed simply connected n-manifolds such that πn−1(Nj) ⊗ Q = 0, j = 1, 2, and DQ(Mi, Nj) = {0}, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then DQ(M1#M2, N1#N2) = DQ(M1, N1) ∩ DQ(M2, N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' According to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1 and the rational version of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2 (which can be proved following the same arguments as in [14, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3]), we get that DQ(M1#M2, N1#N2) ⊂ DQ(M1, N1) ∩ DQ(M2, N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Conversely, let (AMi, [mi]) and (ANi, [ni]) be Sullivan minimal models of (Mi, [Mi]) and (Ni, [Ni]) respectively, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' For d ∈ DQ(M1, N1) ∩ DQ(M2, N2) there exists fi : ANi → AMi with fi(ni) = d · mi + αi and where αi is a coboundary, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Because (AM1, [m1 + α1])#(AM2, [m2 + α2]) and (AN1, [n1])#(AN2, [n2]) are Sullivan minimal models for M1#M2 and N1#N2 respectively, then f1 and f2 give rise to a well defined cdga-morphism f1#f2 : (AN1, [n1])#(AN2, [n2]) → (AM1, [m1 + α1])#(AM2, [m2 + α2]) defined by (f1#f2)(x) = � f1(x), if x ∈ AN1, f2(x), if x ∈ AN2 and whose degree is deg(f1#f2) = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' The previous result can be generalized to an arbitrary finite iterated con- nected sum, as in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Namely, if Mi, Ni, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , r, are oriented closed simply connected n-manifolds such that πn−1(Nj) = 0, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , r, and DQ(Mi, Nj) = {0}, for i ̸= j, then DQ(M1# · · · #Mr, N1# · · · #Nr) = r� i=1 DQ(Mi, Ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Inflexible Sullivan minimal models of inflexible manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Following the same strategy as in Section 3, we consider spherical fibrations over certain elliptic and inflexible Sullivan minimal models (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='5), whose total spaces are the Sullivan minimal models of inflexible manifolds (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' These manifolds will be the building blocks to construct, by means of iterated connected sums, manifolds that realize finite sets of rational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' COSTOYA, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' MU˜NOZ, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' VIRUEL Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let (A, ∂) be an elliptic, inflexible Sullivan minimal model of formal dimension 2m, m ≥ 1, such that πj(A) = 0 for j ≥ 2m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Fix µ ∈ A a representative of its cohomological fundamental class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then for any non-zero q ∈ Q, define the following Sullivan minimal model (Kq(A), ∂) := (A ⊗ Λ(y2m−1), ∂) that extends the differential of A by ∂(y2m−1) = qµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Notice that (Kq(A), ∂) is the total space in the rational S2m−1-fiber sequence: (Λ(y2m−1), 0) ←− (Kq(A), ∂) ←− (A, ∂), whose Euler class is q[µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let (A, ∂) be an elliptic, inflexible Sullivan minimal model of formal di- mension 2m, m ≥ 1, such that πj(A) = 0 for j ≥ 2m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Fix µ ∈ A a representative of the fundamental class of A, and let x ∈ A such that ∂(x) = µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then for any non- zero q ∈ Q, � Kq(A), [y2m−1µ − qx] � is the Sullivan minimal model of an oriented closed inflexible (4m − 1)-manifold MKq, with the same connectivity as (A, ∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' According to [6, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1], (Kq(A), ∂) is an elliptic Sullivan model of formal dimension 4m−1 where y2m−1µ−qx is a representative of its cohomological fundamental class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' By [6, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2], (Kq(A), ∂) is an inflexible algebra because (A, ∂) is so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Now, since its formal dimension is 4m − 1 ≡ 3 mod 4, the obstruction theory of Sullivan [15, Theorem (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2)] and Barge [2, Th´eor`eme 1] guarantees that (Kq(A), [y2m−1µ − qx]) is the Sullivan minimal model of an oriented closed simply-connected manifold MKq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Finally, by [4, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1], MKq and (A, ∂) have the same connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' □ We compute the rational mapping degree set between the manifolds appearing in the previous lemma: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' For any non-zero q ∈ Q, let MKq be the oriented closed manifold from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='7 whose Sullivan minimal model is (Kq(A), ∂) from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then DQ(MKp, MKq) = {0, q/p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' We follow the ideas in [6, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let f : (Kq(A, ∂) → (Kp(A), ∂) be a morphism of non-trivial degree d ∈ Q, that is, f(y2m−1µ − qx) = d(y2m−1µ − px) + α (3) where α is a coboundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' By a degree argument, f induces a non-trivial degree morphism f|A : (A, ∂) → (A, ∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' On the one hand f(µ) = �dµ + β1 and f(x) = �d 2x + β2 where β1, β2 are coboundaries, and �d ∈ {−1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' On the other hand, f(y2m−1) = ay2m−1 + γ where a ∈ Q and γ is a coboundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Because f(∂y2m−1) = ∂f(y2m−1), we get that ap = q �d and β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Hence a = �d (q/p) and f(y2m−1µ − qx) = � (�d q/p y2m−1 + γ � (�dµ) − q(�d 2x + β2) = (�d 2q/p)(y2m−1µ − px) − qβ2 = (q/p)(y2m−1µ − px) − qβ2 (recall �d ∈ {−1, 1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' By comparing this equation to (3), we obtain that d = q/p and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' □ FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS 11 We illustrate the existence of elliptic, inflexible Sullivan minimal models satisfying the conditions from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='5 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='7: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let Γ be a connected finite simple graph with more that one vertex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=', |V (Γ)| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Given an integer k ≥ 1, let (Ak(Γ), ∂) be the (30k +17)-connected elliptic and inflexible Sullivan algebra constructed in [4, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1], whose formal dimension is 2m = 540k2+984k +396+|V (Γ)|(360k2+436k +132) and πj � Ak(Γ) � = 0 for j ≥ 2m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Fix µ ∈ Ak(Γ) a representative of the cohomological fundamental class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then for any nonz-zero q ∈ Q, define the following Sullivan minimal model (Kq(Γ, k), ∂) := (Ak(Γ) ⊗ Λ(y2m−1), ∂) that extends the differential of Ak(Γ) by ∂(y2m−1) = qµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Because conditions from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='7 hold, (Kq(Γ, k), ∂) is a Sullivan model of an oriented closed (30k + 17)-connected inflexible (4m − 1)-manifold MKq(Γ,k), where 2m = 540k2 + 984k + 396 + |V (Γ)|(360k2 + 436k + 132).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let Γ1 and Γ2 be connected finite simple graphs with |V (Γ1)| = |V (Γ2)| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Given a positive integer k ≥ 1, and a non-zero pi ∈ Q, i = 1, 2, consider the manifold MKpi(Γi,k), i = 1, 2 as in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then DQ(MKp1(Γ1,k), MKp2(Γ2,k)) = � {0, p2/p1}, if Γ1 ∼= Γ2, {0}, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let (Kpi(Γi, k), ∂) = (Ak(Γi) ⊗ Λ(yi), ∂), introduced in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='9, be the Sul- livan model of the manifold MKpi(Γi,k), where ∂(yi) = piµi for µi a representative of the cohomological fundamental class of Ak(Γi), i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Recall from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='7 that for xi ∈ Ak(Γi) satisfying ∂(xi) = µ2 i , the element yiµi − pixi is a representative of the cohomological fundamental class of (Kpi(Γi, k), ∂), i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' With these constructions in mind, we follow the ideas from [6, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Consider a morphism of non-trivial degree d ∈ Q: f : (Kp2(Γ2, k), ∂) → (Kp1(Γ1, k), ∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then f(y2µ2 − p2x2) = d(y1µ1 − p1x1) + α with α a coboundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' By a degree argument, the morphism f induces a non-trivial degree morphism f|Ak(Γ2): (Ak(Γ2), ∂) → (Ak(Γ1), ∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Focusing specifically on this former morphism, the arguments in [4, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='12] (see also [6, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='8]), show that it is induced by a graph full monomorphism σ: Γ1 → Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Now, since |V (Γ1)| = |V (Γ2)|, σ is indeed an isomorphism of graphs, and f(µ2) = µ1 + β1 and f(x2) = x1 + β2 with β1, β2 coboundaries, by [4, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Finally, by another degree reasoning argument, one obtains that f(y2) = ay1 + γ where a is a non-zero rational number, and γ is a coboundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' We conclude as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' □ 12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' COSTOYA, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' MU˜NOZ, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' VIRUEL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let A = {0, d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , dn} where d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , dn are pairwise different non-zero rational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Fix an integer k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' According to Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3, there exist finite sequences of not necessarily pairwise distinct non-zero rational numbers B(i), i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n, such that A = n� i=0 SB(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Choose Γ0, Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , Γn, pairwise non-isomorphic connected finite simple graphs, such that |V (Γi)| = |V (Γj)| > 1 for every i, j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' According to Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='10, we define the (30k + 17)-connected manifolds Mi = # b∈B(i) MKb−1(Γi,k) Ni = MK1(Γi,k), for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' By Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='11 and the rational version of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2 (which can be proved following the same arguments as in [14, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3]), we have that DQ(Mi, Ni) = SB(i) , and DQ(Mi, Nj) = {0}, for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Finally, define M = M0#M1# .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' #Mn, N = N0#N1# .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' #Nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' and use Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3 (see also Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='4) to get DQ(M, N) = N� i=0 SB(i) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' From unstable Adams operations to mapping degree sets We recall the basics on unstable Adams operations following Jackowski-McCLure- Oliver’s work [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Given a compact connected Lie group G, a self-map f : BG → BG is called an unstable Adams operation of degree r ≥ 0, if H2i(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Q) is the multiplication by ri for each i > 0 [10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' 183].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' For a given simple Lie group G with Weyl group WG, an unstable Adams operation of degree r > 0 exists if and only if (r, |WG|) = 1, and moreover, this operation is unique [10, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' In particular, when G = SO(2m − 1) or G = SO(2m), m > 1, unstable Adams operations of degree r > 0 exist if r and m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' are coprime numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' In what follows, we denote by ϕr the unstable Adams operation of degree r > 0 on BSO(2m − 1) and BSO(2m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Notice that since they are unique, then ϕs ◦ ϕr = ϕrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Henceforward, (Σ, [Σ]) is a fixed oriented closed connected 2m-manifold whose ratio- nalization (Σ(0), [Σ]Q) is inflexible and πj(Σ(0)) = 0 for j ≥ 2m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let (AΣ, ∂) be a Sullivan minimal model of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Denote by π: Σ → S2m the map obtained by collapsing the (2m − 1)-skeleton of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS 13 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let X2m ∈ H2m(BSO(2m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Z) be the Euler class of the spherical fiber se- quence S2m−1 → BSO(2m − 1) → BSO(2m), thus X2m is a torsion free integral cohomology class [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='5, Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1)], and S2m is thought of as an oriented closed manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' There exists ι: S2m → BSO(2m), a tor- sion free element in π2m(BSO(2m)), and a non-zero integer κ ∈ Z such that H∗(ι;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Z)(X2m) = κ[S2m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Recall that π2m(BSO(2m)) ∼= π2m−1(SO(2m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' By [13, Corollary IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='14] (see also [12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' 161]), π2m−1(SO(2m)) contains a copy of Z inducing the p-local (thus rational) splitting SO(2m) ≃(p) SO(2m − 1) × S2m−1 [13, Corollary IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let ι be a generator of such a copy of Z in π2m(BSO(2m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' By construction, H∗(ι;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Q) is non-trivial on the Euler class of the rational fiber sequence S2m−1 (0) → BSO(2m − 1)(0) → BSO(2m)(0), which is just X2m ⊗Q 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Therefore, H∗(ι;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Z)(X2m) = κ[S2m] for some non-zero κ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' □ Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Given any integers r > 0, m > 1, with r coprime to m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=', we define: (1) The oriented (4m − 1)-manifold Erm as the total space in the principal spherical SO(2m)-fiber bundle S2m−1 S2m−1 Erm BSO(2m − 1) Σ BSO(2m), ⌟ φr (4) where φr = ϕr ◦ ι ◦ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' (2) The oriented (4m−1)–manifold E−rm obtained by reversing the original orientation on the manifold Erm above introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' The Euler class of the spherical fiber bundle over Σ given in diagram (4) is κrm[Σ] by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Recall from the beginning of this section that (Σ, [Σ]) is a fixed oriented closed connected 2m-manifold where (AΣ, ∂) is its Sullivan minimal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let Erm be the manifold introduced in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' A Sullivan mini- mal model of Erm is Kκrm(AΣ) as given in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Therefore Erm is rationally equivalent to MKκrm, the manifold given in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' As it was pointed out in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='6, Kκrm(AΣ) is a Sullivan minimal model for the total space in a rational S2m−1-fiber sequence whose Euler class is κrm[Σ]Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' It coin- cides with the Euler class of the rationalization of the spherical SO(2m)-fiber bundle in diagram (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Therefore Kκrm(AΣ) is a Sullivan minimal model for Erm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' □ 14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' COSTOYA, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' MU˜NOZ, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' VIRUEL Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let i, j, m be positive integers, m > 1, such that (i, m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=') = (j, m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=') = 1, and let Erm, r = i, j, be the (4m − 1)-manifold introduced in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Then D(Eim, Ejm) = � {0, (j/i)m}, if i|j, {0}, if i̸ |j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='4, the manifolds Erm and MKκrm are rationally equivalent, for every 0 < r ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Therefore: D(Eim, Ejm) ⊂ DQ(Eim, Ejm) ∩ Z = DQ(MKκim, MKκjm) ∩ Z = {0, (j/i)m} ∩ Z (by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='8) = � {0, (j/i)m}, if i|j, {0}, if i̸ |j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' The proof will be completed if we construct a map f : Eim → Ejm of degree (j/i)m when i|j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' To this end, let us suppose that j = di, d ∈ Z, and recall that unstable Adams operations satisfy that ϕj = ϕd ◦ ϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Therefore, by construction (see Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2) φj = ϕd ◦ φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let f : Eim → Ejm be the map obtained by the universal property of pullbacks in the following commutative diagram: Eim BSO(2m − 1) Ejm BSO(2m − 1) Σ BSO(2m) BSO(2m) f ϕd ⌟ φi ϕd (5) Diagram (5) gives rise to a commutative diagram of spherical fiber sequences S2m−1 S2m−1 Eim Ejm Σ Σ, �f f (6) whose associated Serre spectral sequences (Sss) can be compared via the edge morphisms given by naturality: the Sss associated to the left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' right) side of diagram (6) is fully determined by the differential d2m([S2m−1]) = κim[Σ] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' d2m([S2m−1]) = κjm[Σ]), and since by naturality d2m � H∗( �f)([S2m−1]) � = H∗(IdΣ) � d2m([S2m−1]) � we obtain that deg( �f) = (j/i)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS 15 Now, the cohomological fundamental class [Eim] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' [Ejm]) is represented by the class [S2m−1]⊗[Σ] in the E2m−1,2m ∞ term of the Sss associated to the left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' right) fiber sequence in diagram (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Hence by naturality H∗(f)([Ejm]) = H∗( �f)([S2m−1]) ⊗ H∗(IdΣ)([Σ])] = � (j/i)m[S2m−1] � ⊗ [Σ] = (j/i)m[Eim] and therefore deg(f) = (j/i)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Notice that manifolds Erm and E−rm differ in just the orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Hence, for any other oriented closed connected (4m − 1)-manifold N, the mapping set degree is D(E−rm, N) = −D(Erm, N) and D(N, E−rm) = −D(N, Erm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Proof of Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let Σ be an oriented closed k-connected 2m-manifold verifying that Σ(0) is inflexible and πj(Σ(0)) = 0 for j ≥ 2m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let A = {0, d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , dn} where d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , dn are pairwise different non-zero integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' According to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2, there exist finite sequences B(i), i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n, of not necessarily pairwise distinct non-zero integers, such that every element in B(i) can be written as ±rm for 0 < r ∈ Z with (r, m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=') = 1, and A = n� i=0 SB(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Choose pairwise distinct prime numbers q0, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , qn, in such a way that qj > max{|b| : b ∈ B(i), i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n} and (qj, m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=') = 1, for j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Let αi = qm i � b∈B(i) b, for every i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Notice that αi and αi/b, b ∈ B(i), are integers that can be written up to a sign as rm for some positive integer r such that (r, m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=') = 1 Following the notation in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='2, we define the following (4m − 1)-manifolds Mi = # b∈B(i) Eαi/b Ni = Eαi for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' According to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='5 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='1, we deduce that D(Mi, Ni) = SB(i) , and D(Mi, Nj) = {0}, for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Finally, we construct M = M0#M1# .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' #Mn, N = N0#N1# .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' #Nn, and, according to Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='4, we obtain that D(M, N) = N � i=0 SB(i) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' 16 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' COSTOYA, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' MU˜NOZ, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' VIRUEL References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Amann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Degrees of self-maps of simply connected manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=', 2015(18):8545– 8589, 2015.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Sup´er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' (4), 9:469–501, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' The cohomology of BSOnand BOnwith integer coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=', 85:283–288, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Costoya, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' M´endez, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Viruel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Homotopically rigid Sullivan algebras and their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' In An alpine bouquet of algebraic topology, volume 708 of Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=', pages 103–121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=', Providence, RI, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Costoya and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Viruel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Every finite group is the group of self-homotopy equivalences of an elliptic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=', 213(1):49–62, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Crowley and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' L¨oh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Functorial seminorms on singular homology and (in)flexible manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Algebr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=', 15(3):1453–1499, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' [8] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' F´elix, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Halperin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Rational homotopy theory, volume 205 of Springer-Verlag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Springer-Verlag, New York, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Halperin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Finiteness in the minimal models of sullivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=', 230:173–199, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Jackowski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' McClure, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Oliver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Homotopy classification of self-maps of BG via G-actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' (2), 135(1):183–226, 1992.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Realising sets of integers as mapping degree sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' (to appear), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' [15] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Sullivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Infinitesimal computations in topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=', Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Hautes ´Etud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=', 47:269–331, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' CITIC, Departamento de Computaci´on, Universidade da Coru˜na, 15071-A Coru˜na, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content=' Email address: cristina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='costoya@udc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='es Departamento de ´Algebra, Geometr´ıa y Topolog´ıa, Universidad Complutense de Madrid, Plaza de las Ciencias, 3, 28040-Madrid, Spain Email address: vicente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='munoz@ucm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='es Departamento de ´Algebra, Geometr´ıa y Topolog´ıa, Universidad de M´alaga, Campus de Teatinos, s/n, 29071-M´alaga, Spain Email address: viruel@uma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} +page_content='es' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFST4oBgHgl3EQfFzg4/content/2301.13719v1.pdf'} diff --git a/f9AzT4oBgHgl3EQfavwP/content/tmp_files/2301.01372v1.pdf.txt b/f9AzT4oBgHgl3EQfavwP/content/tmp_files/2301.01372v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b454fd26c889d6f7f3f2d0e7bb3c3ded1c2bf54 --- /dev/null +++ b/f9AzT4oBgHgl3EQfavwP/content/tmp_files/2301.01372v1.pdf.txt @@ -0,0 +1,2306 @@ +Spatially Varying Anisotropy for Gaussian Random Fields +in Three-Dimensional Space +Martin Outzen Berild∗and Geir-Arne Fuglstad +Department of Mathematical Sciences, +Norwegian University of Science and Technology, Norway +Abstract +Isotropic covariance structures can be unreasonable for phenomena in +three-dimensional spaces. We construct a class of non-stationary anisotropic +Gaussian random fields (GRFs) in three dimensions through stochastic par- +tial differential equations allowing for Gaussian Markov random field approx- +imations. The class is proven in a simulation study where we explore the +amount of data required to estimate these models. Then, we apply it to an +ocean mass outside Trondheim, Norway, based on simulations from a numer- +ical ocean model. And our model outperforms a stationary anisotropic GRF +on predictions using in-situ measurements collected with an autonomous +underwater vehicle. +Keywords: Spatial non-stationarity; spatially-varying anisotropy; stochastic par- +tial differential equations; Gaussian Markov random fields. +∗Corresponding author, martin.o.berild@ntnu.no +1 +arXiv:2301.01372v1 [stat.ME] 3 Jan 2023 + +1 +Introduction +Gaussian random fields (GRFs) are a powerful tool for spatial and spatio-temporal +geostatistical modeling (Diggle et al., 1998; Cressie and Wikle, 2015). When the +key goal is predictions at unobserved locations, i.e., kriging, isotropic covariance +functions often perform well, and more flexible covariance structures should be +used with care (Fuglstad et al., 2015b). However, the screening effect in kriging +(Stein, 2002) is not relevant in other settings where the primary goal is the esti- +mated covariance structure. E.g., to describe internal variability in a climate model +ensemble (Castruccio et al., 2019), or to produce a spatial prior based on numerical +simulations that will later be used to guide autonomous sampling (Fossum et al., +2021; Foss et al., 2021). For the former, Fuglstad and Castruccio (2020); Hu et al. +(2021) demonstrated that flexible covariance structures can perform better than +stationary covariance structures. +There are many approaches to constructing flexible covariance structures (Samp- +son, 2010; Salvaña and Genton, 2021; Schmidt et al., 2011). Some early approaches +are the deformation method (Sampson and Guttorp, 1992) and kernel convolutions +(Paciorek and Schervish, 2006), but they both involve the covariances between any +pair of locations. This means standard implementations are infeasible for large +datasets. There are many ways to overcome such computational issues in spatial +statistics and some are applicable for flexible covariance structures (Heaton et al., +2019). The stochastic partial differential equation (SPDE) approach (Lindgren +et al., 2011) is interesting because it directly gives rise to computationally efficient +models and easily extends to non-stationary covariance models. +However, increasing the degree of flexibility in the covariance structure requires +increasing the number of parameters. The common isotropic Matérn covariance +functions (Stein, 2012) are parametrized through 3 parameters: marginal vari- +ance, range, and smoothness. Flexible models can have 100s or more parameters +2 + +(Fuglstad et al., 2015b). An appealing way to reduce dimensionality is to describe +the covariance structure through covariates (Schmidt et al., 2011; Neto et al., 2014; +Ingebrigtsen et al., 2014, 2015; Risser and Calder, 2015). +The aforementioned works are all considering flexible covariance structures in +two-dimensional space, and while the methods can be extended to three-dimensional +space, the literature is sparse. For example, the SPDE approach has been used +for simple anisotropic covariance structures in the context of fMRI data from the +brain (Sidén et al., 2021), and more complex covariance structures in the context of +astronomy (Lee and Gammie, 2021), though this was two-dimensional space and +time treated as three-dimensional space. However, spatially varying anisotropy +in the SPDE approach (Fuglstad et al., 2015a) has not been extended to three- +dimensional space. +The aim of this paper is to develop a new method for spatially varying anisotropy +in three-dimensional space through the SPDE approach. +A key advantage is +that the formulation as an SPDE guarantees a valid covariance structure, and +the main challenge is how to describe and parametrize non-stationary covariance +structures. Fuglstad et al. (2015a) used one vector field to describe spatially vary- +ing anisotropy, but in three dimensions, two spatially varying orthogonal vector +fields are necessary for full generality. +In a simulation study, we investigate how much data is necessary to recover +parameters for three different model complexities: stationary isotropic, station- +ary anisotropic, and non-stationary anisotropic. We then estimate GRF priors to +encode knowledge about the ocean from a numerical forecast generated by the nu- +merical model SINMOD by SINTEF. A stationary GRF prior and a non-stationary +GRF prior are updated based on in-situ measurements by an autonomous under- +water vehicle (AUV), and we evaluate the predictive ability during a mission in +Trondheimsfjorden, Norway, on May 27, 2021. Improved predictions are key, for +3 + +example, in autonomous sampling of the oceans (Fossum et al., 2019, 2021), but +current approaches in autonomous ocean sampling are limited to stationary GRFs. +In Section 2, we describe how to model anisotropy and non-stationarity in three +dimensions using SPDEs. Then in Section 3, we describe how to perform inference +for the new model in a computationally efficient way. In Section 4, we describe +the simulation study and discuss the results, and continue with the application to +sampling in the ocean in Section 5. We end with a discussion in Section 6. +2 +Constructing SPDEs with spatially varying anisotropy +2.1 +Existing models +The Matérn covariance function on R3 is given by +r(s1, s2) = +σ2 +2ν−1Γ(ν)(κ||s1 − s2||)νKν(κ||s1 − s2||), +s1, s2 ∈ R3, +(1) +where ||·|| is the Euclidean distance in R3, σ > 0 is the marginal standard deviation, +Kν is the modified Bessel function of the second kind and order ν > 0, and κ > 0 +is an inverse spatial scale parameter. As discussed in Lindgren et al. (2011), GRFs +with this covariance function is the stationary solutions of the SPDE +(κ2 − ∇ · ∇)α/2(τu(s)) = W(s), +s ∈ R3, +(2) +where α = ν + 3/2, τ = +√ +8πκ/σ, ∇ · ∇ is the Laplacian, and W is a standard +Gaussian white noise process. +Lindgren et al. (2011) proposed to introduce non-stationarity by allowing κ +and τ to vary in space (Ingebrigtsen et al., 2014, 2015) or by deformations of space +(Hildeman et al., 2021). Fuglstad et al. (2015a,b) consider a version of the SPDE, +where the Laplacian is replaced by an anisotropic Laplacian where the direction +and degree of anisotropy vary spatially. This was further extended to spherical +4 + +geometry in Fuglstad and Castruccio (2020); Hu et al. (2021). However, all of +these works were in two-dimensional base spaces, and only simpler models have +been applied for three-dimensional base spaces (Sidén et al., 2021). +The key idea in Fuglstad et al. (2015a) was to replace ∇ · ∇ by ∇ · H(s)∇, +where H(s) is everywhere a symmetric positive definite 2 × 2 matrix that controls +the strength and direction of anisotropy. The matrix-valued function was specified +as H(s) = γ(s)I2 + v(s)v(s)T, s ∈ R2, where γ(·) is a positive function and v(·) +is a vector field. This allows γ(·) to control the baseline strength of dependence in +all directions, and v(·) to control the strength and direction of additional spatial +dependence. However, the same parametrization in R3 is not sufficiently general +to control anisotropy fully. +2.2 +Stationary anisotropy in R3 +We follow the idea in Fuglstad et al. (2015a) for R2, and change the SPDE in +Equation (2) to +(κ2 − ∇ · H∇)u(s) = W(s), s ∈ R3, +(3) +where ∇·H∇ is an anisotropic Laplacian and the symmetric positive definite 3×3 +matrix H controls the anisotropy. The parameter τ has been dropped since κ and +H together control both marginal variance and correlation. +As shown in Appendix A.1, the resulting marginal variance is +σ2 +m = +1 +8πκ +� +det(H) +(4) +and the covariance function is explicitly known as +r(s1, s2) = +1 +8πκ +� +det(H) +exp +� +−κ||H−1/2(s1 − s2)||) +� +(5) +for s1, s2 ∈ R3. The latter is derived in Appendix A.2. This corresponds to geo- +metric anisotropy in the Matérn covariance function with smoothness ν = 1/2. To +5 + +understand the behavior of the covariance function, it is useful to think about H +in terms of its eigenvalue decomposition. Let ˜v1, ˜v2, and ˜v3 be orthonormal eigen- +vectors corresponding to eigenvalues λ1, λ2 and λ3, respectively. Then Figure 1 +shows an example of the 0.37 level iso-correlation surface that will arise from the +covariance function in Equation (5). The semi-axes of the ellipsoid in the figure +are v1 = (√λ1/κ)˜v1, v2 = (√λ2/κ)˜v2, and v3 = (√λ3/κ)˜v3, which by evaluating +the covariance function with either of these semi-axes will yield the relationship +and the iso-correlation level r(v)/σ2 +m = e−1 ≈ 0.37. +v +v +v +1 +2 +3 +Figure 1: Iso-correlation surface at the ∼0.37 level of Equation (5), where v1, v2, +and v3 are the eigenvectors of H with lengths √λ1/κ, √λ2/κ and √λ3/κ. +We generalize the parametrization described in Section 2.2 and H is decom- +posed as +H = γI3 + vvT + ωωT. +(6) +where v = (vx, vy, vz)T ∈ R3 and w = (ωx, ωy, ωz)T ∈ R3, v ⊥ ω, and γ > 0. +The eigenvalue decomposition of H has eigenvalues λ1 = γ, λ2 = γ + ||v||2 and +6 + +λ3 = γ + ||w||2 with the corresponding eigenvectors v1 = v × ω, v2 = v and +v3 = ω, respectively. We construct ω by a linear combination of two orthogonal +vectors in the plane with v as normal vector. First, let ω1 = (−vy, vx, 0)T, which +satisfies v ⊥ ω1. Second, let ω2 = v × ω1 = (−vzvx, −vzvy, v2 +x + v2 +y)T, which also +satisfies v ⊥ ω2. We parametrize ω through +ω = ρ1 +ω1 +||ω1|| + ρ2 +ω2 +||ω2||, +(7) +where ρ1, ρ2 ∈ R which works whenever vx = vy ̸= 0. An alternative solution is to +use Euler-Rodrigues parametrization (Euler, 1771; Rodrigues, 1840) to obtain both +v and ω; however, in this case, the parameters are less interpretable and the issue +is simply nullified by numerical optimization with appropriate initial parameter +values. +The above parametrization for H uses six parameters, γ, vx, vy, vz, ρ1, and +ρ2, to describe all forms of geometric anisotropy. The parameterization is inter- +pretable: 1) γ controls the isotropic effect, 2) vx, vy, and vz controls one anisotropy +in one direction, and 3) ρ1 and ρ2 controls anisotropy in a second direction orthog- +onal to the first. Lastly, κ simultaneously controls scaling of spatial dependence +equally in all directions, and the variance of the GRF together with the six other +parameters as seen in Equation (4). +2.3 +Spatially varying anisotropy on bounded domain D ⊂ R3 +Non-stationarity and spatially varying anisotropy is achieved by making the coef- +ficients in Equation (3) spatially varying, +(κ(s)2 − ∇ · H(s)∇)u(s) = W(s), +s ∈ R3, +(8) +where κ(·) is a positive function, and H is a spatially varying symmetric positive +definite 3 × 3 matrix. +Heuristically, one can imagine that the SPDE is gluing +7 + +together different local behavior described by ellipsoids, as discussed in Section +2.2, to a valid non-stationary covariance structure. +In practice, we need to limit Equation (8) to a bounded domain to parametrize +the non-stationarity. The SPDE we propose is +(κ(s)2 − ∇ · H(s)∇)u(s) = W(s), +s ∈ D ⊂ R3, +(9) +where D is bounded, and we enforce the boundary condition +(H(s)∇u(s))Tn(s), +s ∈ ∂D, +where n(s) is the outward normal vector of D. This corresponds to no flux through +the boundary. The effect of the boundary conditions is increased marginal variance +on the boundary and increased spatial dependency due to the “reflective” boundary +condition. As discussed in Lindgren et al. (2011); Fuglstad et al. (2015b), one can +extend the domain D outside the area with observations to reduce boundary effects, +or one can consider the boundary effects a feature that the non-stationary model +can adjust for if necessary. +3 +Estimating SPDEs with spatially varying anisotropy +3.1 +Parameterizing the non-stationarity +Before using the SPDE in Equation (8) in inference, we parametrize the non- +stationarity through a finite number of parameters. +This involves expanding +log(κ(·)), log(γ(·)), vx(·), vy(·), vz(·), ρ1(·), and ρ2(·) in basis functions. +The +log-transform is used for κ(·) and γ(·) since they must be positive functions. +Let g : R3 → R denote a generic function that we want to expand in a basis, +and let p > 0 the number of basis functions. +We use basis splines similar to +Fuglstad et al. (2015b), and set +g(s) = f(s)Tαg, +(10) +8 + +where αg ∈ Rp, and f(s) = (f1(s), . . . , fp(s))T is a p-dimensional vector with the +basis functions evaluated at location s. +In this paper, we will use rectangular domains D = [A1, B1]×[A2, B2]×[A3, B3], +and a basis constructed as a tensor product of three one-dimensional B-splines. +This means that p = m3, where m > 0 is the number of basis functions used in each +dimension. We use clamped splines where the derivative is 0 at each boundary, +and the construction of the clamped one-dimensional B-splines is discussed in +Appendix A.3. +Figure 2 shows an example of the resulting basis functions in +1-dimension. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Figure 2: Clamped B-spline basis with three basis functions in 1D. +Let Bx,i denote the i-th basis function of the second-order basis in the x- +dimension, and similarly By,j and Bz,k for the y- and z-dimension. The resulting +tree-dimensional basis is then +fijk (s) = Bx,i(s1) · By,j(s2) · Bz,k(s3), +s = (s1, s2, s3)T ∈ D, +(11) +for all combinations i, j, k ∈ {1, . . . , m}. This means that αg ∈ Rm3, and m3 +parameters must be estimated for each of the seven functions described at the +start of the section. +9 + +Figure 3: Parameterized function representation with B-splines in 3D. +In Sections 4 and 5, we use m = p3 = 33 = 27. For a total of 189 parameters +in the seven functions. When data is sparse, such a model can easily result in +overfitting (Fuglstad et al., 2015b), and it is necessary to introduce penalties on +the seven functions. In Fuglstad et al. (2015b), this was achieved by a hierarchical +model where +τg∆g(s) = Wg(s), +s ∈ D, +together with Neumann boundary conditions of zero derivatives on the boundary +of the domain. +However, this requires selecting a reasonable value for τg > 0 +for each of the seven functions and is computationally expensive if it is done +using cross-validation. However, in the context of this paper, we are constructing +a stochastic model that mimics the behavior of a densely “observed” numerical +simulation model and does not include penalties beyond the restriction of using +27 basis functions. We demonstrate the ability of this model to be estimated in +our context in the simulation study in Section 4, and also investigate the amount +of data needed to estimate the model. +10 + +40 +35 +30 +25 +20 +L +15 +10 +10 +15 +0 +25 +25 +35 +AO3.2 +Hierarchical model and discretization +Consider a bounded domain D ⊂ R3, and observations y = (y1, y2, . . . , yn) made +at locations s1, s2, . . . , sn ∈ D. We assume a Gaussian observation model +yi|η(si), σ2 +N ∼ N(η(si), σ2 +N), +i = 1, . . . , n, +where σ2 +N > 0 is the nugget variance and +η(s) = x(s)Tβ + u(s), +s ∈ D, +describes true spatial variation as a combination of covariates and a GRF. Here +x(·) is a spatially varying vector of k covariates, β ∈ Rk are the coefficients of +the covariates, and u(·) is a GRF with spatially varying anisotropy as presented +in Section 2. +As described in Appendix B, the GRF u(·) is discretized using a regular grid +with l cells, and we get a Gaussian Markov random field w = (w1, . . . , wl)T. Let +θ be the vector of all parameters controlling u(·), then +w|θ ∼ Nl(0, Q−1), +where dependence on θ is suppressed for Q, and Q is a l × l precision matrix +with a three-dimensional spatial sparsity structure. +The vector w is linked to +u(·) through a linear transformation u(s) = a(s)Tw, where a has only one non- +zero entry corresponding to which grid cell location s belongs. This gives u = +(u(s1), . . . , u(sn))T = Aw, where the n × l matrix A only has one non-zero entry +on each row. +The coefficients of the fixed effect, β, is assigned the weak penalty β ∼ +NK(0, V IK) for a fixed V > 0. Thus we can write y as +y = Xβ + Aw + ϵ, +(12) +11 + +where X is the design matrix of covariates, and ϵ ∼ Nn(0, Inσ2 +N) is an n-dimensional +vector of random noise. This gives rise to the hierarchical formulation +y|β, w, σ2 +N ∼ Nn(Xβ + Aw, σ2 +NIn), +β ∼ Nk(0, V Ik), +w|θ ∼ Nl(0, Q−1). +Let s∗ ∈ D be an unobserved location. After parameters ˆθ and ˆ +σ2 +N are esti- +mated, one can predict the underlying value η(s∗) = x(s∗)Tβ + a(s∗)Tw or a new +observation y∗ = x(s∗)Tβ + a(s∗)Tw + ϵ∗, where ϵ∗ ∼ N(0, ˆ +σ2 +N) is a new nugget. +The predictions are made using the conditional distributions η(s∗)|y, θ = ˆθ, σ2 +N = +ˆ +σ2 +N and y∗|y, θ = ˆθ, σ2 +N = ˆ +σ2 +N. The estimation of parameters is detailed in the next +section. +3.3 +Parameter inference +Simplify notation by letting z = (uT, βT)T. Then +z|θ ∼ N(0, Q−1 +z ), where Qz = +� +�Q +0 +0 +V Ik +� +� . +Let S = +� +A +X +� +, then the observation model can be rewritten as +y|z, σ2 +N ∼ Nn(Sz, Inσ2 +N). +(13) +Using this notation the log-likelihood can be expressed as +log π(θ, σ2 +N|y) = Const + log π(θ, σ2 +N) + 1 +2 log det (Qz) − n +2 log(σ2 +N) +− 1 +2 log det (QC) − 1 +2µT +CQCµC − +1 +2σ2 +N +(y − SµC)T(y − SµC). +(14) +Here dependence on θ is suppressed for µC, Qz and QC, and π(θ, σ2 +N) can be used +to assign a penalty on θ, e.g., like the random-walk penalty used in Fuglstad et al. +12 + +(2015b). The conditional precision matrix QC is +QC = Qz + STS/σ2 +N +(15) +and µC is the conditional mean, +µC = Q−1 +C STy/σ2 +N. +(16) +Parameter inference is done by maximizing Equation (14) with respect to θ +and σ2 +N. The parameter vector θ includes all coefficients for the basis functions, +and when using 27 basis functions for each function, +θ = +� +αlog(κ2), αlog γ, αvx, αvy, αvz, αρ1, αρ2 +� +, +has 189 parameters. The parameter space is challenging to search and we use an +analytical expression for the gradient in the optimization algorithm. The deriva- +tion of the analytical gradient involves many nested chain rules and a technique +to calculate a partial inverse of sparse matrices (Rue and Held, 2010), see Ap- +pendix A.5 for a complete description. +4 +Simulation study +In this section, we perform a simulation study to investigate the amount of data +required to acquire reasonable parameter estimates of models with varying com- +plexity that are specified through the SPDE. A comparison of these estimates is +made from simulated data generated from three different parametrizations of the +covariance structures. +The observation model for the different parametrizations is +ymod = Awmod + ϵ, +(17) +13 + +where wmod is the GMRF controlled by the parameters θmod in the respective +models, and ϵ is the independent noise term with mean zero and standard deviation +σN = 0.1 which is identical for all the parametrizations. Furthermore, the models +are discretized on the same domain with a grid of size (M, N, P) = (30, 30, 30) +resulting in a total of 27000 grid nodes where the center of which is our spatial +locations s ∈ D = [A1, B1] × [A2, B2] × [A3, B3] = [0, 40] × [0, 40] × [0, 40]. +The first and simplest model is a Stationary Isotropic (SI) model which has a +covariance structure controlled by the three parameters θSI = (log κ2, log γ, log σ2 +N), +that is assigned to the values κ2 = 0.2, γ = 2.5 and σN = 0.1. The resulting spatial +range is 10.59 with a marginal variance of 0.023. +The second is a Stationary Anisotropic (SA) model composed of the 8 parame- +ters θSA = (log κ2, log γ, vx, vy, vz, ρ1, ρ2, log σ2 +N) set to κ2 = 0.35, γ = 0.5, vx = 1.9, +vy = 1.4, vz = 0.4, ρ1 = 1.4, ρ2 = 0.6 and σN = 0.1. This results in spatial ranges +of 10.08 along the x-dimension, 6.75 along y, and 3.88 along z with a marginal +variance of 0.023. +The parameters of these first two models are simply assigned some reason- +able value; however, the third and most complex model with a non-stationary +anisotropic covariance and a total of 190 parameters, they are much more trou- +blesome to select. Therefore, functions are chosen to assign the parameter val- +ues in θNA throughout the domain D such that the dependency directions im- +itate a vortex. +Using these functions and evaluating them at the spatial lo- +cations in the discretization the parameters of the B-splines, described in Sec- +tion 3.1, are found by optimization. These aforementioned parameters are θNA = +� +αlog(κ2), αlog γ, αvx, αvy, αvz, αρ1, αρ2, log σN +� +with σN = 0.1, and the resulting +covariance structure can be viewed in Figure 4. +We will now examine the extent of data required to fit back the parameters +of the three models described above. First, we simulate multiple datasets from +14 + +(a) Correlation +(b) Marginal Variance +Figure 4: Spatial correlation at location [26,26,20] (a) and variance of the spatial +effect (b) in the non-stationary anisotropic model. +the observation model, Equation (17), with a different number of observed spatial +locations and realizations (replicated observations of these spatial locations). The +number of spatial locations varies between 100, 10000, and 27000 (all), and the +number of realization range between 1, 10, and 100, so nine different combinations +of dataset sizes. Furthermore, we want to perform 100 different trials for each of +these combinations, and thereby have 900 total datasets per model. Also, note +that the observed spatial locations are randomly chosen in each trial. From this, +some statistics can be recovered about the model estimates that can give insight +into the applicability of the different parameterizations. +Table 1 shows the root mean square error (RMSE) between the set parameter +values in each model and their values inferred by the different datasets. This was +obtained using the inference method described in Section 3.3 with the observation +model in Equation (17) for each respective parametrization and trial. The columns +describe the different number of observation locations (No. loc.) and the number +15 + +40 +0.8 +30 +0.6 +- 40 +0.4 +- 30 +20 +X +0.2 +10 +a +0 +y0.09 +40 +0.08 +30 +0.07 +0.06 +20- +0.05 +10 +0.04 +&o - +40 +0.03 +30- +30 +20 +y +20 +x +0.02 +10 +10 +0 +0.01Table 1: The Root Mean Square Error (RMSE) of parameter estimates in the +stationary isotropic, stationary anisotropic, and non-stationary anisotropic model +from 100 independent trials for each combination of dataset sizes; the number of +observed locations (No. loc.) and the number of replicated observations of these +locations (No. real.). +No. loc. +100 +10000 +27000 +No. real. +1 +10 +100 +1 +10 +100 +1 +10 +100 +Stat. Iso. +log κ +0.763 0.168 0.047 0.123 +log γ +0.626 0.164 0.062 0.032 +log τ +2.670 0.674 0.182 0.049 +Stationary Anisotropic +log κ +0.876 +0.195 +0.081 0.094 0.038 +log γ +8.289 +5.601 +0.463 0.228 0.079 +|vx| +1.208 +0.785 +0.440 0.200 0.070 +|vy| +1.040 +0.679 +0.354 0.152 0.035 +|vz| +1.091 +0.498 +0.214 0.075 0.027 +|ρ1| +0.977 +0.801 +0.249 0.129 0.038 +|ρ2| +1.337 +0.489 +0.275 0.078 0.027 +log τ +1.977 +1.352 +0.182 0.189 0.028 +Non-Stationary Anisotropic +log κ +2.572 0.811 0.356 0.269 +log γ +2.615 1.173 0.694 0.585 +|vx| +1.929 0.742 0.531 0.509 +|vy| +2.699 0.668 0.453 0.432 +|vz| +1.591 0.610 0.343 0.296 +|ρ1| +0.144 0.714 0.287 0.210 +|ρ2| +0.420 0.604 0.376 0.344 +log τ +1.152 0.017 0.005 0.005 +16 + +of realizations (No. real.), and the different blocks represent the different models. +The columns highlighted in bold for each respective model are the ones we have +deemed as reasonable parameter estimates. Also, note that some parts of the table +are omitted to simplify the presentation of the results for the reader as the full +table does not affect the conclusion of this study. From Table 1 we observe that the +(simple) stationary models, SI and SA, require very little data. In fact, observing +under 1% of the grid for 10 realizations or more is good enough for the SI and the +SA only requires some more realizations to attain similar parameter accuracy. +On the other hand, the most flexible parameterization, the NA model, requires +much more data and only reaches reasonable parameter accuracy when the whole +grid is observed with 10 or more realizations. Now there is a large discrepancy +between 10000 observed points ( 37%) and 27000 (100%), so it could be interesting +to investigate where in this range reasonable estimates are obtained. However, we +have not chosen to explore this here. We also want to note that these estimates +will change with the complexity of the covariance structure and with the initial +values in the optimization. +5 +GRF prior for statistical sampling of the ocean +5.1 +Aim +Forecasts produced by numerical ocean models describe realistic behavior for the +ocean, but local behavior such as plumes created by freshwater discharge from a +river into the ocean are hard to accurately forecast. However, we can construct +a prior based on the numerical ocean model that informs prior beliefs about the +ocean, which can aid AUVs to more effectively sample the ocean. In this paper, +the goal is to determine the three-dimensional extent of a freshwater plume in the +ocean, and we assume operation time is short enough to justify a purely spatial +17 + +prior that does not assume dynamical changes in time. +There are two steps in our approach. Step 1 is to estimate a stationary GRF +prior and a non-stationary GRF prior based on a simulation from the numerical +ocean model as described in Section 5.2. Step 2 is to combine each of the estimated +priors with an observation model, and evaluate the predictive ability on in-situ +observations from AUV as described in Section 5.3. The GRFs that we estimate +based on the numerical ocean model can be viewed as statistical emulators of the +ocean. +5.2 +The numerical ocean model and the GRF prior +The model training data used in this application is from a forecast produced by +the ocean model SINMOD. Data is provided by SINTEF Ocean which developed +and ran the simulation. SINMOD is a three-dimensional numerical ocean model +based on primitive equations that are solved using finite difference methods on a +regular grid with horizontal cell sizes of 20km×20km and is nested in several steps +down to 32m × 32m. Moreover, it uses z* vertical layers which allow for varying +grid resolutions depending on the depth and help capture the higher variability of +the surface. SINMOD is driven by atmospheric forces, freshwater outflows, and +tides, and it provides numerical simulations of multiple variables such as salinity, +temperature, and currents. +The reader is referred to Slagstad and McClimans +(2005) for a more detailed description of the method. +The area of operation is located in Trondheimsfjorden at Ladehammaren just +outside of Trondheim, Norway, and the operation date, the time measurements are +collected with the AUV, is May 27, 2021, between 10:30 and 14:30. The outlined +area in Figure 5 indicates the operational area which covers 1408m × 1408m in +the horizontal plane. At the southeast side of this field, the Nidelva river flows +into the fjord. +This causes a very dynamic salinity field that is unfeasible to +18 + +Figure 5: The area of operation in Trondheimsfjorden at Ladehammaren just +outside of Trondheim, Norway. The compass shows the cardinal directions relative +to the map. +describe with a stationary covariance model. Therefore, we will use the numerical +simulations from SINMOD to estimate a non-stationary GRF. As demonstrated +in the simulation study, complex covariance structures can reliably be estimated +based on such dense data. +In this application, we will focus on univariate modeling of the salinity and +we choose the fine-scale horizontal grid sizes hx = 32 m hy = 32 m, which in total +gives N = 45 and M = 45 grid nodes for both the numerical and the statistical +model. Moreover, in the vertical plane, we use 1-meter increments between the +depth layers, i.e., hz = 1 m. +To avoid any major effects of the boundaries in +this direction P = 11 depth layers are used resulting in a depth range of 0.5m +19 + +Ostmarkneset +Munkholmen +Korsvika +Ladehammaren +6668 +Trondheim +Traante +6668 +Reina +Brattora +Trondheim sentralstasjon +706 +moen +6692 +706 +Sjobadet +706 +6690 +Skansen +Kuhauoer +6650 +6666 +6650 +6650 +Rosenheto 10.5m. SINMOD outputs zt, t = 0, 1, 2, . . . , 143, which are vectors of salinity +values in all cells in the three-dimensional grid at different time points throughout +the whole May 27, 2021. The timesteps are 10 minutes, and Figure 6 shows five +timesteps from SINMOD for the top six depth layers during the operation. Note +Figure 6: Five timesteps of the dataset simulated with the numerical ocean model +SINMOD for May 27, 2021. The timestamps are displayed over their respective +timesteps. The N-arrow is the cardinal north. +that the varying vertical layers in the numerical model are either with 0.5m or 1m +increments, so the SINMOD simulations don’t require any additional modification +to fit within our statistical model. +We first estimate the model +zt = Φzt−1 + ϵt, +t = 1, . . . , 143, +where Φ is a diagonal matrix of AR(1) coefficients. The diagonal entries of Φ are +estimated with maximum likelihood separately for each spatial location such that +ˆΦii = �143 +t=1 zt,izt−1,i/ �143 +t=1 z2 +t−1,i for i = 1, . . . , NMP, where zt,i is the value in cell +i at time t. We then compute empirical innovations ˆϵt = zt− ˆΦzt−1, t = 1, . . . , 143. +20 + +10:30 +Depth: +0.5 +30 +N +1.5 +25 +20 +2.5 +15 +3.5 +10 +4.5 +5 +5.5 +014:30 +Depth: +0.5 +30 +N +1.5 +25 +20 +2.5 +15 +3.5 +10 +4.5 +5 +5.5 +013:30 +Depth: +0.5 +30 +N +1.5 +25 +20 +2.5 +15 +3.5 +10 +4.5 +5 +5.5 +012:30 +Depth: +0.5 +30 +N +1.5 +25 +20 +2.5 +15 +3.5 +10 +4.5 +5 +5.5 +011:30 +Depth: +0.5 +30 +N +1.5 +25 +20 +2.5 +15 +3.5 +10 +4.5 +5 +5.5 +0These empirical innovations describe the spatial covariance structure for short-term +changes in salinity. +We fit the flexible non-stationary anisotropic model with 190 parameters, ˆθNA = +(αlog κ, αlog γ, αvx, αvy, αvz, αρ1, αρ2, log σ2 +N), and the stationary anisotropic model +with 8 parameters, ˆθSA = (log κ2, log γ, vx, vy, vz, ρ1, ρ2, log σ2 +N), to the assumed in- +dependent realization from a GRF ˆϵ1, . . . .ˆϵ143. Note that there are NMP = 22275 +spatial locations and the 144 empirical innovations cover the whole day of May 27, +2021. Figures 7b show the resulting variance of the spatial effect and Figure 7c +the spatial correlation with location (x, y, z) = (22, 10, 0) of the non-stationary +anisotropic model. The same figures of the stationary anisotropic model can be +found in Appendix C, Figure S3. +(a) SINMOD prior +(b) Marginal Variance +(c) Correlation +Figure 7: +Prior field (a) found from SINMOD simulations, the variance of the +spatial effect (b) and spatial correlation of point [22,10,0] (marked) (c) in the +non-stationary anisotropic model. The N-arrow is the cardinal north. +21 + +Depth: +0.5 +N +0.8 +1.5 +2.5 +0.6 +3.5 +0.4 +4.5 +0.2 +5.5 +0Depth: +0.5 +N +30 +1.5 +25 +2.5 +20 +3.5 +15 +4.5 +10 +5.5 +5Depth: +0.5 +N +0.7 +1.5 +0.6 +0.5 +2.5 +0.4 +3.5 +0.3 +4.5 +0.2 +5.5 +0.1In the next step, we construct the expected value of the GRF using the time +average of the whole day, µ = �143 +t=0 zt/144. The mean is shown in Figure 7a +and shows the overall tendency for freshwater near the river outlet and saltwater +further out in the ocean. We choose the prior +η = µ + e, +(18) +where we combine the fixed mean vector, µ, with a new realization, e, of the +estimated stationary anisotropic model or the non-stationary anisotropic model. +This is a spatial prior on a 32 m × 32 m × 1 m resolution. +5.3 +In-situ data collection and emulator evaluation +In-situ measurements were made with the AUV on May 27, 2021, between 10:30 +and 14:30. The AUV followed 9 pre-planned paths within the area of operation: +two intersects at 0.5m depth one northbound and one north-westbound starting +from the river, two zig-zags in each depth layer (0.5m,2m,5m), and one up and +down pattern in depth ranging from 0.5m to 10.5m moving north-westbound start- +ing from the river. Figure 8 displays the locations of the measurements in the top +5 layers of the field. +The AUV is moving at 1.5 m/s and continuously samples the salinity. This +means that multiple measurements are made within each 32 m × 32 m × 1 m grid +cell. Measurements are represented as yi, i = 1, . . . , nobs, whereby yi is the average +value measured in grid cell i. We combine these measurements with the prior in +Equation (18) using +yi|η, σ2 +N +ind +∼ N(aT +i η, σ2 +meas), +i = 1, . . . , nobs, +η ∼ N(µ, Q−1 +Prior), +where ai selects the correct grid cell, Q−1 +Prior is the estimated precision matrix +for the GMRF, and the Gaussian likelihood with nugget variance σ2 +meas describes +22 + +Figure 8: Measurement locations of the AUV in the top 6 depth layers of the spatial +field on May 27th, 2021, in Trondheimsfjorden at Ladehammaren just outside of +Trondheim, Norway. The N-arrow is the cardinal north. +measurement noise and sub-grid variation. In general, we would estimate σ2 +meas +using a trial run, but in this case, we estimated σ2 +meas using the average empirical +variance over all observed grid cells in the total dataset. Note that we have not +accounted for the uncertainty in the AUVs positions in these models. As the AUV +dive, it loses its GPS signal and only relies on estimated location. +When the +GPS signal is returned a linear interpolation is made to account for drift but no +uncertainty is included. +We evaluated the two priors, or emulators, by randomly ordering the 9 seg- +ments and then sequentially including more and more observations for predicting +the remaining hold-out data. The random permutation of the segments was done +repeatedly to determine the variation in scores over different paths. This scheme +23 + +0.5m +Nevaluates the AUVs’ ability to predict future observations while maintaining the +sequential structure of measurements. +Figure 9 shows that the non-stationary +model provides a better prior for the salinity in the ocean than the stationary +model. The differences are largest when little data is available, which is consistent +with the idea that the prior is most important in this case. The non-stationary +model can leverage knowledge about which areas are most uncertain using the +spatially varying marginal variance and update the prior based on expected simi- +larities from the spatially varying anisotropy. The improvements are seen both in +point predictions through RMSE and in predictive distributions as measured by +CPRS (Gneiting and Raftery, 2007). +6 +Discussion +We extend the class of SPDE-based GRFs introduced in Fuglstad et al. (2015a) +to three-dimensional space by overcoming two key issues: parametrization and +computation. For the former, we developed a specification of spatially varying +anisotropy through a spatially varying baseline isotropic dependence, and two +orthogonal spatially varying vector fields that describe extra dependence. This +allows for an interpretable description of the 3×3 positive definite matrix describing +anisotropy. For the latter, we use a finite volume method to construct a GMRF +that approximates the solution of the SPDE. +The specification of spatially varying marginal variance and spatially varying +anisotropy requires specifying 7 spatially varying real functions. In this paper, +we expand each function with a clamped B-spline basis. If each function uses +P 3 basis functions, this gives in total 7P 3 coefficients. As demonstrated in the +simulation study, an unpenalized estimation of these parameters requires a densely +observed area and multiple realizations. Application of the new models in data- +24 + +Figure 9: The root mean square error (RMSE, top) and the continuous ranked +probability score (CRPS, bottom) of predictions from the stationary anisotropic +(orange) and non-stationary anisotropic models (blue) given different proportions +of observed data (5%, 95%). The error bars are the standard deviations of the +different measures under random permutations of the 9 segments. +sparse situations will require penalties that restrict the regularity of the 7 spatially +varying functions. However, more research is needed to come up with a practical +25 + +2.5 +2.0 +1.0 +0.5 +1.2 +1.0 +CRPS +B'0 +0.6 +0.4 +0.2 +5%10%15%20%25%30%35%40%45%50%55%60%65%70%75%80%85%90%95% +StationaryAnisotropic +Non-stationaryAnisotropicway to determine the appropriate strength of penalization for each of the functions. +While we did not experience any practical issues with the chosen way to de- +scribe the two orthogonal vector fields, the construction has a “gimbal lock” type +issue. If one vector field points exactly along the z-axis, there is no unique choice +for the second vector field. A potential way to avoid this issue is by describing +the orientation of the two orthogonal vector fields through quaternions or Euler- +Rodrigues parameters. +Moving from two-dimensional space to three-dimensional space introduces an +asymptotically higher computation cost as a function of grid size. For a regular +three-dimensional grid with N nodes, the computational cost is O(N 2) compared +to O(N 3/2) in two-dimensional space. This increased computational cost arises +from increased fill-in in the Cholesky factor. However, the application demon- +strates that the use of a grid size of N = 22275 is unproblematic even for real-time +updates on an AUV. +For the predictions of salinity in the Trondheim’s fjord, we see the highest +improvement of the complex GRF prior compared to an isotropic GRF, for sparse +in-situ measurements. As more data is collected, the difference between the models +decreases. This suggests that the key advantage of training the more complex +GRF is to encode prior physical knowledge so that we can more effectively update +knowledge about unobserved locations. Salinity was used as an example, but in +general, the same approach could be used to map other biologically interesting +quantities such as phytoplankton (Fossum et al., 2019). The GRFs developed in +this paper are a step forward in quantifying beliefs about unobserved regions in +the ocean, which is essential for optimal decisions and more effective autonomous +sampling (Fossum et al., 2021). +In future work, it would be interesting to add a dynamic component to the +model to capture physical processes such as diffusion and advection. However, +26 + +this substantially increases computational cost, and it is not clear to which degree +an advection field from a numerical model should be trusted and which boundary +conditions are best in an advection-dominated problem. The new class of GRFs +shows great promise for encoding prior knowledge about a phenomenon in a com- +putationally efficient way. However, overfitting is an important issue, and we must +consider ways to penalize the complexity. In particular, we need to consider ways +to allow flexibility in an area where it is needed such as a river outlet, and restrict +flexibility in areas where we expect stationarity. +Acknowledgments +Berild and Fuglstad are supported by the Research Council of Norway, project +number 305445. The authors are grateful to Ingrid Ellingsen and SINTEF for +providing the simulations from the numerical ocean model SINMOD. +27 + +A. +General properties +A.1 +Marginal Variance +Here, we will derive the expression for the marginal variance in a general sense and +then specify it for three-dimensional spaces with exponential covariance functions. +The SPDE considered in this work is +(κ2 − ∇ · H∇)α/2u(s) = W(s), +(S1) +where s ∈ D ⊆ Rd a spatial location in the domain of dimension d and α = ν +d/2 +where ν > 0 is the smoothness. Any solution of this SPDE is a Matérn field and +let σm > 0 be its marginal standard deviation; then, its covariance function is +r(s1, s2) = +σ2 +m +2ν−1Γ(ν)(κ||H−1/2(s1 − s2)||)νKν(κ||H−1/2(s1 − s2)||). +(S2) +The transfer function of the SPDE is +g(w) = (κ2 + wTHw)−α/2. +Using this and by including the spectral density of standard Gaussian white noise +in Rd is (2π)−d, the spectral density of the solution of the SPDE is +fS(w) = (2π)−d(κ2 + wTHw)−α. +Lastly, to find the marginal variance of the field the integral of the spectral density +is made over Rd as +σ2 +m = +� +Rd fS(w)dw. +Including the change of variables w = κH−1/2z the expression becomes +σ2 +m = (2π)−d +� +Rd(κ2 + κ2zTz)−α det(κH−1/2)dz += (2π)−d +� +Rd κd−2α(1 + zTz)−α det(H)−1/2dz +α=ν+d/2 += +(2π)−dκ−2ν det(H)−1/2 +� +Rd(1 + zTz)−αdz, +(S3) +28 + +which by specifying a exponential covariance in R3 with α = 2, ν = 1/2 and d = 3 +is +σ2 +m = +1 +8πκ +� +det(H) +. +Note that the integral in Equation (S3) is solved by converting to polar coordinates +as +� +R3 +1 +(1 + zTz)2dz = +� π +0 +sin(φ)dφ +� 2π +0 +dθ +� ∞ +0 +ρ2 +(1 + ρ2)2dρ = π2. +A.2 +Covariance function +Evaluating Equation (S2) at ν = 1/2 and including the expression for the marginal +variance the covariance function can be formalized as +r(s1, s2) = +� +2 +π +1 +8πκ +� +det(H) +� +κ||H−1/2(s1 − s2)||K 1 +2(κ||H−1/2(s1 − s2)||). +Then, consider the modified Bessel function of the second kind +Kn(z) = +� π +2z +e−z +(n − 1 +2)! +� ∞ +0 +e−ttn−1/2 +� +1 − t +2z +�n−1/2 +dt, +and evaluate this at order 1/2 gives +K 1 +2(z) = +� π +2ze−z. +The covariance function can then be formalized as +r (s1, s2) = +� +2 +πσ2 +m +� +κ||H−1/2(s1 − s2)|| +× +� +π +2 · κ||H−1/2(s1 − s2)|| exp +� +−κ||H−1/2(s1 − s2)|| +� +=σ2 +m exp +� +−κ||H−1/2(s1 − s2)|| +� +. +(S4) +A.3 +One-dimensional clamped B-splines +We illustrate the construction of 1-dimensional splines B-splines using the interval +[A, B] ∈ R. Let A = t0 < t1 < · · · < tm = B be the knot points. Then the +29 + +zero-order B-splines are constructed recursively as +Bi,0(t) = +� +� +� +� +� +1, +ti ≤ t ≤ ti+1, +0, +otherwise, +, +t ∈ [A, B], +for i = 0, . . . , p − 1. +Let r denote the order of the B-splines. +The first- and +second-order basis splines are constructed as +Bi,r(t) = +t − ti +ti+r − ti +Bi,r−1(t) + +ti+r+1 − t +ti+r+1 − ti+1 +Bi+1,r−1(t), +t ∈ [A, B], +for i = 0, . . . , p − r − 1. +Using the r-order B-spline basis, we construct a function g : [A, B] → R by +g(t) = +p−r−1 +� +i=0 +αiBi,r(t). +where α0, . . . , αp−r−1 ∈ R are coefficients. We use a clamped spline where g′(A) = +g′(B) = 0 and need the additional requirement that α0 = α1 and αp−r−2 = αp−r−1. +A.4 +Integrated likelihood +The distribution of z = (u, β) is given by +z|θ ∼ N(0, Q−1 +z ), +and the observation model is +y|z, θ, σ2 +N ∼ Nn(Sz, Inσ2 +N). +30 + +From this the distribution of z given some observations y is +π(z|θ, σ2 +N, y) ∝ π(z, θ, σ2 +N, y) += π(θ, σ2 +N)π(z|θ)π(y|θ, σ2 +N, z) +∝ exp +� +−1 +2zTQzz − 1 +2(y − Sz)TInσ−2 +N (y − Sz) +� +∝ exp +� +−1 +2 +� +zT � +Qz + σ−2 +N STS +� +z − 2zTSTy · σ−2 +N +�� +∝ exp +� +−1 +2(z − µC)TQC(z − µC) +� +⇓ +z|θ, σ2 +N, y ∼ Nn +� +µC, Q−1 +C +� +Here, QC = Qz +STS·σ−2 +N is the conditional precision matrix and µC = Q−1 +C STy· +σ−2 +N is the conditional mean. +Then, integrating out z from the joint distribution gives +π(θ, σ2 +N, y) = π(θ, z, σ2 +N, y) +π(z|θ, σ2 +N, y) += π(θ, σ2 +N)π(z|θ)π(y|θ, σ2 +N, z) +π(z|θ, σ2 +N, y) +, +where the left-hand side does not depend on z such that it may be evaluated for +any given value. Let us evaluate it for z = µC such that +π(θ, σ2 +N, y) ∝π(θ, σ2 +N)π(z = µC|θ)π(y|θ, σ2 +N, z = µC) +π(z = µC|θ, σ2 +N, y) +∝π(θ)|Qz|1/2|In · σ−2 +N |1/2 +|QC|1/2 +exp +� +−1 +2µT +CQzµC +� +× exp +� +−1 +2(y − SµC)TIn · σ−2 +N (y − SµC) +� +. +The last term π(z|θ, σ2 +N, y) is removed since it is equal to 1. Thereby, conditioning +31 + +on y and taking the log we have the log-likelihood +log(π(θ, σ2 +N|y)) =Constant + log(π(θ, σ2 +N)) + 1 +2 log(det(Qz)) + n +2 log(σ−2 +N ) +− 1 +2 log(det(QC)) − 1 +2µT +CQzµC − +1 +2 · σ2 +N +(y − SµC)T(y − SµC). +(S5) +A.5 +Gradient of the log-likelihood +This section is similar to the derivation of the gradient presented in the supple- +mentary material of Fuglstad et al. (2015b). +log(π(θ, τN|y)) =Constant + log(π(θ, τN)) + 1 +2 log(det(Qz)) + n +2 log(σ−2 +N ) +− 1 +2 log(det(QC)) + 1 +2µT +CQCµC − τN +2 yTy. +Note that the last two terms are rewritten for simplicity in the gradient calculation +and that the variance of the Gaussian noise term, σ2 +N is re-parametrized with its +inverse τN = 1/σ2 +N (precision). Derivatives of the log-likelihood are taken with +respect to θi, the elements of θ, and the precision on log scale as log(τN). +The first term is a constant and therefore its derivative is zero with respect to +any of the parameters. The next term, the penalty or the prior of the parameters, +is not used in this paper and otherwise depends on the choice of penalty so gradient +calculation is not specified for this term. +To continue note the derivatives of the precision matrix +∂QC +∂θi += ∂Qz +∂θi +and +∂QC +∂ log(τN) = STSτN, +which is used in the following derivations. First, the derivatives with respect to θi +are considered. The derivative of the log determinant terms are +∂ +∂θi +(log(det(Q)) − log(det(QC))) =Tr +� +Q−1∂Q +∂θi +� +− Tr +� +Q−1 +C +∂Q +∂θi +� +=Tr +� +(Q−1 − Q−1 +C )∂Q +∂θi +� +, +32 + +and the derivative of the quadratic terms are +∂ +∂θi +�1 +2yTyτN + 1 +2µT +CQCµC +� += ∂ +∂θi +�1 +2µT +CQCµC +� += − 1 +2yTτNSQ−1 +C +�∂QC +∂θi +� +Q−1 +C STτNy += − 1 +2µT +C +�∂Q +∂θi +� +µC. +Then, combining these the derivative of the log-likelihood with respect to θi is +∂ +∂θi +log(π(θ, τN|y)) = ∂ +∂θi +log(π(θ, τN))+Tr +� +(Q−1 − Q−1 +C )∂Q +∂θi +� +−1 +2µT +C +�∂Q +∂θi +� +µC +Next, the derivative with respect to the log precision, log τN, is considered. +The derivative of the log determinant terms are +∂ +∂ log(τN) +�n +2 log(τN) − 1 +2 log(det(QC)) +� +=n +2 − 1 +2Tr +� +Q−1 +C +∂ +∂ log(τN)QC +� +=n +2 − 1 +2Tr +� +Q−1 +C STS · τN +� +Further, the derivative of 1/2yTy · τN with respect to log(τN) is just the same +expression so the remaining quadratic term becomes +∂ 1 +2µT +CQCµC +∂ log(τN) +=∂ 1 +2yTτNSQ−1 +C STτNy +∂ log(τN) +=yTτNSQ−1 +C ST +∂τN +∂ log(τN)y − 1 +2yTτNSQ−1 +C +∂QC +∂ log(τN)Q−1 +C STτNy +=µT +CSTτNy − 1 +2µT +CSTSµCτN, +and then, by adding the last quadratic term, the expression simplifies to +−1/2yTy · τN + µT +CSTy · τN − 1 +2µT +CSTSµC · τN = −1 +2(y − SµC)T(y − SµC) · τN. +Finally, combining all these terms we have the derivative of the log-likelihood with +respect to log(τN): +∂ log(π(θ, τN|y)) +∂ log(τN)) +=∂ log(π(θ, τN) +∂ log(τN) ++ n +2 − 1 +2Tr +� +Q−1 +C STS · τN +� +− 1 +2(y − SµC)T(y − SµC) · τN +33 + +Note that the derivative of QC can be calculated quickly and it is derived from +a series of chain rules; first on QC, then on A and AH, and finally within H. The +most computationally heavy calculation in the gradient of the log-likelihood is to +calculate the inverses in the difference Q−1 − Q−1 +C . However, since this term is +multiplied with the derivative of Q with respect to θi, which carries the non-zero +structure of Q, only elements of Q−1 and Q−1 +C which correspond to the non-zero +structure of Q need to be calculated. This is done by calculating a partial inverse +of two matrices as described in Rue and Held (2010). +B. +Derivation +B.1 +Discretization +To find the local solution of the SPDE the domain D = [A1, B1]×[A2, B2]×[A3, B3] +is divided into equally sized rectangular cubes or cells. We use M cells to divide +[A1, B1] in the x-direction, N cells on [A2, B1] in y-direction and P cells on [A3, B3] +in z-direction. The cells have sides parallel to each axis of size hx = (B1 − A1)/M, +hy = (B2 − A2)/N, and hz = (B3 − A3)/P. The cells are assigned an index with +regards to their cell number along each axes starting from number 0; i ∈ [0, M] +along x, j ∈ [0, N] along y, and k ∈ [0, P] along z. For a specific cell, its domain +can be denoted as +Ei,j,k = [ihx, (i + 1)hx] × [jhy, (j + 1)hy] × [khz, (k + 1)hz], +and Figure S1 shows this cell and its closest neighbors. Furthermore, as a regular +grid is employed the volume of a cell is V = hxhyhz. +To further define the local solution of the SPDE we denote the faces of a grid +cell as σF +i,j,k (front), σB +i,j,k (back), σL +i,j,k (left), σR +i,j,k (right), σU +i,j,k (up) and σD +i,j,k +(down) with their respective face centers si,j−1/2,k, si,j+1/2,k, si−1/2,j,k, si+1/2,j,k, +34 + +Figure S1: One cell Ei,j,k in the discretization with its closest neighbours; Ei+1,j,k, +Ei−1,j,k, Ei,j+1,k, Ei,j−1,k, Ei,j,k+1, and Ei,j,k−1. +si,j,k+1/2 and si,j,k−1/2. Figure S2 describes the different faces of a cell. +B.2 +Local solution of the SPDE +Note that this description is an extension to three dimensions of the derivation +described in Fuglstad et al. (2015a), and the reader is referred to there for fur- +ther details. To locally solve the SPDE a finite volume scheme is derived. First, +35 + +I +1Figure S2: One cell Ei,j,k of the discretization with all its faces; σF +i,j,k (front), +σB +i,j,k (back), σL +i,j,k (left), σR +i,j,k (right), σU +i,j,k (up), and σD +i,j,k (down) each with its +respective face centres. +Equation (S1) is integrated over a cell Ei,j,k as +� +Eijk +κ2(s)ds − +� +Eijk +∇ · H(s)∇u(s)ds = +� +Eijk +W(s)ds, +(S6) +where ds is a volume element. The integral of the Gaussian white noise on the +right-hand side is a Gaussian variable with mean zero and variance equal to the +volume of a cell which is independent of neighboring cells. Let zijk be an standard +Gaussian variable; then, Equation (S6) becomes +� +Eijk +κ2(s)ds − +� +Eijk +∇ · H(s)∇u(s)ds = +√ +V zijk. +36 + +Then, applying the divergence theorem to the second integral with the divergence +operator gives +� +Eijk +κ2(s)ds − +� +∂Eijk +(H(s)∇u(s))Tn(s)dσ = +√ +V zijk. +The first integral is approximated by letting k2 +ijk be the average value of the con- +tinuous function κ2(s) within a cell, i.e. κ2 +ijk = 1/V +� +Eijk κ2(s)ds, resulting in +V κ2 +ijkuijk − +� +∂Eijk +(H(s)∇u(s))Tn(s)dσ = +√ +V zijk. +(S7) +To describe the solution of the second integral it is divided into integrals over +each surface as +� +∂Eijk +(H(s)∇u(s))Tn(s)dσ = W L +ijk + W R +ijk + W B +ijk + W F +ijk + W U +ijk + W D +ijk, +(S8) +or W dir +ijk = +� +σdir +ijk(H(s)∇u(s))Tn(s)dσ, where dir denotes the surface; R (posi- +tive x-direction), L (negative x-direction), B (positive y-direction), F (negative +y-direction), U (positive z-direction), and D (negative z-direction). Now, an ap- +proximation of this surface integral over each face is required. It is assumed that +the gradient of u(s) is constant over each face and equal to the value at the center +of each face. The resulting scheme for the gradient on each face is described in +Table S1. +Furthermore, let H be approximated by its value at the center of the +face, and then, we have the approximation +W dir +ijk = +� +σdir +ijk +∇u(s)TH(s)n(s)dσ +≈∇u(cdir +ijk)TH(cdir +ijk)n(cdir +ijk) +� +σdir +ijk +dσ +=∇u(cdir +ijk)TH(cdir +ijk)n(cdir +ijk)A(σdir +ijk), +(S9) +where cdir +ijk is the center of face dir in the cell Eijk, and A(σdir +ijk) is the area of the +face. Combining Equation (S9) with the scheme of ∇u(cdir +ijk) from Table S1, and +37 + +Face +Scheme +σR +i,j,k +∂ +∂xu(si+1/2,j,k) ≃ +1 +hx (u(si+1,j,k) − u(si,j,k)) +∂ +∂yu(si+1/2,j,k) ≃ +1 +4hy (u(si+1,j+1,k) + u(si,j+1,k) − u(si+1,j−1,k) − u(si,j−1,k)) +∂ +∂zu(si+1/2,j,k) ≃ +1 +4hz (u(si+1,j,k+1) + u(si,j,k+1) − u(si+1,j,k−1) − u(si,j,k−1)) +σL +i,j,k +∂ +∂xu(si−1/2,j,k) ≃ +1 +hx (u(si,j,k) − u(si−1,j,k)) +∂ +∂yu(si−1/2,j,k) ≃ +1 +4hy (u(si,j+1,k) + u(si−1,j+1,k) − u(si,j−1,k) − u(si−1,j−1,k)) +∂ +∂zu(si−1/2,j,k) ≃ +1 +4hz (u(si,j,k+1) + u(si−1,j,k+1) − u(si,j,k−1) − u(si−1,j,k−1)) +σB +i,j,k +∂ +∂xu(si,j+1/2,k) ≃ +1 +4hx (u(si+1,j+1,k) + u(si+1,j,k) − u(si−1,j+1,k) − u(si−1,j,k)) +∂ +∂yu(si,j+1/2,k) ≃ +1 +hy (u(si,j+1,k) − u(si,j,k)) +∂ +∂zu(si,j+1/2,k) ≃ +1 +4hz (u(si,j+1,k+1) + u(si,j,k+1) − u(si,j+1,k−1) − u(si,j,k−1)) +σF +i,j,k +∂ +∂xu(si,j−1/2,k) ≃ +1 +4hx (u(si+1,j,k) + u(si+1,j−1,k) − u(si−1,j,k) − u(si−1,j−1,k)) +∂ +∂yu(si,j−1/2,k) ≃ +1 +hy (u(si,j,k) − u(si,j−1,k)) +∂ +∂zu(si,j−1/2,k) ≃ +1 +4hz (u(si,j,k+1) + u(si,j−1,k+1) − u(si,j,k−1) − u(si,j−1,k−1)) +σU +i,j,k +∂ +∂xu(si,j,k+1/2) ≃ +1 +4hz (u(si+1,j,k+1) + u(si+1,j,k) − u(si−1,j,k+1) − u(si−1,j,k)) +∂ +∂yu(si,j,k+1/2) ≃ +1 +4hy (u(si,j+1,k+1) + u(si,j+1,k) − u(si,j−1,k+1) − u(si,j−1,k)) +∂ +∂zu(si,j,k+1/2) ≃ +1 +hx (u(si,j,k+1) − u(si,j,k)) +σD +i,j,k +∂ +∂xu(si,j,k−1/2) ≃ +1 +4hz (u(si+1,j,k) + u(si+1,j,k−1) − u(si−1,j,k) − u(si−1,j,k−1)) +∂ +∂yu(si,j,k−1/2) ≃ +1 +4hy (u(si,j+1,k) + u(si,j+1,k−1) − u(si,j−1,k) − u(si,j−1,k−1)) +∂ +∂zu(si,j,k−1/2) ≃ +1 +hx (u(si,j,k) − u(si,j,k−1)) +Table S1: Numerical scheme of the partial derivative with respect to x, y and z of +uijk on the different faces of cell Eijk. +denoting the components of H as +H(s) = +� +���� +H11(s) +H12(s) +H13(s) +H21(s) +H22(s) +H23(s) +H31(s) +H32(s) +H33(s) +� +���� +38 + +the approximations for each face become +ˆW R +i,j,k = +hyhz +� +H11(si+1/2,j,k)u(si+1,j,k) − u(si,j,k) +hx +� ++ +hyhz +� +H21(si+1/2,j,k)u(si+1,j+1,k) + u(si,j+1,k) − u(si+1,j−1,k) − u(si,j−1,k) +4hy +� ++ +hyhz +� +H31(si+1/2,j,k)u(si+1,j,k+1) + u(si,j,k+1) − u(si+1,j,k−1) − u(si,j,k−1) +4hz +� +, +ˆW L +i,j,k = +hyhz +� +H11(si−1/2,j,k)u(si−1,j,k) − u(si,j,k) +hx +� ++ +hyhz +� +H21(si−1/2,j,k)u(si,j−1,k) + u(si−1,j−1,k) − u(si,j+1,k) − u(si−1,j+1,k) +4hy +� ++ +hyhz +� +H31(si−1/2,j,k)u(si,j,k−1) + u(si−1,j,k−1) − u(si,j,k+1) − u(si−1,j,k+1) +4hz +� +, +ˆW B +i,j,k = +hxhz +� +H12(si,j+1/2,k)u(si+1,j+1,k) + u(si+1,j,k) − u(si−1,j+1,k) − u(si−1,j,k) +4hx +� ++ +hxhz +� +H22(si,j+1/2,k)u(si,j+1,k) − u(si,j,k) +hy +� ++ +hxhz +� +H32(si,j+1/2,k)u(si,j+1,k+1) + u(si,j,k+1) − u(si,j+1,k−1) − u(si,j,k−1) +4hz +� +, +ˆW F +i,j,k = +hxhz +� +H12(si,j−1/2,k)u(si−1,j,k) + u(si−1,j−1,k) − u(si+1,j,k) − u(si+1,j−1,k) +4hx +� ++ +hxhz +� +H22(si,j−1/2,k)u(si,j−1,k) − u(si,j,k) +hy +� ++ +hxhz +� +H32(si,j−1/2,k)u(si,j,k−1) + u(si,j−1,k−1) − u(si,j,k+1) − u(si,j−1,k+1) +4hz +� +, +39 + +ˆW U +i,j,k = +hxhy +� +H13(si,j,k+1/2)u(si+1,j,k+1) + u(si+1,j,k) − u(si−1,j,k+1) − u(si−1,j,k) +4hx +� ++ +hxhy +� +H23(si,j,k+1/2)u(si,j+1,k+1) + u(si,j+1,k) − u(si,j−1,k+1) − u(si,j−1,k) +4hy +� ++ +hxhy +� +H33(si,j,k+1/2)u(si,j,k+1) − u(si,j,k) +hz +� +, +ˆW D +i,j,k = +hxhy +� +H13(si,j,k−1/2)u(si−1,j,k) + u(si−1,j,k−1) − u(si+1,j,k) − u(si+1,j,k−1) +4hx +� ++ +hxhy +� +H23(si,j,k−1/2)u(si,j−1,k) + u(si,j−1,k−1) − u(si,j+1,k) − u(si,j+1,k−1) +4hy +� ++ +hxhy +� +H33(si,j,k−1/2)u(si,j,k−1) − u(si,j,k) +hz +� +, +ˆW T +i,j,k = +hxhy +� +H13(si,j,k+1/2)u(si+1,j,k+1) + u(si+1,j,k) − u(si−1,j,k+1) − u(si−1,j,k) +4hx +� ++ +hxhy +� +H23(si,j,k+1/2)u(si,j+1,k+1) + u(si,j+1,k) − u(si,j−1,k+1) − u(si,j−1,k) +4hy +� ++ +hxhy +� +H33(si,j,k+1/2)u(si,j,k+1) − u(si,j,k) +hz +� +, +ˆW B +i,j,k = +hxhy +� +H13(si,j,k−1/2)u(si−1,j,k) + u(si−1,j,k−1) − u(si+1,j,k) − u(si+1,j,k−1) +4hx +� ++ +hxhy +� +H23(si,j,k−1/2)u(si,j−1,k) + u(si,j−1,k−1) − u(si,j+1,k) − u(si,j+1,k−1) +4hy +� ++ +hxhy +� +H33(si,j,k−1/2)u(si,j,k−1) − u(si,j,k) +hz +� +. +Next, a vectorization of the discretization is made; first moving along the z- +direction, then along x-direction, and lastly along the y-direction. Let us denote +this with the common index l = j · M · P + i · P + k so sijk = sj·M·P+i·P+k = sl +which gives u(sijk) = ul and κ2(sijk) = κ2 +l , and let the last index be L = (N − +40 + +1)MP + (M − 1)P + P − 1. Further, the vectorization results in the linear system +of equations +(DV Dκ2 − AH)u = D1/2 +V z, +(S10) +where DV = V · IMNP, Dκ2 = [κ2 +0, . . . , κ2 +l , . . . , κ2 +L] IMNP, and z ∼ N(0, IMNP). +For simplicity the indices of the neighbors are denoted kp = k + 1, kn = k − 1, +jp = j + 1, jn = j − 1, ip = i + 1, and in = i − 1. The development of AH is done +by the sum ˆW L +ijk + ˆW R +ijk + ˆW B +ijk + ˆW F +ijk + ˆW U +ijk + ˆW D +ijk and accounting for the index +in uijk to form the linear relationship. In the following, non-zero elements of the +(jMN + iP + k)-th row of AH are formalized, and the index in (AH)_ denotes +the column being assigned. The resulting coefficient with the point itself is +(AH)j·M·P+i·P+k = − hyhz +hx +� +H11(si+1/2,j,k) + H11(si−1/2,j,k) +� +− hxhz +hy +� +H22(si,j+1/2,k) + H22(si,j−1/2,k) +� +− hxhy +hz +� +H33(si,j,k+1/2) + H22(si,j,k−1/2) +� +, +with the six closest neighbors are +(AH)j·M·P+i·P+kp =hxhy +hz +H33(si,j,k+1/2) ++ hy +4 +� +H31(si+1/2,j,k) − H31(si−1/2,j,k) +� ++ hx +4 +� +H32(si,j+1/2,k) − H32(si,j−1/2,k) +� +(AH)j·M·P+i·P+kn =hxhy +hz +H33(si,j,k−1/2) +− hy +4 +� +H31(si+1/2,j,k) − H31(si−1/2,j,k) +� +− hx +4 +� +H32(si,j+1/2,k) − H32(si,j−1/2,k) +� +(AH)j·M·P+ip·P+k =hzhy +hx +H11(si+1/2,j,k) ++ hy +4 +� +H12(si,j,k+1/2) − H12(si,j,k−1/2) +� ++ hz +4 +� +H13(si,j+1/2,k) − H13(si,j−1/2,k) +� +41 + +(AH)j·M·P+in·P+k =hzhy +hx +H11(si−1/2,j,k) +− hy +4 +� +H12(si,j,k+1/2) − H12(si,j,k−1/2) +� +− hz +4 +� +H13(si,j+1/2,k) − H13(si,j−1/2,k) +� +(AH)jp·M·P+i·P+k =hxhz +hy +H22(si,j+1/2,k) ++ hx +4 +� +H23(si,j,k+1/2) − H23(si,j,k−1/2) +� ++ hz +4 +� +H21(si+1/2,j,k) − H21(si−1/2,j,k) +� +(AH)jn·M·P+i·P+k =hxhz +hy +H22(si,j−1/2,k) +− hx +4 +� +H23(si,j,k+1/2) − H23(si,j,k−1/2) +� +− hz +4 +� +H21(si+1/2,j,k) − H21(si−1/2,j,k) +� +, +and with the twelve closest diagonals are +(AH)j·M·P+ip·P+kp = hy +4 +� +H31(si+1/2,j,k) + H13(si,j,k+1/2) +� +, +(AH)j·M·P+in·P+kn = hy +4 +� +H31(si−1/2,j,k) + H13(si,j,k−1/2) +� +, +(AH)j·M·P+in·P+kp = −hy +4 +� +H31(si−1/2,j,k) + H13(si,j,k+1/2) +� +(AH)j·M·P+ip·P+kn = −hy +4 +� +H31(si+1/2,j,k) + H13(si,j,k−1/2) +� +, +(AH)jp·M·P+i·P+kp = hx +4 +� +H32(si,j+1/2,k) + H23(si,j,k+1/2) +� +, +(AH)jn·M·P+i·P+kn = hx +4 +� +H32(si,j−1/2,k) + H23(si,j,k−1/2) +� +, +(AH)jn·M·P+i·P+kp = −hx +4 +� +H32(si,j−1/2,k) + H23(si,j,k+1/2) +� +, +(AH)jp·M·P+i·P+kn = −hx +4 +� +H32(si,j+1/2,k) + H23(si,j,k−1/2) +� +, +(AH)jp·M·P+ip·P+k = hz +4 +� +H21(si+1/2,j,k) + H12(si,j+1/2,k) +� +, +42 + +(AH)jn·M·P+in·P+k = hz +4 +� +H21(si−1/2,j,k) + H12(si,j−1/2,k) +� +, +(AH)jn·M·P+ip·P+k = −hz +4 +� +H21(si+1/2,j,k) + H12(si,j−1/2,k) +� +, +(AH)jp·M·P+in·P+k = −hz +4 +� +H21(si−1/2,j,k) + H12(si,j+1/2,k) +� +. +Note that the corner points are not included in this scheme. +Denoting A = +DV Dκ2 − AH, Equation (S10) can be written as +z = D−1/2 +V +Au, +and thus, the joint distribution of u is +π(u) ∝ π(z) ∝ exp +� +−1 +2zTz +� +π(u) ∝ exp +� +−1 +2uTATD−1 +V Au +� +π(u) ∝ exp +� +−1 +2uTQu +� +. +Here, Q = ATD−1 +V A which is a sparse matrix of 93 non-zero elements per row. This +corresponds to the point, the 18 closest neighbors, and their 18 closest neighbors. +Then removing duplicates results in 93 points. +C. +Additional figures +In the application, Section 5, we estimate the parameters of a non-stationary +anisotropic and stationary anisotropic model on a simulated dataset from the nu- +merical ocean model SINMOD. The resulting properties of the non-stationary +model are presented in Figure 7 in Section 5.2 since this is the main focus of the +applications. The properties of the stationary anisotropic model fit on the same +dataset are presented in Figure S3. The marginal variance in Figure S3b, which +should be constant for this stationary model, shows some variability caused by the +43 + +boundary conditions. Notice that this boundary effect is also bigger in the direc- +tion of the strongest dependency directions seen in the south and north corners. +Notice also the large discrepancies between the correlations in these two models, +Figure S3c and Figure 7c, as the stationary anisotropic model kind of captures an +average correlation within the field. +(a) SINMOD prior +(b) Marginal Variance +(c) Correlation +Figure S3: +Prior field (a) found from SINMOD simulations, the variance of the +spatial effect (b) and spatial correlation of point [22,10,0] (c) in the stationary +anisotropic model. The N-arrow shows the cardinal north. +44 + +Depth: +0.5 +N +0.7 +1.5 +0.6 +2.5 +0.5 +3.5 +0.4 +4.5 +0.3 +5.5 +0.2Depth: +0.5 +N +0.8 +1.5 +2.5 +0.6 +3.5 +0.4 +4.5 +0.2 +5.5 +0Depth: +0.5 +N +30 +1.5 +25 +2.5 +20 +3.5 +15 +4.5 +10 +5.5 +5References +Castruccio, S., Hu, Z., Sanderson, B., Karspeck, A., and Hammerling, D. (2019). +Reproducing internal variability with few ensemble runs. 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Interpolation of spatial data: some theory for kriging. Springer +Science & Business Media. +48 + diff --git a/f9AzT4oBgHgl3EQfavwP/content/tmp_files/load_file.txt b/f9AzT4oBgHgl3EQfavwP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9cb71d62ca6fa3e7cff36681c96cd6e9b0130923 --- /dev/null +++ b/f9AzT4oBgHgl3EQfavwP/content/tmp_files/load_file.txt @@ -0,0 +1,1523 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf,len=1522 +page_content='Spatially Varying Anisotropy for Gaussian Random Fields in Three-Dimensional Space Martin Outzen Berild∗and Geir-Arne Fuglstad Department of Mathematical Sciences, Norwegian University of Science and Technology, Norway Abstract Isotropic covariance structures can be unreasonable for phenomena in three-dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We construct a class of non-stationary anisotropic Gaussian random fields (GRFs) in three dimensions through stochastic par- tial differential equations allowing for Gaussian Markov random field approx- imations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The class is proven in a simulation study where we explore the amount of data required to estimate these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Then, we apply it to an ocean mass outside Trondheim, Norway, based on simulations from a numer- ical ocean model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' And our model outperforms a stationary anisotropic GRF on predictions using in-situ measurements collected with an autonomous underwater vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Keywords: Spatial non-stationarity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' spatially-varying anisotropy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' stochastic par- tial differential equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Gaussian Markov random fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' ∗Corresponding author, martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='berild@ntnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='no 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='01372v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='ME] 3 Jan 2023 1 Introduction Gaussian random fields (GRFs) are a powerful tool for spatial and spatio-temporal geostatistical modeling (Diggle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Cressie and Wikle, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' When the key goal is predictions at unobserved locations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', kriging, isotropic covariance functions often perform well, and more flexible covariance structures should be used with care (Fuglstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' However, the screening effect in kriging (Stein, 2002) is not relevant in other settings where the primary goal is the esti- mated covariance structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', to describe internal variability in a climate model ensemble (Castruccio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2019), or to produce a spatial prior based on numerical simulations that will later be used to guide autonomous sampling (Fossum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Foss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' For the former, Fuglstad and Castruccio (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2021) demonstrated that flexible covariance structures can perform better than stationary covariance structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' There are many approaches to constructing flexible covariance structures (Samp- son, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Salvaña and Genton, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Some early approaches are the deformation method (Sampson and Guttorp, 1992) and kernel convolutions (Paciorek and Schervish, 2006), but they both involve the covariances between any pair of locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This means standard implementations are infeasible for large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' There are many ways to overcome such computational issues in spatial statistics and some are applicable for flexible covariance structures (Heaton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The stochastic partial differential equation (SPDE) approach (Lindgren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2011) is interesting because it directly gives rise to computationally efficient models and easily extends to non-stationary covariance models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' However, increasing the degree of flexibility in the covariance structure requires increasing the number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The common isotropic Matérn covariance functions (Stein, 2012) are parametrized through 3 parameters: marginal vari- ance, range, and smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Flexible models can have 100s or more parameters 2 (Fuglstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' An appealing way to reduce dimensionality is to describe the covariance structure through covariates (Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Neto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Ingebrigtsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2014, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Risser and Calder, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The aforementioned works are all considering flexible covariance structures in two-dimensional space, and while the methods can be extended to three-dimensional space, the literature is sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' For example, the SPDE approach has been used for simple anisotropic covariance structures in the context of fMRI data from the brain (Sidén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2021), and more complex covariance structures in the context of astronomy (Lee and Gammie, 2021), though this was two-dimensional space and time treated as three-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' However, spatially varying anisotropy in the SPDE approach (Fuglstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2015a) has not been extended to three- dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The aim of this paper is to develop a new method for spatially varying anisotropy in three-dimensional space through the SPDE approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' A key advantage is that the formulation as an SPDE guarantees a valid covariance structure, and the main challenge is how to describe and parametrize non-stationary covariance structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Fuglstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2015a) used one vector field to describe spatially vary- ing anisotropy, but in three dimensions, two spatially varying orthogonal vector fields are necessary for full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' In a simulation study, we investigate how much data is necessary to recover parameters for three different model complexities: stationary isotropic, station- ary anisotropic, and non-stationary anisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We then estimate GRF priors to encode knowledge about the ocean from a numerical forecast generated by the nu- merical model SINMOD by SINTEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' A stationary GRF prior and a non-stationary GRF prior are updated based on in-situ measurements by an autonomous under- water vehicle (AUV), and we evaluate the predictive ability during a mission in Trondheimsfjorden, Norway, on May 27, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Improved predictions are key, for 3 example, in autonomous sampling of the oceans (Fossum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2019, 2021), but current approaches in autonomous ocean sampling are limited to stationary GRFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' In Section 2, we describe how to model anisotropy and non-stationarity in three dimensions using SPDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Then in Section 3, we describe how to perform inference for the new model in a computationally efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' In Section 4, we describe the simulation study and discuss the results, and continue with the application to sampling in the ocean in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We end with a discussion in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 2 Constructing SPDEs with spatially varying anisotropy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='1 Existing models The Matérn covariance function on R3 is given by r(s1, s2) = σ2 2ν−1Γ(ν)(κ||s1 − s2||)νKν(κ||s1 − s2||), s1, s2 ∈ R3, (1) where ||·|| is the Euclidean distance in R3, σ > 0 is the marginal standard deviation, Kν is the modified Bessel function of the second kind and order ν > 0, and κ > 0 is an inverse spatial scale parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' As discussed in Lindgren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2011), GRFs with this covariance function is the stationary solutions of the SPDE (κ2 − ∇ · ∇)α/2(τu(s)) = W(s), s ∈ R3, (2) where α = ν + 3/2, τ = √ 8πκ/σ, ∇ · ∇ is the Laplacian, and W is a standard Gaussian white noise process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Lindgren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2011) proposed to introduce non-stationarity by allowing κ and τ to vary in space (Ingebrigtsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2014, 2015) or by deformations of space (Hildeman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Fuglstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2015a,b) consider a version of the SPDE, where the Laplacian is replaced by an anisotropic Laplacian where the direction and degree of anisotropy vary spatially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This was further extended to spherical 4 geometry in Fuglstad and Castruccio (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' However, all of these works were in two-dimensional base spaces, and only simpler models have been applied for three-dimensional base spaces (Sidén et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The key idea in Fuglstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2015a) was to replace ∇ · ∇ by ∇ · H(s)∇, where H(s) is everywhere a symmetric positive definite 2 × 2 matrix that controls the strength and direction of anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The matrix-valued function was specified as H(s) = γ(s)I2 + v(s)v(s)T, s ∈ R2, where γ(·) is a positive function and v(·) is a vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This allows γ(·) to control the baseline strength of dependence in all directions, and v(·) to control the strength and direction of additional spatial dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' However, the same parametrization in R3 is not sufficiently general to control anisotropy fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2 Stationary anisotropy in R3 We follow the idea in Fuglstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2015a) for R2, and change the SPDE in Equation (2) to (κ2 − ∇ · H∇)u(s) = W(s), s ∈ R3, (3) where ∇·H∇ is an anisotropic Laplacian and the symmetric positive definite 3×3 matrix H controls the anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The parameter τ has been dropped since κ and H together control both marginal variance and correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' As shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='1, the resulting marginal variance is σ2 m = 1 8πκ � det(H) (4) and the covariance function is explicitly known as r(s1, s2) = 1 8πκ � det(H) exp � −κ||H−1/2(s1 − s2)||) � (5) for s1, s2 ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The latter is derived in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This corresponds to geo- metric anisotropy in the Matérn covariance function with smoothness ν = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' To 5 understand the behavior of the covariance function, it is useful to think about H in terms of its eigenvalue decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Let ˜v1, ˜v2, and ˜v3 be orthonormal eigen- vectors corresponding to eigenvalues λ1, λ2 and λ3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Then Figure 1 shows an example of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='37 level iso-correlation surface that will arise from the covariance function in Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The semi-axes of the ellipsoid in the figure are v1 = (√λ1/κ)˜v1, v2 = (√λ2/κ)˜v2, and v3 = (√λ3/κ)˜v3, which by evaluating the covariance function with either of these semi-axes will yield the relationship and the iso-correlation level r(v)/σ2 m = e−1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' v v v 1 2 3 Figure 1: Iso-correlation surface at the ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='37 level of Equation (5), where v1, v2, and v3 are the eigenvectors of H with lengths √λ1/κ, √λ2/κ and √λ3/κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We generalize the parametrization described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2 and H is decom- posed as H = γI3 + vvT + ωωT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (6) where v = (vx, vy, vz)T ∈ R3 and w = (ωx, ωy, ωz)T ∈ R3, v ⊥ ω, and γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The eigenvalue decomposition of H has eigenvalues λ1 = γ, λ2 = γ + ||v||2 and 6 λ3 = γ + ||w||2 with the corresponding eigenvectors v1 = v × ω, v2 = v and v3 = ω, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We construct ω by a linear combination of two orthogonal vectors in the plane with v as normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' First, let ω1 = (−vy, vx, 0)T, which satisfies v ⊥ ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Second, let ω2 = v × ω1 = (−vzvx, −vzvy, v2 x + v2 y)T, which also satisfies v ⊥ ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We parametrize ω through ω = ρ1 ω1 ||ω1|| + ρ2 ω2 ||ω2||, (7) where ρ1, ρ2 ∈ R which works whenever vx = vy ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' An alternative solution is to use Euler-Rodrigues parametrization (Euler, 1771;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Rodrigues, 1840) to obtain both v and ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' however, in this case, the parameters are less interpretable and the issue is simply nullified by numerical optimization with appropriate initial parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The above parametrization for H uses six parameters, γ, vx, vy, vz, ρ1, and ρ2, to describe all forms of geometric anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The parameterization is inter- pretable: 1) γ controls the isotropic effect, 2) vx, vy, and vz controls one anisotropy in one direction, and 3) ρ1 and ρ2 controls anisotropy in a second direction orthog- onal to the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Lastly, κ simultaneously controls scaling of spatial dependence equally in all directions, and the variance of the GRF together with the six other parameters as seen in Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='3 Spatially varying anisotropy on bounded domain D ⊂ R3 Non-stationarity and spatially varying anisotropy is achieved by making the coef- ficients in Equation (3) spatially varying, (κ(s)2 − ∇ · H(s)∇)u(s) = W(s), s ∈ R3, (8) where κ(·) is a positive function, and H is a spatially varying symmetric positive definite 3 × 3 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Heuristically, one can imagine that the SPDE is gluing 7 together different local behavior described by ellipsoids, as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2, to a valid non-stationary covariance structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' In practice, we need to limit Equation (8) to a bounded domain to parametrize the non-stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The SPDE we propose is (κ(s)2 − ∇ · H(s)∇)u(s) = W(s), s ∈ D ⊂ R3, (9) where D is bounded, and we enforce the boundary condition (H(s)∇u(s))Tn(s), s ∈ ∂D, where n(s) is the outward normal vector of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This corresponds to no flux through the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The effect of the boundary conditions is increased marginal variance on the boundary and increased spatial dependency due to the “reflective” boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' As discussed in Lindgren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Fuglstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2015b), one can extend the domain D outside the area with observations to reduce boundary effects, or one can consider the boundary effects a feature that the non-stationary model can adjust for if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 3 Estimating SPDEs with spatially varying anisotropy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='1 Parameterizing the non-stationarity Before using the SPDE in Equation (8) in inference, we parametrize the non- stationarity through a finite number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This involves expanding log(κ(·)), log(γ(·)), vx(·), vy(·), vz(·), ρ1(·), and ρ2(·) in basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The log-transform is used for κ(·) and γ(·) since they must be positive functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Let g : R3 → R denote a generic function that we want to expand in a basis, and let p > 0 the number of basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We use basis splines similar to Fuglstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2015b), and set g(s) = f(s)Tαg, (10) 8 where αg ∈ Rp, and f(s) = (f1(s), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , fp(s))T is a p-dimensional vector with the basis functions evaluated at location s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' In this paper, we will use rectangular domains D = [A1, B1]×[A2, B2]×[A3, B3], and a basis constructed as a tensor product of three one-dimensional B-splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This means that p = m3, where m > 0 is the number of basis functions used in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We use clamped splines where the derivative is 0 at each boundary, and the construction of the clamped one-dimensional B-splines is discussed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Figure 2 shows an example of the resulting basis functions in 1-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='0 Figure 2: Clamped B-spline basis with three basis functions in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Let Bx,i denote the i-th basis function of the second-order basis in the x- dimension, and similarly By,j and Bz,k for the y- and z-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The resulting tree-dimensional basis is then fijk (s) = Bx,i(s1) · By,j(s2) · Bz,k(s3), s = (s1, s2, s3)T ∈ D, (11) for all combinations i, j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This means that αg ∈ Rm3, and m3 parameters must be estimated for each of the seven functions described at the start of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 9 Figure 3: Parameterized function representation with B-splines in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' In Sections 4 and 5, we use m = p3 = 33 = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' For a total of 189 parameters in the seven functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' When data is sparse, such a model can easily result in overfitting (Fuglstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2015b), and it is necessary to introduce penalties on the seven functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' In Fuglstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2015b), this was achieved by a hierarchical model where τg∆g(s) = Wg(s), s ∈ D, together with Neumann boundary conditions of zero derivatives on the boundary of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' However, this requires selecting a reasonable value for τg > 0 for each of the seven functions and is computationally expensive if it is done using cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' However, in the context of this paper, we are constructing a stochastic model that mimics the behavior of a densely “observed” numerical simulation model and does not include penalties beyond the restriction of using 27 basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We demonstrate the ability of this model to be estimated in our context in the simulation study in Section 4, and also investigate the amount of data needed to estimate the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 10 40 35 30 25 20 L 15 10 10 15 0 25 25 35 AO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2 Hierarchical model and discretization Consider a bounded domain D ⊂ R3, and observations y = (y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , yn) made at locations s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , sn ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We assume a Gaussian observation model yi|η(si), σ2 N ∼ N(η(si), σ2 N), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , n, where σ2 N > 0 is the nugget variance and η(s) = x(s)Tβ + u(s), s ∈ D, describes true spatial variation as a combination of covariates and a GRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Here x(·) is a spatially varying vector of k covariates, β ∈ Rk are the coefficients of the covariates, and u(·) is a GRF with spatially varying anisotropy as presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' As described in Appendix B, the GRF u(·) is discretized using a regular grid with l cells, and we get a Gaussian Markov random field w = (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , wl)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Let θ be the vector of all parameters controlling u(·), then w|θ ∼ Nl(0, Q−1), where dependence on θ is suppressed for Q, and Q is a l × l precision matrix with a three-dimensional spatial sparsity structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The vector w is linked to u(·) through a linear transformation u(s) = a(s)Tw, where a has only one non- zero entry corresponding to which grid cell location s belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This gives u = (u(s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , u(sn))T = Aw, where the n × l matrix A only has one non-zero entry on each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The coefficients of the fixed effect, β, is assigned the weak penalty β ∼ NK(0, V IK) for a fixed V > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Thus we can write y as y = Xβ + Aw + ϵ, (12) 11 where X is the design matrix of covariates, and ϵ ∼ Nn(0, Inσ2 N) is an n-dimensional vector of random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This gives rise to the hierarchical formulation y|β, w, σ2 N ∼ Nn(Xβ + Aw, σ2 NIn), β ∼ Nk(0, V Ik), w|θ ∼ Nl(0, Q−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Let s∗ ∈ D be an unobserved location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' After parameters ˆθ and ˆ σ2 N are esti- mated, one can predict the underlying value η(s∗) = x(s∗)Tβ + a(s∗)Tw or a new observation y∗ = x(s∗)Tβ + a(s∗)Tw + ϵ∗, where ϵ∗ ∼ N(0, ˆ σ2 N) is a new nugget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The predictions are made using the conditional distributions η(s∗)|y, θ = ˆθ, σ2 N = ˆ σ2 N and y∗|y, θ = ˆθ, σ2 N = ˆ σ2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The estimation of parameters is detailed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='3 Parameter inference Simplify notation by letting z = (uT, βT)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Then z|θ ∼ N(0, Q−1 z ), where Qz = � �Q 0 0 V Ik � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Let S = � A X � , then the observation model can be rewritten as y|z, σ2 N ∼ Nn(Sz, Inσ2 N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (13) Using this notation the log-likelihood can be expressed as log π(θ, σ2 N|y) = Const + log π(θ, σ2 N) + 1 2 log det (Qz) − n 2 log(σ2 N) − 1 2 log det (QC) − 1 2µT CQCµC − 1 2σ2 N (y − SµC)T(y − SµC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (14) Here dependence on θ is suppressed for µC, Qz and QC, and π(θ, σ2 N) can be used to assign a penalty on θ, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', like the random-walk penalty used in Fuglstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 12 (2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The conditional precision matrix QC is QC = Qz + STS/σ2 N (15) and µC is the conditional mean, µC = Q−1 C STy/σ2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (16) Parameter inference is done by maximizing Equation (14) with respect to θ and σ2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The parameter vector θ includes all coefficients for the basis functions, and when using 27 basis functions for each function, θ = � αlog(κ2), αlog γ, αvx, αvy, αvz, αρ1, αρ2 � , has 189 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The parameter space is challenging to search and we use an analytical expression for the gradient in the optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The deriva- tion of the analytical gradient involves many nested chain rules and a technique to calculate a partial inverse of sparse matrices (Rue and Held, 2010), see Ap- pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 for a complete description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 4 Simulation study In this section, we perform a simulation study to investigate the amount of data required to acquire reasonable parameter estimates of models with varying com- plexity that are specified through the SPDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' A comparison of these estimates is made from simulated data generated from three different parametrizations of the covariance structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The observation model for the different parametrizations is ymod = Awmod + ϵ, (17) 13 where wmod is the GMRF controlled by the parameters θmod in the respective models, and ϵ is the independent noise term with mean zero and standard deviation σN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='1 which is identical for all the parametrizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Furthermore, the models are discretized on the same domain with a grid of size (M, N, P) = (30, 30, 30) resulting in a total of 27000 grid nodes where the center of which is our spatial locations s ∈ D = [A1, B1] × [A2, B2] × [A3, B3] = [0, 40] × [0, 40] × [0, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The first and simplest model is a Stationary Isotropic (SI) model which has a covariance structure controlled by the three parameters θSI = (log κ2, log γ, log σ2 N), that is assigned to the values κ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2, γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 and σN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The resulting spatial range is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='59 with a marginal variance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The second is a Stationary Anisotropic (SA) model composed of the 8 parame- ters θSA = (log κ2, log γ, vx, vy, vz, ρ1, ρ2, log σ2 N) set to κ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='35, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5, vx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='9, vy = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='4, vz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='4, ρ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='4, ρ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='6 and σN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This results in spatial ranges of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='08 along the x-dimension, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='75 along y, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='88 along z with a marginal variance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The parameters of these first two models are simply assigned some reason- able value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' however, the third and most complex model with a non-stationary anisotropic covariance and a total of 190 parameters, they are much more trou- blesome to select.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Therefore, functions are chosen to assign the parameter val- ues in θNA throughout the domain D such that the dependency directions im- itate a vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Using these functions and evaluating them at the spatial lo- cations in the discretization the parameters of the B-splines, described in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='1, are found by optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' These aforementioned parameters are θNA = � αlog(κ2), αlog γ, αvx, αvy, αvz, αρ1, αρ2, log σN � with σN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='1, and the resulting covariance structure can be viewed in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We will now examine the extent of data required to fit back the parameters of the three models described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' First, we simulate multiple datasets from 14 (a) Correlation (b) Marginal Variance Figure 4: Spatial correlation at location [26,26,20] (a) and variance of the spatial effect (b) in the non-stationary anisotropic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' the observation model, Equation (17), with a different number of observed spatial locations and realizations (replicated observations of these spatial locations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The number of spatial locations varies between 100, 10000, and 27000 (all), and the number of realization range between 1, 10, and 100, so nine different combinations of dataset sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Furthermore, we want to perform 100 different trials for each of these combinations, and thereby have 900 total datasets per model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Also, note that the observed spatial locations are randomly chosen in each trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' From this, some statistics can be recovered about the model estimates that can give insight into the applicability of the different parameterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Table 1 shows the root mean square error (RMSE) between the set parameter values in each model and their values inferred by the different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This was obtained using the inference method described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='3 with the observation model in Equation (17) for each respective parametrization and trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The columns describe the different number of observation locations (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=') and the number 15 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='8 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='6 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='4 30 20 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2 10 a 0 y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='09 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='08 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='06 20- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='05 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='04 &o - 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='03 30- 30 20 y 20 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='02 10 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='01Table 1: The Root Mean Square Error (RMSE) of parameter estimates in the stationary isotropic, stationary anisotropic, and non-stationary anisotropic model from 100 independent trials for each combination of dataset sizes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' the number of observed locations (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=') and the number of replicated observations of these locations (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 100 10000 27000 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 1 10 100 1 10 100 1 10 100 Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' log κ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='123 log γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='626 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='032 log τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='670 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='049 Stationary Anisotropic log κ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='876 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='038 log γ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='289 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='601 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='463 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='079 |vx| 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='785 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='440 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='070 |vy| 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='679 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='354 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='035 |vz| 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='498 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='027 |ρ1| 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='977 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='038 |ρ2| 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='337 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='489 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='027 log τ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='977 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='352 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='028 Non-Stationary Anisotropic log κ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='572 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='356 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='269 log γ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='615 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='173 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='585 |vx| 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='929 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='742 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='509 |vy| 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='699 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='668 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='453 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='432 |vz| 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='591 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='610 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='296 |ρ1| 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='287 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='210 |ρ2| 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='420 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='376 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='344 log τ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='005 16 of realizations (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' ), and the different blocks represent the different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The columns highlighted in bold for each respective model are the ones we have deemed as reasonable parameter estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Also, note that some parts of the table are omitted to simplify the presentation of the results for the reader as the full table does not affect the conclusion of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' From Table 1 we observe that the (simple) stationary models, SI and SA, require very little data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' In fact, observing under 1% of the grid for 10 realizations or more is good enough for the SI and the SA only requires some more realizations to attain similar parameter accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' On the other hand, the most flexible parameterization, the NA model, requires much more data and only reaches reasonable parameter accuracy when the whole grid is observed with 10 or more realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Now there is a large discrepancy between 10000 observed points ( 37%) and 27000 (100%), so it could be interesting to investigate where in this range reasonable estimates are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' However, we have not chosen to explore this here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We also want to note that these estimates will change with the complexity of the covariance structure and with the initial values in the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 5 GRF prior for statistical sampling of the ocean 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='1 Aim Forecasts produced by numerical ocean models describe realistic behavior for the ocean, but local behavior such as plumes created by freshwater discharge from a river into the ocean are hard to accurately forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' However, we can construct a prior based on the numerical ocean model that informs prior beliefs about the ocean, which can aid AUVs to more effectively sample the ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' In this paper, the goal is to determine the three-dimensional extent of a freshwater plume in the ocean, and we assume operation time is short enough to justify a purely spatial 17 prior that does not assume dynamical changes in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' There are two steps in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Step 1 is to estimate a stationary GRF prior and a non-stationary GRF prior based on a simulation from the numerical ocean model as described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Step 2 is to combine each of the estimated priors with an observation model, and evaluate the predictive ability on in-situ observations from AUV as described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The GRFs that we estimate based on the numerical ocean model can be viewed as statistical emulators of the ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2 The numerical ocean model and the GRF prior The model training data used in this application is from a forecast produced by the ocean model SINMOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Data is provided by SINTEF Ocean which developed and ran the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' SINMOD is a three-dimensional numerical ocean model based on primitive equations that are solved using finite difference methods on a regular grid with horizontal cell sizes of 20km×20km and is nested in several steps down to 32m × 32m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Moreover, it uses z* vertical layers which allow for varying grid resolutions depending on the depth and help capture the higher variability of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' SINMOD is driven by atmospheric forces, freshwater outflows, and tides, and it provides numerical simulations of multiple variables such as salinity, temperature, and currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The reader is referred to Slagstad and McClimans (2005) for a more detailed description of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The area of operation is located in Trondheimsfjorden at Ladehammaren just outside of Trondheim, Norway, and the operation date, the time measurements are collected with the AUV, is May 27, 2021, between 10:30 and 14:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The outlined area in Figure 5 indicates the operational area which covers 1408m × 1408m in the horizontal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' At the southeast side of this field, the Nidelva river flows into the fjord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This causes a very dynamic salinity field that is unfeasible to 18 Figure 5: The area of operation in Trondheimsfjorden at Ladehammaren just outside of Trondheim, Norway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The compass shows the cardinal directions relative to the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' describe with a stationary covariance model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Therefore, we will use the numerical simulations from SINMOD to estimate a non-stationary GRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' As demonstrated in the simulation study, complex covariance structures can reliably be estimated based on such dense data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' In this application, we will focus on univariate modeling of the salinity and we choose the fine-scale horizontal grid sizes hx = 32 m hy = 32 m, which in total gives N = 45 and M = 45 grid nodes for both the numerical and the statistical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Moreover, in the vertical plane, we use 1-meter increments between the depth layers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', hz = 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' To avoid any major effects of the boundaries in this direction P = 11 depth layers are used resulting in a depth range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5m 19 Ostmarkneset Munkholmen Korsvika Ladehammaren 6668 Trondheim Traante 6668 Reina Brattora Trondheim sentralstasjon 706 moen 6692 706 Sjobadet 706 6690 Skansen Kuhauoer 6650 6666 6650 6650 Rosenheto 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' SINMOD outputs zt, t = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , 143, which are vectors of salinity values in all cells in the three-dimensional grid at different time points throughout the whole May 27, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The timesteps are 10 minutes, and Figure 6 shows five timesteps from SINMOD for the top six depth layers during the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Note Figure 6: Five timesteps of the dataset simulated with the numerical ocean model SINMOD for May 27, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The timestamps are displayed over their respective timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The N-arrow is the cardinal north.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' that the varying vertical layers in the numerical model are either with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5m or 1m increments, so the SINMOD simulations don’t require any additional modification to fit within our statistical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We first estimate the model zt = Φzt−1 + ϵt, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , 143, where Φ is a diagonal matrix of AR(1) coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The diagonal entries of Φ are estimated with maximum likelihood separately for each spatial location such that ˆΦii = �143 t=1 zt,izt−1,i/ �143 t=1 z2 t−1,i for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , NMP, where zt,i is the value in cell i at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We then compute empirical innovations ˆϵt = zt− ˆΦzt−1, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 20 10:30 Depth: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 30 N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 25 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 014:30 Depth: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 30 N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 25 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 013:30 Depth: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 30 N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 25 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 012:30 Depth: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 30 N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 25 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 011:30 Depth: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 30 N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 25 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0These empirical innovations describe the spatial covariance structure for short-term changes in salinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We fit the flexible non-stationary anisotropic model with 190 parameters, ˆθNA = (αlog κ, αlog γ, αvx, αvy, αvz, αρ1, αρ2, log σ2 N), and the stationary anisotropic model with 8 parameters, ˆθSA = (log κ2, log γ, vx, vy, vz, ρ1, ρ2, log σ2 N), to the assumed in- dependent realization from a GRF ˆϵ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='ˆϵ143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Note that there are NMP = 22275 spatial locations and the 144 empirical innovations cover the whole day of May 27, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Figures 7b show the resulting variance of the spatial effect and Figure 7c the spatial correlation with location (x, y, z) = (22, 10, 0) of the non-stationary anisotropic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The same figures of the stationary anisotropic model can be found in Appendix C, Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (a) SINMOD prior (b) Marginal Variance (c) Correlation Figure 7: Prior field (a) found from SINMOD simulations, the variance of the spatial effect (b) and spatial correlation of point [22,10,0] (marked) (c) in the non-stationary anisotropic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The N-arrow is the cardinal north.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 21 Depth: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0Depth: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 N 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 5Depth: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='1In the next step, we construct the expected value of the GRF using the time average of the whole day, µ = �143 t=0 zt/144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The mean is shown in Figure 7a and shows the overall tendency for freshwater near the river outlet and saltwater further out in the ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We choose the prior η = µ + e, (18) where we combine the fixed mean vector, µ, with a new realization, e, of the estimated stationary anisotropic model or the non-stationary anisotropic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This is a spatial prior on a 32 m × 32 m × 1 m resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='3 In-situ data collection and emulator evaluation In-situ measurements were made with the AUV on May 27, 2021, between 10:30 and 14:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The AUV followed 9 pre-planned paths within the area of operation: two intersects at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5m depth one northbound and one north-westbound starting from the river, two zig-zags in each depth layer (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5m,2m,5m), and one up and down pattern in depth ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5m to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5m moving north-westbound start- ing from the river.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Figure 8 displays the locations of the measurements in the top 5 layers of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The AUV is moving at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 m/s and continuously samples the salinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This means that multiple measurements are made within each 32 m × 32 m × 1 m grid cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Measurements are represented as yi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , nobs, whereby yi is the average value measured in grid cell i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We combine these measurements with the prior in Equation (18) using yi|η, σ2 N ind ∼ N(aT i η, σ2 meas), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , nobs, η ∼ N(µ, Q−1 Prior), where ai selects the correct grid cell, Q−1 Prior is the estimated precision matrix for the GMRF, and the Gaussian likelihood with nugget variance σ2 meas describes 22 Figure 8: Measurement locations of the AUV in the top 6 depth layers of the spatial field on May 27th, 2021, in Trondheimsfjorden at Ladehammaren just outside of Trondheim, Norway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The N-arrow is the cardinal north.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' measurement noise and sub-grid variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' In general, we would estimate σ2 meas using a trial run, but in this case, we estimated σ2 meas using the average empirical variance over all observed grid cells in the total dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Note that we have not accounted for the uncertainty in the AUVs positions in these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' As the AUV dive, it loses its GPS signal and only relies on estimated location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' When the GPS signal is returned a linear interpolation is made to account for drift but no uncertainty is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We evaluated the two priors, or emulators, by randomly ordering the 9 seg- ments and then sequentially including more and more observations for predicting the remaining hold-out data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The random permutation of the segments was done repeatedly to determine the variation in scores over different paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This scheme 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5m Nevaluates the AUVs’ ability to predict future observations while maintaining the sequential structure of measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Figure 9 shows that the non-stationary model provides a better prior for the salinity in the ocean than the stationary model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The differences are largest when little data is available, which is consistent with the idea that the prior is most important in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The non-stationary model can leverage knowledge about which areas are most uncertain using the spatially varying marginal variance and update the prior based on expected simi- larities from the spatially varying anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The improvements are seen both in point predictions through RMSE and in predictive distributions as measured by CPRS (Gneiting and Raftery, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 6 Discussion We extend the class of SPDE-based GRFs introduced in Fuglstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2015a) to three-dimensional space by overcoming two key issues: parametrization and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' For the former, we developed a specification of spatially varying anisotropy through a spatially varying baseline isotropic dependence, and two orthogonal spatially varying vector fields that describe extra dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This allows for an interpretable description of the 3×3 positive definite matrix describing anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' For the latter, we use a finite volume method to construct a GMRF that approximates the solution of the SPDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The specification of spatially varying marginal variance and spatially varying anisotropy requires specifying 7 spatially varying real functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' In this paper, we expand each function with a clamped B-spline basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' If each function uses P 3 basis functions, this gives in total 7P 3 coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' As demonstrated in the simulation study, an unpenalized estimation of these parameters requires a densely observed area and multiple realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Application of the new models in data- 24 Figure 9: The root mean square error (RMSE, top) and the continuous ranked probability score (CRPS, bottom) of predictions from the stationary anisotropic (orange) and non-stationary anisotropic models (blue) given different proportions of observed data (5%, 95%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The error bars are the standard deviations of the different measures under random permutations of the 9 segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' sparse situations will require penalties that restrict the regularity of the 7 spatially varying functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' However, more research is needed to come up with a practical 25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content="0 CRPS B'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2 5%10%15%20%25%30%35%40%45%50%55%60%65%70%75%80%85%90%95% StationaryAnisotropic Non-stationaryAnisotropicway to determine the appropriate strength of penalization for each of the functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' While we did not experience any practical issues with the chosen way to de- scribe the two orthogonal vector fields, the construction has a “gimbal lock” type issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' If one vector field points exactly along the z-axis, there is no unique choice for the second vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' A potential way to avoid this issue is by describing the orientation of the two orthogonal vector fields through quaternions or Euler- Rodrigues parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Moving from two-dimensional space to three-dimensional space introduces an asymptotically higher computation cost as a function of grid size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' For a regular three-dimensional grid with N nodes, the computational cost is O(N 2) compared to O(N 3/2) in two-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This increased computational cost arises from increased fill-in in the Cholesky factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' However, the application demon- strates that the use of a grid size of N = 22275 is unproblematic even for real-time updates on an AUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' For the predictions of salinity in the Trondheim’s fjord, we see the highest improvement of the complex GRF prior compared to an isotropic GRF, for sparse in-situ measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' As more data is collected, the difference between the models decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This suggests that the key advantage of training the more complex GRF is to encode prior physical knowledge so that we can more effectively update knowledge about unobserved locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Salinity was used as an example, but in general, the same approach could be used to map other biologically interesting quantities such as phytoplankton (Fossum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The GRFs developed in this paper are a step forward in quantifying beliefs about unobserved regions in the ocean, which is essential for optimal decisions and more effective autonomous sampling (Fossum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' In future work, it would be interesting to add a dynamic component to the model to capture physical processes such as diffusion and advection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' However, 26 this substantially increases computational cost, and it is not clear to which degree an advection field from a numerical model should be trusted and which boundary conditions are best in an advection-dominated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The new class of GRFs shows great promise for encoding prior knowledge about a phenomenon in a com- putationally efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' However, overfitting is an important issue, and we must consider ways to penalize the complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' In particular, we need to consider ways to allow flexibility in an area where it is needed such as a river outlet, and restrict flexibility in areas where we expect stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Acknowledgments Berild and Fuglstad are supported by the Research Council of Norway, project number 305445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The authors are grateful to Ingrid Ellingsen and SINTEF for providing the simulations from the numerical ocean model SINMOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 27 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' General properties A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='1 Marginal Variance Here, we will derive the expression for the marginal variance in a general sense and then specify it for three-dimensional spaces with exponential covariance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The SPDE considered in this work is (κ2 − ∇ · H∇)α/2u(s) = W(s), (S1) where s ∈ D ⊆ Rd a spatial location in the domain of dimension d and α = ν +d/2 where ν > 0 is the smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Any solution of this SPDE is a Matérn field and let σm > 0 be its marginal standard deviation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' then, its covariance function is r(s1, s2) = σ2 m 2ν−1Γ(ν)(κ||H−1/2(s1 − s2)||)νKν(κ||H−1/2(s1 − s2)||).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (S2) The transfer function of the SPDE is g(w) = (κ2 + wTHw)−α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Using this and by including the spectral density of standard Gaussian white noise in Rd is (2π)−d, the spectral density of the solution of the SPDE is fS(w) = (2π)−d(κ2 + wTHw)−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Lastly, to find the marginal variance of the field the integral of the spectral density is made over Rd as σ2 m = � Rd fS(w)dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Including the change of variables w = κH−1/2z the expression becomes σ2 m = (2π)−d � Rd(κ2 + κ2zTz)−α det(κH−1/2)dz = (2π)−d � Rd κd−2α(1 + zTz)−α det(H)−1/2dz α=ν+d/2 = (2π)−dκ−2ν det(H)−1/2 � Rd(1 + zTz)−αdz, (S3) 28 which by specifying a exponential covariance in R3 with α = 2, ν = 1/2 and d = 3 is σ2 m = 1 8πκ � det(H) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Note that the integral in Equation (S3) is solved by converting to polar coordinates as � R3 1 (1 + zTz)2dz = � π 0 sin(φ)dφ � 2π 0 dθ � ∞ 0 ρ2 (1 + ρ2)2dρ = π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2 Covariance function Evaluating Equation (S2) at ν = 1/2 and including the expression for the marginal variance the covariance function can be formalized as r(s1, s2) = � 2 π 1 8πκ � det(H) � κ||H−1/2(s1 − s2)||K 1 2(κ||H−1/2(s1 − s2)||).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Then, consider the modified Bessel function of the second kind Kn(z) = � π 2z e−z (n − 1 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' � ∞ 0 e−ttn−1/2 � 1 − t 2z �n−1/2 dt, and evaluate this at order 1/2 gives K 1 2(z) = � π 2ze−z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The covariance function can then be formalized as r (s1, s2) = � 2 πσ2 m � κ||H−1/2(s1 − s2)|| × � π 2 · κ||H−1/2(s1 − s2)|| exp � −κ||H−1/2(s1 − s2)|| � =σ2 m exp � −κ||H−1/2(s1 − s2)|| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (S4) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='3 One-dimensional clamped B-splines We illustrate the construction of 1-dimensional splines B-splines using the interval [A, B] ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Let A = t0 < t1 < · · · < tm = B be the knot points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Then the 29 zero-order B-splines are constructed recursively as Bi,0(t) = � � � � � 1, ti ≤ t ≤ ti+1, 0, otherwise, , t ∈ [A, B], for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Let r denote the order of the B-splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The first- and second-order basis splines are constructed as Bi,r(t) = t − ti ti+r − ti Bi,r−1(t) + ti+r+1 − t ti+r+1 − ti+1 Bi+1,r−1(t), t ∈ [A, B], for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , p − r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Using the r-order B-spline basis, we construct a function g : [A, B] → R by g(t) = p−r−1 � i=0 αiBi,r(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' where α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , αp−r−1 ∈ R are coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We use a clamped spline where g′(A) = g′(B) = 0 and need the additional requirement that α0 = α1 and αp−r−2 = αp−r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='4 Integrated likelihood The distribution of z = (u, β) is given by z|θ ∼ N(0, Q−1 z ), and the observation model is y|z, θ, σ2 N ∼ Nn(Sz, Inσ2 N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 30 From this the distribution of z given some observations y is π(z|θ, σ2 N, y) ∝ π(z, θ, σ2 N, y) = π(θ, σ2 N)π(z|θ)π(y|θ, σ2 N, z) ∝ exp � −1 2zTQzz − 1 2(y − Sz)TInσ−2 N (y − Sz) � ∝ exp � −1 2 � zT � Qz + σ−2 N STS � z − 2zTSTy · σ−2 N �� ∝ exp � −1 2(z − µC)TQC(z − µC) � ⇓ z|θ, σ2 N, y ∼ Nn � µC, Q−1 C � Here, QC = Qz +STS·σ−2 N is the conditional precision matrix and µC = Q−1 C STy· σ−2 N is the conditional mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Then, integrating out z from the joint distribution gives π(θ, σ2 N, y) = π(θ, z, σ2 N, y) π(z|θ, σ2 N, y) = π(θ, σ2 N)π(z|θ)π(y|θ, σ2 N, z) π(z|θ, σ2 N, y) , where the left-hand side does not depend on z such that it may be evaluated for any given value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Let us evaluate it for z = µC such that π(θ, σ2 N, y) ∝π(θ, σ2 N)π(z = µC|θ)π(y|θ, σ2 N, z = µC) π(z = µC|θ, σ2 N, y) ∝π(θ)|Qz|1/2|In · σ−2 N |1/2 |QC|1/2 exp � −1 2µT CQzµC � × exp � −1 2(y − SµC)TIn · σ−2 N (y − SµC) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The last term π(z|θ, σ2 N, y) is removed since it is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Thereby, conditioning 31 on y and taking the log we have the log-likelihood log(π(θ, σ2 N|y)) =Constant + log(π(θ, σ2 N)) + 1 2 log(det(Qz)) + n 2 log(σ−2 N ) − 1 2 log(det(QC)) − 1 2µT CQzµC − 1 2 · σ2 N (y − SµC)T(y − SµC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (S5) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 Gradient of the log-likelihood This section is similar to the derivation of the gradient presented in the supple- mentary material of Fuglstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' log(π(θ, τN|y)) =Constant + log(π(θ, τN)) + 1 2 log(det(Qz)) + n 2 log(σ−2 N ) − 1 2 log(det(QC)) + 1 2µT CQCµC − τN 2 yTy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Note that the last two terms are rewritten for simplicity in the gradient calculation and that the variance of the Gaussian noise term, σ2 N is re-parametrized with its inverse τN = 1/σ2 N (precision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Derivatives of the log-likelihood are taken with respect to θi, the elements of θ, and the precision on log scale as log(τN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The first term is a constant and therefore its derivative is zero with respect to any of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The next term, the penalty or the prior of the parameters, is not used in this paper and otherwise depends on the choice of penalty so gradient calculation is not specified for this term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' To continue note the derivatives of the precision matrix ∂QC ∂θi = ∂Qz ∂θi and ∂QC ∂ log(τN) = STSτN, which is used in the following derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' First, the derivatives with respect to θi are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The derivative of the log determinant terms are ∂ ∂θi (log(det(Q)) − log(det(QC))) =Tr � Q−1∂Q ∂θi � − Tr � Q−1 C ∂Q ∂θi � =Tr � (Q−1 − Q−1 C )∂Q ∂θi � , 32 and the derivative of the quadratic terms are ∂ ∂θi �1 2yTyτN + 1 2µT CQCµC � = ∂ ∂θi �1 2µT CQCµC � = − 1 2yTτNSQ−1 C �∂QC ∂θi � Q−1 C STτNy = − 1 2µT C �∂Q ∂θi � µC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Then, combining these the derivative of the log-likelihood with respect to θi is ∂ ∂θi log(π(θ, τN|y)) = ∂ ∂θi log(π(θ, τN))+Tr � (Q−1 − Q−1 C )∂Q ∂θi � −1 2µT C �∂Q ∂θi � µC Next, the derivative with respect to the log precision, log τN, is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The derivative of the log determinant terms are ∂ ∂ log(τN) �n 2 log(τN) − 1 2 log(det(QC)) � =n 2 − 1 2Tr � Q−1 C ∂ ∂ log(τN)QC � =n 2 − 1 2Tr � Q−1 C STS · τN � Further,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' the derivative of 1/2yTy · τN with respect to log(τN) is just the same expression so the remaining quadratic term becomes ∂ 1 2µT CQCµC ∂ log(τN) =∂ 1 2yTτNSQ−1 C STτNy ∂ log(τN) =yTτNSQ−1 C ST ∂τN ∂ log(τN)y − 1 2yTτNSQ−1 C ∂QC ∂ log(τN)Q−1 C STτNy =µT CSTτNy − 1 2µT CSTSµCτN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' and then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' by adding the last quadratic term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' the expression simplifies to −1/2yTy · τN + µT CSTy · τN − 1 2µT CSTSµC · τN = −1 2(y − SµC)T(y − SµC) · τN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Finally, combining all these terms we have the derivative of the log-likelihood with respect to log(τN): ∂ log(π(θ, τN|y)) ∂ log(τN)) =∂ log(π(θ, τN) ∂ log(τN) + n 2 − 1 2Tr � Q−1 C STS · τN � − 1 2(y − SµC)T(y − SµC) · τN 33 Note that the derivative of QC can be calculated quickly and it is derived from a series of chain rules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' first on QC, then on A and AH, and finally within H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The most computationally heavy calculation in the gradient of the log-likelihood is to calculate the inverses in the difference Q−1 − Q−1 C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' However, since this term is multiplied with the derivative of Q with respect to θi, which carries the non-zero structure of Q, only elements of Q−1 and Q−1 C which correspond to the non-zero structure of Q need to be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This is done by calculating a partial inverse of two matrices as described in Rue and Held (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Derivation B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='1 Discretization To find the local solution of the SPDE the domain D = [A1, B1]×[A2, B2]×[A3, B3] is divided into equally sized rectangular cubes or cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' We use M cells to divide [A1, B1] in the x-direction, N cells on [A2, B1] in y-direction and P cells on [A3, B3] in z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The cells have sides parallel to each axis of size hx = (B1 − A1)/M, hy = (B2 − A2)/N, and hz = (B3 − A3)/P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The cells are assigned an index with regards to their cell number along each axes starting from number 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' i ∈ [0, M] along x, j ∈ [0, N] along y, and k ∈ [0, P] along z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' For a specific cell, its domain can be denoted as Ei,j,k = [ihx, (i + 1)hx] × [jhy, (j + 1)hy] × [khz, (k + 1)hz], and Figure S1 shows this cell and its closest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Furthermore, as a regular grid is employed the volume of a cell is V = hxhyhz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' To further define the local solution of the SPDE we denote the faces of a grid cell as σF i,j,k (front), σB i,j,k (back), σL i,j,k (left), σR i,j,k (right), σU i,j,k (up) and σD i,j,k (down) with their respective face centers si,j−1/2,k, si,j+1/2,k, si−1/2,j,k, si+1/2,j,k, 34 Figure S1: One cell Ei,j,k in the discretization with its closest neighbours;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Ei+1,j,k, Ei−1,j,k, Ei,j+1,k, Ei,j−1,k, Ei,j,k+1, and Ei,j,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' si,j,k+1/2 and si,j,k−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Figure S2 describes the different faces of a cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2 Local solution of the SPDE Note that this description is an extension to three dimensions of the derivation described in Fuglstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2015a), and the reader is referred to there for fur- ther details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' To locally solve the SPDE a finite volume scheme is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' First, 35 I 1Figure S2: One cell Ei,j,k of the discretization with all its faces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' σF i,j,k (front), σB i,j,k (back), σL i,j,k (left), σR i,j,k (right), σU i,j,k (up), and σD i,j,k (down) each with its respective face centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Equation (S1) is integrated over a cell Ei,j,k as � Eijk κ2(s)ds − � Eijk ∇ · H(s)∇u(s)ds = � Eijk W(s)ds, (S6) where ds is a volume element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The integral of the Gaussian white noise on the right-hand side is a Gaussian variable with mean zero and variance equal to the volume of a cell which is independent of neighboring cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Let zijk be an standard Gaussian variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' then, Equation (S6) becomes � Eijk κ2(s)ds − � Eijk ∇ · H(s)∇u(s)ds = √ V zijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 36 Then, applying the divergence theorem to the second integral with the divergence operator gives � Eijk κ2(s)ds − � ∂Eijk (H(s)∇u(s))Tn(s)dσ = √ V zijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The first integral is approximated by letting k2 ijk be the average value of the con- tinuous function κ2(s) within a cell, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' κ2 ijk = 1/V � Eijk κ2(s)ds, resulting in V κ2 ijkuijk − � ∂Eijk (H(s)∇u(s))Tn(s)dσ = √ V zijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (S7) To describe the solution of the second integral it is divided into integrals over each surface as � ∂Eijk (H(s)∇u(s))Tn(s)dσ = W L ijk + W R ijk + W B ijk + W F ijk + W U ijk + W D ijk, (S8) or W dir ijk = � σdir ijk(H(s)∇u(s))Tn(s)dσ, where dir denotes the surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' R (posi- tive x-direction), L (negative x-direction), B (positive y-direction), F (negative y-direction), U (positive z-direction), and D (negative z-direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Now, an ap- proximation of this surface integral over each face is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' It is assumed that the gradient of u(s) is constant over each face and equal to the value at the center of each face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The resulting scheme for the gradient on each face is described in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Furthermore, let H be approximated by its value at the center of the face, and then, we have the approximation W dir ijk = � σdir ijk ∇u(s)TH(s)n(s)dσ ≈∇u(cdir ijk)TH(cdir ijk)n(cdir ijk) � σdir ijk dσ =∇u(cdir ijk)TH(cdir ijk)n(cdir ijk)A(σdir ijk), (S9) where cdir ijk is the center of face dir in the cell Eijk, and A(σdir ijk) is the area of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Combining Equation (S9) with the scheme of ∇u(cdir ijk) from Table S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' and 37 Face Scheme σR i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k ∂ ∂xu(si+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) ≃ 1 hx (u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)) ∂ ∂yu(si+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) ≃ 1 4hy (u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)) ∂ ∂zu(si+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) ≃ 1 4hz (u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) + u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) − u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1)) σL i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k ∂ ∂xu(si−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) ≃ 1 hx (u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)) ∂ ∂yu(si−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) ≃ 1 4hy (u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)) ∂ ∂zu(si−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) ≃ 1 4hz (u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) + u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1) − u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1)) σB i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k ∂ ∂xu(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) ≃ 1 4hx (u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)) ∂ ∂yu(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) ≃ 1 hy (u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)) ∂ ∂zu(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) ≃ 1 4hz (u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) + u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1)) σF i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k ∂ ∂xu(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) ≃ 1 4hx (u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)) ∂ ∂yu(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) ≃ 1 hy (u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)) ∂ ∂zu(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) ≃ 1 4hz (u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) + u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1)) σU i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k ∂ ∂xu(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1/2) ≃ 1 4hz (u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) + u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) − u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)) ∂ ∂yu(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1/2) ≃ 1 4hy (u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) + u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)) ∂ ∂zu(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1/2) ≃ 1 hx (u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)) σD i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k ∂ ∂xu(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1/2) ≃ 1 4hz (u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1) − u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1)) ∂ ∂yu(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1/2) ≃ 1 4hy (u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1)) ∂ ∂zu(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1/2) ≃ 1 hx (u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1)) Table S1: Numerical scheme of the partial derivative with respect to x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' y and z of uijk on the different faces of cell Eijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' denoting the components of H as H(s) = � ���� H11(s) H12(s) H13(s) H21(s) H22(s) H23(s) H31(s) H32(s) H33(s) � ���� 38 the approximations for each face become ˆW R i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k = hyhz � H11(si+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) hx � + hyhz � H21(si+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) + u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) − u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1) 4hz � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' ˆW L i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k = hyhz � H11(si−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) hx � + hyhz � H21(si−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) 4hy � + hyhz � H31(si−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1) + u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) − u(si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) 4hz � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' ˆW B i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k = hxhz � H12(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1/2,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) hy � + hxhz � H32(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) + u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1) − u(si,' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) 4hx � + hxhz � H22(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k)u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1) 4hx � + hxhy � H23(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1/2)u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1) 4hy � + hxhy � H33(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1/2)u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1) − u(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) hz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Next, a vectorization of the discretization is made;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' first moving along the z- direction, then along x-direction, and lastly along the y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Let us denote this with the common index l = j · M · P + i · P + k so sijk = sj·M·P+i·P+k = sl which gives u(sijk) = ul and κ2(sijk) = κ2 l , and let the last index be L = (N − 40 1)MP + (M − 1)P + P − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Further, the vectorization results in the linear system of equations (DV Dκ2 − AH)u = D1/2 V z, (S10) where DV = V · IMNP, Dκ2 = [κ2 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , κ2 l , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' , κ2 L] IMNP, and z ∼ N(0, IMNP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' For simplicity the indices of the neighbors are denoted kp = k + 1, kn = k − 1, jp = j + 1, jn = j − 1, ip = i + 1, and in = i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The development of AH is done by the sum ˆW L ijk + ˆW R ijk + ˆW B ijk + ˆW F ijk + ˆW U ijk + ˆW D ijk and accounting for the index in uijk to form the linear relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' In the following, non-zero elements of the (jMN + iP + k)-th row of AH are formalized, and the index in (AH)_ denotes the column being assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The resulting coefficient with the point itself is (AH)j·M·P+i·P+k = − hyhz hx � H11(si+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + H11(si−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) � − hxhz hy � H22(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + H22(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) � − hxhy hz � H33(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1/2) + H22(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1/2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' with the six closest neighbors are (AH)j·M·P+i·P+kp =hxhy hz H33(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1/2) + hy 4 � H31(si+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − H31(si−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) � + hx 4 � H32(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − H32(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) � (AH)j·M·P+i·P+kn =hxhy hz H33(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1/2) − hy 4 � H31(si+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − H31(si−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) � − hx 4 � H32(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − H32(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) � (AH)j·M·P+ip·P+k =hzhy hx H11(si+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + hy 4 � H12(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1/2) − H12(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1/2) � + hz 4 � H13(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − H13(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) � 41 (AH)j·M·P+in·P+k =hzhy hx H11(si−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − hy 4 � H12(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1/2) − H12(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1/2) � − hz 4 � H13(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − H13(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) � (AH)jp·M·P+i·P+k =hxhz hy H22(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + hx 4 � H23(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1/2) − H23(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1/2) � + hz 4 � H21(si+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − H21(si−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) � (AH)jn·M·P+i·P+k =hxhz hy H22(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − hx 4 � H23(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1/2) − H23(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1/2) � − hz 4 � H21(si+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) − H21(si−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' and with the twelve closest diagonals are (AH)j·M·P+ip·P+kp = hy 4 � H31(si+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + H13(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1/2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (AH)j·M·P+in·P+kn = hy 4 � H31(si−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + H13(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1/2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (AH)j·M·P+in·P+kp = −hy 4 � H31(si−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + H13(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1/2) � (AH)j·M·P+ip·P+kn = −hy 4 � H31(si+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + H13(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1/2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (AH)jp·M·P+i·P+kp = hx 4 � H32(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + H23(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1/2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (AH)jn·M·P+i·P+kn = hx 4 � H32(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + H23(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1/2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (AH)jn·M·P+i·P+kp = −hx 4 � H32(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + H23(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k+1/2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (AH)jp·M·P+i·P+kn = −hx 4 � H32(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + H23(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k−1/2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (AH)jp·M·P+ip·P+k = hz 4 � H21(si+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + H12(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 42 (AH)jn·M·P+in·P+k = hz 4 � H21(si−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + H12(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (AH)jn·M·P+ip·P+k = −hz 4 � H21(si+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + H12(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (AH)jp·M·P+in·P+k = −hz 4 � H21(si−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) + H12(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='j+1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Note that the corner points are not included in this scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Denoting A = DV Dκ2 − AH, Equation (S10) can be written as z = D−1/2 V Au, and thus, the joint distribution of u is π(u) ∝ π(z) ∝ exp � −1 2zTz � π(u) ∝ exp � −1 2uTATD−1 V Au � π(u) ∝ exp � −1 2uTQu � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Here, Q = ATD−1 V A which is a sparse matrix of 93 non-zero elements per row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' This corresponds to the point, the 18 closest neighbors, and their 18 closest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Then removing duplicates results in 93 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Additional figures In the application, Section 5, we estimate the parameters of a non-stationary anisotropic and stationary anisotropic model on a simulated dataset from the nu- merical ocean model SINMOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The resulting properties of the non-stationary model are presented in Figure 7 in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2 since this is the main focus of the applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The properties of the stationary anisotropic model fit on the same dataset are presented in Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The marginal variance in Figure S3b, which should be constant for this stationary model, shows some variability caused by the 43 boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Notice that this boundary effect is also bigger in the direc- tion of the strongest dependency directions seen in the south and north corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Notice also the large discrepancies between the correlations in these two models, Figure S3c and Figure 7c, as the stationary anisotropic model kind of captures an average correlation within the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (a) SINMOD prior (b) Marginal Variance (c) Correlation Figure S3: Prior field (a) found from SINMOD simulations, the variance of the spatial effect (b) and spatial correlation of point [22,10,0] (c) in the stationary anisotropic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' The N-arrow shows the cardinal north.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 44 Depth: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2Depth: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 0Depth: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 N 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content='5 5References Castruccio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', Hu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', Sanderson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', Karspeck, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=', and Hammerling, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Reproducing internal variability with few ensemble runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Journal of Climate, 32(24):8511–8522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Cressie, N.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Stein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Interpolation of spatial data: some theory for kriging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' Springer Science & Business Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} +page_content=' 48' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQfavwP/content/2301.01372v1.pdf'} diff --git a/gdE2T4oBgHgl3EQfyQgz/content/tmp_files/2301.04118v1.pdf.txt b/gdE2T4oBgHgl3EQfyQgz/content/tmp_files/2301.04118v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f6c8753798847abea05347d6b9d559c4b1a1113a --- /dev/null +++ b/gdE2T4oBgHgl3EQfyQgz/content/tmp_files/2301.04118v1.pdf.txt @@ -0,0 +1,3955 @@ +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL +DEFERRED TERM INSURANCE PURCHASING POLICY +YUQI LI∗ AND LIHUA ZHANG† +Abstract. +This paper researches the problem of purchasing deferred term insurance in the +context of financial planning to maximize the probability of achieving a personal financial goal. +Specifically, our study starts from the perspective of hedging death risk and longevity risk, and +considers the purchase of deferred term life insurance and deferred term pure endowment to achieve a +given financial goal for the first time in both deterministic and stochastic framework. In particular, we +consider income, consumption and risky investment in the stochastic framework, extending previous +results in Bayraktar et al. (2016). The time cutoff m and n make the work more difficult. However, by +establishing new controls,“quasi-ideal value” and“ideal value”, we solve the corresponding ordinary +differential equations or stochastic differential equations, and give the specific expressions for the +maximum probability. +Then we provide the optimal life insurance purchasing strategies and the +optimal risk investment strategies. In general, when m ⩾ 0, n > 0, deferred term insurance or term +life insurance is a better choice for those who want to achieve their financial or bequest goals but +are not financially sound. In particular, if m > 0, n → ∞, our viewpoint also sheds light on reaching +a bequest goal by purchasing deferred whole life insurance. It is worth noting that when m = 0, +n → ∞, our problem is equivalent to achieving the just mentioned bequest goal by purchasing whole +life insurance, at which point the maximum probability and the life insurance purchasing strategies +we provide are consistent with those in Bayraktar et al. (2014, 2016). +Key words. Deferred term life insurance, deterministic control, variational inequality, optimal +strategy, personal financial planning, financial goal, stochastic differential equations. +1. Introduction. “Financial management” is an important issue in a person’s +life. +Therefore, how to achieve financial goals has become one of the topics that +everyone pays more and more attention to, some researches refer to Michael et al. +(2021), Subbakrishna and Murali (2018). In order to achieve financial goals, many +scholars have previously studied various investment strategies among different groups +of people. For example, Huang (2016) studied for college students how to do their +financial planning and gave the specific steps for financial management. Topa et al. +(2018) subdivided the age of retirees and made a corresponding financial plan by ana- +lyzing financial capabilities and investment goals. In fact, there are many factors that +affect personal financial goals. For instance, Pietrzyk and Rokita (2015) examined a +model of household financial planning that takes into account factors such as family +survival, investment returns, labor income, health status, and life insurance to achieve +financial goals. More about investments and financial management, please refer to +Arup (2017), Biradar et al. (2021), Bender et al. (2022), Bajtelsmit and Wang (2018), +Deimena (2014), Dhanasekaran and Kumar (2016). +In last several years, Drive et al. (2018), Scriven (2008), Weedige et al. (2019) +pointed that life insurance has attracted much attention as one of the investment +methods due to its insurance characteristics, as it helps to reduce the financial bur- +den of adverse events such as premature death, terminal illness, incapacity to work, +or incapacity due to injury or disability by transferring personal losses to insurance +companies. Recently, an increasing number of scholars have addressed the question of +how to link the purchase of life insurance with investments. Richard (1975) integrated +life insurance with consumption and investment based on the optimal consumption +and optimal investment problems first solved by Merton (1969, 1971). Since then, +∗School of Sciences, Beijing University of Posts and Telecommunications, Beijing, 100876, China +(Llyq@bupt.edu.cn). +†Corresponding author. School of Sciences, Beijing University of Posts and Telecommunications, +Beijing, 100876, China (zhlh@bupt.edu.cn). +1 +This manuscript is for review purposes only. +arXiv:2301.04118v1 [q-fin.PM] 9 Dec 2022 + +2 +Y. Q. LI AND L. H. ZHANG +most scholars have begun to follow Richard’s research methods but based on the +principle of maximizing consumption utility. Bayraktar and Young (2013) considered +the problem about how to purchasing life insurance to maximize utility of household +consumption. Liang and Zhao (2016) studied the stochastic optimal control problem +of whole life insurance purchasing strategy with consumption and investment under +the CAAR utility. Lee (2021) considered the utility maximization problem under the +exponential utility functions, and then gave the optimal portfolio, consumption and +whole life insurance strategies. +Other than the criterion of maximizing consumption utility, many scholars have +also incorporated the idea of probability into the whole life insurance purchasing +strategy. They combined the minimum probability of lifetime ruin or the maximum +probability of reaching a bequest goal with the whole life insurance purchasing strat- +egy. In the early years, there were predictions that in the ten years between 2020 and +2030, the cost of living for retirees would exceed their financial capital by 400 billion +dollars. Under this plan, individuals (rather than employers) must bear all investment +and living risks. In this context, Young (2004) gave the optimal investment strategies +to minimize the probability of lifetime ruin. Bayraktar and Young (2006) used the +stochastic optimal control technique to determine the optimal investment strategies +for the minimum probability of lifetime ruin under the given two consumption rates +and credit constraints. To make the model more complete and realistic, Wang and +Young (2012) introduced the annuity, and determined the minimum probability of +lifetime ruin when buying a convertible life annuity and investing in a risky market. +Building on the aforementioned body of prior work, Bayraktar et al examined a num- +ber of issues related to purchasing whole life insurance policy in order to maximize the +probability of reaching a bequest goal. Bayraktar et al. (2014, 2015) first considered +this issue without consumption and investment, and further assumed that the force +of death changes over time. Based on the above researches, Bayraktar et al. (2016) +considered the model with investment and consumption, and indicated the optimal +strategies for whole life insurance purchase. Recently, Liang and Young (2019) de- +termined the optimal robust strategy for maximizing the probability of reaching a +bequest goal under moral hazard rate uncertainty and risky asset drift. They ex- +tended the work of Bayraktar et al by allowing that financial markets and personal +mortality are ambiguous. +Through the study of this series of questions, previous researches have focused on +the whole life insurance. However, Michael et al. (2021), Subbakrishna and Murali +(2018) found that due to the higher premiums of whole life insurance, it is not suit- +able for young or middle-aged people who need protection for a certain period of time +but have only average economic power. Therefore, it is necessary to consider term life +insurance. As can be seen in Xiong and Shen (2020), Promislow (2011), deferred term +life insurance is a more general type of insurance than term life insurance, and from +an economic perspective, it further reduces the financial burden. Thus, it is natural +to include deferred term life insurance in personal financial planning. Therefore, we +explore the problem of purchasing deferred term insurance in personal financial plan- +ning to maximize the probability of achieving a financial goal, and provide the optimal +purchasing strategies in the deterministic framework. Furthermore, we consider the +problem under the stochastic framework which include the consumption, income and +risky investment, and then, give the optimal life insurance purchasing strategies and +the optimal investment strategies. +The specific structure and innovation of this paper are as follows. +In Section +2, the policyholder purchases m-year deferred n-year term life insurance through a +This manuscript is for review purposes only. + +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY 3 +single premium or a continuously paid premium to take out a death benefit plan. +Although the time cutoff m and n complicates the work, we still make progress by +establishing new deterministic controls, “quasi-ideal value” and “ideal value”. After +solving the associated variational inequalities, we finally obtain the optimal purchas- +ing strategies. When m ⩾ 0, n > 0, deferred term insurance is a better choice for +those who want to achieve their financial or bequest goals but are not financially +sound. In particular, if m > 0, n → ∞, our viewpoint also sheds light on reaching a +bequest goal by purchasing deferred whole life insurance. It is worth noting that if +m = 0, n → ∞, our problem is equivalent to achieving the just mentioned bequest +goal by purchasing whole life insurance. In this case, the maximum probability and +life insurance purchasing strategies we provide are consistent with those in Bayraktar +et al. (2014). Moreover, whether or not the policyholder has enough time to reach +the “ideal value” has an obvious impact on the maximum probability of achieving a +given financial goal. For the case of continuous premium payment, we also study this +situation by comparing the force of interest r and the force of death λ. Therefore, +we give the maximum probability of achieving a given financial goal and the corre- +sponding strategies for purchasing life insurance in the following two cases, when the +policyholder has enough time to reach the “ideal value” (i.e. λ ⩽ r) or when there +is not enough time to reach the “ideal value”(i.e. λ > r). In addition to the risk of +death, there is another risk that deserves attention, namely the risk of longevity. To +address this risk, we provide the relevant conclusions in Section 3. +For the stochastic framework, we assume that the policyholder’s account includes +consumption, income and risky investment to achieve a financial goal under two dif- +ferent models in Section 4. We introduce the approximation functions and functional +operators, then by the Itˆo’s formula and Legendre transform, we solve the maximal +probability and give the optimal n-year term life insurance purchasing strategies and +the optimal investment strategies. When n → ∞ and when the income process van- +ishes, our results are also consistent with those in Bayraktar et al. (2016). Finally, +Section 5 concludes the paper. From the above analyses and our results, it is clear that +our research findings are more general in comparison with whole life insurance. At +the same time, it lays a new direction and research foundation for personal financial +planning issues. +2. Purchasing m-year deferred n-year term life insurance in personal +financial planning under the deterministic framework. Assume the policy- +holder has an investment account to achieve a financial goal, and this account doesn’t +include the consuming and risky investment. He/She may invest in a risk-free market +to earn interest, which has the force of interest r, or purchase m-year deferred n-year +term life insurance. +This insurance can help the policyholder to achieve financial +motivation. Let δd be the future lifetime, it follows an exponential distribution with +parameter λ, that is, the probability density function of δd is f(x) = λe−λx, x > 0. +Equivalently, the policyholder is subject to a constant force of mortality λ. What +needs to be explained here is that, in this paper, we always assume that the moment +when the policyholder purchases insurance is the initial moment. +2.1. m-year deferred n-year term life insurance purchased by a single +premium. The policyholder buys m-year deferred n-year term life insurance by a +single premium with no cash value available. We define that a dollar death benefit +payable immediately at time δd which between m and m+n costs m�Kn. The premium +is payable at the moment of the contract, so m�Kn is the single premium per dollar of +This manuscript is for review purposes only. + +4 +Y. Q. LI AND L. H. ZHANG +death benefit. We give the single premium as follows: +m�Kn = +� +1 + θ +� � m+n +m +e−rtλe−λtdt = +� +1 + θ +� +λ +λ + r +� +e−(r+λ)m − e−(r+λ)(m+n)� +, +in which θ is the proportional risk loading. +Denote the wealth in this investment account at time t ⩾ 0 by W(t), and denote +the amount of death benefit payable at time δd purchased at or before time t by D(t), +in which 0 ⩽ t < δd. Therefore, with single-premium m-year deferred n-year term life +insurance, the wealth follows the dynamics +� +� +� +dW(t) = rW(t−)dt − m�KndD(t), +0 ⩽ t < δd, +W(δd) = W(δd−) + D(δd−)1{m<δd⩽m+n}, +in which 1{m<δd⩽m+n} equals 1, otherwise, it equals 0. The similar definition in the +latter sections that we omit the explanation. +Remark 2.1. We define insurance purchasing strategy D = {D(t)}t⩾0. In this +section, D is called admissible if D is a non-decreasing, non-negative process, inde- +pendent of δd; and if wealth under this process is non-negative for all t ⩾ 0. +We define the maximum probability of achieving the given financial goal (id est +f) as follows: +m�ϕn(w, D) = sup +D +� +Pw,D(W(δd) ⩾ f | δd < m)P(δd < m) ++ Pw,D(W(δd) ⩾ f | m ⩽ δd ⩽ m + n)P(m ⩽ δd ⩽ m + n) ++ Pw,D(W(δd) ⩾ f | δd > m + n)P(δd > m + n) +� +:= sup +D +� +Pw,D +1 ++ Pw,D +2 ++ Pw,D +3 +� +in which Pw,D denotes the conditional probability given W(0−) = w and D(0−) = D. +Pw,D +1 += Pw,D(W(δd) ⩾ f | δd < m)P(δd < m), and the definitions of Pw,D +i +, i = 2, 3 +are similar. +D represents the compensation available for other types of financial +products that the policyholder owns before purchasing m-year deferred n-year term +life insurance, and D < f, 0 ⩽ w < (m�Kn + 1)(f − D) := w∗. For the convenience of +later discussions, we introduce the following notations +m�ϕn(w, D) := m�ϕ1 +n(w, D) + m�ϕ2 +n(w, D) + m�ϕ3 +n(w, D), +m�ϕ2,3 +n (w, D) := m�ϕ2 +n(w, D) + m�ϕ3 +n(w, D), +in which +m�ϕi +n(w, D) = sup +D +Pw,D +i +, i = 1, 2, 3. +Remark 2.2. (1). In this paper, we assume that D < f since the policyholder +would achieve the financial goal if D ⩾ f. +(2). If wealth equals m�Kn(f − D) := K∗, then the policyholder will spend all wealth +to buy m-year deferred n-year term life insurance of f − D, and if he/she survives +This manuscript is for review purposes only. + +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY 5 +more than m years after purchasing insurance, but dies within m + n years, then the +policyholder’s total death benefit becomes (f − D) + D = f, we simply call K∗ the +“quasi-ideal value”. If the policyholder dies after m+n years of purchasing insurance, +he/she may not receive the death benefit. So, we need to find the real “ideal value”. +If wealth equals w∗ and policyholder doesn’t receive the death benefit, then only when +the remained wealth is greater than or equal to f − D, the policyholder may achieve +the financial goal. Therefore, w∗ is called the “ideal value”, at this time, it’s optimal +for the policyholder to purchase m-year deferred n-year term life insurance of f − D, +whether or not he/she can receive the death benefit, the total wealth at δd will reach +the given financial target f. +Under the purchasing strategies discussed above, we obtain m�ϕn(w, D) = 1 for +w ⩾ w∗, 0 ⩽ D < f. Thus, all that remains is to calculate the maximum probability +of achieving the given financial goal f for �L = {(w, D) : 0 ⩽ w < w∗, 0 ⩽ D < f}. +Proposition 2.3. The maximum probability of achieving the financial goal on �L +is given by +m�ϕn(w, D) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� w +K∗ +� λ +r � +e−λm − e−λ(m+n) +� +, +0 ⩽ w < K∗, +� +e−λm − e−λ(m+n) +� ++ e−λ(m+n) +�w − K∗ +f − D +� λ +r +, +K∗ ⩽ w < w0, +� +1 + e−λ(m+n) +� �w − K∗ +f − D +� λ +r +− e−λ(m+n), +w0 ⩽ w < w∗, +in which w0 = (e−rm + m�Kn)(f − D). +The related optimal insurance purchasing strategy is not to purchase additional +insurance until wealth reaches w∗, at which point, it is optimal to buy additional m- +year deferred n-year term life insurance of f − D. +To prove the Proposition 2.3, we first give several auxiliary lemmas. +Lemma 2.4. Let φ = φ(w, D) be a function that is non-decreasing, continuous, +and piecewise differentiable with respect to both w and D on �L, except that φ might +have infinite derivative with respect to w at w = 0. Suppose φ satisfies the following +variational inequality on �L, except possibly when w = 0. +When 0 < w < K∗, +(2.1) +max +� +rwφw − λφ, φD − m�Knφw +� += 0. +When w ⩾ K∗, +(2.2) max +� +rwφw − λφ + (λ(e−λm − e−λ(m+n)) − rK∗φw), φD − (m�Kn + 1)φw +� += 0. +Additionally, suppose +φ +� +K∗, D +� += e−λm − e−λ(m+n), +φ +� +w∗, D +� += e−λm. +Then, on �L, m�ϕ2,3 +n (w, D) = φ(w, D). +Proof. First, we consider the case when 0 < w < K∗. In this case, +m�ϕ3 +n(w, D) = 0, +m�ϕ2,3 +n (w, D) = m�ϕ2 +n(w, D). +This manuscript is for review purposes only. + +6 +Y. Q. LI AND L. H. ZHANG +Then, we rewrite the expression for m�ϕ2 +n(w, D) as follows: +m�ϕ2 +n(w, D) = sup +D +Ew,D +�� ∞ +0 +λe−λt1{W (t)+D(t)⩾f} +� +e−λm − e−λ(m+n)� +dt +� +, +in which Ew,D denotes the conditional expectation given W(0−) = w, D(0−) = D. +Define the functional operator L D: +L Dφ = rwφw − λφ + λ(e−λm − e−λ(m+n))1{w+D⩾f}. +Since we are considering a single premium, we obtain L Dφ ⩽ 0 from the lemma +hypothesis. Define τn = inf {s ⩾ 0 : Ds ⩾ n}, then applying the Itˆo’s formula for +e−λτnφ(W(τn), D(τn)), we have +e−λτnφ(W(τn), D(τn)) = φ(w, D) ++ +� τn +0 +e−λt(L Dφ − λ(e−λm − e−λ(m+n))1{W (t)+D(t)⩾f})dt ++ +� τn +0 +e−λt(φD − m�Knφw)dD(t). +(2.3) +By Wang and Young (2012), we can first assume φ is bounded from below and after +removing this assumption, the conclusion still holds. Suppose φ ⩾ φ∗. By taking +expectations of (2.3), we obtain +Ew,D � +e−λτnφ∗� +⩽ Ew,D � +e−λτnφ +� += φ(w, D) ++ Ew,D +�� τn +0 +e−λt(L Dφ − λ(e−λm − e−λ(m+n))1{W (t)+D(t)⩾f})dt +� ++ Ew,D +�� τn +0 +e−λt(φD − m�Knφw)dD(t) +� +⩽ φ(w, D) − Ew,D +�� τn +0 +λ(e−λm − e−λ(m+n))e−λt1{W (t)+D(t)⩾f}dt +� +. +(2.4) +Let τn → ∞ and apply the monotonic convergence theorem to (2.4) +φ(w, D) ⩾ m�ϕ2 +n(w, D). +By Wang and Young (2012), because φ(w, D) satisfies the boundary conditions, then +φ(w, D) = m�ϕ2 +n(w, D). +For w ⩾ K∗, let ψ(w, D) = m�ϕ2,3 +n (w, D) − (e−λm − e−λ(m+n)) and w0 = w − K∗. +Then repeat the above steps, we also can obtain φ(w, D) = m�ϕ2,3 +n (w, D). +Lemma 2.5. The maximum probability of achieving the financial goal on �L is +given by +(2.5) +m�ϕ2,3 +n (w, D) = +� +� +� +� +� +� +� +� +� +� w +K∗ +� λ +r � +e−λm − e−λ(m+n) +� +, +0 ⩽ w < K∗, +� +e−λm − e−λ(m+n) +� ++ e−λ(m+n) +�w − K∗ +f − D +� λ +r +, +K∗ ⩽ w < w∗. +This manuscript is for review purposes only. + +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY 7 +The related optimal insurance purchasing strategy is not to purchase additional +insurance until wealth reaches w∗, at which point, it is optimal to buy additional m- +year deferred n-year term life insurance of f − D. +Proof. In order to help us solve the variational inequality in verification Lemma +2.4, we recall the similar problems in Milevsky et al. (2006). Then in our situation, +m�ϕ2,3 +n +solves the following boundary-value problem on �L, except possibly at w = 0. +(2.6)� +� +� +� +� +� +� +� +� +rw(m�ϕ2,3 +n )w − λ m�ϕ2,3 +n += 1{w⩾K∗} +� +rK∗(m�ϕ2,3 +n )w − λ(e−λm − e−λ(m+n)) +� +, +m�ϕ2,3 +n +� +K∗, D +� += e−λm − e−λ(m+n), +m�ϕ2,3 +n +� +w∗, D +� += e−λm. +Our general guideline is to use verification Lemma 2.4 to prove Lemma 2.5. +We notice that m�ϕ2,3 +n +in (2.5) is increasing and differentiable with respect to both +w and D on �L, except possibly at w = 0. m�ϕ2,3 +n +solves the boundary-value problem +(2.6), which means +(2.7) rw(m�ϕ2,3 +n )w − λ m�ϕ2,3 +n ++ 1{w⩾K∗} +� +λ(e−λm − e−λ(m+n)) − rK∗(m�ϕ2,3 +n )w +� += 0 +on �L, except possibly at w = 0. We give the method of solving m�ϕ2,3 +n +as follows and it +will be omitted that the similar method of solving the maximum probability in latter +sections. +Firstly, we solve the equation +λ m�ϕ2,3 +n += rw(m�ϕ2,3 +n )w. +By rewriting (m�ϕ2,3 +n )w as ∂ m�ϕ2,3 +n +∂w +, equivalently +∂ m�ϕ2,3 +n +m�ϕ2,3 +n += λ +r +∂w +w . +Then we integrate both sides and obtain +ln m�ϕ2,3 +n += λ +r lnw + C(D), +in which C(D) is a function about D. +Thus, m�ϕ2,3 +n += ˜C(D)w +λ +r , in which ˜C(D) +is also a function about D. +Substituting m�ϕ2,3 +n (K∗, D) = e−λm − e−λ(m+n) into +m�ϕ2,3 +n += ˜C(D)w +λ +r , then +˜C(D) = +� +K∗ +�− λ +r (e−λm − e−λ(m+n)). +Therefore +m�ϕ2,3 +n (w, D) = +� w +K∗ +� λ +r +(e−λm − e−λ(m+n)). +Secondly, we solve the equation +rw(m�ϕ2,3 +n )w − λ m�ϕ2,3 +n += rK∗(m�ϕ2,3 +n )w − λ(e−λm − e−λ(m+n)). +This manuscript is for review purposes only. + +8 +Y. Q. LI AND L. H. ZHANG +Equivalently, we have +r(w − K∗)(m�ϕ2,3 +n )w = λ(m�ϕ2,3 +n +− (e−λm − e−λ(m+n))). +Then we make variable substitution. Let y = w − K∗ and m�φ2,3 +n += m�ϕ2,3 +n − (e−λm − +e−λ(m+n)). Then the equation changes to the type which we have proved, and repeat +above proof, we can obtain +m�ϕ2,3 +n +� +w, D +� += +� +e−λm − e−λ(m+n)� ++ e−λ(m+n) +�w − K∗ +f − D +� λ +r +. +Next, we prove that +(m�ϕ2,3 +n )D − m�Kn(m�ϕ2,3 +n )w < 0, +0 < w < K∗, +and +(m�ϕ2,3 +n )D − (m�Kn + 1)(m�ϕ2,3 +n )w < 0, +K∗ ⩽ w < w∗. +If 0 < w < K∗, then +(m�ϕ2,3 +n )D = λ +r +� w +K∗ +� λ +r −1 +(e−λm − e−λ(m+n)) +w +m�Kn(f − D)2 , +and +(m�ϕ2,3 +n )w = λ +r +� w +K∗ +� λ +r −1 +(e−λm − e−λ(m+n)) 1 +K∗ +. +Thus +(m�ϕ2,3 +n )D − m�Kn(m�ϕ2,3 +n )w = +λ +r m�Kn +� w +K∗ +� λ +r −1 +(e−λm − e−λ(m+n)) w − K∗ +(f − D)2 < 0. +If K∗ ⩽ w < w∗, then analogy to the above method, we have +(m�ϕ2,3 +n )D − (m�Kn + 1)(m�ϕ2,3 +n )w = λ +r +�w − K∗ +f − D +� λ +r −1 +e−λ(m+n) w − w∗ +(f − D)2 < 0. +Therefore, it’s shown that the expression for m�ϕ2,3 +n +in (2.5) satisfies the variational +inequality(2.1) and (2.2). When w = 0, obviously, m�ϕ2,3 +n (w, D) = 0. The optimal +insurance purchasing strategy is to buy additional m-year deferred n-year term life +insurance of f − D when wealth reaches w∗. +Proof of Proposition 2.3: Step1. We first calculate m�ϕ1 +n(w, D). +(i). If w ⩾ K∗, then we set t∗ satisfies +� +w−K∗ +� +ert∗ = f −D, the maximum probability +of achieving the financial goal f is as follows: +m�ϕ1 +n(w, D) = +� m +t∗ +λe−λtdt = +�w − K∗ +f − D +� λ +r +− e−λm, +in which w satisfies w ⩾ (e−rm + +m�Kn)(f − D) because of m�ϕ1 +n(w, D) ⩾ 0. +If +w < (e−rm + m�Kn)(f − D), then m�ϕ1 +n(w, D) = 0. +This manuscript is for review purposes only. + +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY 9 +(ii). If w < K∗, then we set t∗ satisfies wert∗ = f − D, the maximum probability of +achieving the financial goal f is as follows: +m�ϕ1 +n(w, D) = +� m +t∗ +λe−λtdt = +� +w +f − D +� λ +r +− e−λm, +in which w satisfies w ⩾ e−rm(f −D) because of m�ϕ1 +n(w, D) ⩾ 0. If w < e−rm(f −D), +then m�ϕ1 +n(w, D) = 0. +Step2. For m�ϕ2,3 +n , the relevant conclusions have been given in the previous Lemma +2.2. Having said all of above, we restrict the premium m�Kn < e−rm, then by Lemma +2.4 and Lemma 2.5, the Proposition 2.3 is proved. +■ +Remark 2.6. We now give another explanation of Lemma 2.5. If the policyholder +dies after m years of purchasing insurance and the wealth reaches the “quasi-ideal +value”, the probability of the policyholder receiving the death benefit is e−λm − +e−λ(m+n). If the policyholder dies after m + n years of purchasing insurance, then +he/she is only dependent on the remaining assets to achieve the financial goal by +investing in a risk-free market with interest returns. The time when the wealth reaches +the “ideal value”, denoted by t0, is given by +� +w − K∗ +� +ert0 = f − D. +Thus t0 = +1 +rln +� f−D +w−K∗ +� +. Therefore the policyholder will achieve the financial goal if he/she dies +after t0, and this occurs with probability e−λt0. In summary, the maximum probability +of achieving the financial goal is e−λm − e−λ(m+n) + e−λt0e−λ(m+n). +If the policyholder purchases n-year term life insurance with a single premium, +the maximum probability and optimal strategy can be obtained if m = 0 is assumed, +apparently P(δd > m) = 1, a.s.. +Corollary 2.7. When m = 0, n > 0, the maximum probability of achieving the +financial goal f on �L is given by +0�ϕn(w, D) = +� +� +� +� +� +� +� +� +� +� w +K∗ +� λ +r � +1 − e−λn� +, +0 ⩽ w < K∗, +� +1 − e−λn� ++ e−λn +�w − K∗ +f − D +� λ +r +, +K∗ ⩽ w < w∗, +in which K∗ = 0�Kn(f − D), w∗ = (0�Kn + 1)(f − D). +The related optimal insurance purchasing strategy is not to purchase additional +insurance until wealth reaches w∗, at which point, it is optimal to buy n-year term life +insurance of f − D. +When m = 0 and n → ∞, our problem changes to how the policyholder reach a +given bequest goal by purchasing whole life insurance, the maximum probability and +life insurance purchasing strategies we give are consistent with the main results in +Bayraktar et al. (2014). +Corollary 2.8. When m = 0 and n → ∞, then the maximum probability of +achieving the financial goal f is given by +0�ϕ∞(w, D) = +� +w +0�K∞(f − D) +� λ +r +, +0 ⩽ w < 0�K∞(f − D). +The related optimal insurance purchasing strategy is not to purchase additional +insurance until wealth reaches 0�K∞(f − D), at which point, it’s optimal to buy whole +This manuscript is for review purposes only. + +10 +Y. Q. LI AND L. H. ZHANG +life insurance of f − D. +In particular, when m > 0, n → ∞, the policyholder purchases m-year deferred +whole life insurance to achieve the financial goal, our viewpoint also sheds light on +reaching a given bequest goal by purchasing deferred whole life insurance. +Corollary 2.9. When m > 0, n → ∞, then the maximum probability of achiev- +ing the financial goal is given by +m�ϕ∞(w, D) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� w +K∗ +� λ +r +e−λm, +0 ⩽ w < K∗, +e−λm, +K∗ ⩽ w < w1, +�w − K∗ +f − D +� λ +r +, +w1 ⩽ w < w∗, +in which K∗ = m�K∞(f −D), w1 = (e−rm + m�K∞)(f −D) , w∗ = (m�K∞ +1)(f −D) +and m�K∞ < e−rm. +The related optimal purchasing strategy is not to purchase additional insurance +until wealth reaches w∗, at which point, it’s optimal to buy m-year deferred whole life +insurance of f − D. +2.2. m-year deferred n-year term life insurance purchased by a contin- +uously paid premium. Assume that the policyholder buys m-year deferred n-year +term life insurance via a premium paid continuously at the rate of m�Hn per dollar of +insurance, +m�Hn = (1 + θ)m| ¯A1x:n +¯ax:m +, +m| ¯A1x:n = +� m+n +m +λe−(r+λ)tdt, +¯A1x:m = +� m +0 +λe−(r+λ)tdt, +¯ax:m = 1 − ¯Ax:m +r += 1 − ¯A1x:m − Ax: +1 +m +r +, +Ax: +1 +m = +� ∞ +m +λe−rne−λtdt, +in which θ is the proportional risk loading, the explanation of the corresponding +symbols see Promislow (2011), Xiong and Shen (2020). +The proportional loading +includes costs, profit, and risk margin, and reserves are established. The policyholder +may change the amount of his/her insurance coverage at any time. +In our time- +homogeneous scenario, the policyholder in this section purchases instantaneous m- +year deferred n-year term life insurance. In this case, the wealth follows the dynamics +� +� +� +dW(t) = (rW(t) − m�HnD(t)1{t⩽m})dt, +0 ⩽ t < δd, +W(δd) = W(δd−) + D(δd−)1{m<δd⩽m+n}. +An admissible insurance strategy D = {D(t)}t⩾0 is any non-negative process, but +for all t ⩾ 0, the probability of W(t) ⩾ 0 doesn’t equal one because of the negative +drift term m�HnD(t). Thus, we define δ0 = inf {t ⩾ 0 : W(t) ⩾ 0} and the maximum +This manuscript is for review purposes only. + +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY11 +probability of achieving the financial goal f +m�Φn(w) = sup +D +� +Pw(W(δd ∧ δ0) ⩾ f | δd < m)P(δd < m) ++ Pw(W(δd ∧ δ0) ⩾ f | m ⩽ δd ⩽ m + n)P(m ⩽ δd ⩽ m + n) ++ Pw(W(δd ∧ δ0) ⩾ f | δd > m + n)P(δd > m + n) +� +:= sup +D +� +Pw +1 + Pw +2 + Pw +3 +� +, +in which Pw denotes the conditional probability given W(0−) = w. Pw +1 = Pw(W(δd∧ +δ0) ⩾ f | δd < m)P(δd < m), and the definitions of Pw +i , i = 2, 3 are similar. Analogous +to the previous section, we give the following notations +m�Φn(w) := m�Φ1 +n(w) + m�Φ2 +n(w) + m�Φ3 +n(w), +m�Φ2,3 +n (w) := m�Φ2 +n(w) + m�Φ3 +n(w), +in which +m�Φi +n(w) = sup +D +Pw +i , i = 1, 2, 3. +Remark 2.10. (1). We first define the “quasi-ideal value” and the “ideal value”. +If the wealth is equal to +m�Hnf +r+ m�Hn := H∗ which follows from the equation rw = +m�Hn(f − w), and H∗ is called the “quasi-ideal value”. That’s, when wealth reaches +H∗, the policyholder purchases m-year deferred n-year term life insurance of f − H∗ +via a premium paid continuously, and if he/she survives more than m years after +purchasing insurance, but dies within m + n years, then his/her total death benefit is +f. However, if the policyholder dies within m + n years after purchasing insurance, +then he/she cannot receive the death benefit, so in this case, the policyholder cannot +achieve the financial goal f. We therefore call H∗ the “quasi-ideal value”. +(2). Assume the “ideal value” is w∗, if wealth equals w∗, it’s optimal for the policy- +holder to purchase m-year deferred n-year term life insurance of f −H∗, then whether +or not he/she can receive the death benefit, he/she will achieve the financial goal f. +Thus, m�Φn(w) = 1 for w ⩾ w∗. Derived from our setting, we obtain w∗ by the +following equation +rw∗ − m�Hn +� +f − H∗� += f − w∗. +Thus, we get +w∗ = (r + m�Hn + r m�Hn)f +(r + m�Hn)(r + 1) +. +Proposition 2.11. (1). If λ ⩽ r, then the maximum probability of achieving the +This manuscript is for review purposes only. + +12 +Y. Q. LI AND L. H. ZHANG +financial goal f before ruining is given by +m�Φn(w) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� w +H∗ +� λ +r � +e−λm − e−λ(m+n) +� +, +0 ⩽ w < H∗, +� +e−λm − e−λ(m+n) +� ++ e−λ(m+n) +� w − H∗ +w∗ − H∗ +� λ +r +, +H∗ ⩽ w < w0, +� +1 + e−λ(m+n) +�� w − H∗ +w∗ − H∗ +� λ +r +− e−λ(m+n), +w0 ⩽ w < w∗, +in which w0 = e−rm(w∗ − H∗) + H∗ and the initial wealth w ∈ [0, w∗). +The related optimal insurance purchasing strategy is not to purchase until wealth +reaches w∗, at which point, it’s optimal to buy m-year deferred n-year term life in- +surance of f − H∗ = +rf +r+ m�Hn . +(2). If λ > r, then the maximum probability of achieving the financial goal f +before ruining is given by +m�Φn(w) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 − +� +1 − w +H∗ +� +λ +r+ m�Hn �� +e−λm − e−λ(m+n)� +, +0 ⩽ w < w0, +� w +H∗ +� λ +r � +e−λm − e−λ(m+n) +� +, +w0 ⩽ w < H∗, +� +e−λm − e−λ(m+n) +� ++ e−λ(m+n) +� w − H∗ +w∗ − H∗ +� λ +r +, +H∗ ⩽ w < w1, +� +1 + e−λ(m+n) +�� w − H∗ +w∗ − H∗ +� λ +r +− e−λ(m+n), +w1 ⩽ w < w∗, +in which w1 = e−rm(w∗ − H∗) + H∗, and the initial wealth w ∈ [0, w∗), where w0 is +the unique zero in (0, H∗) of the following equations +1 − +� +1 − w +H∗ +� +λ +r+ m�Hn += +� w +H∗ +� λ +r . +The related optimal purchasing strategy is: +• If wealth w is less than w0, then the policyholder purchase m-year deferred +n-year term life insurance of f − w; +• If wealth w is greater than or equal to w0, then the policyholder doesn’t pur- +chase insurance until the wealth reaches w∗, at which point, it’s optimal to +buy m-year deferred n-year term life insurance of f − H∗ = +rf +r+ m�Hn . +To prove Proposition 2.11, we first give some auxiliary lemmas. +Lemma 2.12. Let F = F(w) be a function that is non-decreasing, continuous, +and piecewise differentiable on [0, w∗), except that F might not be differentiable at 0. +Suppose F satisfies the following variational inequality on (0, w∗) +λF = rwFw + +� +max +� +λ(e−λm − e−λ(m+n)) − m�Hn(f − w)Fw, 0 +�� +1{w 0. +Secondly, we suppose that in a neighborhood of H∗, the policyholder buys no +insurance, then we need to solve the equation +� +λ m�Φ2,3 +n += rw(m�Φ2,3 +n )w, +m�Φ2,3 +n +� +H∗� += e−λm − e−λ(m+n). +We denote the solution of this equation by m�Φ2,3 +b,n, then m�Φ2,3 +b,n is given as follows: +m�Φ2,3 +b,n(w) = +� +e−λm − e−λ(m+n) +�� w +H∗ +� λ +r +. +(ii). If wealth is greater than or equal to H∗, then the optimal insurance is that the +policyholder buys m-year deferred n-year term life insurance of f − H∗, we just need +This manuscript is for review purposes only. + +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY15 +to solve the equation(2.9). +(iii). By the above analyses, we need to determine which of m�Φ2,3 +a,n and m�Φ2,3 +b,n is +greater for w near H∗, spontaneously, there are two cases to discuss: λ ⩽ r and +λ > r. +• If λ ⩽ r, that means the policyholder has enough time to reach the “ideal +value”, then m�Φ2,3 +a,n ⩽ m�Φ2,3 +b,n for all 0 ⩽ w ⩽ w∗. If wealth is less than H∗, +the maximum probability of achieving the financial goal is equal to +� w +H∗ +� λ +r +multiply P(m < δd ⩽ m + n). The method for solving +� w +H∗ +� λ +r as in Bayrak- +tar et al. (2014). If wealth is greater than or equal to H∗, the policyholder +purchases m-year deferred n-year term life insurance via a premium paid +continuously, and if the policyholder survives more than m years after pur- +chasing insurance, but dies within m+n years, then he/she can get the death +benefit and achieve the financial goal. But in another case where the pol- +icyholder cannot receive the death benefit, we need to find time t1 which +satisfies(w − H∗)ert1 = w∗ − H∗, and the probability of reaching the ”ideal +value” before dying is equal to e−λt1, so, in this case, the probability of +achieving the financial goal is e−λt1e−λ(m+n); +• If λ > r, then there’s a wealth level w0 such that m�Φ2,3 +n +includes m�Φ2,3 +a,n and +m�Φ2,3 +b,n, and if the initial wealth w is small enough (of course, w < H∗), we +set w < w0. In this situation, the policyholder has no enough time to reach +the ”ideal value”. Therefore, the optimal purchasing strategy is to purchase +m-year deferred n-year term life insurance of f − w. The wealth at time t +satisfies the following equation +H∗ − W(t) = +� +H∗ − w +� +e(r+ m�Hn)t. +Let t2 satisfy that W(t2) = 0, then the probability that the policyholder +achieves the financial goal equals (1 − e−λt2)(e−λm − e−λ(m+n)). If the initial +wealth w ⩾ w0, then the results as same as λ ⩽ r. +For more details, see Lemma 2.13 and Lemma 2.14. +Lemma 2.13. If λ ⩽ r, then the maximum probability of achieving the financial +goal f before ruining is given by +(2.12) +m�Φ2,3 +n (w) = +� +� +� +� +� +� +� +� +� +� w +H∗ +� λ +r � +e−λm − e−λ(m+n)� +, +0 ⩽ w < H∗, +� +e−λm − e−λ(m+n)� ++ e−λ(m+n) +� w − H∗ +w∗ − H∗ +� λ +r +, +H∗ ⩽ w < w∗, +for initial wealth w ∈ [0, w∗). +The related optimal insurance purchasing strategy is not to purchase until wealth +reaches w∗, at which point, it’s optimal to buy m-year deferred n-year term life in- +surance of f − H∗ = +rf +r+ m�Hn . +Proof. Our general guideline is to use verification Lemma 2.12 to prove this +lemma. +Firstly, we notice that m�Φ2,3 +n +in (2.12) is continuous and increasing on [0, w∗), +and is piecewise differentiable on (0, w∗), obviously, we can verify that m�Φ2,3 +n +satisfies +the variational inequality (2.8) when H∗ ⩽ w < w∗. +Then, we prove that the inequality λ(e−λm−e−λ(m+n))− m�Hn(f −w)(m�Φ2,3 +n )w ⩽ +This manuscript is for review purposes only. + +16 +Y. Q. LI AND L. H. ZHANG +0 holds on (0, H∗). By calculation we have +λ(e−λm − e−λ(m+n)) − m�Hn(f − w)(m�Φ2,3 +n )w = λ(e−λm − e−λ(m+n)) +− λ(r + m�Hn)(f − w) m�Hn +r m�Hnf +(e−λm − e−λ(m+n)) +� w +H∗ +� λ +r −1 +, +thus, λ(e−λm − e−λ(m+n)) − m�Hn(f − w)(m�Φ2,3 +n )w ⩽ 0 is equivalent to +1 − r + m�Hn +r +� w +H∗ +� λ +r −1 ++ m�Hn +r +� w +H∗ +� λ +r ⩽ 0. +Let y = +w +H∗ , obviously y ∈ (0, 1). The above inequality is equivalent to +1 − r + m�Hn +r +y +λ +r −1 + m�Hn +r +y +λ +r ⩽ 0. +Referring to Bayraktar et al. (2014), this inequality holds on y ∈ (0, 1) obviously. +Therefore, we have proved that m�Φ2,3 +n +in (2.12) satisfies the variational inequality +(2.8). When w = 0, obviously, m�Φ2,3 +n (w, D) = 0. The optimal insurance strategy is +not to purchase until wealth reaches w∗, at which time, it’s optimal to buy m-year +deferred n-year term life insurance of f − H∗ = +rf +r+ m�Hn . +Lemma 2.14. If λ > r, then the maximum probability of achieving the financial +goal f before ruining is given by +(2.13) +m�Φ2,3 +n (w) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 − +� +1 − w +H∗ +� +λ +r+ m�Hn �� +e−λm − e−λ(m+n) +� +, +0 ⩽ w < w0, +� w +H∗ +� λ +r � +e−λm − e−λ(m+n) +� +, +w0 ⩽ w < H∗, +� +e−λm − e−λ(m+n) +� ++ e−λ(m+n) +� +w − H∗ +w∗ − H∗ +� λ +r +, +H∗ ⩽ w < w∗, +for initial wealth w ∈ [0, w∗), where w0 is the unique zero in (0, H∗) of the following +equations +(2.14) +1 − +� +1 − w +H∗ +� +λ +r+ m�Hn += +� w +H∗ +� λ +r +. +The associated optimal purchasing strategy is: +• If wealth w is less than w0, then the policyholder purchases m-year deferred +n-year term life insurance of f − w; +• If wealth w is greater than or equal to w0, then the policyholder doesn’t pur- +chase insurance until the wealth reaches w∗, at which point, it’s optimal to +buy m-year deferred n-year term life insurance of f − H∗ = +rf +r+ m�Hn . +Proof. As similar to our previous proof, our general guideline is to use verification +Lemma 2.12 to prove this proposition. Firstly, we notice that m�Φ2,3 +n +in (2.13) is con- +tinuous and increasing on [0, w∗] and is piecewise differentiable on (0, w∗), obviously, +we can verify that m�Φ2,3 +n +satisfies the variational inequality (2.8) when H∗ ⩽ w < w∗. +Next, we refer to the “Lemma 3.4 and Proposition 3.5” in Bayraktar et al. (2014) to +This manuscript is for review purposes only. + +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY17 +prove the following three things: +(a). The equation (2.14) has a unique zero on (0, H∗) that is w0; +(b). On [0, w0), λm�Φ2,3 +n += rw(m�Φ2,3 +n )w+λ(e−λm−e−λ(m+n))− m�Hn(f −w)(m�Φ2,3 +n )w +and the inequality λ(e−λm − e−λ(m+n)) − m�Hn(f − w)(m�Φ2,3 +n )w ⩾ 0 holds, then the +optimal purchasing strategy is to buy m-year deferred n-year term life insurance of +f − w; +(c). On [w0, H∗) , λ m�Φ2,3 +n += rw(m�Φ2,3 +n )w and the inequality λ(e−λm − e−λ(m+n)) − +m�Hn(f − w)(m�Φ2,3 +n )w +⩽ 0 holds, then the optimal strategy is not to buy insurance until the wealth reaches +w∗. +The details of above three things are the straightforward application of Bayraktar +et al. (2014). +Therefore, m�Φ2,3 +n +in (2.13) satisfies the variational inequality (2.8). +When w = 0, obviously, m�Φ2,3 +n (w, D) = 0, and our conclusions hold. +Proof of Proposition 2.11: Step1. We first calculate m�Φ1 +n. +(i). If λ ⩽ r, w ⩾ H∗, then we set t∗ satisfies +� +w −H∗� +ert∗ = w∗ −H∗, the maximum +probability of achieving the financial goal f is as follows: +m�Φ1 +n(w) = +� m +t∗ +λe−λtdt = +� w − H∗ +w∗ − H∗ +� λ +r +− e−λm, +in which w satisfies w ⩾ e−rm(w∗ − H∗) + H∗ because of m�Φ1 +n(w) ⩾ 0. If w < +e−rm(w∗ − H∗) + H∗, then m�Φ1 +n(w) = 0. +(ii). If λ ⩽ r, w < H∗, then we set t∗ satisfies wert∗ = w∗, the maximum probability +of achieving the financial goal f is as follows: +m�Φ1 +n(w) = +� m +t∗ +λe−λtdt = +� w +w∗ +� λ +r +− e−λm, +in which w satisfies w ⩾ e−rmw∗ because of m�Φ1 +n(w) ⩾ 0. If w < e−rmw∗, then +m�Φ1 +n(w) = 0, when w ⩾ +(iii). If λ > r, w < H∗, in this case, obviously, m�Φ1 +n(w) = 0. +(iv). If λ > r, w ⩾ H∗, the results are same as (i). +Step2. For m�Φ2,3 +n , the relevant conclusions have been given in the previous Lemma +2.12, Lemma 2.13 and Lemma 2.14, as same as the previous section, we generally +assume e−rm > +m�Hn+r m�Hn +r+ m�Hn+r m�Hn , then the Proposition 2.11 is proved. +■ +If the policyholder purchases n-year term life insurance through a continuously +paid premium, the maximum probability and the optimal strategies can be obtained +by taking m = 0, apparently P(δd > m) = 1, a.s.. We have the following Corollary +2.15. +Corollary 2.15. (1). When m = 0, n > 0 and if λ ⩽ r, then the maximum +probability of achieving the financial goal f before ruining is given by +0�Φn(w) = +� +� +� +� +� +� +� +� +� +� w +H∗ +� λ +r � +1 − e−λn +� +, +0 ⩽ w < H∗, +� +1 − e−λn +� ++ e−λn +� w − H∗ +w∗ − H∗ +� λ +r +, +H∗ ⩽ w < w∗, +for initial wealth w ∈ [0, w∗), in which H∗ = +0�Hnf +r+ 0�Hn , w∗ = (r+ 0�Hn+r 0�Hn)f +(r+ 0�Hn)(r+1) . +The related optimal insurance purchasing strategy is not to purchase until wealth +This manuscript is for review purposes only. + +18 +Y. Q. LI AND L. H. ZHANG +reaches w∗, at which point, it’s optimal to buy n-year term life insurance of f −H∗ = +rf +r+ 0�Hn . +(2). When m = 0, n > 0 and if λ > r, then the maximum probability of achieving the +financial goal f before ruining is given by +0�Φn(w) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 − +� +1 − w +H∗ +� +λ +r+ 0�Hn �� +1 − e−λn +� +, +0 ⩽ w < w0, +� w +H∗ +� λ +r � +1 − e−λn +� +, +w0 ⩽ w < H∗, +1 − e−λn + e−λn +� w − H∗ +w∗ − H∗ +� λ +r +, +H∗ ⩽ w < w∗, +for initial wealth w ∈ [0, w∗), where w0 is the unique zero in (0, H∗) of the following +equations +1 − +� +1 − w +H∗ +� +λ +r+ 0�Hn += +� w +H∗ +� λ +r +. +The related optimal purchasing strategy is +• If wealth w is less than w0, then the policyholder purchases n-year term life +insurance of f − w; +• If wealth w is greater than or equal to w0, then the policyholder doesn’t pur- +chase insurance until the wealth reaches w∗, at which time, it’s optimal to +buy n-year term life insurance of f − H∗ = +rf +r+ 0�Hn . +When m = 0 and n → ∞, our problem is equivalent to the problem of maximizing +probability of reaching a given bequest goal in Bayraktar et al. (2014), the concrete +results in Corollary 2.16 below. +Corollary 2.16. (1). When m = 0, n → ∞ and if λ ⩽ r, then the maximum +probability of achieving the financial goal f before ruining is given by +0�Φ∞(w) = +� w +H∗ +� λ +r +, +for initial wealth w ∈ [0, H∗), in which H∗ = +0�H∞f +r+ 0�H∞ . +The related optimal purchasing strategy is not to purchase until wealth reaches +H∗, at which point, it’s optimal to buy life insurance of f − H∗ = +rf +r+ 0�H∞ . +(2). When m = 0, n → ∞ and if λ > r, then the maximum probability of achieving +the financial goal f before ruining is given by +0�Φ∞(w) = +� +� +� +� +� +� +� +� +� +1 − +� +1 − w +H∗ +� +λ +r+ 0�H∞ +, +0 ⩽ w < w0, +� w +H∗ +� λ +r +, +w0 ⩽ w < H∗, +for initial wealth w ∈ [0, H∗), where w0 is the unique zero in (0, H∗) of the following +This manuscript is for review purposes only. + +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY19 +equations +1 − +� +1 − w +H∗ +� +λ +r+ 0�H∞ += +� w +H∗ +� λ +r +. +The related optimal purchasing strategy is +• If wealth w is less than w0, then the policyholder purchases life insurance of +f − w; +• If wealth w is greater than or equal to w0, then the policyholder doesn’t pur- +chase insurance until the wealth reaches H∗, at which point, it’s optimal to +buy life insurance of f − H∗ = +rf +r+ 0�H∞ . +When m > 0, n → ∞, the problem turns to consider buying m-year deferred whole +life insurance, assume +m�H∞ +r+ m�H∞ < e−rm, the corresponding results are as follows. +Corollary 2.17. (1). When m > 0, n → ∞ and if λ ⩽ r, then the maximum +probability of achieving the financial goal f before ruining is given by +m�Φ∞(w) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� w +H∗ +� λ +r +e−λm, +0 ⩽ w < H∗, +e−λm, +H∗ ⩽ w < w0, +� w − H∗ +w∗ − H∗ +� λ +r +, +w0 ⩽ w < w∗, +in which w0 = e−rm(w∗ − H∗) + H∗, H∗ = +m�H∞f +r+ m�H∞ , w∗ = (r+ m�H∞+r m�H∞)f +(r+ m�H∞)(r+1) +and +the initial wealth w ∈ [0, w∗). +The related optimal purchasing strategy is not to purchase until wealth reaches w∗, +at which point, it’s optimal to buy m-year deferred whole life insurance of f − H∗ = +rf +r+ m�H∞ . +(2). When m > 0, n → ∞ and if λ > r, then the maximum probability of achieving +the financial goal f before ruining is given by +m�Φ∞(w) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 − +� +1 − w +H∗ +� +λ +r+ m�H∞ � +e−λm, +0 ⩽ w < w0, +� w +H∗ +� λ +r +e−λm, +w0 ⩽ w < H∗, +e−λm, +H∗ ⩽ w < w1, +� w − H∗ +w∗ − H∗ +� λ +r +, +w1 ⩽ w < w∗, +in which w1 = e−rm(w∗ − H∗) + H∗ and the initial wealth w ∈ [0, w∗), where w0 is +the unique zero in (0, H∗) of the following equations +1 − +� +1 − w +H∗ +� +λ +r+ m�H∞ += +� w +H∗ +� λ +r +. +The related optimal purchasing strategy is: +This manuscript is for review purposes only. + +20 +Y. Q. LI AND L. H. ZHANG +• If wealth w is less than w0, then the policyholder purchase m-year deferred +whole life insurance of f − w; +• If wealth w is greater than or equal to w0, then the policyholder doesn’t pur- +chase insurance until the wealth reaches w∗, at which point, it’s optimal to +buy m-year deferred whole life insurance of f − H∗ = +rf +r+ m�H∞ . +3. Purchasing m-year deferred n-year term pure endowment in per- +sonal financial planning under the deterministic framework. The policy- +holder purchases m-year deferred n-year term pure endowment in financial planning +to make a longevity risk protection plan. Assume he/she has the financial goal f to +ensure adequate pension at time τ which is the retirement moment, and τ follows an +exponential distribution with parameter λ. Assumptions and methods are as similar +as that in Section 2, and we also add the time cutoff n and m, but because of the +policyholder must live for m + n years, so the discussion about n , m and τ changes. +By the same way as that in previous Section 2, we will directly give the relevant +conclusions in this section. +3.1. m-year deferred n-year term pure endowment purchased by a +single premium. The policyholder purchases m-year deferred n-year term pure en- +dowment by a single premium for m�Rn per dollar of insurance, in which +m�Rn = (1 + θ) +� ∞ +m+n +e−(m+n)rλe−λtdt = (1 + θ)e−(m+n)(r+λ). +Similar to Section 2, we restrict premium m�Rn < e−rm. +The wealth follows the +dynamics +� +� +� +dW(t) = rW(t−)dt − m�RndD(t), +0 ⩽ t < τ, +W(τ) = W(τ−) + D(τ−)1{τ>m+n}. +We define the maximum probability of achieving the financial goal f is m�φn as follows: +m�φn(w, D) = sup +D +� +Pw,D(W(τ) ⩾ f | τ ⩽ m + n)P(τ ⩽ m + n) ++ Pw,D(W(τ) ⩾ f | τ > m + n)P(τ > m + n) +� +. +Proposition 3.1. The maximum probability of achieving the financial goal is +given by +m�φn(w, D) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +w +R∗ +� λ +r e−λ(m+n), +0 ⩽ w < R∗, +e−λ(m+n) + +� +e−λm − e−λ(m+n) +� � +w−R∗ +f−D +� λ +r , +R∗ ⩽ w < w0, +� +1 + e−λm − e−λ(m+n) +� � +w−R∗ +f−D +� λ +r − e−λm + e−λ(m+n), +w0 ⩽ w < w∗, +in which R∗ = m�Rn(f −D), w0 = (e−rm+ m�Rn)(f −D) and w∗ = (m�Rn+1)(f −D). +The related optimal purchasing strategy is not to purchase until wealth reaches +w∗, at which point, it’s optimal to buy additional m-year deferred n-year term pure +endowment of f − D. +This manuscript is for review purposes only. + +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY21 +Corollary 3.2. When m = 0, n ⩾ 0, the maximum probability of achieving the +financial goal f on L is given by +0�φn(w, D) = +� +� +� +� +� +� +� +� +� +� w +R∗ +� λ +r +e−λn, +0 ⩽ w < R∗, +e−λn + +� +1 − e−λn +� �w − R∗ +f − D +� λ +r +, +R∗ ⩽ w < w∗, +in which R∗ = 0�Rn(f − D), w∗ = (0�Rn + 1)(f − D). +The related optimal purchasing strategy is not to purchase until wealth reaches w∗, +at which point, it’s optimal to buy additional n-year term pure endowment of f − D. +3.2. m-year deferred n-year term pure endowment purchased by a con- +tinuously paid premium. We work out the problem about buying instantaneous +m-year deferred n-year term pure endowment via a premium paid continuously at the +rate of m�Mn per dollar of insurance and +m�Mn+r m�Mn +r+ m�Mn+r m�Mn < e−rm, in which +m�Mn = (1 + θ)m|¯ax:n +¯ax:m +. +The wealth satisfies the following dynamics +� +� +� +dW(t) = (rW(t) − m�MnD(t)1{t⩽m})dt, +0 ⩽ t < τ, +W(τ) = W(τ−) + D(τ−)1{τ>m+n}. +The maximum probability of achieving the financial goal f is as follows: +m�Φn(w) = sup +D +� +Pw(W(τ ∧ δ0) ⩾ f | τ ⩽ m + n)P(τ ⩽ m + n) ++ Pw(W(τ ∧ δ0) ⩾ f | τ > m + n)P(τ > m + n) +� +. +Proposition 3.3. (1). If λ ⩽ r, then the maximum probability of achieving the +financial goal f before ruining is given by +m�Φn(w) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� w +M ∗ +� λ +r +e−λ(m+n), +0 ⩽ w < M ∗, +e−λ(m+n) + +� +e−λm − e−λ(m+n) +�� w − M ∗ +w∗ − M ∗ +� λ +r +, +M ∗ ⩽ w < w0, +� +1 + e−λm − e−λ(m+n) +�� w − M ∗ +w∗ − M ∗ +� λ +r +− e−λm + e−λ(m+n), +w0 ⩽ w < w∗, +in which M ∗ = +m�Mnf +r+ m�Mn , w0 = e−rm(w∗ − M ∗) + M ∗, w∗ = (r+ m�Mn+r m�Mn)f +(r+ m�Mn)(r+1) +and +the initial wealth w ∈ [0, w∗). +The related optimal purchasing strategy is not to purchase until wealth reaches +w∗, at which point, it’s optimal to buy m-year deferred n-year term pure endowment +of f − M ∗ = +rf +r+ m�Mn . +(2). If λ > r, then the maximum probability of achieving the financial goal f before +This manuscript is for review purposes only. + +22 +Y. Q. LI AND L. H. ZHANG +ruining is given by +m�Φn(w) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 − +� +1 − w +M ∗ +� +λ +r+ m�Mn � +e−λ(m+n), +0 ⩽ w < w0, +� w +M ∗ +� λ +r +e−λ(m+n), +w0 ⩽ w < M ∗, +e−λ(m+n) + +� +e−λm − e−λ(m+n) +�� w − M ∗ +w∗ − M ∗ +� λ +r +, +M ∗ ⩽ w < w1, +� +1 + e−λm − e−λ(m+n) +�� w − M ∗ +w∗ − M ∗ +� λ +r +− e−λm + e−λ(m+n), +w1 ⩽ w < w∗, +in which w1 = e−rm(w∗ − M ∗) + M ∗, the initial wealth w ∈ [0, w∗), where w0 is the +unique zero in (0, M ∗) of the following equations +1 − +� +1 − w +M ∗ +� +λ +r+ m�Mn += +� w +M ∗ +� λ +r +. +The related optimal purchasing strategy is +• If wealth w is less than w0, then the policyholder purchases m-year deferred +n-year term pure endowment of f − w; +• If wealth w is greater than or equal to w0, then the policyholder doesn’t pur- +chase until the wealth reaches w∗, at which point, it’s optimal to buy m-year +deferred n-year term pure endowment of f − M ∗ = +rf +r+ m�Mn . +Corollary 3.4. (1). When m = 0, n ⩾ 0 and if λ ⩽ r, then the maximum +probability of achieving the financial goal f before ruining is given by +0�Φn(w) = +� +� +� +� +� +� +� +� +� +� w +M ∗ +� λ +r +e−λn, +0 ⩽ w < M ∗, +e−λn + +� +1 − e−λn +� � w − M ∗ +w∗ − M ∗ +� λ +r +, +M ∗ ⩽ w < w∗, +in which M ∗ = +0�Mnf +r+ 0�Mn , w∗ = (r+ 0�Mn+r 0�Mn)f +(r+ 0�Mn)(r+1) +, for initial wealth w ∈ [0, w∗). +The related optimal n-year term pure endowment purchasing strategy is not to +purchase until wealth reaches w∗, at which point, it’s optimal to buy n-year term pure +endowment of f − M ∗ = +rf +r+ 0�Mn . +(2). When m = 0, n ⩾ 0 and if λ > r, then the maximum probability of achieving the +financial goal f before ruining is given by +0�Φn(w) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 − +� +1 − w +M ∗ +� +λ +r+ 0�Mn � +e−λn, +0 ⩽ w < w0, +� w +M ∗ +� λ +r +e−λn, +w0 ⩽ w < M ∗, +e−λn + +� +1 − e−λn +� � w − M ∗ +w∗ − M ∗ +� λ +r +, +M ∗ ⩽ w < w∗, +This manuscript is for review purposes only. + +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY23 +for initial wealth w ∈ [0, w∗), where w0 is the unique zero in (0, M ∗) of the following +equations +1 − +� +1 − w +M ∗ +� +λ +r+ 0�Mn += +� w +M ∗ +� λ +r +. +The related optimal purchasing strategy is +• If wealth w is less than w0, then the policyholder purchases n-year term pure +endowment of f − w; +• If wealth w is greater than or equal to w0, then the policyholder doesn’t pur- +chase until the wealth reaches w∗, at which point, it’s optimal to buy n-year +term pure endowment of f − M ∗ = +rf +r+ 0�Mn . +4. Purchasing n-year term life insurance in personal financial planning +under the stochastic framework. Assume the policyholder has an account to +achieve a financial goal, and this account include the consuming, incoming and risky +investment. Specifically, he/she invest in a financial market which including two parts, +one is investing risk-free asset to earn interest, we denote the force of interest is r, +another is investing in a risky market whose process S = {S(t)}t⩾0 follows geometric +Brownian motion. The individual buys n-year term life insurance via a premium paid +continuously at the rate of Hn per dollar of insurance. In this section, the definitions +of δd, δ0, W(t), D(t) are same as the previous sections. +4.1. n-year term life insurance purchased by a continuously paid pre- +mium under the model I. The specific components of model I are as follows +� +� +� +� +� +dS(t) = µS(t)dt + σS(t)dB(t), +dY (t) = aW(t)dt + ldB(t), +c(W) = cW(t), +in which Y = {Y (t)}t⩾0 represents the price process of income, c = {cW(t)}t⩾0 +represents the consumption rate. B = {B(t)}t⩾0 is a standard Brownian motion on +a filtered probability space (Ω, F, F = {Ft}t⩾0 , P), µ, σ, a, l, c are all constants, in +particular, we assume µ > r and c < r + a. +Denote πt is the amount invested in the risky market at time t ⩾ 0. An investment +strategy Π = {πt}t⩾0 is admissible if it is an F-progressively measurable process +satisfying +� t +0 π2 +sds < ∞. An admissible insurance strategy D = {D(t)}t⩾0 is any non- +negative and F-progressively measurable process, but for all t ⩾ 0, the probability of +W(t) ⩾ 0 doesn’t equal one because of the negative drift term HnD(t). Therefore, +the wealth follows the dynamics +� +dW(t) = +� +(r + a − c)W(t) + (µ − r)πt − HnD(t)1{t⩽n} +� +dt + (σπt + l)dBt, +0 ⩽ t < δd, +W(δd) = W(δd−) + D(δd−)1{δd⩽n}. +The maximum probability of achieving the financial goal f +Φn(w) = sup +D +� +Pw(W(δd ∧ δ0) ⩾ f | δd ⩽ n)P(δd ⩽ n) ++ Pw(W(δd ∧ δ0) ⩾ f | δd > n)P(δd > n) +� +:= sup +D +� +Pw +1 + Pw +2 +� +, +This manuscript is for review purposes only. + +24 +Y. Q. LI AND L. H. ZHANG +in which Pw denotes conditional probability given W(0) = w ⩾ 0. +Remark 4.1. (1). Firstly, we define“quasi-ideal value” and“ideal value”. If wealth +equals wq0 = +Hnf +r+a−c+Hn which is solved from the equation (r+a−c)wq0 = Hn(f−wq0), +then it’s called the “quasi-ideal value”. It means that if wealth reaches +Hnf +r+a−c+Hn , then +the policyholder purchases n-year term life insurance of f − +Hnf +r+a−c+Hn via a premium +paid continuously, and if he/she dies within n years after purchasing insurance, then +the individual’s total death benefit becomes f. If the policyholder dies after n years +of purchasing insurance, he/she may not receive the death benefit, so in this case, +he/she can’t achieve the financial goal f. Thus, we call +Hnf +r+a−c+Hn is the “quasi-ideal +value”. +(2). Supposing the “ideal value” is wi0, if wealth equals wi0, then it’s optimal for the +policyholder to purchase n-year term life insurance of f − +Hnf +r+a−c+Hn , then whether or +not he/she can get the death benefit, he/she will achieve the financial goal f. Deriving +from our setting, we obtain wi0 by following equation +(r + a − c)wi0 − Hn +� +f − +Hnf +r + a − c + Hn +� += f − wi0, +thus, we get wi0 = +� +r + a − c + Hn + (r + a − c)Hn +� +f +(r + a − c + Hn)(r + a − c + 1) +. +To motivate the verification lemma for this problem, we first denote wb0 which is +called the buy level, and it’s value will be given in subsequent discussions. We give +the control equation about Φn as follows: +λ +� +Φn−(1 − e−λn)1{w⩾wb0} +� += +� +(r + a − c)w − Hn +� +f − w1� +w< +Hnf +r+a−c+Hn + +(µ−r)l +σ(r+a−c+Hn) +� +− +Hnf +r + a − c + Hn 1� +w⩾ +Hnf +r+a−c+Hn + +(µ−r)l +σ(r+a−c) +�� +1{wb0 ⩽w⩽wi0} +� +(Φn)w ++ max +π +� +(µ − r)π(Φn)w + 1 +2(σπ + l)2(Φn)ww +� +(4.1) +Step1. We simplify the last term in above control equation. +max +π +� +(µ − r)π(Φn)w + 1 +2(σπ + l)2(Φn)ww +� +(4.2) += max +π +� +(µ − r)π(Φn)w + 1 +2(σ2π2 + l2 + 2σlπ)(Φn)ww +� += max +π +�1 +2σ2(Φn)wwπ2 + [(µ − r)(Φn)w + σl(Φn)ww]π + 1 +2l2(Φn)ww +� += − 1 +2 +�µ − r +σ +�2 (Φn)2 +w +(Φn)ww +− (µ − r)l +σ +(Φn)w +Step2. We simplify the above control equation into three parts. +(i). If w < wb0 (it can be seen from the solution of the latter equation that in fact +(µ−r)l +σ(r+a−c) ⩽ w < wb0), then the control equation can be replaced with the following +equivalent expression +(4.3) +λΦn = +� +(r + a − c)w − (µ − r)l +σ +� +(Φn)w − 1 +2 +�µ − r +σ +�2 (Φn)2 +w +(Φn)ww +. +(ii). If wb0 ⩽ w < +Hnf +r+a−c+Hn + +(µ−r)l +σ(r+a−c+Hn), then the control equation can be replaced +This manuscript is for review purposes only. + +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY25 +with the following equivalent expression +(4.4) +λ +� +Φn − (1 − e−λn) +� += +� +(r + a − c + Hn)w − (Hnf + (µ − r)l +σ +) +� +(Φn)w − 1 +2 +�µ − r +σ +�2 (Φn)2 +w +(Φn)ww . +(iii). If +Hnf +r+a−c+Hn + +(µ−r)l +σ(r+a−c) ⩽ w < wi0, then the control equation can be replaced +with the following equivalent expression +(4.5) +λ +� +Φn−(1−e−λn) +� += +� +(r+a−c)w−((r + a − c)Hnf +r + a − c + Hn +(µ − r)l +σ +) +� +(Φn)w−1 +2 +�µ − r +σ +�2 (Φn)2 +w +(Φn)ww . +These observations lead to the following verification Lemma 4.2. +Lemma 4.2. Define a differential operator L π,D +1 +: +L π,D +1 +˜F = +� +(r+a−c)w+(µ−r)π−HnD +� ˜Fw + 1 +2(σπ+l)2 ˜Fww −λ +� ˜F −λ(1−e−λn)1{w+D⩾f} +� +. +Let F = F(w) be a function that is non-decreasing, continuous, and piecewise dif- +ferentiable on [0, wi0), except that F might not be differentiable at 0, w0, w1 and w2. +Suppose F satisfies Fw > 0, Fww < 0 on (0, wi0) \ +� +(0, w0) ∪ (w1, w2) +� +. If F satisfies +the following boundary-value problem on [0, wi0]: +� +maxπ,D⩾0 L π,D +1 +F(w) = 0, +F(0) = F(w0) = 0, +F(w1) = F(w2) = 1 − e−λn, +F(wi0) = 1. +Then, on [0, wi0], +Φn = F. +The optimal investment amount in risk market is +π∗ +t = −µ − r +σ +2 (Φn)w(w∗ +t ) +(Φn)ww(w∗ +t ), +in which w∗ +t is the optimal control wealth value at time t. The optimal n-year term +life insurance purchase amount is +D∗ +t = +� +f − w1{w −∞; +(2). Fw(0) < +∞. +Denote δa +n := inf +� +s > 0 : +� s +0 π2 +t dt ⩾ n +� +, let δn = δ0 ∧ δa +n, then applying the Itˆo’s +formula for e−λτnF(Wτn), we have +e−λτnF(Wτn) = F(w)+ +� δn +0 +e−λtFw(Wt)(σπt + l)dBt ++ +� δn +0 +e−λt[L π,D +1 +F(Wt) − λ(1 − e−λn)1{Wt+Dt⩾f}]dt ++ +� +0⩽t⩽δn +e−λt[F(Wt) − F(Wt−)]. +Due to +F 2 +w(w) ⩽ F 2 +w(0), w ⩾ 0, +Ew +�� δn +0 +e−2λtF 2 +w(Wt)(σπt + l)2dt +� +< ∞, +therefore +Ew +�� δn +0 +e−λtFw(Wt)(σπt + l)dBt +� += 0. +Then we have +Ew[e−λδnV ] ⩽ Ew[e−λδnF(Wδn)] ⩽ F(w) − Ew[ +� δn +0 +e−λtλ(1 − e−λn)1{Wt+Dt⩾f}dt]. +When n → ∞, then δn → ∞, naturally +F(w) ⩾ Ew +�� δ0 +0 +λe−λt(1 − e−λn)1{W (t)+D(t)⩾f}dt +� +, +that means F ⩾ Φn. +Next we prove that the conclusion F ⩾ Φn still holds when the above additional +conditions (1) and (2) are removed. Suppose εn monotonically decreases to 0, denote +F εn(w) = F(w + εn), then +0 ⩾ L π,D +1 +F(w + εn) = L π,D +1 +F εn(w) + (r + a − c + 1)εnF εn +w (w), +due to (r + a − c + 1)εnF εn +w (w) ⩾ 0, then = L π,D +1 +F εn(w) ⩽ 0, F εn satisfied the +conditions in Lemma, repeat the above steps to get F εn ⩾ Φn. +From F(w) = +lim +n→∞ F εn(w) ⩾ Φn(w), we can obtain +F ⩾ Φn. +By Wang and Young (2012), for F satisfies the boundary conditions, then F = Φn. +Then, π∗ +t can be obtained from Step1, and D∗ +t naturally be given by the control +equation (4.1). +This manuscript is for review purposes only. + +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY27 +Step3. We separately solve the above three control equations. Since the three equa- +tions are basically the same in form, we take the first equation as an example for a +specific solution. +Lemma 4.3. The solution to equation (4.3) is of the form as follows +(4.6) +Φn(w) = D1 +� +w − +(µ − r)l +σ(r + a − c) +�p +in which D1 is a constant to be determined and p is a known constant. +Proof. Let m = 1 +2 +� +µ−r +σ +�2 +, A = r + a − c, B = µ−r +σ . Then +λΦn = (Aw − B)(Φn)w − m (Φn)2 +w +(Φn)ww +. +We consider the Legendre transform: +ϕ(y) = sup +w (Φn(w) − wy). +Then, +Φn(w) = inf +y (ϕ(y) + wy) = ϕ(y) − yϕ′(y), +ϕ′(y) = −w, +ϕ′′(y) = − +1 +Φ′′n(w). +Therefore, +λ[ϕ(y) − −yϕ′(y)] = (−Aϕ′(y) − B)y + my2ϕ′′(y). +Equivalently, we obtain +(4.7) +my2ϕ′′(y) + (λ − A)yϕ′(y) − λϕ(y) − By = 0. +Firstly, we solve the Euler equation as follows +y2ϕ′′(y) + (λ − A) +m +yϕ′(y) − λ +mϕ(y) = 0. +The corresponding characteristic equation is +x2 + λ − A − m +m +x − λ +m = 0. +Solve the above characteristic equation to get the characteristic root as follows +x1 = A − λ + m − +� +(A − λ + m)2 + 4λm +2m +< 0, +x2 = A − λ + m + +� +(A − λ + m)2 + 4λm +2m +> 1, +So the general solution of this Euler equation is +˜ϕ(y) = D1yx1 + D2yx2, +This manuscript is for review purposes only. + +28 +Y. Q. LI AND L. H. ZHANG +in which D1 is a constant and D2 = 0 because ϕ′(y) ⩽ 0 holds for any y. +A particular solution to the original equation (4.7) is obtained as follows +ϕ∗(y) = ˜Dyx1 − B +Ay. +in which ˜D is a constant. +So we obtain the solution of original equation (4.7) +ϕ(y) = 2 ˜ +D1yx1 − B +Ay. +in which ˜ +D1 is a constant. +Due to the Legendre transform, we have +Φn(w) = 2 ˜ +D1(Φ′ +n(w))x1 − B +AΦ′ +n(w) + wΦ′ +n(w). +Let p = +x1 +x1−1 ∈ (0, 1), the solution Φn(w) should be of the form as follows +Φn(w) = D1 +� +w − B +A +�p +. +Substitute the value of A and B, we have +Φn(w) = D1 +� +w − +(µ − r)l +σ(r + a − c) +�p +. +in which D1 is a constant to be determined. +Similarly, we have the Lemma 4.4 and Lemma 4.5. +Lemma 4.4. The solution to equation (4.4) is of the form as follows +(4.8) +Φn(w) = 1 − e−λn + D2 +� +w − (µ − r)l + σHnf +σ(r + a − c + Hn) +�q +in which D1 is a constant to be determined, q = +k1 +k1−1 > 1, +k1 = A1 − λ − m + +� +(A1 − λ + m)2 + 4λm +2m +, +A1 = r + a − c + Hn +Lemma 4.5. The solution to equation (4.5) is of the form as follows +(4.9) +Φn(w) = 1 − e−λn + D3 +� +w − +� +Hnf +r + a − c + Hn ++ +(µ − r)l +σ(r + a − c) +��p +in which D3 is a constant to be determined, the value of p is equal to that in the +Lemma 4.3. +D1, D2, D3 can be determined according to continuity and critical value, the +specific method for their determination will be presented in the appendix. We will +directly give the concrete values of this parameters in the following propositions. +This manuscript is for review purposes only. + +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY29 +Proposition 4.6. The maximum probability of achieving the financial goal f be- +fore ruining is given by +Φn(w) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +0, +0 ⩽ w < w0, +(1 − e−λn)q(1 − p) +q − p +� w − w0 +wb0 − w0 +�p +, +w0 ⩽ w < wb0, +(1 − e−λn) +� +1 − p(q − 1) +q − p +� +w − w1 +wb0 − w1 +�q� +, +wb0 ⩽ w < w1, +1 − e−λn, +w1 ⩽ w < w2, +1 − e−λn + e−λn +� w − w2 +wi0 − w2 +�p +, +w2 ⩽ w < wi0, +in which +w0 = +(µ − r)l +σ(r + a − c) , w1 = +Hnf +r + a − c + Hn ++ +(µ − r)l +σ(r + a − c + Hn) , w2 = +Hnf +r + a − c + Hn ++ +(µ − r)l +σ(r + a − c) , +p = +k1 +k1 − 1, +k1 = r + a − c − λ + m − +� +(r + a − c − λ + m)2 + 4λm +2m +q = +k2 +k2 − 1, +k2 = r + a − c + Hn − λ + m + +� +(r + a − c + Hn − λ + m)2 + 4λm +2m +, +wb0 = (1 − p)w1 − (1 − q)w0 +q − p +, +The associated optimal n-year term life insurance purchasing strategy is not to +purchase until the wealth reaches wb0, and if wb0 ⩽ w < w1, it’s optimal to buy n-year +term life insurance of f − w; if w1 ⩽ w < wi0, it’s optimal to buy n-year term life +insurance of f − +Hnf +r+a−c+Hn . +The optimal investment strategy in risky market is not to invest until the wealth +reaches w0 + σl(1−p) +µ−r +and when the wealth is in a small range, i.e. w1 + σl(1−q) +µ−r +⩽ w < +w2 + σl(1−p) +µ−r , the individual also doesn’t invest. In other cases, the optimal investment +amount is equal to +π(w) = +� +� +� +� +� +� +� +� +� +� +� +� +� +(µ − r)(w − w0) + σl(p − 1) +σ2(1 − p) +, +w0 + σl(1−p) +µ−r +⩽ w < wb0, +(µ − r)(w − w1) + σl(q − 1) +σ2(1 − q) +, +wb0 ⩽ w < w1 + σl(1−q) +µ−r , +(µ − r)(w − w2) + σl(p − 1) +σ2(1 − p) +, +w2 + σl(1−p) +µ−r +⩽ w < wi0, +Proof. Our general guideline is to use Lemma 4.2 to prove this proposition. In +fact, it’s a direct application of Lemma 4.2. +By Lemma 4.2 and Step 2, we can get three equations (4.3), (4.4), (4.5), by +Lemma 4.3, 4.4, 4.5 and the continuity at critical points, we obtain Φn in Proposition +4.6. +Notice that Φn(w) is non-decreasing continuous, and piecewise differentiable +on [0, wi0), except at 0, w0, w1 and w2. Φn satisfies (Φn)w > 0, (Φn)ww < 0 on +(0, wi0) \ +� +(0, w0) ∪ (w1, w2) +� +, and obviously, F(0) = F(w0) = 0, F(w1) = F(w2) = +1−e−λn, F(wi0) = 1. After the above analysis, the proposition is the direct application +This manuscript is for review purposes only. + +30 +Y. Q. LI AND L. H. ZHANG +of Lemma 4.2. It is important to note that, since the investment amount in risky +market is non-negative, to ensure that it’s meaningful, we make adjustments to the +range of π(w). +4.2. n-year term life insurance purchased by a continuously paid pre- +mium under the model II. The specific components of model II are as follows. +� +� +� +� +� +dS(t) = µS(t)dt + σS(t)dB(t), +dY (t) = adt + ldB(t), +c(W) = c, +in which Y = {Y (t)}t⩾0 represents the price process of income, c represents the con- +sumption rate. B = {B(t)}t⩾0 is a standard Brownian motion on a filtered probability +space (Ω, F, F = {Ft}t⩾0 , P), µ, σ, a, l, c are all constants, µ > r. Therefore, the +wealth follows the dynamics +� +dW(t) = +� +rW(t) + (µ − r)πt − HnD(t)1{t⩽n} + a − c +� +dt + (σπt + l)dBt, +0 ⩽ t < δd, +W(δd) = W(δd−) + D(δd−)1{δd⩽n}. +4.2.1. The case for which a−c = 0. In this case, the problem is equivalent to +the case where a − c = 0 in previous section, so we directly give the conclusion. The +“quasi-ideal value” wq1 = +Hnf +r+Hn and the “ideal value” wi1 = +� +r+Hn+rHn +� +f +(r+Hn)(r+1) . +Proposition 4.7. The maximum probability of achieving the financial goal f be- +fore ruining is given by +Φn(w) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +0, +0 ⩽ w < w3, +(1 − e−λn)q(1 − p) +q − p +� w − w3 +wb1 − w3 +�p +, +w3 ⩽ w < wb1, +(1 − e−λn) +� +1 − p(q − 1) +q − p +� +w − w4 +wb1 − w4 +�q� +, +wb1 ⩽ w < w4, +1 − e−λn, +w4 ⩽ w < w5, +1 − e−λn + e−λn +� w − w2 +wi1 − w2 +�p +, +w5 ⩽ w < wi1, +in which +w3 = (µ − r)l +σr +, w4 = +Hnf +r + Hn ++ +(µ − r)l +σ(r + Hn), w5 = +Hnf +r + Hn ++ (µ − r)l +σr +, +p = +k3 +k3 − 1, +k3 = r − λ + m − +� +(r − λ + m)2 + 4λm +2m +q = +k4 +k4 − 1, +k4 = r + Hn − λ + m + +� +(r + Hn − λ + m)2 + 4λm +2m +, +wb1 = (1 − p)w4 − (1 − q)w3 +q − p +, +This manuscript is for review purposes only. + +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY31 +The associated optimal n-year term life insurance purchasing strategy is not to +purchase until the wealth reaches wb1, and if wb1 ⩽ w < w4, it’s optimal to buy n-year +term life insurance of f − w; if w4 ⩽ w < wi1, it’s optimal to buy n-year term life +insurance of f − +Hnf +r+Hn . +The optimal investment strategy in risky market is not to invest until the wealth +reaches w3 + σl(1−p) +µ−r +and when the wealth is in a small range, i.e. w4 + σl(1−q) +µ−r +⩽ w < +w5 + σl(1−p) +µ−r , the individual also doesn’t invest. In other cases, the optimal investment +amount is equal to +π(w) = +� +� +� +� +� +� +� +� +� +� +� +� +� +(µ − r)(w − w3) + σl(p − 1) +σ2(1 − p) +, +w3 + σl(1−p) +µ−r +⩽ w < wb1, +(µ − r)(w − w4) + σl(q − 1) +σ2(1 − q) +, +wb1 ⩽ w < w4 + σl(1−q) +µ−r , +(µ − r)(w − w5) + σl(p − 1) +σ2(1 − p) +, +w5 + σl(1−p) +µ−r +⩽ w < wi1, +4.2.2. The case for which a−c > 0. In this case, we firstly give the new “quasi- +ideal value” wq2 = Hnf+c−a +r+Hn +and the new “ideal value” wi2 = (r+Hn+rHn)f−(c−a)Hn +(r+Hn)(r+1) +, +which solving methods are the same as that in the model I. Denote wb2 which is called +the buy level, then the control equation about Φn as follows: +λ +� +Φn − (1 − e−λn)1{w⩾wb2} +� += +� +rw − Hn +� +f − w1� +w< Hnf+c−a +r+Hn ++ +(µ−r)l +σ(r+Hn) +� +− +Hnf +r + a − c + Hn 1� +w⩾ Hnf+c−a +r+Hn ++ (µ−r)l +σr +�� +1{wb2 ⩽w⩽wi2} + (c − a) +� +(Φn)w ++ max +π +� +(µ − r)π(Φn)w + 1 +2(σπ + l)2(Φn)ww +� +Similarly, we simplify the above control equation into three parts. +(i). If w < wb2 (it can be seen from the solution of the latter equation that in fact +(µ−r)l +σr ++ c−a +r +⩽ w < wb2), then the control equation can be replaced with the following +equivalent expression +(4.10) +λΦn = +� +rw − (µ − r)l +σ ++ a − c +� +(Φn)w − 1 +2 +�µ − r +σ +�2 (Φn)2 +w +(Φn)ww +. +(ii). If wb2 ⩽ w < Hnf+c−a +r+Hn ++ +(µ−r)l +σ(r+Hn), then the control equation can be replaced +with the following equivalent expression +(4.11) +λ +� +Φn − (1 − e−λn) +� += +� +(r + Hn)w − (Hnf + (µ − r)l +σ +) + a − c +� +(Φn)w − 1 +2 +�µ − r +σ +�2 (Φn)2 +w +(Φn)ww . +(iii). If Hnf+c−a +r+Hn ++ (µ−r)l +σr +⩽ w < wi2, then the control equation can be replaced with +the following equivalent expression +(4.12) +λ +� +Φn−(1−e−λn) +� += +� +rw−((r + a − c)Hnf +r + a − c + Hn + (µ − r)l +σ +)+a−c +� +(Φn)w− 1 +2 +�µ − r +σ +�2 (Φn)2 +w +(Φn)ww . +These observations lead to the following verification Lemma 4.8, which proof is same +as the Lemma 4.2. +Lemma 4.8. Define a differential operator L π,D +2 +: +L π,D +2 +˜F = +� +(r+a−c)w+(µ−r)π−HD +� ˜Fw + 1 +2(σπ+l)2 ˜Fww −λ +� ˜F −λ(1−e−λn)1{w+D⩾f} +� +. +This manuscript is for review purposes only. + +32 +Y. Q. LI AND L. H. ZHANG +Let F = F(w) be a function that is non-decreasing, continuous, and piecewise dif- +ferentiable on [0, wi2), except that F might not be differentiable at 0, w6, w7 and w8. +Suppose F satisfies Fw > 0, Fww < 0 on (0, wi2) \ +� +(0, w6) ∪ (w7, w8) +� +. If F satisfies +the following boundary-value problem on [0, wi2]: +� +maxπ,D⩾0 L π,D +2 +F(w) = 0, +F(0) = F(w6) = 0, +F(w7) = F(w8) = 1 − e−λn, +F(wi2) = 1. +Then, on [0, wi2], +Φn = F. +The optimal investment amount in risk market is +π∗ +t = −µ − r +σ +2 (Φn)w(w∗ +t ) +(Φn)ww(w∗ +t ), +in which w∗ +t is the optimal control wealth value at time t. The optimal n-year term +life insurance purchase amount is +D∗ +t = +� +f − w1{w C0, that means the consumption is too high, the problem +is going to get more complicated and we won’t discuss it in this paper. +(2). If c ⩽ C0, due to the existence of noise terms l in the income process, we need to +consider two cases l = 0 and l ̸= 0. +• If l ̸= 0, wb2 can’t be equal to zero, then the associated conclusions are same +as Proposition 4.9; +• If l = 0, wb2 can be equal to zero, in this case, if the consumption rate is +large enough and the premium rate is small enough, assume c ⩾ C1, then +the individual purchase n-year term life insurance at all wealth level. The +associated conclusions can be obtained by taking wb2 = 0 and l = 0 in the +Proposition 4.9. For specific conclusions, see the following Proposition 4.11. +This manuscript is for review purposes only. + +34 +Y. Q. LI AND L. H. ZHANG +Proposition 4.11. If l = 0, C1 + a ⩽ c ⩽ C0, Hn ⩽ ˜H, in which +C1 = Hnf +�(r + Hn)q +λ +− 1 +� +, +Hn ⩽ ˜H is the solution of the following inequality +Hnf +�(r + Hn)q +λ +− 1 +� +⩽ +� +r + Hn + rHn +� +rf +(r + Hn)(r + 1) + rHn +. +Then, the maximum probability of achieving the financial goal f before ruining is given +by +Φn(w) = +� +� +� +� +� +� +� +� +� +(1 − e−λn) +� +1 − +� +1 − w +w9 +�q� +, +0 ⩽ w < w9, +1 − e−λn + e−λn +� w − w9 +wi2 − w9 +�p +, +w9 ⩽ w < wi2, +in which +w9 = Hnf + c − a +r + Hn +, +p = +k3 +k3 − 1, +k3 = r − λ + m − +� +(r − λ + m)2 + 4λm +2m +q = +k4 +k4 − 1, +k4 = r + Hn − λ + m + +� +(r + Hn − λ + m)2 + 4λm +2m +. +The associated optimal n-year term life insurance purchasing strategy is purchas- +ing n-year term life insurance of f − w if 0 ⩽ w < w9; if w9 ⩽ w < wi2, it’s optimal +to buy n-year term life insurance of f − Hnf+c−a +r+Hn +. +The optimal investment amount is equal to +π(w) = +� +� +� +� +� +� +� +(µ − r)(w − w9) +σ2(1 − q) +, +0 ⩽ w < w9, +(µ − r)(w − w9) +σ2(1 − p) +, +w9 ⩽ w < wi2, +Proof. Our general guideline is to use lemma 4.8 to prove this proposition. Obvi- +ousely, (Φn)w > 0, (Φn)ww < 0 on (0, wi2) and Φn satisfies the boundary conditions, +then we only need to prove +λ(1 − e−λn) − Hn(f − w)(Φn)w(w) ⩾ 0 +for all 0 ⩽ w ⩽ wi2 . As same as Bayraktar et al. (2016), when c ⩾ C1, the above +inequality holds. +Remark 4.12. We give an explanation of C1 from another point of view. +If wb2 = 0, then p = 1, q = +λ +r+Hn , and +1 − +� +1 − +w +C1+Hnf +r+Hn +�q += 1 − +� +1 − +w +Hnf +r+Hn +� +λ +r+Hn +, +This manuscript is for review purposes only. + +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY35 +� +1 − +w +Hnf +r+Hn +� +1 +r+Hn += +� +1 − +w +Hnfq +λ +� q +λ +. +Combining the above two equations, we can get +C1 = Hnf +�(r + Hn)q +λ +− 1 +� +. +5. Summary and conclusions. We researched the problem about purchasing +m-year deferred n-year term life insurance or m-year deferred n-year term pure endow- +ment by a single premium or via a premium paid continuously in financial planning, +to maximize the probability of achieving the given financial goal f. The problems +were solved by giving the optimal purchasing strategies through establishing new de- +terministic control equations, “quasi-ideal value” and ”ideal value”, and solving the +associated variational inequalities. +In Section 2, it was shown that the optimal purchasing strategy in the case where +the policyholder purchases insurance by paying a single premium, is to buy it only +when the asset has reached “ideal value”. At that point, it is optimal to buy m-year +deferred n-year term life insurance of f − D, in which D represents the compensa- +tion available for other types of financial products that the policyholder owns before +purchasing deferred term insurance. For the case of continuous premium payment, +we also examined this situation by comparing the force of interest r and the force +of death λ. We respectively gave the maximum probability of achieving the given +financial goal and the related strategies for purchasing life insurance in the following +two cases, if the policyholder has enough time to reach the “ideal value” (i.e. λ ⩽ r) +and if he/she does not have enough time to reach the “ideal value”(i.e. λ > r). When +λ ⩽ r, we got the similar conclusions as for the single premium; when λ > r, the +optimal purchasing strategy is to purchase insurance of f − w if w is less than the +critical value w0 ∈ (0, H∗), otherwise, the policyholder does not buy insurance until +the wealth reaches then “ideal value”. In particular, if m > 0, n → ∞, our viewpoint +also shed light on reaching a bequest goal by purchasing deferred whole life insurance. +It is worth noting that if m = 0, n → ∞, our problem is equivalent to achieving the +just mentioned bequest goal by purchasing whole life insurance. In this case, the max- +imum probability and life insurance purchasing strategies we provided are consistent +with those in Bayraktar et al. (2014). +Besides the risk of death, there is another risk that deserves attention, namely +the risk of longevity. To deal with this risk, we thought about the situation for which +the policyholder purchased m-year deferred n-year term pure endowment in personal +financial planning in Section 3. We used some similar methods as in the previous +section and got the corresponding optimal purchasing strategies, but the maximum +probability of achieving the financial goal has changed. The associated results when +m = 0 which mean the policyholder purchases n-year term pure endowment were +given in the end of this section. +In Section 4, we considered two models under the stochastic framework. In model +I, the drift term about price process of income and the consumption rate correlated +with the wealth value, the amount of investment in risky market followed the geo- +metric Brownian motion. In this situation, we found the optimal n-year term life +insurance purchasing strategy is not to purchase until the wealth reaches the “quasi- +ideal value”, and the associated optimal investment strategy in risky market is not to +This manuscript is for review purposes only. + +36 +Y. Q. LI AND L. H. ZHANG +invest until the wealth reaches a relatively large value. We also found an interesting +thing, when the wealth reaches a relatively large value but is in a small range, the +individual also doesn’t invest, see Proposition 4.6. In model II, the amount of in- +vestment in risky market also followed the geometric Brownian motion, but the drift +term about price process of income and the consumption rate were constants a and c. +Due to the change of the model, we discussed the above two constants in three cases. +Specifically, the following results were obtained: +• If a−c = 0, the problem was equivalent to the case where a−c = 0 in Section +2, see Proposition 4.7; +• If a − c > 0, in this case, the idea of the problem and the solution methods +were similar to those in the previous sections, but the “quasi-ideal value”, +“ideal value” and the relevant critical values had changed, see Proposition +4.9; +• If a − c < 0, we considered the case when the consumption rate was large +enough and the premium rate was small enough. +If l ̸= 0, wb2 can’t be +equal to zero, then the associated conclusions are same as Proposition 4.9. If +l = 0, wb2 can be equal to zero, in this case, if the consumption rate is large +enough,then the individual purchase n-year term life insurance at all wealth +level, see Proposition 4.11. +In our future work, on the one hand, we expect to complete the results in Section +4 when c > C0. On the other hand, we will extend our work to consider the individual +purchasing irreversible insurance or m-year deferred n-year term life insurance with +income and consumption to maximize the probability of achieving the financial goal, +then find the optimal purchasing strategies in above cases. We expect this to be an +important guide for personal financial management. +Appendix. The specific method of determination about D1, D2, D3. +We first give the form of the solution of the following equation, the method refers to +Andrei and Valentin (2018), Arnold (2020): +y = xy′ +x + a(y′ +x)n, +y = Ax +n +n−1 , +aAn−1nn = −(n − 1)n−1, +n ̸= 1. +Therefore, by Lemma 4.3 and Lemma 4.4, we have +(5.1) +Dx1(D1p)x1−1 = Ck1(D2q)k1−1 = −1, +in which C, D are the constants to be determined. By the continuous of wb0, +D1(wb0 − w0)p = 1 − e−λn + D2(wb0 − w1)q, +thus, +(5.2) +D2 = D1(wb0 − w0)p − (1 − e−λn) +(wb0 − w1)q +Rewrite (5.1) and by (5.2), we obtain +D(x1 − 1)Dx1−1 +1 +px1 = C(k1 − 1)qk1 +� +D1(wb0 − w0)p − (1 − e−λn) +�k1−1 +(wb0 − w1)k1 +, +This manuscript is for review purposes only. + +ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY37 +i.e. +Dx1−1 +1 += k1 − 1 +x1 − 1 +qk1 +px1 +� +D1(wb0 − w0)p − (1 − e−λn) +�k1−1 C +D +1 +(wb0 − w1)k1 . +Therefore, we have the following form about D1 and C +D +D1 = R +1 +(wb0 − w0)p , +C +D = (wb0 − w1)k1(wb0 − w0)x1, +in which R is the solution of equation +Rx1−1 = k1 − 1 +x1 − 1 +� +R − (1 − e−λn) +�k1−1 qk1 +px1 . +Then, we have +D1 = (1 − e−λn)q(1 − p) +q − p +� +1 +wb0 − w0 +�p +, +D2 = (1 − e−λn)p(1 − q) +q − p ( +1 +wb0 − w1 +)q, +D3 can be obtained in the same way as above +D3 = e−λn +� +1 +wi0 − w2 +�p +. +■ +References. +D. P. Andrei and F. Z. Valentin. Handbook of ordinary differential equations. CRC, +2018. +V. I. Arnold. Mathematical methods of classical mechanics (second edition). Springer- +Verlag, 2020. +S. K. Arup. Analysis of individual investors behavior of stock market. IJTSRD, 1(5): +922–931, 2017. +V. L. Bajtelsmit and T. Y. Wang. Household financial planning strategies for man- +aging longevity risk. FPR, 1(2):e1007, 2018. +E. Bayraktar and V. R. Young. Minimizing the probability of lifetime ruin under +borrowing constraints. Insurance: Math. Econ., 41:196–221, 2006. +E. Bayraktar and V. R. Young. +Life insurance purchasing to maximize utility of +household consumption. North Amer. Actuarial J, 17(2):114–135, 2013. +E. Bayraktar, S. D. Promislow, and V. R. Young. Purchasing life insurance to reach +a bequest goal. Insurance : Math. Econ., 58:204–216, 2014. +E. Bayraktar, S. D. Promislow, and V. R. Young. Purchasing life insurance to reach +a bequest goal:time-dependent case. North Amer. 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Actuarial J, 8(4):106–126, 2004. +This manuscript is for review purposes only. + diff --git a/gdE2T4oBgHgl3EQfyQgz/content/tmp_files/load_file.txt b/gdE2T4oBgHgl3EQfyQgz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..befbdb3a19836cee7c86334905e4898efdef2963 --- /dev/null +++ b/gdE2T4oBgHgl3EQfyQgz/content/tmp_files/load_file.txt @@ -0,0 +1,1252 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf,len=1251 +page_content='ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DEFERRED TERM INSURANCE PURCHASING POLICY YUQI LI∗ AND LIHUA ZHANG† Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' This paper researches the problem of purchasing deferred term insurance in the context of financial planning to maximize the probability of achieving a personal financial goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Specifically, our study starts from the perspective of hedging death risk and longevity risk, and considers the purchase of deferred term life insurance and deferred term pure endowment to achieve a given financial goal for the first time in both deterministic and stochastic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In particular, we consider income, consumption and risky investment in the stochastic framework, extending previous results in Bayraktar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The time cutoff m and n make the work more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' However, by establishing new controls,“quasi-ideal value” and“ideal value”, we solve the corresponding ordinary differential equations or stochastic differential equations, and give the specific expressions for the maximum probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Then we provide the optimal life insurance purchasing strategies and the optimal risk investment strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In general, when m ⩾ 0, n > 0, deferred term insurance or term life insurance is a better choice for those who want to achieve their financial or bequest goals but are not financially sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In particular, if m > 0, n → ∞, our viewpoint also sheds light on reaching a bequest goal by purchasing deferred whole life insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' It is worth noting that when m = 0, n → ∞, our problem is equivalent to achieving the just mentioned bequest goal by purchasing whole life insurance, at which point the maximum probability and the life insurance purchasing strategies we provide are consistent with those in Bayraktar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2014, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Deferred term life insurance, deterministic control, variational inequality, optimal strategy, personal financial planning, financial goal, stochastic differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' “Financial management” is an important issue in a person’s life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Therefore, how to achieve financial goals has become one of the topics that everyone pays more and more attention to, some researches refer to Michael et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2021), Subbakrishna and Murali (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In order to achieve financial goals, many scholars have previously studied various investment strategies among different groups of people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' For example, Huang (2016) studied for college students how to do their financial planning and gave the specific steps for financial management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Topa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2018) subdivided the age of retirees and made a corresponding financial plan by ana- lyzing financial capabilities and investment goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In fact, there are many factors that affect personal financial goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' For instance, Pietrzyk and Rokita (2015) examined a model of household financial planning that takes into account factors such as family survival, investment returns, labor income, health status, and life insurance to achieve financial goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' More about investments and financial management, please refer to Arup (2017), Biradar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2021), Bender et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2022), Bajtelsmit and Wang (2018), Deimena (2014), Dhanasekaran and Kumar (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In last several years, Drive et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2018), Scriven (2008), Weedige et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2019) pointed that life insurance has attracted much attention as one of the investment methods due to its insurance characteristics, as it helps to reduce the financial bur- den of adverse events such as premature death, terminal illness, incapacity to work, or incapacity due to injury or disability by transferring personal losses to insurance companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Recently, an increasing number of scholars have addressed the question of how to link the purchase of life insurance with investments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Richard (1975) integrated life insurance with consumption and investment based on the optimal consumption and optimal investment problems first solved by Merton (1969, 1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Since then, ∗School of Sciences, Beijing University of Posts and Telecommunications, Beijing, 100876, China (Llyq@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' School of Sciences, Beijing University of Posts and Telecommunications, Beijing, 100876, China (zhlh@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' 1 This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='04118v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='PM] 9 Dec 2022 2 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' LI AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' ZHANG most scholars have begun to follow Richard’s research methods but based on the principle of maximizing consumption utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Bayraktar and Young (2013) considered the problem about how to purchasing life insurance to maximize utility of household consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Liang and Zhao (2016) studied the stochastic optimal control problem of whole life insurance purchasing strategy with consumption and investment under the CAAR utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Lee (2021) considered the utility maximization problem under the exponential utility functions, and then gave the optimal portfolio, consumption and whole life insurance strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Other than the criterion of maximizing consumption utility, many scholars have also incorporated the idea of probability into the whole life insurance purchasing strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' They combined the minimum probability of lifetime ruin or the maximum probability of reaching a bequest goal with the whole life insurance purchasing strat- egy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In the early years, there were predictions that in the ten years between 2020 and 2030, the cost of living for retirees would exceed their financial capital by 400 billion dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Under this plan, individuals (rather than employers) must bear all investment and living risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In this context, Young (2004) gave the optimal investment strategies to minimize the probability of lifetime ruin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Bayraktar and Young (2006) used the stochastic optimal control technique to determine the optimal investment strategies for the minimum probability of lifetime ruin under the given two consumption rates and credit constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' To make the model more complete and realistic, Wang and Young (2012) introduced the annuity, and determined the minimum probability of lifetime ruin when buying a convertible life annuity and investing in a risky market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Building on the aforementioned body of prior work, Bayraktar et al examined a num- ber of issues related to purchasing whole life insurance policy in order to maximize the probability of reaching a bequest goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Bayraktar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2014, 2015) first considered this issue without consumption and investment, and further assumed that the force of death changes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Based on the above researches, Bayraktar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2016) considered the model with investment and consumption, and indicated the optimal strategies for whole life insurance purchase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Recently, Liang and Young (2019) de- termined the optimal robust strategy for maximizing the probability of reaching a bequest goal under moral hazard rate uncertainty and risky asset drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' They ex- tended the work of Bayraktar et al by allowing that financial markets and personal mortality are ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Through the study of this series of questions, previous researches have focused on the whole life insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' However, Michael et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2021), Subbakrishna and Murali (2018) found that due to the higher premiums of whole life insurance, it is not suit- able for young or middle-aged people who need protection for a certain period of time but have only average economic power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Therefore, it is necessary to consider term life insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' As can be seen in Xiong and Shen (2020), Promislow (2011), deferred term life insurance is a more general type of insurance than term life insurance, and from an economic perspective, it further reduces the financial burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Thus, it is natural to include deferred term life insurance in personal financial planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Therefore, we explore the problem of purchasing deferred term insurance in personal financial plan- ning to maximize the probability of achieving a financial goal, and provide the optimal purchasing strategies in the deterministic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Furthermore, we consider the problem under the stochastic framework which include the consumption, income and risky investment, and then, give the optimal life insurance purchasing strategies and the optimal investment strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The specific structure and innovation of this paper are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In Section 2, the policyholder purchases m-year deferred n-year term life insurance through a This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY 3 single premium or a continuously paid premium to take out a death benefit plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Although the time cutoff m and n complicates the work, we still make progress by establishing new deterministic controls, “quasi-ideal value” and “ideal value”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' After solving the associated variational inequalities, we finally obtain the optimal purchas- ing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' When m ⩾ 0, n > 0, deferred term insurance is a better choice for those who want to achieve their financial or bequest goals but are not financially sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In particular, if m > 0, n → ∞, our viewpoint also sheds light on reaching a bequest goal by purchasing deferred whole life insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' It is worth noting that if m = 0, n → ∞, our problem is equivalent to achieving the just mentioned bequest goal by purchasing whole life insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In this case, the maximum probability and life insurance purchasing strategies we provide are consistent with those in Bayraktar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Moreover, whether or not the policyholder has enough time to reach the “ideal value” has an obvious impact on the maximum probability of achieving a given financial goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' For the case of continuous premium payment, we also study this situation by comparing the force of interest r and the force of death λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Therefore, we give the maximum probability of achieving a given financial goal and the corre- sponding strategies for purchasing life insurance in the following two cases, when the policyholder has enough time to reach the “ideal value” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' λ ⩽ r) or when there is not enough time to reach the “ideal value”(i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' λ > r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In addition to the risk of death, there is another risk that deserves attention, namely the risk of longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' To address this risk, we provide the relevant conclusions in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' For the stochastic framework, we assume that the policyholder’s account includes consumption, income and risky investment to achieve a financial goal under two dif- ferent models in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' We introduce the approximation functions and functional operators, then by the Itˆo’s formula and Legendre transform, we solve the maximal probability and give the optimal n-year term life insurance purchasing strategies and the optimal investment strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' When n → ∞ and when the income process van- ishes, our results are also consistent with those in Bayraktar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Finally, Section 5 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' From the above analyses and our results, it is clear that our research findings are more general in comparison with whole life insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' At the same time, it lays a new direction and research foundation for personal financial planning issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Purchasing m-year deferred n-year term life insurance in personal financial planning under the deterministic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Assume the policy- holder has an investment account to achieve a financial goal, and this account doesn’t include the consuming and risky investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' He/She may invest in a risk-free market to earn interest, which has the force of interest r, or purchase m-year deferred n-year term life insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' This insurance can help the policyholder to achieve financial motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Let δd be the future lifetime, it follows an exponential distribution with parameter λ, that is, the probability density function of δd is f(x) = λe−λx, x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Equivalently, the policyholder is subject to a constant force of mortality λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' What needs to be explained here is that, in this paper, we always assume that the moment when the policyholder purchases insurance is the initial moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' m-year deferred n-year term life insurance purchased by a single premium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The policyholder buys m-year deferred n-year term life insurance by a single premium with no cash value available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' We define that a dollar death benefit payable immediately at time δd which between m and m+n costs m�Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The premium is payable at the moment of the contract, so m�Kn is the single premium per dollar of This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' 4 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' LI AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' ZHANG death benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' We give the single premium as follows: m�Kn = � 1 + θ � � m+n m e−rtλe−λtdt = � 1 + θ � λ λ + r � e−(r+λ)m − e−(r+λ)(m+n)� , in which θ is the proportional risk loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Denote the wealth in this investment account at time t ⩾ 0 by W(t), and denote the amount of death benefit payable at time δd purchased at or before time t by D(t), in which 0 ⩽ t < δd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Therefore, with single-premium m-year deferred n-year term life insurance, the wealth follows the dynamics � � � dW(t) = rW(t−)dt − m�KndD(t), 0 ⩽ t < δd, W(δd) = W(δd−) + D(δd−)1{m<δd⩽m+n}, in which 1{m<δd⩽m+n} equals 1, otherwise, it equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The similar definition in the latter sections that we omit the explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' We define insurance purchasing strategy D = {D(t)}t⩾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In this section, D is called admissible if D is a non-decreasing, non-negative process, inde- pendent of δd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' and if wealth under this process is non-negative for all t ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' We define the maximum probability of achieving the given financial goal (id est f) as follows: m�ϕn(w, D) = sup D � Pw,D(W(δd) ⩾ f | δd < m)P(δd < m) + Pw,D(W(δd) ⩾ f | m ⩽ δd ⩽ m + n)P(m ⩽ δd ⩽ m + n) + Pw,D(W(δd) ⩾ f | δd > m + n)P(δd > m + n) � := sup D � Pw,D 1 + Pw,D 2 + Pw,D 3 � in which Pw,D denotes the conditional probability given W(0−) = w and D(0−) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Pw,D 1 = Pw,D(W(δd) ⩾ f | δd < m)P(δd < m), and the definitions of Pw,D i , i = 2, 3 are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' D represents the compensation available for other types of financial products that the policyholder owns before purchasing m-year deferred n-year term life insurance, and D < f, 0 ⩽ w < (m�Kn + 1)(f − D) := w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' For the convenience of later discussions, we introduce the following notations m�ϕn(w, D) := m�ϕ1 n(w, D) + m�ϕ2 n(w, D) + m�ϕ3 n(w, D), m�ϕ2,3 n (w, D) := m�ϕ2 n(w, D) + m�ϕ3 n(w, D), in which m�ϕi n(w, D) = sup D Pw,D i , i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In this paper, we assume that D < f since the policyholder would achieve the financial goal if D ⩾ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' If wealth equals m�Kn(f − D) := K∗, then the policyholder will spend all wealth to buy m-year deferred n-year term life insurance of f − D, and if he/she survives This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY 5 more than m years after purchasing insurance, but dies within m + n years, then the policyholder’s total death benefit becomes (f − D) + D = f, we simply call K∗ the “quasi-ideal value”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' If the policyholder dies after m+n years of purchasing insurance, he/she may not receive the death benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' So, we need to find the real “ideal value”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' If wealth equals w∗ and policyholder doesn’t receive the death benefit, then only when the remained wealth is greater than or equal to f − D, the policyholder may achieve the financial goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Therefore, w∗ is called the “ideal value”, at this time, it’s optimal for the policyholder to purchase m-year deferred n-year term life insurance of f − D, whether or not he/she can receive the death benefit, the total wealth at δd will reach the given financial target f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Under the purchasing strategies discussed above, we obtain m�ϕn(w, D) = 1 for w ⩾ w∗, 0 ⩽ D < f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Thus, all that remains is to calculate the maximum probability of achieving the given financial goal f for �L = {(w, D) : 0 ⩽ w < w∗, 0 ⩽ D < f}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The maximum probability of achieving the financial goal on �L is given by m�ϕn(w, D) = � � � � � � � � � � � � � � � � � � � � � � w K∗ � λ r � e−λm − e−λ(m+n) � , 0 ⩽ w < K∗, � e−λm − e−λ(m+n) � + e−λ(m+n) �w − K∗ f − D � λ r , K∗ ⩽ w < w0, � 1 + e−λ(m+n) � �w − K∗ f − D � λ r − e−λ(m+n), w0 ⩽ w < w∗, in which w0 = (e−rm + m�Kn)(f − D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The related optimal insurance purchasing strategy is not to purchase additional insurance until wealth reaches w∗, at which point, it is optimal to buy additional m- year deferred n-year term life insurance of f − D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' To prove the Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='3, we first give several auxiliary lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Let φ = φ(w, D) be a function that is non-decreasing, continuous, and piecewise differentiable with respect to both w and D on �L, except that φ might have infinite derivative with respect to w at w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Suppose φ satisfies the following variational inequality on �L, except possibly when w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' When 0 < w < K∗, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='1) max � rwφw − λφ, φD − m�Knφw � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' When w ⩾ K∗, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='2) max � rwφw − λφ + (λ(e−λm − e−λ(m+n)) − rK∗φw), φD − (m�Kn + 1)φw � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Additionally, suppose φ � K∗, D � = e−λm − e−λ(m+n), φ � w∗, D � = e−λm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Then, on �L, m�ϕ2,3 n (w, D) = φ(w, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' First, we consider the case when 0 < w < K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In this case, m�ϕ3 n(w, D) = 0, m�ϕ2,3 n (w, D) = m�ϕ2 n(w, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' 6 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' LI AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' ZHANG Then, we rewrite the expression for m�ϕ2 n(w, D) as follows: m�ϕ2 n(w, D) = sup D Ew,D �� ∞ 0 λe−λt1{W (t)+D(t)⩾f} � e−λm − e−λ(m+n)� dt � , in which Ew,D denotes the conditional expectation given W(0−) = w, D(0−) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Define the functional operator L D: L Dφ = rwφw − λφ + λ(e−λm − e−λ(m+n))1{w+D⩾f}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Since we are considering a single premium, we obtain L Dφ ⩽ 0 from the lemma hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Define τn = inf {s ⩾ 0 : Ds ⩾ n}, then applying the Itˆo’s formula for e−λτnφ(W(τn), D(τn)), we have e−λτnφ(W(τn), D(τn)) = φ(w, D) + � τn 0 e−λt(L Dφ − λ(e−λm − e−λ(m+n))1{W (t)+D(t)⩾f})dt + � τn 0 e−λt(φD − m�Knφw)dD(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='3) By Wang and Young (2012), we can first assume φ is bounded from below and after removing this assumption, the conclusion still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Suppose φ ⩾ φ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' By taking expectations of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='3), we obtain Ew,D � e−λτnφ∗� ⩽ Ew,D � e−λτnφ � = φ(w, D) + Ew,D �� τn 0 e−λt(L Dφ − λ(e−λm − e−λ(m+n))1{W (t)+D(t)⩾f})dt � + Ew,D �� τn 0 e−λt(φD − m�Knφw)dD(t) � ⩽ φ(w, D) − Ew,D �� τn 0 λ(e−λm − e−λ(m+n))e−λt1{W (t)+D(t)⩾f}dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='4) Let τn → ∞ and apply the monotonic convergence theorem to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='4) φ(w, D) ⩾ m�ϕ2 n(w, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' By Wang and Young (2012), because φ(w, D) satisfies the boundary conditions, then φ(w, D) = m�ϕ2 n(w, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' For w ⩾ K∗, let ψ(w, D) = m�ϕ2,3 n (w, D) − (e−λm − e−λ(m+n)) and w0 = w − K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Then repeat the above steps, we also can obtain φ(w, D) = m�ϕ2,3 n (w, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The maximum probability of achieving the financial goal on �L is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='5) m�ϕ2,3 n (w, D) = � � � � � � � � � � w K∗ � λ r � e−λm − e−λ(m+n) � , 0 ⩽ w < K∗, � e−λm − e−λ(m+n) � + e−λ(m+n) �w − K∗ f − D � λ r , K∗ ⩽ w < w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY 7 The related optimal insurance purchasing strategy is not to purchase additional insurance until wealth reaches w∗, at which point, it is optimal to buy additional m- year deferred n-year term life insurance of f − D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In order to help us solve the variational inequality in verification Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='4, we recall the similar problems in Milevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Then in our situation, m�ϕ2,3 n solves the following boundary-value problem on �L, except possibly at w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='6)� � � � � � � � � rw(m�ϕ2,3 n )w − λ m�ϕ2,3 n = 1{w⩾K∗} � rK∗(m�ϕ2,3 n )w − λ(e−λm − e−λ(m+n)) � , m�ϕ2,3 n � K∗, D � = e−λm − e−λ(m+n), m�ϕ2,3 n � w∗, D � = e−λm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Our general guideline is to use verification Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='4 to prove Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' We notice that m�ϕ2,3 n in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='5) is increasing and differentiable with respect to both w and D on �L, except possibly at w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' m�ϕ2,3 n solves the boundary-value problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='6), which means (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='7) rw(m�ϕ2,3 n )w − λ m�ϕ2,3 n + 1{w⩾K∗} � λ(e−λm − e−λ(m+n)) − rK∗(m�ϕ2,3 n )w � = 0 on �L, except possibly at w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' We give the method of solving m�ϕ2,3 n as follows and it will be omitted that the similar method of solving the maximum probability in latter sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Firstly, we solve the equation λ m�ϕ2,3 n = rw(m�ϕ2,3 n )w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' By rewriting (m�ϕ2,3 n )w as ∂ m�ϕ2,3 n ∂w , equivalently ∂ m�ϕ2,3 n m�ϕ2,3 n = λ r ∂w w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Then we integrate both sides and obtain ln m�ϕ2,3 n = λ r lnw + C(D), in which C(D) is a function about D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Thus, m�ϕ2,3 n = ˜C(D)w λ r , in which ˜C(D) is also a function about D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Substituting m�ϕ2,3 n (K∗, D) = e−λm − e−λ(m+n) into m�ϕ2,3 n = ˜C(D)w λ r , then ˜C(D) = � K∗ �− λ r (e−λm − e−λ(m+n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Therefore m�ϕ2,3 n (w, D) = � w K∗ � λ r (e−λm − e−λ(m+n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Secondly, we solve the equation rw(m�ϕ2,3 n )w − λ m�ϕ2,3 n = rK∗(m�ϕ2,3 n )w − λ(e−λm − e−λ(m+n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' 8 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' LI AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' ZHANG Equivalently, we have r(w − K∗)(m�ϕ2,3 n )w = λ(m�ϕ2,3 n − (e−λm − e−λ(m+n))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Then we make variable substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Let y = w − K∗ and m�φ2,3 n = m�ϕ2,3 n − (e−λm − e−λ(m+n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Then the equation changes to the type which we have proved, and repeat above proof, we can obtain m�ϕ2,3 n � w, D � = � e−λm − e−λ(m+n)� + e−λ(m+n) �w − K∗ f − D � λ r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Next, we prove that (m�ϕ2,3 n )D − m�Kn(m�ϕ2,3 n )w < 0, 0 < w < K∗, and (m�ϕ2,3 n )D − (m�Kn + 1)(m�ϕ2,3 n )w < 0, K∗ ⩽ w < w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' If 0 < w < K∗, then (m�ϕ2,3 n )D = λ r � w K∗ � λ r −1 (e−λm − e−λ(m+n)) w m�Kn(f − D)2 , and (m�ϕ2,3 n )w = λ r � w K∗ � λ r −1 (e−λm − e−λ(m+n)) 1 K∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Thus (m�ϕ2,3 n )D − m�Kn(m�ϕ2,3 n )w = λ r m�Kn � w K∗ � λ r −1 (e−λm − e−λ(m+n)) w − K∗ (f − D)2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' If K∗ ⩽ w < w∗, then analogy to the above method, we have (m�ϕ2,3 n )D − (m�Kn + 1)(m�ϕ2,3 n )w = λ r �w − K∗ f − D � λ r −1 e−λ(m+n) w − w∗ (f − D)2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Therefore, it’s shown that the expression for m�ϕ2,3 n in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='5) satisfies the variational inequality(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' When w = 0, obviously, m�ϕ2,3 n (w, D) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The optimal insurance purchasing strategy is to buy additional m-year deferred n-year term life insurance of f − D when wealth reaches w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='3: Step1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' We first calculate m�ϕ1 n(w, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' If w ⩾ K∗, then we set t∗ satisfies � w−K∗ � ert∗ = f −D, the maximum probability of achieving the financial goal f is as follows: m�ϕ1 n(w, D) = � m t∗ λe−λtdt = �w − K∗ f − D � λ r − e−λm, in which w satisfies w ⩾ (e−rm + m�Kn)(f − D) because of m�ϕ1 n(w, D) ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' If w < (e−rm + m�Kn)(f − D), then m�ϕ1 n(w, D) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY 9 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' If w < K∗, then we set t∗ satisfies wert∗ = f − D, the maximum probability of achieving the financial goal f is as follows: m�ϕ1 n(w, D) = � m t∗ λe−λtdt = � w f − D � λ r − e−λm, in which w satisfies w ⩾ e−rm(f −D) because of m�ϕ1 n(w, D) ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' If w < e−rm(f −D), then m�ϕ1 n(w, D) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Step2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' For m�ϕ2,3 n , the relevant conclusions have been given in the previous Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Having said all of above, we restrict the premium m�Kn < e−rm, then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='4 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='5, the Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='3 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' ■ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' We now give another explanation of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' If the policyholder dies after m years of purchasing insurance and the wealth reaches the “quasi-ideal value”, the probability of the policyholder receiving the death benefit is e−λm − e−λ(m+n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' If the policyholder dies after m + n years of purchasing insurance, then he/she is only dependent on the remaining assets to achieve the financial goal by investing in a risk-free market with interest returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The time when the wealth reaches the “ideal value”, denoted by t0, is given by � w − K∗ � ert0 = f − D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Thus t0 = 1 rln � f−D w−K∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Therefore the policyholder will achieve the financial goal if he/she dies after t0, and this occurs with probability e−λt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In summary, the maximum probability of achieving the financial goal is e−λm − e−λ(m+n) + e−λt0e−λ(m+n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' If the policyholder purchases n-year term life insurance with a single premium, the maximum probability and optimal strategy can be obtained if m = 0 is assumed, apparently P(δd > m) = 1, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='. Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' When m = 0, n > 0, the maximum probability of achieving the financial goal f on �L is given by 0�ϕn(w, D) = � � � � � � � � � � w K∗ � λ r � 1 − e−λn� , 0 ⩽ w < K∗, � 1 − e−λn� + e−λn �w − K∗ f − D � λ r , K∗ ⩽ w < w∗, in which K∗ = 0�Kn(f − D), w∗ = (0�Kn + 1)(f − D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The related optimal insurance purchasing strategy is not to purchase additional insurance until wealth reaches w∗, at which point, it is optimal to buy n-year term life insurance of f − D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' When m = 0 and n → ∞, our problem changes to how the policyholder reach a given bequest goal by purchasing whole life insurance, the maximum probability and life insurance purchasing strategies we give are consistent with the main results in Bayraktar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' When m = 0 and n → ∞, then the maximum probability of achieving the financial goal f is given by 0�ϕ∞(w, D) = � w 0�K∞(f − D) � λ r , 0 ⩽ w < 0�K∞(f − D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The related optimal insurance purchasing strategy is not to purchase additional insurance until wealth reaches 0�K∞(f − D), at which point, it’s optimal to buy whole This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' 10 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' LI AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' ZHANG life insurance of f − D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In particular, when m > 0, n → ∞, the policyholder purchases m-year deferred whole life insurance to achieve the financial goal, our viewpoint also sheds light on reaching a given bequest goal by purchasing deferred whole life insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' When m > 0, n → ∞, then the maximum probability of achiev- ing the financial goal is given by m�ϕ∞(w, D) = � � � � � � � � � � � � � � � � � � � � w K∗ � λ r e−λm, 0 ⩽ w < K∗, e−λm, K∗ ⩽ w < w1, �w − K∗ f − D � λ r , w1 ⩽ w < w∗, in which K∗ = m�K∞(f −D), w1 = (e−rm + m�K∞)(f −D) , w∗ = (m�K∞ +1)(f −D) and m�K∞ < e−rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The related optimal purchasing strategy is not to purchase additional insurance until wealth reaches w∗, at which point, it’s optimal to buy m-year deferred whole life insurance of f − D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' m-year deferred n-year term life insurance purchased by a contin- uously paid premium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Assume that the policyholder buys m-year deferred n-year term life insurance via a premium paid continuously at the rate of m�Hn per dollar of insurance, m�Hn = (1 + θ)m| ¯A1x:n ¯ax:m , m| ¯A1x:n = � m+n m λe−(r+λ)tdt, ¯A1x:m = � m 0 λe−(r+λ)tdt, ¯ax:m = 1 − ¯Ax:m r = 1 − ¯A1x:m − Ax: 1 m r , Ax: 1 m = � ∞ m λe−rne−λtdt, in which θ is the proportional risk loading, the explanation of the corresponding symbols see Promislow (2011), Xiong and Shen (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The proportional loading includes costs, profit, and risk margin, and reserves are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The policyholder may change the amount of his/her insurance coverage at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In our time- homogeneous scenario, the policyholder in this section purchases instantaneous m- year deferred n-year term life insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' In this case, the wealth follows the dynamics � � � dW(t) = (rW(t) − m�HnD(t)1{t⩽m})dt, 0 ⩽ t < δd, W(δd) = W(δd−) + D(δd−)1{m<δd⩽m+n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' An admissible insurance strategy D = {D(t)}t⩾0 is any non-negative process, but for all t ⩾ 0, the probability of W(t) ⩾ 0 doesn’t equal one because of the negative drift term m�HnD(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Thus, we define δ0 = inf {t ⩾ 0 : W(t) ⩾ 0} and the maximum This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' ACHIEVING A GIVEN FINANCIAL GOAL WITH OPTIMAL DTI PURCHASING POLICY11 probability of achieving the financial goal f m�Φn(w) = sup D � Pw(W(δd ∧ δ0) ⩾ f | δd < m)P(δd < m) + Pw(W(δd ∧ δ0) ⩾ f | m ⩽ δd ⩽ m + n)P(m ⩽ δd ⩽ m + n) + Pw(W(δd ∧ δ0) ⩾ f | δd > m + n)P(δd > m + n) � := sup D � Pw 1 + Pw 2 + Pw 3 � , in which Pw denotes the conditional probability given W(0−) = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Pw 1 = Pw(W(δd∧ δ0) ⩾ f | δd < m)P(δd < m), and the definitions of Pw i , i = 2, 3 are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Analogous to the previous section, we give the following notations m�Φn(w) := m�Φ1 n(w) + m�Φ2 n(w) + m�Φ3 n(w), m�Φ2,3 n (w) := m�Φ2 n(w) + m�Φ3 n(w), in which m�Φi n(w) = sup D Pw i , i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' We first define the “quasi-ideal value” and the “ideal value”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' If the wealth is equal to m�Hnf r+ m�Hn := H∗ which follows from the equation rw = m�Hn(f − w), and H∗ is called the “quasi-ideal value”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' That’s, when wealth reaches H∗, the policyholder purchases m-year deferred n-year term life insurance of f − H∗ via a premium paid continuously, and if he/she survives more than m years after purchasing insurance, but dies within m + n years, then his/her total death benefit is f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' However, if the policyholder dies within m + n years after purchasing insurance, then he/she cannot receive the death benefit, so in this case, the policyholder cannot achieve the financial goal f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' We therefore call H∗ the “quasi-ideal value”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Assume the “ideal value” is w∗, if wealth equals w∗, it’s optimal for the policy- holder to purchase m-year deferred n-year term life insurance of f −H∗, then whether or not he/she can receive the death benefit, he/she will achieve the financial goal f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Thus, m�Φn(w) = 1 for w ⩾ w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Derived from our setting, we obtain w∗ by the following equation rw∗ − m�Hn � f − H∗� = f − w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Thus, we get w∗ = (r + m�Hn + r m�Hn)f (r + m�Hn)(r + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' If λ ⩽ r, then the maximum probability of achieving the This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' 12 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' LI AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' ZHANG financial goal f before ruining is given by m�Φn(w) = � � � � � � � � � � � � � � � � � � � � � � w H∗ � λ r � e−λm − e−λ(m+n) � , 0 ⩽ w < H∗, � e−λm − e−λ(m+n) � + e−λ(m+n) � w − H∗ w∗ − H∗ � λ r , H∗ ⩽ w < w0, � 1 + e−λ(m+n) �� w − H∗ w∗ − H∗ � λ r − e−λ(m+n), w0 ⩽ w < w∗, in which w0 = e−rm(w∗ − H∗) + H∗ and the initial wealth w ∈ [0, w∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The related optimal insurance purchasing strategy is not to purchase until wealth reaches w∗, at which point, it’s optimal to buy m-year deferred n-year term life in- surance of f − H∗ = rf r+ m�Hn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' If λ > r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' then the maximum probability of achieving the financial goal f before ruining is given by m�Φn(w) = � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1 − � 1 − w H∗ � λ r+ m�Hn �� e−λm − e−λ(m+n)� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' 0 ⩽ w < w0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' � w H∗ � λ r � e−λm − e−λ(m+n) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' w0 ⩽ w < H∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' � e−λm − e−λ(m+n) � + e−λ(m+n) � w − H∗ w∗ − H∗ � λ r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' H∗ ⩽ w < w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' � 1 + e−λ(m+n) �� w − H∗ w∗ − H∗ � λ r − e−λ(m+n),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' w1 ⩽ w < w∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' in which w1 = e−rm(w∗ − H∗) + H∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' and the initial wealth w ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' w∗),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' where w0 is the unique zero in (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' H∗) of the following equations 1 − � 1 − w H∗ � λ r+ m�Hn = � w H∗ � λ r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' The related optimal purchasing strategy is: If wealth w is less than w0, then the policyholder purchase m-year deferred n-year term life insurance of f − w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' If wealth w is greater than or equal to w0, then the policyholder doesn’t pur- chase insurance until the wealth reaches w∗, at which point, it’s optimal to buy m-year deferred n-year term life insurance of f − H∗ = rf r+ m�Hn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' To prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='11, we first give some auxiliary lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Let F = F(w) be a function that is non-decreasing, continuous, and piecewise differentiable on [0, w∗), except that F might not be differentiable at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfyQgz/content/2301.04118v1.pdf'} +page_content=' Suppose F satisfies the following variational inequality on (0, w∗) λF = rwFw + � max � λ(e−λm − e−λ(m+n)) − m�Hn(f − w)Fw, 0 �� 1{w0, we have � +i +′S i = 0, where the +sum is over the terms S i with Supp(S i) = W and p-ord(S i) = N, with the convention that the +sum over an empty set is zero. +Following [GPZ3, Definition 3.2], +(i) a family of (convex) cones is called properly positioned if every pair of cones in the +family intersect along their faces, including the zero dimensional face at 0, and their +union does not contain a straight line; +(ii) a family of polar germs properly positioned if, for each of the polar germs, there is +a choice of a supporting cone such that the resulting family of cones is properly posi- +tioned; +(iii) a family of polar germs is called projectively properly positioned if it is properly posi- +tioned and none of the denominators of the polar germs is proportional to another. +Here is a useful criterion for the linear independence of polar germs. +Proposition 2.3. [GPZ3, Proposition 3.6] A finite family of polar germs with projectively prop- +erly positioned supporting cones is linearly independent. +2.2. Laurent expansions and the induced decompositions. There are several decompositions +of meromorphic germs with linear poles. Recall that a convex cone is called simplicial if it is +spanned by a set of linearly independent vectors. +Proposition 2.4. [GPZ3, Theorem 4.13.] For any f in MK, there exist a properly positioned +family C of simplicial cones together with a family of K-polar germs {S j} j∈J supported on C (in +the sense that a supporting cone of each S j is in C), and a holomorphic germ h, such that +(6) +f = +� +j∈J +S j + h. + +8 +LI GUO, SYLVIE PAYCHA, AND BIN ZHANG +Eq. (6) is called a Laurent expansion of f supported on C and it is unique up to subdivisions +of the properly positioned family of simplicial cones. +For p ∈ Z≥0, d ∈ Z≥0 and a finite dimensional K-subspace U ⊂ R∞, let +• Mp +K denote the linear span of K-polar germs of p-order p; +• MK,d denote the linear span of K-polar germs whose supporting cones have dimension +d; +• MK,U denote the linear span of K-polar germs with supporting space U. +Theorem 2.5. [GPZ3, Theorem 5.3] We have the decompositions +MK += +� +p≥0 +Mp +K, +(7) +MK += +� +d≥0 +MK,d, +(8) +MK += +� +U⊂R∞ +MK,U. +(9) +In particular, there is a decomposition (see also [BV1]): +MK = MK+ ⊕ MQ +K−. +We give further notations. +(i) Corresponding to the decomposition in Eq. (7), let q be the highest p-order of the polar +germs in a (thus every) Laurent expansion of f . Define the p-residue of f [GPZ3, +Definition 6.1] by +(10) +pRes(f ) : = +� +p-ord(S i)=q +hi(0) +⃗L⃗si +i +. +(ii) Corresponding to the decomposition in Eq. (8), let e be the largest among the dimensions +of the supporting spaces of the polar germs in a (thus every) Laurent expansion of f . +Define the d-residue of f by +(11) +dRes(f ) : = +� +dim(Supp(S i))=e +hi(0) +⃗L⃗si +i +. +Remark 2.6. As proved in [GPZ3, Proposition 6.2], the p-residue of a meromorphic germ with +linear poles depends neither on the choice of a Laurent expansion nor on the choice of the inner +product. The d-residue does not depend on the choice of a Laurent expansion, but it does depend +on the choice of the inner product. +2.3. Dependence subspaces. For any subset U of MQ, let QU denote the Q-subspace of MK +spanned by U. +A simplex fraction is a fraction of the form +1 +Ls1 +1 ···L +sk +k , where L1, . . . , Lk ∈ LQ are linearly +independent and si ∈ Z>0, i = 1, . . . , k. Let F be the set of all simplex fractions over Q. Then +for any inner product Q in (R∞, Z∞), we trivially have QF ⊂ MQ +Q−. +Example 2.7. In the Euclidean filtered lattice space (R∞, Z∞, Q), let B ≔ (ei)i∈Z>0 be an or- +thonormal basis. Let zi be the coordinate function corresponding to ei. A fraction of the form +(12) f +� s1,...,sk +u1,...,uk +� +≔ +1 +zs1 +u1(zu1 + zu2)s2 · · · (zu1 + zu2 + · · · + zuk)sk , ui, si ∈ Z>0, k ∈ N, ui � u j if i � j, + +GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES +9 +is called a Chen fraction. The set of Chen fractions is denoted by +FCh ≔ FCh,Q,B. +We borrow the following definitions from [GPZ3]. A meromorphic function f on Ck of the +form f = g(L1, . . . , Ln), where L1, . . . , Ln are linear forms on Ck and g a meromorphic function +on Cn, is said to depend on the linear subspace of (Ck)∗ spanned by L1, . . . , Ln. One can check +that if f depends on V1 and V2, then it depends on V1 ∩ V2. Thus it makes sense to set the +following definition. +Definition 2.8. The dependence subspace Dep(f ) of f is the smallest linear subspace of (Ck)∗ +on which f depends. +Example 2.9. Let (e1, e2, . . .) be an orthonormal basis of (R∞, Z∞, Q). +Dep +� +1 +z1(z1 + z2) + +1 +z2(z1 + z2) − +2 +z1(z1 + 2z2) − +1 +z2(z1 + 2z2) + 1 +z3 +� += {e3}, +since the sum of the first four terms is zero. +Definition 2.10. Two meromorphic germs f and g in MQ are called Q-orthogonal, which we +write f ⊥Q g, if their dependence subspaces are orthogonal. +Example 2.11. +(i) Let (e1, e2, . . .) be an orthonormal basis of (R∞, Z∞, Q). We have +1 +z1 + z2 +⊥Q (z1 − z2). +(ii) Polar germs are precisely germs of the form h/M for h in MQ+(C∞) and M given by +products of powers of linearly independent linear forms, such that h ⊥Q M. +An element f ∈ MQ is of the form f = +h +ℓ1···ℓr for a holomorphic germ h and linear forms +ℓ1, . . . , ℓr. The next lemma shows that the factors in the fraction can be chosen to have their +dependence subspaces contained in the dependence subspace of f . +Lemma 2.12. For any f in MQ, there are linear forms ℓi = ℓi(L1, . . . , Ln), i = 1, . . . , p, and a +holomorphic germ h = h(L1, . . . , Ln) for a basis L1, . . . , Ln of Dep(f ), such that f = +h +ℓ1 · · · ℓp +. +Proof. Let f be in MQ(Ck) for some k ≥ 1. We extend a basis L1, . . . , Ln of Dep(f ) to a basis +L1, . . . , Ln, . . . , Lk of (Ck)∗. Since f is in MQ(Ck), there are linear combinations ℓ1, . . . , ℓm of +L1, . . . , Lk such that the product ℓ1 · · · ℓm f is in MQ+(Ck), that is, +(13) +ℓ1 · · · ℓm f (L1, . . . , Ln) ∈ MQ+(Ck). +By rearrangement, we can assume that ℓ1, . . . , ℓp are linear combinations of L1, . . . , Ln only; +while ℓp+1, . . . , ℓm have nontrivial linear contributions from the extra linear forms Ln+1, . . . , Lk. +Then we can choose a tuple (an+1, . . . , ak) ∈ Ck−n such that the maps +(L1, . . . , Ln) �→ λi(L1, . . . , Ln) ≔ ℓi(L1, . . . , Ln, an+1, . . . , ak), +i = p + 1, . . ., m, +are affine with λi(0, . . . , 0) � 0. Consequently, the maps (L1, . . . , Ln) �−→ +1 +λi(L1,...,Ln) are holomor- +phic germs. +Thus setting Ln+1 = an+1, . . . , Lk = ak in Eq. (13) yields a holomorphic germ +h : (L1, . . . , Ln) �−→ ℓ1 · · · ℓp λp+1 · · · λm f, +from which we define another holomorphic germ + +10 +LI GUO, SYLVIE PAYCHA, AND BIN ZHANG +˜h(L1, . . . , Ln) ≔ h(L1, . . . , Ln) +λp+1 · · · λm +. +Hence, +f = f (L1, . . . , Ln) = +h(L1, . . . , Ln) +ℓ1 · · · ℓpλp+1 · · · λm += +˜h(L1, . . . , Ln) +ℓ1(L1, . . . , Ln) · · · ℓp(L1, . . . , Ln) +is of the desired form. +□ +Theorem 2.13. Write a germ f in MQ according to the decomposition in Eq. (9): +(14) +f = +� +U∈U +fU + f0, +where U is a finite set of nonzero finite-dimensional subspaces of R∞, 0 � fU is a sum of polar +germs with supporting space U and f0 lies in MQ+. We have +Dep(f ) = +� +U∈U +Dep(fU) + Dep(f0). +Proof. Since f = � fU + f0, clearly we have +Dep(f ) ⊂ +� +U∈U +Dep(fU) + Dep(f0). +It remains to show that Dep(fU) ⊂ Dep(f ) for all U ∈ U and Dep(f0) ⊂ Dep(f ). +By Lemma 2.12, there are linear forms ℓ1, . . . , ℓp and a homomorphic germ h, all with depen- +dent spaces in Dep(f ) such that +f = +h +ℓ1 · · · ℓp += +h(L1, . . . , Ln) +ℓ1(L1, . . . , Ln) · · · ℓp(L1, . . . , Ln), +for a basis L1, . . . , Ln of Dep(f ). Then we can take the Laurent expansion of f in Dep(f ) +by [GPZ3, Theorem 2.11]. Thus for all the polar germs in this Laurent expansion of f , their +linear poles and holomorphic numerators have dependence space in Dep(f ). This gives another +decomposition +f = g0 + +� +V⊂Dep(f) +gV +according to Eq. (9). Comparing with the decomposition of f in Eq. (14) as a sum of a holo- +morphic germ and polar germs, we have +f = f0 + +� +U⊂U +fU = g0 + +� +V⊂Dep(f) +gV. +Using the uniqueness of the decomposition in Eq. (9), we infer that for any subspace U ⊂ U +(resp. U = 0), there is a space V ⊂ Dep(f ) (resp. V = 0) such that fU = gV. This implies that +Dep(fU) is contained in Dep(f ). Thus the proof is completed. +□ +Lemma 2.14. For a nonzero rational linear combination f = � +i∈I αiS i ∈ MQ(Ck) of simplex +fractions S i, i ∈ I, with the same supporting space U, we have Dep(f ) = U. +Proof. Clearly, Dep(f ) ⊆ U. Suppose Dep(f ) ⊊ U. Theorem 2.13 gives +f = g0 + +� +V⊂Dep(f) +gV + +GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES +11 +where the terms in gV have supporting space V. Hence +0 = f − +� +i∈I +αi S i = g0 + +� +V⊂Dep(f) +gV − +� +i∈I +αi S i. +From Dep(f ) ⊊ U, we have V ⊊ U in the above sum. Thus the above sum is the decomposition +of 0 according to the supporting spaces in Eq. (9), in which � +i∈I αi S i is the component with +supporting space U. Thus � +i∈I αi S i = 0, meaning f = 0. This is a contradiction. +□ +This leads to the following statement on sums of polar germs. +Proposition 2.15. If f = � fi is a nonzero sum of polar germs fi with the same supporting +space U, then U is a subset of Dep(f ). +Proof. This follows from Lemma 2.14 after evaluation of the numerators of the polar germs fi +at appropriate arguments in the spirit of the proof of [GPZ3, Theorem 3.7]. Indeed, let us write +the polar germs fi = hi S i where +S i = +1 +Ls1 +1 · · · L +sni +n +, +s j ∈ Z≥0, +are simplex fractions with the same supporting space and hi are holomorphic germs in some +common set of variables ℓn+1, . . . , ℓk which complete the independent linear forms L1, . . . , Ln +arising in the S i’s to an orthonormal basis of Rk. Since f � 0 we can assume without loss of +generality that none of the holomorphic germs hi is identically zero. Hence, there is some tuple +(ℓ0 +n+1, . . . , ℓ0 +k) such that αi ≔ hi(ℓ0 +n+1, . . . , ℓ0 +k) � 0 for all i. We write f = f (L1, . . . , Ln, ℓn+1, . . . , ℓk) +and take the specialisation g(L1, . . . , Ln) ≔ f (L1, . . . , Ln, ℓ0 +n+1, . . . , ℓ0 +k). Then Dep(f ) ⊃ Dep(g). +Applying Lemma 2.14 to g = � +i hi(ℓ0 +n+1, . . . , ℓ0 +k) S i with αi = hi(ℓ0 +n+1, . . . , ℓ0 +k), we obtain Dep(g) = +Supp(g). This completes the proof. +□ +The following result shows that without loss of generality, we can assume that a sum f of +polar germs with the same supporting space can be written as a sum of polar germs whose +numerators are holomorphic germs with dependence space in Dep(f ). +Proposition 2.16. Let f = � hi S i be a nonzero sum of polar germs with the same supporting +space, where S i is a simplex fraction and hi is a holomorphic germ. Then f can be written as +a sum f = � ˜hi S i of polar germs where the ˜hi’s are now holomorphic germs with dependence +spaces in Dep(f ). +Proof. By Proposition 2.15, the common supporting space U lies in Dep(f ). Let L1, . . . , Ln be +a basis of U which we extend to a basis L1, . . . , Ln, ℓn+1, . . . , ℓm of Dep(f ) and then further to a +basis L1, . . . , Ln, ℓn+1, . . . , ℓk of (Ck)∗ with Q(Li, ℓ j) = 0, 1 ≤ i ≤ n, n + 1 ≤ j ≤ k. Thus, S i is a +simplex fraction in the variables L1, . . . , Ln and +f = f (L1, . . . , Ln, ℓn+1, . . . , ℓk) = +� +hi(ℓn+1, . . . , ℓk) S i(L1, . . . , Ln). +By the definition of dependence space, f does not depends on ℓm+1, . . . , ℓk, so +f = f (L1, . . . , Ln, ℓn+1, . . . , ℓm, 0, . . . , 0) = +� +hi(ℓn+1, . . . , ℓm, 0, . . . , 0) S i(L1, . . . , Lk) +as announced. +□ +3. Locality transformation groups on meromorphic germs +In this section, we study meromorphic germs with linear poles in the context of locality +algebras. We then introduce the locality Galois group defined as a group of automorphisms of +meromorphic germs in this locality framework. + +12 +LI GUO, SYLVIE PAYCHA, AND BIN ZHANG +3.1. Locality algebras of meromorphic germs. We give general background on locality al- +gebras and then focus on locality subalgebras of meromorphic germs. +3.1.1. Locality algebras. We recall notations on locality structures from [CGPZ1]. +Definition 3.1. A locality set is a couple (X, ⊤) where X is a set and +⊤ ≔ X ×⊤ X ⊆ X × X +is a binary symmetric relation, called a locality relation, on X. For x1, x2 ∈ X, denote x1⊤x2 if +(x1, x2) ∈ ⊤. +For a subset U ⊂ X, the polar subset of U is +U⊤ ≔ {x ∈ X | (x, U) ⊆ ⊤}. +For locality sets (X, ⊤X) and (Y, ⊤Y), a map f : X → Y is called a locality map if +(15) +x1⊤Xx2 =⇒ f (x1)⊤Y f (x2), +∀x1, x2 ∈ X. +We give some examples that will be further explored in the sequel. +Example 3.2. +(i) For any nonempty set X, being distinct: x1⊤x2 if x1 � x2, defines a +locality relation on X; +(ii) The Q-orthogonality relation ⊥Q⊂ MQ×MQ of Definition 2.10, turns MQ into a locality +set. +(iii) Let (X, ⊤) = (Z>0, ⊤) be the locality set in (i) and (MQ, ⊥Q) the locality set in (ii). With +the notation in Example 2.7, the map +f : X → MQ, +n �→ f +� 1 +n +� +≔ 1 +zn +, +n > 1, +is a locality map. +Other algebraic structures can be generalised to the locality setting. +Definition 3.3. +(i) A locality vector space is a vector space V equipped with a locality +relation ⊤ which is compatible with the linear structure on V in the sense that, for any +subset X of V, X⊤ is a linear subspace of V. +(ii) A (nonunitary) locality algebra over K is a locality vector space (A, ⊤) over K together +with a map +mA : A ×⊤ A → A, (u, v) �→ u · v = mA(x, y) +for all (u, v) ∈ A ×⊤ A +satisfying the following variations of the associativity and distributivity. +(a) For u, v, w ∈ A with u⊤v, u⊤w, v⊤w, we have +(16) +(u · v)⊤w, +u⊤(v · w), +(u · v) · w = u · (v · w). +(b) For u, v, w ∈ A with u⊤w, v⊤w (and hence (u + v)⊤w, w⊤(u + v)), we have +(u + v) · w = u · w + v · w, +w · (u + v) = w · u + w · v, +(ku) · w = k(u · w), +u · (kw) = k(u · w), k ∈ K. +(iii) A unitary locality algebra is a locality algebra (A, ⊤, mA) with a unit 1A such that, for +each u ∈ A, we have 1A⊤u and +1A · u = u · 1A = u. +We shall omit explicitly mentioning the unit 1A unless doing so generates ambiguity. + +GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES +13 +(iv) Let (A, ⊤A) be a locality algebra. A subspace B of A is called a (resp. unitary) locality +subalgebra of A if, with the restricted relation +⊤B ≔ ⊤A ∩ (B × B) +of ⊤A to B, the pair (B, ⊤B) is a (resp. unitary) locality algebra. +(v) Let (A, ⊤A) be a commutative locality algebra and C a unitary locality subalgebra of A. +A subspace B of A is called a (resp. unitary) locality C-subalgebra of A if B is a (resp. +unitary) locality subalgebra of A that contains C. +Given two locality algebras (Ai, ⊤i), i = 1, 2, a (resp. unitary) locality algebra homomor- +phism is a linear map ϕ : A1 −→ A2 such that a⊤1b implies ϕ(a)⊤2ϕ(b) and ϕ(a·b) = ϕ(a)·ϕ(b) +(resp. and ϕ(1A1) = 1A2). +Example 3.4. [CGPZ1, Corollary 3.23] With the relation ⊥Q of Definition 2.10, the pair (MQ, ⊥Q +) is a unitary locality algebra and the projection +(17) +πQ ++ : MQ = MQ+ ⊕ MQ +Q− → MQ+ +along MQ +Q− is a unitary locality algebra homomorphism. +Definition 3.5. For a unitary locality algebra (A, ⊤), a unitary locality endomorphism of A is a +locality automorphism if it is invertible, preserves the unit, and the inverse map is a locality +algebra homomorphism. Let Aut⊤(A) denote the set of locality automorphisms of A. +We note that Aut⊤(A) forms a group for the composition, called the locality automorphism +group of A. There are counter examples that a bijective locality homomorphism needs not be a +locality automorphism. +3.1.2. Locality subalgebras of meromorphic germs. In the sequel, we consider locality subal- +gebras (A, ⊥Q) of the locality algebra (MQ, ⊥Q) in Example 3.4. For a subset U of MQ, let +(18) +� +U ≔ U ∪ {1}, +with 1 being the constant function. +We first give the structure of locality subalgebras of MQ generated by a set, with rational co- +efficients or MQ+ coefficients. As we shall see, a careful analysis using tools such as supporting +and dependent spaces is needed when extending the notion of subalgebra to the locality setting. +Given a subset U of MQ, let +ΠQ(U) ≔ +� � +i +si +����� si ∈ U, ∀i, si ⊥Q s j, ∀i � j +� +be the set of meromorphic germs locality generated by U. With the notation of Eq. (18), we +have +ΠQ(�U) = ΠQ(U) ∪ {1}. +Proposition 3.6. Given a set S of simplex fractions, the subspace of Q�F +QΠQ(�S) ≔ +� � +i +ci S i +����� ci ∈ Q, S i ∈ ΠQ(�S) +� +spanned by ΠQ(�S) is a unitary locality subalgebra of QF. +Thus QΠQ(�S) is the unitary locality subalgebra of Q�F generated by S. +Proof. Since c0 = 1 serves as the unit, we just need to prove that for f, g in ΠQ(S) with f ⊥Q g, +f g lies in QΠQ(S). According to the grading in Eq. (9), we write +f = +� +U +fU, +g = +� +V +gV, + +14 +LI GUO, SYLVIE PAYCHA, AND BIN ZHANG +where fU is the sum of simplex fractions with the same supporting space U and gV is the sum +of simplex fractions with the same supporting space V. By Theorem 2.13, we have +Dep(f ) = +� +U +Dep(fU), +Dep(g) = +� +V +Dep(gV) +from which we infer that, for any U and V appearing in the decompositions of f and g, +f ⊥Q g =⇒ fU ⊥Q gV. +By Lemma 2.14, each fU � 0 (resp. gV � 0), being a sum of simple fractions with the same +supporting space U (resp. V), gives U = Dep(fU) (resp. V = Dep(gV)). Thus we have +U ⊥Q V. +Hence the products fUgV arising in the decomposition f g = � +U,V fUgV lie in QΠQ(S). Conse- +quently the product f g also lies in QΠQ(S). +□ +Proposition 3.7. Given a set S of simplex fractions, the set +MQ +Q+ +�ΠQ(�S)� ≔ +� � +i +hiS i, +����� hi ∈ MQ+, S i ∈ ΠQ(�S), hi ⊥Q S i +� +is a unitary locality subalgebra of MQ. +Thus MQ +Q+ +�ΠQ(�S)� is the unitary locality MQ+-subalgebra of MQ generated by S. +Proof. Clearly, MQ +Q+(ΠQ(�S)) is a vector space. As in the proof of Proposition 3.6, we only need +to verify that, for a, b ∈ MQ +Q+(ΠQ(�S)) with a ⊥Q b, the locality product ab is well defined and +lies in MQ +Q+(ΠQ(�S)). To complete this, in accordance with the grading in Eq. (9), write +a = +� +U +� +i +aUiS Ui = +� +U +aU, +b = +� +V +� +j +bVjTVj = +� +V +bV, +where aUi, bVj ∈ MQ+, S Ui, TVj ∈ ΠQ(S) and aUi ⊥Q S Ui, bVj ⊥Q TVj, Supp(S Ui) = U, +Supp(TVj) = V (with the convention that Supp(h) = 0 for h ∈ MQ+). +Theorem 2.13 gives Dep(aU) ⊂ Dep(a) and Dep(bV) ⊂ Dep(b). Since a ⊥Q b means +Dep(a) ⊥Q Dep(b), we have Dep(aU) ⊥Q Dep(bV), that is aU ⊥Q bV. +Now let aU � 0 and bV � 0. +(i) By Proposition 2.15, we have U ⊂ Dep(a) and V ⊂ Dep(b). So U ⊥Q V and S Ui ⊥Q TVj; +(ii) From Dep(aU) ⊂ Dep(a) and V ⊂ Dep(b), we obtain aU ⊥Q TVj. Similarly, bV ⊥Q S Ui; +(iii) By Proposition 2.16, there exist holomorphic germs ˜aUi and ˜bVj with +Dep(˜aUi) ⊂ Dep(aU), ˜aUi ⊥Q S Ui, Dep(˜bVj) ⊂ Dep(bU), ˜bVj ⊥Q TVj, +such that +aU = +� +i +˜aUiS Ui, +bV = +� +j +˜bVjTVj. +From aU ⊥Q TVj and Dep(˜aUi) ⊂ Dep(aU), we have ˜aUi ⊥Q TVj. Likewise, ˜bVj ⊥Q S Ui. +In summary, ˜aUi, ˜bVj, S Ui, TVj are mutually Q-orthogonal. Thus aUbV = � +i, j ˜aUi ˜bVjS UiTVj, is de- +fined and gives an element in MQ +Q+(ΠQ(�S))). Therefore, ab = � +U,V aUbV is defined and lies in +MQ +Q+(ΠQ(�S)). +□ +In the subsequent examples, we implicitly fix an orthonormal basis E with respect to an inner +product Q in R∞, only referring to these choices in the notation when necessary. + +GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES +15 +Example 3.8. The set FCh = FCh,Q,E of Chen fractions in Example 2.7 generates the unitary +locality subalgebra MCh +Q ≔ MQ +Q+ +� +ΠQ(�FCh) +� +. +Example 3.9. For a finite subset J of Z>0, we set zJ ≔ � +i∈J zi. As considered by Speer [Sp2] +(see § 5.2), a Feynman fraction is a simplex fraction +(19) +1 +� +J∈J zsJ +J +, sJ > 0, +for a finite collection J of finite subsets of N. With similar notations of Example 3.8, the set of +Feynman fractions and the locality subalgebra it generates are denoted by +FFe ≔ FFe,Q,E, +MFe +Q ≔ MQ +Q+ +� +ΠQ(�FFe) +� +. +3.2. Automorphism groups of simplex locality algebras. For a locality subalgebra A of +(MQ, ⊥Q), let AutQ(A) denote the group of locality automorphisms of A, following Defini- +tion 3.5. +Proposition 3.10. Let S be a set of simplex fractions and let QΠQ(�S) be the unitary locality +subalgebra of Q�FQ generated by S. Let AutQ +Res(QΠQ(�S)) be the set of unitary locality algebra +homomorphisms ϕ : QΠQ(�S) → QΠQ(�S) with the property that, for any fraction S in ΠQ(S), +(20) +ϕ (S ) = S + +� +i +ai S i, +where +ai ∈ Q, S i ∈ ΠQ(S), p-ord(S i) < p-ord(S ), Supp(S i) ⊊ Supp (S ) . +Then AutQ +Res(QΠQ(�S)) is a subgroup of AutQ(QΠQ(�S)). +Remark 3.11. As a consequence of Theorem 3.15 yet to come, we have +(21) +AutQ +Res(QΠQ(�S)) = +� +ϕ ∈ AutQ(QΠQ(�S)) +���� ϕ preserves the p-residue and d-residue +� +which justifies the notation with subscript “Res”. +Proof. Denote R ≔ RS ≔ QΠQ(�S). We first prove that ϕ is one-to-one. Let +f = +� +i +aiS i ∈ R, ai ∈ Q, S i ∈ ΠQ(S) ∪ {1}, +be nonzero. If f is a constant in Q, then ϕ(f ) = f � 0. If f is not a constant, we group the terms +of f according to the gradation in Eq. (9): +f = c0 + +� +U∈U +fU, +where U is a finite nonempty set of nonzero subspaces of R∞ and 0 � fU ∈ QΠQ(�S), for any U +in U, is a sum of fractions with supporting space U. Applying ϕ yields +ϕ(f ) = c0 + +� +U∈U +� +fU + +� +V⊊U +gV +� +, +where gV is a sum (possibly zero) of fractions with supporting space V. For a maximal element +U0 in U, fU0 is the only contribution in the above sum arising in ϕ(f ) to the component with +supporting space U0 in the decomposition in Eq. (9). So ϕ(f ) = 0 implies fU0 = 0. This is a +contradiction. It follows that ϕ(f ) � 0, which ends the proof of the injectivity. +To prove the surjectivity of ϕ, by the linearity of ϕ, we only need to show that every element +f of ΠQ(�S) lies in the range Im ϕ of ϕ. Suppose this is not the case and let U0 � 0 be minimal +among the supporting spaces of elements in ΠQ(�S)\Imϕ. Let f0 be one of the simplex fractions +in ΠQ(�S)\Imϕ with supporting space U0. Then by Eq. (20) we have + +16 +LI GUO, SYLVIE PAYCHA, AND BIN ZHANG +ϕ(f0) = f0 + +� +V⊊U0 +fV, +where fV ∈ Q(ΠQ(�S)) is the sum of simplex fractions with supporting space V. The space U0 +being minimal, each fV lies in the image of ϕ. Therefore, f0 = ϕ(f0) − � +V⊊U0 +fV also lies in the +image of ϕ. This is a contradiction, showing that ϕ is surjective. +We finally prove that ϕ−1 is a locality algebra homomorphism. We first show that ϕ−1 has the +property in Eq. (20), that is, for any S ∈ ΠQ(�S), we have +ϕ−1(S ) = S + +� +i +hiS i, +where each S i ∈ ΠQ(�S) has smaller supporting space and p-order than those of S . Assume that +this were not the case, and let S have a minimal supporting space U0 among the counterexam- +ples. Then +ϕ(S ) = S + +� +j +b jT j, +where each simplex fraction T j ∈ ΠQ(�S) has it supporting space and p-order smaller than those +of S . Applying ϕ−1 gives +(22) +S = ϕ−1(S ) + +� +j +b jϕ−1(T j). +The minimality of the supporting space of S yields +ϕ−1(T j) = T j + +� +jk +d jkT jk, +where each T jk ∈ ΠQ(�S) has its supporting space and p-order smaller than those of T j and hence +of S . Therefore, Eq. (22) gives +ϕ−1(S ) = S − +� +j +ϕ−1(T j) = S − +� +j +b j +T j + +� +jk +d jkT jk + , +which shows that ϕ−1(S ) has the form in Eq. (20). This gives the desired contradiction. +To check that ϕ−1 is a locality map, we consider two linear combinations f, g in R = Q(ΠQ(�S)) +and group the terms +f = c + +� +U∈U +fU +and +g = d + +� +V∈V +gV, +with c, d in Q and nonzero sums fU, gV ∈ Q(ΠQ(�S)) of fractions with supporting spaces U ∈ U +and V ∈ V respectively. We proceed to show that f ⊥Q g implies ϕ−1(f ) ⊥Q ϕ−1(g). +Let U be an element in U and V an element in V. By Theorem 2.13, Dep fU ⊂ Dep f and +Dep gV ⊂ Dep g so that Dep f ⊥Q Dep g implies Dep fU ⊥Q Dep gV and hence fU ⊥Q gV. +By Lemma 2.14, for the linear combination fU (resp. gV) of simplex fractions with the same +supporting space U (resp. V), we have U = Dep(fU) (resp. V = Dep(gV)). Thus fU ⊥Q gV +implies U ⊥Q V. +Since ϕ−1 has the property in Eq. (20), we have Dep(ϕ−1(fU)) ⊆ Dep(fU) and Dep(ϕ−1(gV) ⊆ +Dep(gV). Then ϕ−1(fU) ⊥Q ϕ−1(gV). Therefore, by the linearity of ϕ−1, we obtain ϕ−1(f ) ⊥Q +ϕ−1(g), as needed. +Finally for f, g ∈ R with f ⊥Q g, we have ϕ−1(f ) ⊥Q ϕ−1(g). Hence +ϕ(ϕ−1(f ) ϕ−1(g)) = ϕ(ϕ−1(f )) ϕ(ϕ−1(g)) = f g. +Therefore, applying ϕ−1, we obtain ϕ−1(f ) ϕ−1(g) = ϕ−1(f g), showing that ϕ−1 is a locality +homomorphism. +□ + +GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES +17 +3.3. Locality Galois groups. In this part we consider a locality subalgebra A of (MQ, ⊥Q) +containing MQ+. +Proposition 3.12. For any locality morphism ϕ : A → A with ϕ|MQ+ = Id, we have +Dep(ϕ(f )) ⊂ Dep(f ), +∀f ∈ A. +Proof. For any ℓ in LQ viewed as an element of (Rk)∗ for some k ≥ 1, if ℓ ⊥Q Dep(f ), then +ℓ ⊥Q f , which implies that ϕ(ℓ) ⊥Q ϕ(f ) since ϕ is a locality map. Since ℓ is in MQ+, we have +ϕ(ℓ) = ℓ. Thus ℓ ⊥Q Dep(ϕ(x)). So Dep(f )⊤ ⊆ Dep(ϕ(x))⊤ which yields the statement. +□ +In the sequel, we fix a set S ⊆ F of simplex fractions and let +(23) +B ≔ B(S) ≔ Q(ΠQ(�S)), +A ≔ A(S) ≔ MQ +Q+ +� +ΠQ(�S) +� +be the unitary locality subalgebra and MQ +Q+-subalgebra of MQ generated by S, defined in Propo- +sitions 3.6 and 3.7 respectively. +Definition 3.13. Define a subset of AutQ(A) by +GalQ(A/MQ+) ≔ +ϕ ∈ AutQ(A) +�������� +ϕ|MQ+ = Id +ϕ preserves the p-residue, d-residue +and the locality subalgebra B + +It will be called the locality Galois group of A over MQ+, thanks to Theorem 3.15. +We next give a locality tensor product property of MQ. +Proposition 3.14. Let S ⊆ F and, as in Eq. (23), define +B ≔ B(S) ≔ Q(ΠQ(�S)), +A ≔ A(S) ≔ MQ+ +� +ΠQ(�S) +� +. +(i) For each Q-subspace U of R∞, let AU ≔ A ∩ MQ,U and let BU denote the linear +span of simplex fractions in B with supporting space U. Let MU +Q+ denote the space of +holomorphic germs whose dependent space is contained in +U⊥Q ≔ +� +y ∈ L(C∞) +��� y ⊥Q u, ∀u ∈ U +� +. +Then we have the (inner) tensor product +AU = MU +Q+ ⊗ BU, +that is, MU +Q+ and BU are linearly disjoint. +(ii) Let (V, ⊤) be a locality vector space. Any pair of locality linear maps ϕ : B → V and +ψ : MQ+ → V uniquely extends to a locality linear map +ϕ ⊗Q ψ : A → V. +Proof. (i) By Proposition 3.7, +(24) +f = +� +i +hiS i , +where S i ∈ ΠQ(�S) with Dep(S i) = U, hi is holomorphic with dependent space contained in U⊥Q +and hence hi ⊥Q S i. So AU = MU +Q+BU as a product of subsets. +To prove the disjointness, we more generally consider a linear combination +(25) +� +i +hiS i = 0, +where Dep(S i) = U, hi is holomorphic with dependent space contained in U⊥Q. Suppose that +{S i}i is linearly independent, but hi � 0 for all i in the sum. Denote V = � +i Dep(hi) which is a +finite-dimensional subspace of U⊥Q. Then hi is defined on V. Thus we can choose disjoint sets + +18 +LI GUO, SYLVIE PAYCHA, AND BIN ZHANG +of variables {zk} of U and {wℓ} of V respectively. From hi � 0, there is {w0 +ℓ} such that hi({w0 +ℓ}) � 0 +for all i. Then Eq. (25) gives +� +i +hi({w0 +ℓ}) S i = 0, +showing that {S i}i is linearly dependent. This gives the desired contradiction. +(ii) By Proposition 3.7, A is linearly spanned by homogeneous elements with respect to the +grading by supporting space in the grading Eq. (9). Thus A has the restricted grading +A = +� +U⊂R∞ +AU. +Then we just need to show that ϕ and ψ uniquely define a locality linear map +(ϕ ⊗Q ψ)U : AU → V +for each subspace U of R∞. +By Item (i), for any linear map ϕ : BU → V and ψ : MU +Q+ :→ V, there is a unique linear map +(26) +(ϕ ⊗Q ψ)U : AU → V, +f �→ +� +i +ψ(hi)ϕ(S i) +for any element f = � +i hiS i in AU, expressed in the form in Eq. (24). Indeed, (ϕ ⊗Q ψ)U is +simply the tensor product of the restriction of ψ to MU +Q+ and the restriction of ϕ to BU. Taking +the sum over all subspaces U of L(R∞) including U = 0, we have an extension ϕ ⊗Q ψ of ϕ and +ψ to A. +□ +The remaining part of the section is devoted to the proof of the following theorem which +extends an element of AutQ +Res(B) to an element of GalQ(A/MQ+). +Theorem 3.15. Let S ⊆ F and let A and B be as defined in Eq. (23). +(i) Any element ϕ ∈ AutQ +Res(B) (see Proposition 3.10) uniquely extends to an element of +GalQ(A/MQ+) defined by +(27) +˜ϕ + +� +i +hiS i + ≔ +� +i +hiϕ(S i) +for +(28) +f = +� +i +hiS i ∈ A, hi ∈ MQ+, S i ∈ ΠQ(�S) +as in Proposition 3.7. +(ii) The subset GalQ(A/MQ+) ⊆ AutQ(A) is a subgroup. Restricting to B gives rise to a +group isomorphism +GalQ(A/MQ+) � AutQ +Res(B). +Proof. (i) Applying Proposition 3.14 with ψ the identity map, for any linear map ϕ : BU → BU, +there is a unique linear map +(29) +˜ϕ : MQ,U → MQ,U, +f �→ +� +i +hiϕ(S i) +for any element f = � +i hiS i in MQ,U, expressed in the form in Eq. (24). +We next show that the ˜ϕ obtained this way has the form in Eq. (27). Let f = � +i hiS i as in +Eq. (28). By grouping the terms according to the supporting spaces of S i as in Eq. (9), we have +f = +� +U∈U +fU with fU = +� +j +aU jS U j, +where U is a set of subspaces U of R∞ for which fU � 0, and for each U ∈ U, we have +S U j ∈ ΠQ(S), +Dep(S U j) = Supp(S U j) = U, +Dep(aU j) ⊥Q Dep(S U j) + +GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES +19 +and each term aU jS U j is one of the terms in f = � +i hiS i. Thus fU is in MQ,U and we can apply +Eq. (29) and obtain +˜ϕ(fU) = +� +i +aU jϕ(S U j), +which takes the form in Eq. (27). Hence so is ˜ϕ(f ). This is what we want. +The fact that ˜ϕ preserves the p-residue in Eq. (10) and the d-residue in Eq. (11) follows from +the definition and the special form of ϕ on B. +For f = � +i hiS i ∈ A = MQ+(ΠQ(S)) with hi ∈ MQ+, S i ∈ S, the p-residue p-res(f ) of f is of +the form �′ +i hi(0)S i where the sum is over simplex fractions S i in f with the highest order. By +the definition of ˜ϕ, the sum �′ +i hiS i is still the part of ˜ϕ(f ) with the highest order. Therefore, +p-res(˜ϕ(f )) = p-res(f ). +The same argument, applied to the dimensions of supporting spaces of the polar germs, shows +that ˜ϕ preserves the d-residues. +We next check that �ϕ is a locality MQ+-algebra homomorphism. For a, b ∈ A with a ⊥Q b, +as in the proof of Proposition 3.6, we can write them as +a = +� +U +� +i +hUiS Ui, +b = +� +V +� +j +gVjTVj +such that +hUi � 0, gVj � 0, {hUi, S Ui} ⊥ {gVj, TVj}. +By the special form of ϕ, we have +Dep(ϕ(S Ui)) = U, +Dep(ϕ(TVi)) = V. +Then it follows from the definition of ˜ϕ that ˜ϕ(a) ⊥Q ˜ϕ(b). Moreover (treating h0 as hUS U for +U = 0 and the same for g0), +˜ϕ(ab) = ˜ϕ + +� +U,V +hUigVjS UiTVj + = +� +U,V +hUigVjϕ(S UiTVj) = +� +U,V +hUigVjϕ(S Ui)ϕ(TVj) = ˜ϕ(a)˜ϕ(b). +By construction, the extension ϕ �→ ˜ϕ is functorial: +� +ϕψ = ˜ϕ ˜ψ, +�idB = idA. +So for any ϕ ∈ AutQ +Res(B), ˜ϕ is a linear bijection. The functorial property also shows that +˜ϕ � +ϕ−1 = idA. So ˜ϕ−1 = � +ϕ−1. Thus �ϕ−1 is also a locality MQ+-algebra homomorphism. Thus ˜ϕ is +in GalQ(A/MQ+) for all ϕ ∈ AutQ +Res(B). By � +ϕ ψ = ˜ϕ ˜ψ, +˜ϕ−1 = � +ϕ−1, the image of the map +(30) +Ψ : AutQ +Res(B) → GalQ(A/MQ+) : ϕ �→ ˜ϕ, +is a subgroup of AutQ(A). +(ii) The map Ψ defined in Eq. (30) is clearly injective and its image is in GalQ(A/MQ+). Now +for any g ∈ GalQ(A/MQ+), since it preserves B, for any S ∈ ΠQ(S), +g(S ) = +� +i +aiS i, +with ai ∈ Q, S i ∈ ΠQ(S). +Note that the p-residue of g(S ) equals to the p-residue of S which is just S . So we can write +g(S ) = +� +aiS i = +� +p-ord(S i)=p-ord(S ) +aiS i + +� +p-ord(S i)lex · · · >lex wk are Lyndon words and i1, . . . , ik ≥ 1. + +GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES +21 +(ii) (Radford) Let K be a Q-algebra. The set Lyn(X) of Lyndon words on X is an alge- +braically independent generating set of Sh(KX). Thus Sh(KX) � K[Lyn(X)] is a poly- +nomial algebra. In fact, with the factorisation w = wi1 +1 · · · wik +k in Eq. (31), we have +(32) +w = +1 +i1! · · · ik!w +X i1 +1 +X · · · X w +X ik +k ++ smaller terms. +From a set U we build a set +U ≔ {x0} ⊔ {xu | u ∈ U}. +For a well-ordered set (U, ≤), we then define a well-order ≤ on U by imposing +(33) +x0 < xu, and xu ≤ xv ⇔ u ≤ v, ∀u, v ∈ U. +Denote +W1(U) ≔ {1} ∪ +� +u∈U +W(U)xu and Sh1(KU) ≔ KW1(U) = K +�  +� +u∈U +Sh(KU)xu + . +Note that Sh1(KU) is closed under the shuffle product, and the factorisation of w in W1(U) +given in Eq. (31) has its factors in W1(U). Thus we obtain (see also [Zh, § 3.3.1]) +Proposition 4.3. Let Lyn1(U) ≔ Lyn(U)\{x0}, that is, the set of Lyndon words in U that do not +end with x0. Then Sh1(KU) is a subalgebra of Sh(KU) and is a polynomial algebra generated +by Lyn1(U). +We now extend these constructions to the locality setting. +For a locality set (X, ⊤), let W⊤(X) denote the subset of W(X) consisting of locality words, +namely the words w = w1 · · · wk in which wi⊤wj, 1 ≤ i � j ≤ k, plus the empty word. Let +Sh⊤(KX) ≔ KW⊤(X). +For w = w1 · · · wk and v = v1 · · · vℓ in W⊤(X), define +(34) +w⊤v ⇐⇒ wi⊤vj, ∀1 ≤ i ≤ k, 1 ≤ j ≤ ℓ. +Thus for w, v in W⊤(X) with w⊤v, the word wv also lies in W⊤(X). Since a shuffle of w and v +is obtained from wv by permuting the factors and hence still lies in W⊤(X), the shuffle product +w X v lies in Sh⊤(KX). It follows that Sh⊤(KX) is a locality algebra. +For a well-ordered set (X, ≤) equipped with a locality relation ⊤, the Lyndon words in W⊤(X) +are called locality Lyndon words. The following statement enhances Theorem 4.2 to a locality +setting. +Theorem 4.4. Let (X, ≤) be a well-ordered set equipped with a locality relation ⊤. +(i) (Locality Chen-Fox-Lyndon Theorem) Any word w in W⊤(X) has a unique factorisation +(35) +w = wi1 +1 · · · wik +k +where w1 >lex · · · >lex wk are locality Lyndon words and i1, . . . , ik ≥ 1. +(ii) (Locality Radford Theorem) Let K be a Q-algebra. The set Lyn⊤(X) of locality Lyndon +words on X is a locality algebraically independent generating set (in the sense of Defi- +nition 4.1) of the locality algebra Sh⊤(KX). Thus Sh⊤(KX) � K⊤[Lyn⊤(X)] is a locality +polynomial algebra. + +22 +LI GUO, SYLVIE PAYCHA, AND BIN ZHANG +Proof. (i) In the factorisation w = wi1 +1 · · · wik +k of a locality word w into Lyndon words in Theo- +rem 4.2(i), each wi is still a locality word, giving us the existence of the factorisation in Eq. (35). +The uniqueness of the factorisation follows from the uniqueness of the factorisation in Eq. (31). +(ii) If w is local, then by Item (i) and Theorem 4.2(ii), +w = wi1 +1 X wi2 +2 X · · · X wik +k + smaller terms, +that is, w can be generated by locality Lyndon words modulo smaller terms. The smaller terms +are obtained from w by permuting the letters in w, and hence are again local. Thus as in the +nonlocality case, an induction can be applied to show that Sh⊤(KX) is spanned by Lyn⊤(X). +Locality algebraic independence of the set Lyn⊤(X) is automatic since it is a subset of the +algebraically independent set Lyn(X) and locality algebraic independence is weaker than alge- +braic independence. Therefore, locality Lyndon words are locality polynomial generators of +Sh⊤(KX). +□ +Now let (U, ≤) be a well-ordered set equipped with an irreflexive locality relation ⊥, i.e., +u ̸⊥ u for any u in U. +Then (U, ≤) is a well-ordered set by Eq. (33), and is equipped with a locality relation ⊤ +defined by +(36) +x0⊤xu, ∀u ∈ U ∪ {x0} and xu⊤xv whenever u ⊥ v, +∀u, v ∈ U. +Denote +W1,⊤(U) ≔ W1(U) ∩ W⊤(U), +Sh1,⊤(KU) ≔ KW1,⊤(U). +Corollary 4.5. Let (U, ≤) be a well-ordered set equipped with an irreflexive locality relation ⊥. +(i) Any word w in W⊤(U) admits a unique factorisation +(37) +w = w1 · · · wkxr +0, +where w1 >lex · · · >lex wk >lex x0 are locality Lyndon words in W⊤(U), and k, r ≥ 0. +(ii) Given a Q-algebra K, the locality algebra Sh⊤(KU) is a locality polynomial algebra +generated by the set Lyn⊤(U) of locality Lyndon words on U : Sh⊤(KU) � K⊤[Lyn⊤(U)]. +(iii) The subspace Sh1,⊤(KU) of Sh⊤(KU) is a locality subalgebra. It is also a locality +polynomial algebra generated by Lyn1,⊤(U) ≔ Lyn1(U) ∩ Lyn⊤(U). +Proof. (i) Since x0 is the smallest locality Lyndon word in W⊤(U), it must appear at the end of +factorisation in Eq. (35), giving us w = wi1 +1 · · · wik +k xr +0 with w1 >lex w2 >lex · · · >lex wk >lex x0. +Further, a locality Lyndon word wi >lex x0 must have a factor xu for some u ∈ U. So the +irreflexivitiy of the locality and the locality of w implies that i1 = · · · = ik = 1 and wi⊤wj +for 1 ≤ i � j ≤ k. The uniqueness of this factorisation follows from the uniqueness of the +factorisation in Theorem 4.4.(ii). +(ii) This follows from Theorem 4.4.(i). +(iii) The proof goes as for Proposition 4.3. +□ +4.2. Applications to ordered fractions. For the rest of this section, we fix an orthonormal +basis E of R∞ with respect to the inner product Q. For the sake of simplicity, this choice, which +we make once for all, is suppressed in most of the notations. +Let U be a countable set and let +(38) +L : U → LQ, +u �→ Lu, u ∈ U, + +GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES +23 +be a map with values in the space LQ of linear forms defined as in Eq. (4) with K = Q, which +defines a family of linear forms parameterised by U. For ui in U, si ≥ 1, 1 ≤ i ≤ k, define the +ordered fraction (with respect to L) +fL� s1,...,sk +u1,...,uk +� +≔ +1 +Ls1 +u1(Lu1 + Lu2)s2 · · · (Lu1 + · · · + Luk)sk . +Define the set of ordered fractions (with respect to L) +(39) +FL ≔ +� +fL� s1,...,sk +u1,...,uk +� �����si ≥ 1, ui ∈ U, 1 ≤ i ≤ k, k ≥ 0 +� +⊆ F. +Note that QFL is the Q-subspace spanned by im(L). +Example 4.6. When L = LwCh : Z>0 → LQ is given by L(u) = zu, then +(40) +fL� s1,...,sk +u1,...,uk +� += fwCh� s1,...,sk +u1,...,uk +� +from Example 3.8 and FL = FwCh is the set of weak Chen fractions (called MZV fractions +in [GX] for applications to multiple zeta values). +Note that in contrast with a weak Chen fraction, a Chen fraction requires ui � u j for i � j. +On the other hand, the fraction +1 +(z1+z2)(z2+z3) is not an ordered fraction with respect to this L. +Proposition 4.7. Let U be countable and let L : U → LQ be a map. Then the map +(41) +Φ = ΦL : (Sh1(QU), X ) → Q�FL, +xs1−1 +0 +xu1 · · · xsk−1 +0 +xuk �→ fL� s1,...,sk +u1,...,uk +� +, +is an algebra homomorphism. Here the multiplication on Q�FL is the natural one in MQ. +Proof. When the map L is given by +LwCh : Z>0 → LQ, +u �→ zu, +then FL = FwCh with FwCh defined in Eq. (40) is the set of MZV fractions, in which case +the conclusion, for Φ = ΦwCh : Sh1(QZ>0) → Q�FwCh follows from [GX, Eqs. (7),(8) and +Theorem 2.1]. For a general U and L, on the grounds of the countability of U, we can fix a +bijection θ : U → Z>0 and thus an algebra isomorphism +θ : Sh1(QU) → Sh1(QZ>0). +Also note that the change of variables zi �→ Li, i ∈ Z>0, gives rise to an algebra homomorphism +η : Q�FwCh → Q�FL, +fwCh� s1,...,sk +u1,...,uk +� +�−→ fL� s1,...,sk +u1,...,uk +� += +1 +Ls1 +u1(Lu1+Lu2)s2···(Lu1+···+Luk)sk . +Then ΦL is just the composition η ◦ ΦwCh ◦ θ. +□ +Remark 4.8. The algebra homomorphism Φ is not injective. For example, f +� +2 +u1 +� += +1 +L2u1 and +f +� +1,1 +u1,u1 +� += 1 +2 +1 +L2u1 . Consequently, although the shuffle algebra Sh1(QU) is a polynomial algebra +on the Lyndon words, the same cannot be said of Q�FL. As we will see below, this defect can be +remedied under a locality condition. + +24 +LI GUO, SYLVIE PAYCHA, AND BIN ZHANG +Let (U, ⊤) be a countable locality set and let L : U → LQ be as above. Consider the subset +(42) +FL +⊤ ≔ +� +fL� s1,...,sk +u1,...,uk +� ������ si ≥ 1, ui ∈ U, ui⊤u j ∈ U, 1 ≤ i � j ≤ k, k ≥ 0 +� +⊆ FL +and the Q-subspace QFL +⊤ of QFL. +Theorem 4.9. Let (U, ≤, ⊤) be a countable well-ordered set with an irreflexive locality relation +⊥. Suppose that the map L : (U, ⊤) → (LQ, ⊥Q) defined in Eq. (38) is a locality map in the +sense of Eq. (15): for x, y ∈ U, if x⊤y, then Lx ⊥Q Ly. Then +(i) the set FL +⊤ is linearly independent; +(ii) the algebra homomorphism Φ in Eq. (41) restricts to an isomorphism of locality alge- +bras +(43) +Φ⊤ : Sh1,⊤(U) � QFL +⊤, +w = xs1−1 +0 +xu1 · · · xsk−1 +0 +xuk �→ fL� s1,...,sk +u1,...,uk +� +. +(iii) The locality algebra QFL +⊤ is a locality polynomial algebra. +Proof. (i) The first assertion follows from the facts that the supporting cones of all ordered +fractions are projectively properly positioned and that by [GPZ3, Proposition 3.6] recalled in +Proposition 2.3, a projectively properly positioned family of simplicial fractions is linearly in- +dependent. +(ii) By the assumption on (U, ≤, ⊤), a word w = xs1−1 +0 +xu1 · · · xsk−1 +0 +xuk in Sh(QU) is local if and +only if ui⊤u j, i � j. So w is local if and only if Φ⊤(w) = fL� s1,...,sk +u1,...,uk +� +lies in FL +⊤. Thus by +Item (i), Φ⊤ sends a linear basis of Sh1,⊤(QU) to a linear basis of QFL +⊤ and is therefore a linear +isomorphism. +The linear map also preserves the locality. Indeed, for +w1 = xs1−1 +0 +xu1 · · · xsk−1 +0 +xuk, w2 = xt1−1 +0 +xv1 · · · xtℓ−1 +0 +xvℓ ∈ W1(U), +we have +w1⊤w2 ⇔ {u1, . . . , uk}⊤{v1, . . . , vℓ} ⇔ {Lu1, . . . , Luk}⊤{Lv1, . . . , Lvℓ} +⇔ Supp(Φ(w1)) ⊥Q Supp(Φ(w2)) ⇔ Φ(w1) ⊥Q Φ(w2). +Therefore Φ restricts to a locality linear bijection Φ⊤ : Sh1,⊤(QU) → QFL +⊤. Finally, the multi- +plicativity for Φ in Eq. (41) restricts to one for Φ⊤. Hence Φ⊤ is a locality algebra isomorphism. +(iii) This last assertion follows from Item (ii) and Corollary 4.5.(iii). +□ +We consider two special instances of maps L : U → LQ. +Example 4.10. +(i) Let U be Z>0 equipped with the natural order and the locality relation +n⊤m ⇔ n � m. Define +L : Z>0 → LQ, i �→ zi, i ∈ Z>0. +The corresponding set FL +⊤ of ordered fractions is the set FCh of Chen fractions in Exam- +ple 2.7. +(ii) Let U be the set Pfin(Z>0) of nonempty finite subsets of Z>0. The order is the lexico- +graphic order: for elements +I ≔ {i1 > i2 > · · · > ir}, +J ≔ { j1 > j2 > · · · > js} +in Pfin(Z>0), define I ≥ J if either the first nonzero element in the sequence +i1 − j1, i2 − j2, . . . , imin{r,s} − jmin{r,s} + +GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES +25 +is positive, or the above sequence of numbers are all zero and r > s. The locality relation +in Pfin(Z>0) is: +I⊤J ⇔ I ∩ J = ∅. +Define +(44) +L : Pfin(Z>0) → LQ, I �→ zI ≔ +� +i∈I +zi, +∀I ∈ Pfin(Z>0). +The corresponding fractions fL� s1,...,sk +I1,...,Ik +� +are of the form +(45) +1 +zs1 +I1(zI1 + zI2)s2 · · · (zI1 + · · · + zIk)sk , si ∈ Z>0, k ∈ N, Ij ∩ Ij′ = ∅ if j � j′ +which we will call Speer fractions. Let FSp denote the set of Speer fractions and let +MSp := MQ+ +� +ΠQ(� +FSp) +� +. +The term Speer fractions is to recognize that such fractions first appeared in Speer’s work [Sp2] +in the context of renormalisation. Notice that the linear forms in a Speer fraction correspond to +faces of a Chen cone. See Section 5 for details. +Remark 4.11. We note that MFe +Q ⊃ MSp +Q , yet whether they actually differ is an open question. +As a direct consequence of Theorems 4.9, we obtain +Corollary 4.12. The locality algebras QFCh +⊤ and QFSp +⊤ are locality polynomial algebras on the +set of fractions in FCh and FSp respectively corresponding to the locality Lyndon words. +5. Locality characters and analytic renormalisation +In this last section, as an application of our previous results, we address Problems 1.1-1.3 +raised in the introduction on Speer’s analytic renormalisation. We first show that the pole struc- +tures of the generalised Feynman amplitudes in Speer’s analytic renormalisation are of the form +introduced earlier. We then compare our constructions of locality generalised evaluators with +those of Speer, and obtain a transitivity group action on locality generalised evaluators. +5.1. Speer’s s-families and Speer fractions. In his work on analytic renormalisation [Sp2, +Sp3, Sp4] (see also [BR, DZ]), Speer determined the possible pole structure of generalised +Feynman amplitudes (or regularised Feynman amplitudes). We show that these linear poles are +spanned by Speer fractions described in Eq. (45). +We first summarise Speer’s work, mostly following [Sp3]. A Feynman graph is called 2- +connected if it cannot be disconnected by removing a vertex. A family E of subgraphs of a +Feynman graph G is called a singularity family or simply an s-family [Sp3, Definition 2] if +(i) every element in E is either 2-connected or a single line. Let E′ denote the subset of +2-connected elements in E; +(ii) E is nonoverlapping, that is, for H1, H2 ∈ E, either H1 ⊂ H2 or H2 ⊂ H1 or H1 ∩ H2 = ∅; +(iii) no union of two or more disjoint elements of E is 2-connected; +(iv) E is maximal with these properties. + +26 +LI GUO, SYLVIE PAYCHA, AND BIN ZHANG +For a given Feynman graph G, the generalised Feynman amplitude TG [Sp3, Definition 1]) +is built from products of propagators assigned to each edge ℓ of G, regularised by means of +a complex number λℓ. It enjoys a decomposition [Sp3, Formula (2.16)] as a sum over all s- +families E of G, of meromorphic functions TE +TG = +� +E +TE, +resulting from writing the closed cone � +ℓ∈E(G){αℓ ≥ 0} as a union ∪ED(E) of closed cones D(E) +associated with each s-family E of subgraphs of G. +For an s-family E of G, and H in E, let +Λ(H) ≔ +� +ℓ∈L(H) +(λℓ − 1), +where L(H) is the set of edges of H. Let µ(H) be the superficial divergence of H. According to +Speer [Sp3] (see Eq. (2.21), Theorem 3 and the remark that follows, see also [Sp4], Lemma 1.4 +and its proof), the possible poles of TE are simple poles given by +Λ(H) − 1 +2µ(H) = 0, −1, −2, . . .. +Since we are only renormalising generalised Feynman amplitudes at λℓ = 1, ℓ ∈ L(G), the +possible singularities we need to deal with are of the form +� � +H∈E′ Λ(H) +�−1 +. By a change of +variables zi = λℓi − 1, with an ordering ℓ1, . . . , ℓ|L(G)| of L(G), the Λ(H) corresponds to the linear +form zI(H) in Eq. (44), for I(H) = {i | ℓi ∈ L(H)}. So we only need to deal with germs of the form +� � +H∈E′ +zI(H) +�−1 +h +with h a holomorphic germ. +We now address Problem 1.1. +Proposition 5.1. For any s-family E of G, the fraction +(46) +� � +H∈E′ +zI(H) +�−1 +lies in QFSp. Thus the germs of the generalised Feynman amplitudes at z = 0 are in MSp +Q . +Proof. Since there is no overlaps between any two 2-connected subgraphs in an s-family E, the +Hasse diagram of E′ is a rooted forest. The flattening procedure used in [CGPZ2, Theorem +5.11] which involves the flattening morphism defined in [CGPZ2, Definition 2.10], transforms +this rooted forest into a linear combination of ladder trees. Correspondingly, the fraction in +Eq. (46) is a linear combination of (ordered) Speer fractions. +□ +5.2. Generalised evaluators on locality subalgebras of meromorphic germs. Let A be a +locality subalgebra of the algebra MQ equipped with the locality relation ⊥Q of Definition 2.10. +Definition 5.2. A locality generalised evaluator E on the locality algebra (A, ⊥Q) is a linear +form E : A → C, such that +(i) E(h) = h(0) for h ∈ MQ+. +(ii) E(f1 · f2) = E(f1) · E(f2) for f1, f2 ∈ A with f1 ⊥Q f2. +We use E(A) = EQ(A) to denote the set of locality generalised evaluators on (A, ⊥Q). + +GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES +27 +Remark 5.3. Notice that a locality subalgebra A ⊂ MQ is defined as a direct limit A = lim +−→ Ak +with Ak ⊂ MQ(Ck), so that a linear form E : A → C amounts to a family of linear forms +Ek : Ak → C, k ∈ N such that +Ek|Ak−1 = Ek−1, +k ≥ 1. +Example 5.4. Let ev0 : MQ+ → Q be the evaluation at 0 defined as ev0(h) = h(0). Using the +map πQ ++ defined in Eq. (17), we build a locality generalised evaluator on A as the composition +(47) +EQ +MS ≔ ev0 ◦ πQ ++, +which we call the locality minimal subtraction map since it is a generalisation to higher +dimensions of the one variable minimal subtraction map obtained by projection onto the holo- +morphic part of the Laurent series ring C[ε−1, ε]] followed by evaluation at zero. +In his work [Sp1, Sp2, Sp3, Sp4], Speer achieved analytic renormalisation by means of linear +forms he called generalised evaluators. Let us compare these generalised evaluators with those +defined in Definition 5.2. +Let us first observe that, as in Example 2.7, a function is given in the variables zi means that +we have chosen a basis {ei} of the space, and every element in the space is written in the form +� ziei. In the presence of an inner product we choose {ei} to be an orthonormal basis. +Following Speer (and see Example 3.9), define MFe(Ck) to be the spaces of meromorphic +germs (at zero) f with the property that +f : Ck → C such that f (z1, . . . , zk) · +� +I⊆[k] +zI is holomorphic at zero. +Speer defines a generalised evaluator [Sp1] as a family of linear maps +E ≔ +� +Ek : MFe(Ck) → C +� +k∈N +satisfying the following conditions. +(i) (compatibility with the filtration) Ek|MFe(Ck−1) = Ek−1, k ≥ 1; +(ii) (extension of ev0) E is the usual evaluation ev0 at zero on holomorphic functions; +(iii) (partial multiplicativity) E(f1 · f2) = E(f1) · E(f2) if f1 and f2 depend on disjoint sets of +variables zi; +(iv) (Σk-invariance) E is invariant under permutations of the variables Ek ◦ σ∗ = Ek for any +σ ∈ Σk, with σ∗ f (z1, . . . , zk) ≔ f (zσ(1), . . . , zσ(k)); +There are also reality and continuity conditions which we do not discuss here. Continuity of +generalised evaluators requires enhancing the constructions carried out here to a topological +setting, which is the object of a joint work [DPS] of the second author. +In practice, Speer builds such a generalised evaluator by setting +(48) +Eiter +k +≔ 1 +k! +� +σ∈Σk +ev +reg,zσ(1) +0 +◦ · · · ◦ ev +reg,zσ(k) +0 +, +where, for 1 ≤ i ≤ k, evreg,zi +0 +(f ) is defined by ev0 ◦ πzi ++(f ) when viewing f as a meromorphic +function in the variable zi, where πzi ++ is defined as in Eq. (2). +Example 5.5. We give some examples in the case of k = 2. +a) For f (u, v) = u +v, g(u, v) = +� +u +v +�2, we have Eiter +2 (f ) = Eiter +2 (g) = 0; + +28 +LI GUO, SYLVIE PAYCHA, AND BIN ZHANG +b) A change of variable u = z1 − z2, v = z1 + z2 in f and g gives +˜f(z1, z2) = z1 − z2 +z1 + z2 +, +˜g(z1, z2) = +�z1 − z2 +z1 + z2 +�2 +, +and we have Eiter +2 ( ˜f) = 0 whereas Eiter +2 (˜g) = 1. +We now show that our locality generalised evaluators are the generalised evaluators `a la Speer +and that it provides a useful alternative to the generalised evaluator Eiter originally used by Speer. +Proposition 5.6. Given any inner product Q, a locality generalised evaluator E in Definition 5.2 +satisfies Conditions (i) – (iv) defining generalised evaluators `a la Speer. +Proof. Indeed, conditions (i) follows from Remark 5.3, and (ii) is the same as condition (i) in +Definition 5.2. The locality multiplicative condition (Definition 5.2.(ii)) for E is stronger than +the partial multiplicativity condition (iii) since disjointness of variables of two functions implies +the orthogonality for Q when the variables correspond to coordinates in an orthonormal basis +for Q. Condition (iv) follows from Theorem 5.1(ii) in [CGPZ3]. +□ +In order to address Problem 1.2, combining Corollary 5.8 and Proposition 5.6, we observe +that every generalised evaluator `a la Speer factors through the minimal subtraction scheme EQ +MS. +We compare the two generalised evaluators Eiter +k +and EQ +MS. +(i) We first observe a problem in the inductive procedure proposed by Speer since the func- +tion (z2, · · · , zk) �→ evreg,z1 +0 +(f (z1, · · · , zk)) might be non-meromorphic, which is an obsta- +cle to implementing the composition ev +reg,zj +0 +◦ evreg,z1 +0 +on f for j � 1. +For example, the discontinuity at zero +evreg,z1 +0 +( ˜f ) = +� +1, +z2 = 0, +−1, +z2 � 0, +is an obstacle for the next step which gives the expression evreg,z2 +0 +◦ evreg,z1 +1 +( ˜f) arising in +the definition Eiter +2 ( ˜f). This also suggests that there is no natural way to interpret Speer’s +generalised evaluator as a minimal subtraction scheme in multiple variables. In contrast, +our generalised evaluator EQ +MS is defined for meromorphic germs with linear poles, so +this does not pose a problem. +(ii) The partial multiplicativity property (ii) for Speer’s generalised evaluator applies to a +smaller set of pairs of functions than the one allowed by our locality multiplicativity. +For ˜g1(z1, z2) = (z1 − z2)2 and ˜g2(z1, z2) = (z1 + z2)−2 we have +Eiter +2 (˜g1 ˜g2) � Eiter +2 (˜g1)Eiter +2 (˜g2) +since Eiter +2 (˜g1 ˜g2) = 1 and Eiter +2 (˜g1) = Eiter +2 (˜g2) = 0. Yet the multiplicativity holds for EQ +MS +since the linear forms z1 − z2 and z1 + z2 are orthogonal (as before, the parameters zi’s +correspond to coordinates in an orthonormal basis for Q) and we have +EQ +MS(˜g1 ˜g2) = 0 = EQ +MS(˜g1) EQ +MS(˜g2). +(iii) As illustrated by Example 5.5, Speer’s generalised evaluator Eiter +k +depends on a choice of +basis since Eiter +2 (g) � Eiter +2 (˜g). It is not invariant even under an orthogonal transformation +of the variables. In contrast, our generalised evaluator EQ +MS does not depend on such an +orthogonal transformation. For the function g in Example 5.5 with standard basis z1, z2 +under an inner product Q, EQ +MS(g) = EQ +MS(˜g) = 0. + +GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES +29 +To conclude, the above observations speak in favour of the use of the global minimal sub- +traction scheme EQ +MS, since it is globally defined on MFe +Q (Ck) and is multiplicative on a large set +of pairs. +5.3. Locality Galois actions on generalised evaluators. We consider the action of the locality +Galois group on generalised evaluators. Clearly, the group GalQ(A/MQ+) acts on EQ(A): +(49) +EQ(A) × GalQ(A/MQ+) → EQ(A), +(E, g) �→ E ◦ g. +The subsequent theorem shows that the automorphism group AutQ(A) � GalQ(A/MQ+) acts +transitively on EQ(A), and thus relates any locality generalised evaluator to the locality minimal +subtraction scheme EQ +MS. In this respect, GalQ(A/MQ+) can be regarded as a renormalisation +group on generalised evaluators. +Theorem 5.7. For S ⊆ F, we consider the unitary locality subalgebra A ≔ MQ +Q+(ΠQ(�S)) of MQ. +Suppose that QΠQ(�S) is a locality polynomial subalgebra of �F with a locality polynomial basis +S ⊂ ΠQ(S). Then every locality generalised evaluator E on A factorises through the minimimal +subtraction scheme EQ +MS, that is, there is some ˜ϕ in GalQ(A/MQ+) such that +(50) +EQ +MS ◦ ˜ϕ = E. +Proof. Let E ∈ EQ(A) be given. By assumption, B ≔ QΠQ(�S) is a locality polynomial sub- +algebra of MQ with S ≔ {sα} ⊂ ΠQ(S) as a set of locality polynomial generators. Then the +map +(51) +ϕ : S → B, +sα �→ sα + E(sα) +extends to a locality algebra homomorphism ϕ on B. +For any t in ΠQ(S), we now determine the form of ϕ(t). Write +t = P(S ) +as a polynomial in S . Then P is a sum of monomials of the form Παsα with sα all distinct due +to the locality algebraic independence. Further, let W = Supp(t) and N = p-ord(t). Then by +Lemma 2.2, we can assume that all these monomials have supporting space W and p-order N. +For each of these monomials, by the locality multilplicativity, we have +ϕ(Παsα) = Πα(sα + E(sα)) = Παsα + lower order terms. +Here the lower order terms come from Παsα by replacing one or more factors sα by E(sα), so +they indeed have lower p-orders and smaller supporting spaces. Since all the monomials Παsα +have the same supporting space and p-order, we have +ϕ(t) = ϕ(P(S )) = P(S ) + lower order terms. +The map ϕ therefore defines an element in AutQ +Res(B). The extension ˜ϕ ∈ GalQ(A/MQ+) of ϕ +obtained from Theorem 3.15 is a locality algebra homomorphism. Further we have +EQ +MS ◦ ˜ϕ(sα) = ev0 ◦ πQ ++ ◦ ˜ϕ(sα) = ev0 ◦ πQ ++(sα + E(sα)) = ev0(E(sα)) = E(sα). +Hence, the locality algebra homomorphisms EQ +MS ◦ ˜ϕ and E agree on the locality polynomial +generating set S of A. Therefore EQ +MS ◦ ˜ϕ = E on B. Since they also agree on MQ+, by +Proposition 3.14, they agree on MQ+(ΠQ(S)). +□ +As a direct consequence of Theorem 4.9 and Corollary 4.12, we obtain a statement which +addresses Problem 1.3 + +30 +LI GUO, SYLVIE PAYCHA, AND BIN ZHANG +Corollary 5.8. The space EQ(MCh +Q ) �resp. EQ(MSp +Q )� of locality generalised evaluators on the +locality algebra +� +MCh +Q , ⊥Q� � +resp. +� +MSp +Q , ⊥Q� � +is a homogeneous space of GalQ � +MCh +Q /MQ+ +� +� +resp. GalQ � +MSp +Q /MQ+ +�� +. In other words, these groups act transitively on EQ(MCh +Q ) �resp. +EQ(MSp +Q )�. +To finish the paper, we use multiple zeta values to build an example of locality generalized +evaluators and elements of the locality Galois group. +Recall that for s1, . . . , sk in Z>0 with s1 ≥ 2, the multiple zeta value (also called a multizeta +value) at (s1, · · · , sk) is +ζ (s1, · · · , sk) := +� +nk>···>n1≥1 +n−s1 +1 +· · · n−sk +k += +k +� +i=1 +∞ +� +mi=1 +(m1 + · · · + mi)−si. +For a locality Lyndon word xs1−1 +0 +xu1 . . . xsk−1 +0 +xuk ∈ Sh1,⊤(U) and for the corresponding “Lyn- +don Chen fraction” +f +� s1,...,sk +u1,...,uk +� +≔ +1 +zs1 +u1(zu1 + zu2)s2 · · · (zu1 + zu2 + · · · + zuk)sk , ui, si ∈ Z>0, k ∈ N, ui � u j if i � j, +in FCh +⊤ (see Example 4.10.(i)), define +(52) +Eζ +� +f +� s1,...,sk +u1,...,uk +�� +:= +� +ζ (s1, · · · , sk) , +s1 ≥ 2, +0, +s1 = 1. +Then by Corollary 4.12, this assignment extends to a unique locality algebra homomorphism +Eζ : AChen → R. +Here R is equipped with the full locality condition R × R, implying the locality of the homo- +morphism Eζ. +Note that the map on AChen defined by Eq. (52) for all locality Chen fractions is also a locality +algebra homomorphism, following [IKZ]. It therefore coincides with Eζ. Hence we conclude +that assigning multiple zeta values to locality Chen fractions as in (52) defines a locality gen- +eralized evaluator Eζ. Then thanks to Theorem 5.7, there is a transformation ˜ϕ in the locality +Galois group GalQ(AChen/MQ+), such that +EQ +MS ◦ ˜ϕ = Eζ. +Acknowledgments. The second author is grateful to the Perimeter Institute in Waterloo where +she was hosted on an Emmy Noether fellowship. This research is supported by the National +Natural Science Foundation of China (11890663 and 11821001). +Declaration of interests. The authors have no conflicts of interest to disclose. +Data availability. Data sharing is not applicable to this article as no new data were created or +analyzed in this study. +References +[BR] M. P. Bellon and E. I. Russo, Ward-Schwinger-Dyson equations in ϕ3 +6 quantum field theory, Lett. Math. 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Phys. 60 (2019), +122301. 3 +[Sp1] E. Speer, Analytic renormalization, J. Math. Phys. 9 (1968), 1040. 4, 27 +[Sp2] E. Speer, On the structure of analytic renormalization, Comm. Math. Phys. 23 (1971), 23-36. Added note: +Comm. Math. Phys. 25 (1972), 336. 3, 4, 5, 15, 25, 27 +[Sp3] E. Speer, Lectures on analytic renormalisation, Technical Report No. 73-067, 1972. 4, 5, 25, 26, 27 +[Sp4] E. Speer, Analytic renormalization using many space-time dimensions, Comm. Math. Phys. 37 (1974), 83- +92. 3, 4, 5, 25, 26, 27 +[Zh] J. Zhao, Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values, World Scientific, 2016. +21 + +32 +LI GUO, SYLVIE PAYCHA, AND BIN ZHANG +[Zi] W. Zimmermann, Convergence of Bogoliubov’s method of renormalization in momentum space, Comm. +Math. Phys. 15 (1969), 208-234. 2 +Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102, USA +Email address: liguo@rutgers.edu +Institute of Mathematics, University of Potsdam, D-14469 Potsdam, Germany +Email address: paycha@math.uni-potsdam.de +School of Mathematics, Sichuan University, Chengdu, 610064, China +Email address: zhangbin@scu.edu.cn + diff --git a/iNE0T4oBgHgl3EQfYAA8/content/tmp_files/load_file.txt b/iNE0T4oBgHgl3EQfYAA8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9249a1f905c80a7a08e9a7cc8b42f8328cd076c0 --- /dev/null +++ b/iNE0T4oBgHgl3EQfYAA8/content/tmp_files/load_file.txt @@ -0,0 +1,1587 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf,len=1586 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='02300v1 [math-ph] 5 Jan 2023 LOCALITY GALOIS GROUPS OF MEROMORPHIC GERMS IN SEVERAL VARIABLES LI GUO, SYLVIE PAYCHA, AND BIN ZHANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Meromorphic germs in several variables with linear poles naturally arise in math- ematics in various disguises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We investigate their rich structures under the prism of locality, including locality subalgebras, locality transformation groups and locality characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The key technical tool is the dependence subspace for a meromorphic germ with which we define a local- ity orthogonal relation between two meromorphic germs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We describe the structure of locality subalgebras generated by classes of meromorphic germs with certain types of poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We also define and determine their group of locality transformations which fix the holomorphic germs and preserve multivariable residues, a group we call the locality Galois group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We then specialise to two classes of meromorphic germs with prescribed types of nested poles, arising from multiple zeta functions in number theory and Feynman integrals in perturbative quantum field theory respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We show that they are locality polynomial subalgebras with locality polynomial bases given by the locality counterpart of Lyndon words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This enables us to explicitly describe their locality Galois group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' As an application, we propose a mathematical interpretation of Speer’s analytic renormalisation for Feynman amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We study a class of locality characters, called generalised evaluators after Speer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We show that the locality Galois group acts transitively on generalised evaluators by composition, thus providing a candidate for a renormalisation group in this multivariable approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' From one to multivariable renormalisation 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality for multivariable meromorphic germs 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality Galois groups and locality Lyndon words 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Generalised evaluators and locality characters 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Outline of the paper 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Meromorphic germs with linear poles 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Spaces of meromorphic germs with linear poles 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Laurent expansions and the induced decompositions 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Dependence subspaces 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality transformation groups on meromorphic germs 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality algebras of meromorphic germs 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Automorphism groups of simplex locality algebras 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality Galois groups 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality polynomial algebras generated by Chen and Speer fractions 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality shuffle algebras as locality polynomial algebras 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Applications to ordered fractions 22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality characters and analytic renormalisation 25 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' 32A20, 13B05, 08A55, 32A27, 81T15, 15A63, 52C07, 81T17, 11M32, 05A05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' meromorphic germ, locality, Galois group, Lyndon word, renormalisation, evaluator, convex cone, renormalisation group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' 1 2 LI GUO, SYLVIE PAYCHA, AND BIN ZHANG 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Speer’s s-families and Speer fractions 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Generalised evaluators on locality subalgebras of meromorphic germs 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality Galois actions on generalised evaluators 29 References 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Introduction This work seeks to reveal the rich structure of meromorphic germs in several variables with linear poles, to describe subalgebras, to explore the structure of transformation group, to eval- uate them at poles in a consistent manner, and to compare different evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The locality framework developed by the authors [CGPZ1], appears to be well suited to achieve these goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' From one to multivariable renormalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Renormalisation is a procedure used to evaluate divergent expressions in various areas of physics and mathematics, ranging from the classical instance of Feynman amplitudes in perturbative quantum field theory [t’H, t’HV] to multiple zeta functions at poles (see [GZ, MP] for example) and Todd functions for toric vari- eties see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' [BV2, P].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' In either case, a preliminary step is a regularisation procedure, after which one can extract divergences and evaluate at the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' There is a great freedom in the choice of regularisation and the method of extracting divergences, specifically, a regularisation can involve one or multiple parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The algebraic structure underlying single parameter renormalisation has attracted great in- terest in mathematics since the groundbreaking work of Connes and Kreimer [CK1, CK2] to tackle Feynman integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' In their approach, the combinatorics of the divergent expressions are organized into a con- nected Hopf algebra H, while the regularisations of the divergent expressions have their Laurent series expansions in the Rota-Baxter algebra M(C) = C[z−1, z]], characterised by its linear de- composition (1) M(C) = M−(C) ⊕ M+(C), into subalgebras M−(C) := z−1C[z−1], M+(C) := C[[z]] and with the induced projection (2) π+ : M(C) −→ M+(C), f (z) = ∞ � k=−K ak zk �→ ∞ � k=0 ak zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The regularisation map is enriched to an algebra homomorphism (3) φ : H → M(C), which then factorises according to the algebraic Birkhoff factorisation, as the convolution prod- uct φ = φ−1 − ⋆ φ+ of a holomorphic part φ+ with values in the subalgebra M+(C) and a polar part φ− with values in M−(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The renormalised map φ+ which is then evaluated at the poles, is built inductively with the recursion encoded in the coproduct, reflecting the celebrated BPHZ procedure in perturbative quantum field theory [BP, Hep, Zi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Extending the Connes-Kreimer approach to multiple parameter regularisations leads to an algebra homomorphism φ : H −→ M(C∞) with values in M(C∞), the algebra of multivariable meromorphic germs at zero with linear poles for the Hopf algebra of convex polyhedral cones [GPZ] and then later in a more general locality framework [CGPZ1, CGPZ2, CGPZ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This approach uses in an essential way a locality version of the Rota-Baxter algebraic structure on M(C∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality for multivariable meromorphic germs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' In physics, the principle of locality is a key feature of field theory which states that an object is influenced directly only by its imme- diate surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We interpret locality more generally as certain binary relations [CGPZ1], enhance algebras to locality algebras and call locality morphisms, the morphisms that preserve such locality relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Its relation to causality in quantum field theory was discussed in [Re].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' In this paper the locality relation on meromorphic germs is induced by an inner product Q on the underlying vector space in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' A pair of meromorphic germs lies in the graph of the locality relation if the linear spaces spanned by the sets of variables they respectively depend on, called their dependence subspaces, are mutually orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='This locality provides a natural splitting M(C∞) = M+(C∞) ⊕ MQ −(C∞) into the subspace M+(C∞) of holomorphic germs and a space MQ −(C∞) of what we call “polar germs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' While MQ −(C∞) is not a subalgebra of MQ(C∞), it bares the remarkable property of being a locality ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' In applications [CGPZ2, CGPZ3], we equip the Hopf algebra with a locality structure, turning it into a locality Hopf algebra, and φ becomes a locality morphism of algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thanks to the fact that MQ(C∞) is a locality ideal, not only the algebraic Birkhoff factorisation of φ can be recovered in the locality setting, moreso, the renormalisation procedure φ+ simplifies to the post composition πQ + ◦ φ with the projection πQ + : M(C∞) → M+(C∞) on the holomorphic part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Its composition ev0◦πQ + with the evaluation at zero ev0 : M+(C∞) → C on the resulting holomorphic germs, can be viewed as a multivariable minimal substraction scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' A prototype of this approach was proposed in the pioneering work of Speer [Sp2, Sp4] on an- alytic renormalisation in quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This multivariable renormalisation method was later implemented for the baby model of Riemann integrals indexed by rooted trees in [CGPZ3] and further discussed in [DZ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus while passing from the classical approach of one parameter renormalisation to the lo- cality approach of multiple parameter renormalisation, the focus of analysis is shifted from the source Hopf algebra H of φ to the target algebra φ(H) in M(C∞) discussed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality Galois groups and locality Lyndon words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' In practice, we consider locality subalgebras φ(H) ⊂ A ⊂ M(C∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' A linear transformation T on A induces another morphism T ◦ φ : H −→ A ⊂ M(C∞), from which we can again build a map πQ + ◦ T ◦ φ : H −→ M+(C∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality isomorphisms of locality subalgebras A ⊂ M(C∞), which restrict to the identity map on M+(C∞) form a group GalQ(A/M+) we call locality Galois groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' It plays the role of a renormalisation group [CK2, CM] in relating different renormalisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Evaluating at zero by the map ev0, gives rise to a locality character ET := ev0 ◦ πQ + ◦ T ◦ φ : H −→ C, which after Speer, we call an evaluator, depending on the choice of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' To describe the action of the locality Galois group, we give a careful study of the structure of locality subalgebras of M(C∞), defined by prescribed types of linear poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We focus on Chen 4 LI GUO, SYLVIE PAYCHA, AND BIN ZHANG type poles which typically arise from multiple zeta functions and on the more general class of Speer fractions which arise in Feynman integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We show that they both form locality poly- nomial algebras, with a locality polynomial basis given a locality version of Lyndon words, we call locality Lyndon words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For this purpose, we enhance to the locality setup, the realisation of shuffle product algebras as polynomial algebras on Lyndon words in the classical work of Chen-Fox-Lyndon and Radford [CFL, Ra].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' As an application, transitivity is established for the action of the locality Galois groups on the generalised evaluators on these two classes of locality subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Generalised evaluators and locality characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' In his seminal work [Sp1, Sp2, Sp3, Sp4] on analytic renormalisation, Speer gives an axiomatic formulation for the evaluation of regularised quantities, called generalised evaluators which he applies to spaces of meromorphic germs in several variables singled out by the regularisation step in his study of Feynman inte- grals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' In the final renormalisation step, he proposes a generalised evaluator defined by averaging over iterated one dimensional evaluators successively applied in each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Speer’s pioneering multivariable approach nevertheless lacks a covariance property since it is coordinate dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' To circumvent this coordinate dependence, we require the evaluator to be multiplicative on products of germs depending on variables which span mutually perpendicular spaces (the dependence subspaces mentioned above) instead of them having disjoint sets of variables as in Speer’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This paper provides a covariant counterpart of Speer’s approach in a sound mathematical framework, with the aim of setting up a general framework to tackle divergences in various contexts and with the following three goals in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Equip the space of germs arising in Speer’s and other multiparameter regulari- sations with appropriate locality polynomial algebra structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Interpret Speer’s generalised evaluators as locality characters on the corre- sponding (locality) algebras, leading to a general concept of locality generalised evaluator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Build a transformation group which relates different locality evaluators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' In the coalgebraic approach to renormalisation in one variable `a la Connes and Kreimer, one calls upon an inductive procedure to deal with mutual compensations of divergences among different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Instead, here we want to avoid the occurence of such compensations by means of a multiparameter regularisation, which enables us to regularise each subdivergence in an autonomous way by introducing a different parameter at each level as we go deeper in the subdivergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This is possible using locality structures, which take care of keeping the different levels separate in requiring the regularisation map to be a locality algebra homomorphism with range in MQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We first provide some background in Section 2 on the space MQ of meromorphic germs with linear poles and rational coefficients on the filtered lattice space (R∞, Z∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For a given inner product Q on the underlying space, a complement of the holomorphic germs MQ+ is given by the subspace MQ Q− of polar germs defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This gives rise to Laurent expansions and various decompositions and invariants (residues) in MQ (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='5), which serve as the building blocks of our further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We further give a detailed study of dependence spaces of meromorphic germs and their decompositions in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='3 (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='13), in order to define the orthogonality of meromorphic germs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' With the above orthogonality of meromorphic germs at hand, we carry out a careful study of locality algebras in Section 3, focusing on subalgebras of MQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1 gives a description of the GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES 5 structures of a locality subalgebra MQ Q+(ΠQ(�S)) of MQ, which contains MQ+ and is generated by a set of fractions S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The automorphisms of such a locality subalgebra, which fix the holomorphic germs and preserve residue type invariants of polar germs, are shown to form a group in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Inspired by Cartier’s cosmic Galois group [B, C, CM], we call it the locality Galois group of the locality subalgebra (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' A reduction theorem (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='15) of locality Galois groups is obtained, which shows that there is a subgroup of the locality Galois group that can be described by special automorphisms of the locality subalgebra generated over Q by the same set of fractions S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' To obtain the structure of locality Galois groups, in Section 4 we first give a locality variant of polynomial algebras (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1) and then extend the polynomial generation of shuf- fle product algebras by Lyndon words to the locality setting (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We finally show (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='9) that the locality subalgebras generated by certain classes of fractions are locality polynomial algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' These include Chen fractions arising in multiple zeta functions, described in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='7 and Speer fractions described in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='10, named after Speer in acknowl- edgment of his work on analytic renormalisation [Sp2, Sp3, Sp4] (see also [BR, DZ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Finally in Section 5, we apply the developed results to revisit Speer’s approach in the locality framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We first show (Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1) that the space spanned by the fractions arising from Speer’s s-families is precisely the space of the aforementioned Speer fractions, and hence it is a locality polynomial algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This addresses Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' As an analog of Speer’s generalised evaluators, in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2 we introduce the notion of locality generalised evaluators (Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2) on a locality subalgebra of MQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' They are linear forms E extending the usual evaluation ev0 at 0 defined on holomorphic germs and, in accordance with the locality principle, they are required to obey the following locality multiplicativity: f1 ⊥Q f2 ⇒ E(f1 f2) = E(f1) E(f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' A proptotype is the minimal subtraction evaluator EQ MS ≔ ev0 ◦ πQ +, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We further show in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='6, that our locality generalised evaluators satisfy the conditions required by Speer for generalised evaluators modulo a topological requirement (which lies out of the scope of this paper, and is discussed in [DPS]) and when adapting the locality relation appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This addresses Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Along the way, we discuss the difference between Speer’s coordinate dependent generalised evaluator Eiter and our covariant minimal subtraction scheme EQ MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The locality Galois group naturally acts on locality generalised evaluators by composition (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (49)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' When the locality subalgebra is generated by a locality polynomial algebra of frac- tions, the action is shown to be transitive (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Consequently, every locality gener- alised evaluator on such a locality polynomial algebra factors through the minimal subtraction evaluator EQ MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Applying these results to Chen fractions and Speer fractions, we obtain Corol- lary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='8, addressing Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Finally we show that multiple zeta values naturally give rise to a locality generalised evaluator Eζ on the locality polynomial algebra AChen := MQ Q+(ΠQ(�FCh)) of meromorphic germs at zero with Chen type poles (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (52)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Comparing Eζ with the minimal subtraction evaluator EQ MS gives a natural element in the locality Galois group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Meromorphic germs with linear poles This section first summarises definitions and results of [GPZ3] on the space of meromorphic germs with linear poles, and then studies the dependence space of meromorphic germs and their decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' 6 LI GUO, SYLVIE PAYCHA, AND BIN ZHANG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Spaces of meromorphic germs with linear poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We work in a filtered lattice Eu- clidean space (R∞, Z∞, Q), consisting of (i) a filtered lattice space (R∞, Z∞) defined by direct limits R∞ ≔ lim −−→ Rk, Z∞ ≔ lim −−→ Zk, under the standard embeddings ik : Rk → Rk+1, (ii) an inner product Q on the filtered lattice space, defined by a family Q = (Qk)k≥1 of inner products Qk : Rk ⊗ Rk → R, such that Qk+1|Rk×Rk = Qk, Qk(Zk ⊗ Zk) ⊂ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For a field K with Q ⊂ K ⊂ R, we denote by LK(Ck) = LK(Kk ⊗ C) the space of linear forms on Ck which take K-values on Kk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' On the filtered lattice space (R∞, Z∞), a meromorphic germ f at zero on Rk ⊗ C is said to be K-holomorphic if it is a holomorphic germ at zero whose power series expansion for any dual basis of Zk has coefficients in K, and to have K-linear poles if there are vectors L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Lk ∈ (Zk)∗ ⊗ K (possibly with repetitions) such that f Πk i=1Li is a K-holomorphic germ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For the rest of the paper, all meromorphic germs are taken to be at zero unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let MK(Ck) = MK(Rk⊗C) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' MK+(Ck) = MK+(Rk⊗C)) denote the space of meromorphic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' K-holomorphic) germs with K-linear poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The inner product Q induces a family of linear bijections Qk : Kk → (Kk)∗, u �→ Qk(u, ·) and Q−1 k : (Kk)∗ → Kk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This gives rise to maps pk ≔ Q−1 k i∗ kQk+1 : Ck+1 → Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Explicitly, let Wk be the orthogonal complement of Ck in Ck+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Then the map pk is the projection of Ck+1 to Ck along Wk and give rise to the directed systems: pk : MK(Ck) → MK(Ck+1), pk : MK+(Ck) → MK+(Ck+1), and the direct limits MK := MK(C∞) := lim −−→ MK(Ck), MK+ := MK+(C∞) := lim −−→ MK+(Ck) of spaces of meromorphic germs with K-linear poles (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' K-holomorphic germs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' By restriction, we also let (4) LK ≔ LK(C∞) := lim −−→ LK(Ck) be the direct limit of spaces of K-linear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Notice that Q induces an inner product in LK(C∞) which we still denote by Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' As in [GPZ3], on a filtered lattice Euclidean space (R∞, Z∞, Q), we define a polar germ in Ck with K-coefficients to be a germ of meromorphic functions of the form (5) h(ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓm) Ls1 1 · · · Lsn n , where GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES 7 (i) h lies in MK+(Cm), (ii) ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓm, L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln lie in LK(Ck), with L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln linearly independent, such that Q(ℓi, Lj) = 0 ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=', m}, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , n}, (iii) m is a nonnegative integer and n, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , sn are positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The convex cone (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' subspace space) spanned by L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln is called the supporting cone (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' supporting space) of the polar germ f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The supporting space, independent of the presen- tation of the germ in the form of the fraction (see [GPZ3, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='9]), is denoted by Supp(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Recall from [GPZ3, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1] that, for a polar germ h(ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=',ℓm) Ls1 1 ···Lsn n , the integer p-ord �h(ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓm) Ls1 1 · · · Lsn n � ≔ |(s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , sn)| ≔ s1 + · · · + sn is well defined, called the p-order of the polar germ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let MQ K−(Ck) denote the linear space spanned by polar germs on Ck with K-linear poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Again we have a directed system pk : MQ K−(Ck) → MQ K−(Ck+1) and the direct limit MQ K− ≔ MQ K−(C∞) ≔ lim −−→ MQ K−(Ck) ⊆ MK(C∞) of polar germs with K-linear poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Polar germs split according to their supporting subspaces and p-orders as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' [GPZ3, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='5] Suppose r� i=1 S i = 0 for a sum of holomorphic germs and K- polar germs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For any linear K-subspace W of V and N ∈ Z>0, we have � i ′S i = 0, where the sum is over the terms S i with Supp(S i) = W and p-ord(S i) = N, with the convention that the sum over an empty set is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Following [GPZ3, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2], (i) a family of (convex) cones is called properly positioned if every pair of cones in the family intersect along their faces, including the zero dimensional face at 0, and their union does not contain a straight line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (ii) a family of polar germs properly positioned if, for each of the polar germs, there is a choice of a supporting cone such that the resulting family of cones is properly posi- tioned;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (iii) a family of polar germs is called projectively properly positioned if it is properly posi- tioned and none of the denominators of the polar germs is proportional to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Here is a useful criterion for the linear independence of polar germs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' [GPZ3, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='6] A finite family of polar germs with projectively prop- erly positioned supporting cones is linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Laurent expansions and the induced decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' There are several decompositions of meromorphic germs with linear poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Recall that a convex cone is called simplicial if it is spanned by a set of linearly independent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' [GPZ3, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='] For any f in MK, there exist a properly positioned family C of simplicial cones together with a family of K-polar germs {S j} j∈J supported on C (in the sense that a supporting cone of each S j is in C), and a holomorphic germ h, such that (6) f = � j∈J S j + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' 8 LI GUO, SYLVIE PAYCHA, AND BIN ZHANG Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (6) is called a Laurent expansion of f supported on C and it is unique up to subdivisions of the properly positioned family of simplicial cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For p ∈ Z≥0, d ∈ Z≥0 and a finite dimensional K-subspace U ⊂ R∞, let Mp K denote the linear span of K-polar germs of p-order p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' MK,d denote the linear span of K-polar germs whose supporting cones have dimension d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' MK,U denote the linear span of K-polar germs with supporting space U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' [GPZ3, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='3] We have the decompositions MK = � p≥0 Mp K, (7) MK = � d≥0 MK,d, (8) MK = � U⊂R∞ MK,U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (9) In particular, there is a decomposition (see also [BV1]): MK = MK+ ⊕ MQ K−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We give further notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (i) Corresponding to the decomposition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (7), let q be the highest p-order of the polar germs in a (thus every) Laurent expansion of f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Define the p-residue of f [GPZ3, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1] by (10) pRes(f ) : = � p-ord(S i)=q hi(0) ⃗L⃗si i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (ii) Corresponding to the decomposition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (8), let e be the largest among the dimensions of the supporting spaces of the polar germs in a (thus every) Laurent expansion of f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Define the d-residue of f by (11) dRes(f ) : = � dim(Supp(S i))=e hi(0) ⃗L⃗si i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' As proved in [GPZ3, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2], the p-residue of a meromorphic germ with linear poles depends neither on the choice of a Laurent expansion nor on the choice of the inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The d-residue does not depend on the choice of a Laurent expansion, but it does depend on the choice of the inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Dependence subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For any subset U of MQ, let QU denote the Q-subspace of MK spanned by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' A simplex fraction is a fraction of the form 1 Ls1 1 ···L sk k , where L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Lk ∈ LQ are linearly independent and si ∈ Z>0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let F be the set of all simplex fractions over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Then for any inner product Q in (R∞, Z∞), we trivially have QF ⊂ MQ Q−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' In the Euclidean filtered lattice space (R∞, Z∞, Q), let B ≔ (ei)i∈Z>0 be an or- thonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let zi be the coordinate function corresponding to ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' A fraction of the form (12) f � s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=',sk u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=',uk � ≔ 1 zs1 u1(zu1 + zu2)s2 · · · (zu1 + zu2 + · · · + zuk)sk , ui, si ∈ Z>0, k ∈ N, ui � u j if i � j, GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES 9 is called a Chen fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The set of Chen fractions is denoted by FCh ≔ FCh,Q,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We borrow the following definitions from [GPZ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' A meromorphic function f on Ck of the form f = g(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln), where L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln are linear forms on Ck and g a meromorphic function on Cn, is said to depend on the linear subspace of (Ck)∗ spanned by L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' One can check that if f depends on V1 and V2, then it depends on V1 ∩ V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus it makes sense to set the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The dependence subspace Dep(f ) of f is the smallest linear subspace of (Ck)∗ on which f depends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let (e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=') be an orthonormal basis of (R∞, Z∞, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Dep � 1 z1(z1 + z2) + 1 z2(z1 + z2) − 2 z1(z1 + 2z2) − 1 z2(z1 + 2z2) + 1 z3 � = {e3}, since the sum of the first four terms is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Two meromorphic germs f and g in MQ are called Q-orthogonal, which we write f ⊥Q g, if their dependence subspaces are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (i) Let (e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=') be an orthonormal basis of (R∞, Z∞, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We have 1 z1 + z2 ⊥Q (z1 − z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (ii) Polar germs are precisely germs of the form h/M for h in MQ+(C∞) and M given by products of powers of linearly independent linear forms, such that h ⊥Q M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' An element f ∈ MQ is of the form f = h ℓ1···ℓr for a holomorphic germ h and linear forms ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The next lemma shows that the factors in the fraction can be chosen to have their dependence subspaces contained in the dependence subspace of f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For any f in MQ, there are linear forms ℓi = ℓi(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , p, and a holomorphic germ h = h(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln) for a basis L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln of Dep(f ), such that f = h ℓ1 · · · ℓp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let f be in MQ(Ck) for some k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We extend a basis L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln of Dep(f ) to a basis L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Lk of (Ck)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Since f is in MQ(Ck), there are linear combinations ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓm of L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Lk such that the product ℓ1 · · · ℓm f is in MQ+(Ck), that is, (13) ℓ1 · · · ℓm f (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln) ∈ MQ+(Ck).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' By rearrangement, we can assume that ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓp are linear combinations of L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' while ℓp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓm have nontrivial linear contributions from the extra linear forms Ln+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Then we can choose a tuple (an+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ak) ∈ Ck−n such that the maps (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln) �→ λi(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln) ≔ ℓi(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln, an+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ak), i = p + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=', m, are affine with λi(0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , 0) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Consequently, the maps (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln) �−→ 1 λi(L1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=',Ln) are holomor- phic germs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus setting Ln+1 = an+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Lk = ak in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (13) yields a holomorphic germ h : (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln) �−→ ℓ1 · · · ℓp λp+1 · · · λm f, from which we define another holomorphic germ 10 LI GUO, SYLVIE PAYCHA, AND BIN ZHANG ˜h(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln) ≔ h(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln) λp+1 · · · λm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Hence, f = f (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln) = h(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln) ℓ1 · · · ℓpλp+1 · · · λm = ˜h(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln) ℓ1(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln) · · · ℓp(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln) is of the desired form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Write a germ f in MQ according to the decomposition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (9): (14) f = � U∈U fU + f0, where U is a finite set of nonzero finite-dimensional subspaces of R∞, 0 � fU is a sum of polar germs with supporting space U and f0 lies in MQ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We have Dep(f ) = � U∈U Dep(fU) + Dep(f0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Since f = � fU + f0, clearly we have Dep(f ) ⊂ � U∈U Dep(fU) + Dep(f0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' It remains to show that Dep(fU) ⊂ Dep(f ) for all U ∈ U and Dep(f0) ⊂ Dep(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='12, there are linear forms ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓp and a homomorphic germ h, all with depen- dent spaces in Dep(f ) such that f = h ℓ1 · · · ℓp = h(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln) ℓ1(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln) · · · ℓp(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln), for a basis L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln of Dep(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Then we can take the Laurent expansion of f in Dep(f ) by [GPZ3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus for all the polar germs in this Laurent expansion of f , their linear poles and holomorphic numerators have dependence space in Dep(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This gives another decomposition f = g0 + � V⊂Dep(f) gV according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Comparing with the decomposition of f in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (14) as a sum of a holo- morphic germ and polar germs, we have f = f0 + � U⊂U fU = g0 + � V⊂Dep(f) gV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Using the uniqueness of the decomposition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (9), we infer that for any subspace U ⊂ U (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' U = 0), there is a space V ⊂ Dep(f ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' V = 0) such that fU = gV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This implies that Dep(fU) is contained in Dep(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For a nonzero rational linear combination f = � i∈I αiS i ∈ MQ(Ck) of simplex fractions S i, i ∈ I, with the same supporting space U, we have Dep(f ) = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Clearly, Dep(f ) ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Suppose Dep(f ) ⊊ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='13 gives f = g0 + � V⊂Dep(f) gV GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES 11 where the terms in gV have supporting space V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Hence 0 = f − � i∈I αi S i = g0 + � V⊂Dep(f) gV − � i∈I αi S i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' From Dep(f ) ⊊ U, we have V ⊊ U in the above sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus the above sum is the decomposition of 0 according to the supporting spaces in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (9), in which � i∈I αi S i is the component with supporting space U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus � i∈I αi S i = 0, meaning f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' □ This leads to the following statement on sums of polar germs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' If f = � fi is a nonzero sum of polar germs fi with the same supporting space U, then U is a subset of Dep(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='14 after evaluation of the numerators of the polar germs fi at appropriate arguments in the spirit of the proof of [GPZ3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Indeed, let us write the polar germs fi = hi S i where S i = 1 Ls1 1 · · · L sni n , s j ∈ Z≥0, are simplex fractions with the same supporting space and hi are holomorphic germs in some common set of variables ℓn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓk which complete the independent linear forms L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln arising in the S i’s to an orthonormal basis of Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Since f � 0 we can assume without loss of generality that none of the holomorphic germs hi is identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Hence, there is some tuple (ℓ0 n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓ0 k) such that αi ≔ hi(ℓ0 n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓ0 k) � 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We write f = f (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln, ℓn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓk) and take the specialisation g(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln) ≔ f (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln, ℓ0 n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓ0 k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Then Dep(f ) ⊃ Dep(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='14 to g = � i hi(ℓ0 n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓ0 k) S i with αi = hi(ℓ0 n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓ0 k), we obtain Dep(g) = Supp(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' □ The following result shows that without loss of generality, we can assume that a sum f of polar germs with the same supporting space can be written as a sum of polar germs whose numerators are holomorphic germs with dependence space in Dep(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let f = � hi S i be a nonzero sum of polar germs with the same supporting space, where S i is a simplex fraction and hi is a holomorphic germ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Then f can be written as a sum f = � ˜hi S i of polar germs where the ˜hi’s are now holomorphic germs with dependence spaces in Dep(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='15, the common supporting space U lies in Dep(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln be a basis of U which we extend to a basis L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln, ℓn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓm of Dep(f ) and then further to a basis L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln, ℓn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓk of (Ck)∗ with Q(Li, ℓ j) = 0, 1 ≤ i ≤ n, n + 1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus, S i is a simplex fraction in the variables L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln and f = f (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln, ℓn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓk) = � hi(ℓn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓk) S i(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' By the definition of dependence space, f does not depends on ℓm+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓk, so f = f (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Ln, ℓn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓm, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , 0) = � hi(ℓn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , ℓm, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , 0) S i(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' , Lk) as announced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality transformation groups on meromorphic germs In this section, we study meromorphic germs with linear poles in the context of locality algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We then introduce the locality Galois group defined as a group of automorphisms of meromorphic germs in this locality framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' 12 LI GUO, SYLVIE PAYCHA, AND BIN ZHANG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality algebras of meromorphic germs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We give general background on locality al- gebras and then focus on locality subalgebras of meromorphic germs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We recall notations on locality structures from [CGPZ1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' A locality set is a couple (X, ⊤) where X is a set and ⊤ ≔ X ×⊤ X ⊆ X × X is a binary symmetric relation, called a locality relation, on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For x1, x2 ∈ X, denote x1⊤x2 if (x1, x2) ∈ ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For a subset U ⊂ X, the polar subset of U is U⊤ ≔ {x ∈ X | (x, U) ⊆ ⊤}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For locality sets (X, ⊤X) and (Y, ⊤Y), a map f : X → Y is called a locality map if (15) x1⊤Xx2 =⇒ f (x1)⊤Y f (x2), ∀x1, x2 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We give some examples that will be further explored in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (i) For any nonempty set X, being distinct: x1⊤x2 if x1 � x2, defines a locality relation on X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (ii) The Q-orthogonality relation ⊥Q⊂ MQ×MQ of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='10, turns MQ into a locality set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (iii) Let (X, ⊤) = (Z>0, ⊤) be the locality set in (i) and (MQ, ⊥Q) the locality set in (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' With the notation in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='7, the map f : X → MQ, n �→ f � 1 n � ≔ 1 zn , n > 1, is a locality map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Other algebraic structures can be generalised to the locality setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (i) A locality vector space is a vector space V equipped with a locality relation ⊤ which is compatible with the linear structure on V in the sense that, for any subset X of V, X⊤ is a linear subspace of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (ii) A (nonunitary) locality algebra over K is a locality vector space (A, ⊤) over K together with a map mA : A ×⊤ A → A, (u, v) �→ u · v = mA(x, y) for all (u, v) ∈ A ×⊤ A satisfying the following variations of the associativity and distributivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (a) For u, v, w ∈ A with u⊤v, u⊤w, v⊤w, we have (16) (u · v)⊤w, u⊤(v · w), (u · v) · w = u · (v · w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (b) For u, v, w ∈ A with u⊤w, v⊤w (and hence (u + v)⊤w, w⊤(u + v)), we have (u + v) · w = u · w + v · w, w · (u + v) = w · u + w · v, (ku) · w = k(u · w), u · (kw) = k(u · w), k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (iii) A unitary locality algebra is a locality algebra (A, ⊤, mA) with a unit 1A such that, for each u ∈ A, we have 1A⊤u and 1A · u = u · 1A = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We shall omit explicitly mentioning the unit 1A unless doing so generates ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES 13 (iv) Let (A, ⊤A) be a locality algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' A subspace B of A is called a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' unitary) locality subalgebra of A if, with the restricted relation ⊤B ≔ ⊤A ∩ (B × B) of ⊤A to B, the pair (B, ⊤B) is a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' unitary) locality algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (v) Let (A, ⊤A) be a commutative locality algebra and C a unitary locality subalgebra of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' A subspace B of A is called a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' unitary) locality C-subalgebra of A if B is a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' unitary) locality subalgebra of A that contains C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Given two locality algebras (Ai, ⊤i), i = 1, 2, a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' unitary) locality algebra homomor- phism is a linear map ϕ : A1 −→ A2 such that a⊤1b implies ϕ(a)⊤2ϕ(b) and ϕ(a·b) = ϕ(a)·ϕ(b) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' and ϕ(1A1) = 1A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' [CGPZ1, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='23] With the relation ⊥Q of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='10, the pair (MQ, ⊥Q ) is a unitary locality algebra and the projection (17) πQ + : MQ = MQ+ ⊕ MQ Q− → MQ+ along MQ Q− is a unitary locality algebra homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For a unitary locality algebra (A, ⊤), a unitary locality endomorphism of A is a locality automorphism if it is invertible, preserves the unit, and the inverse map is a locality algebra homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let Aut⊤(A) denote the set of locality automorphisms of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We note that Aut⊤(A) forms a group for the composition, called the locality automorphism group of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' There are counter examples that a bijective locality homomorphism needs not be a locality automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality subalgebras of meromorphic germs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' In the sequel, we consider locality subal- gebras (A, ⊥Q) of the locality algebra (MQ, ⊥Q) in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For a subset U of MQ, let (18) � U ≔ U ∪ {1}, with 1 being the constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We first give the structure of locality subalgebras of MQ generated by a set, with rational co- efficients or MQ+ coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' As we shall see, a careful analysis using tools such as supporting and dependent spaces is needed when extending the notion of subalgebra to the locality setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Given a subset U of MQ, let ΠQ(U) ≔ � � i si ����� si ∈ U, ∀i, si ⊥Q s j, ∀i � j � be the set of meromorphic germs locality generated by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' With the notation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (18), we have ΠQ(�U) = ΠQ(U) ∪ {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Given a set S of simplex fractions, the subspace of Q�F QΠQ(�S) ≔ � � i ci S i ����� ci ∈ Q, S i ∈ ΠQ(�S) � spanned by ΠQ(�S) is a unitary locality subalgebra of QF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus QΠQ(�S) is the unitary locality subalgebra of Q�F generated by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Since c0 = 1 serves as the unit, we just need to prove that for f, g in ΠQ(S) with f ⊥Q g, f g lies in QΠQ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' According to the grading in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (9), we write f = � U fU, g = � V gV, 14 LI GUO, SYLVIE PAYCHA, AND BIN ZHANG where fU is the sum of simplex fractions with the same supporting space U and gV is the sum of simplex fractions with the same supporting space V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='13, we have Dep(f ) = � U Dep(fU), Dep(g) = � V Dep(gV) from which we infer that, for any U and V appearing in the decompositions of f and g, f ⊥Q g =⇒ fU ⊥Q gV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='14, each fU � 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' gV � 0), being a sum of simple fractions with the same supporting space U (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' V), gives U = Dep(fU) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' V = Dep(gV)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus we have U ⊥Q V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Hence the products fUgV arising in the decomposition f g = � U,V fUgV lie in QΠQ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Conse- quently the product f g also lies in QΠQ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Given a set S of simplex fractions, the set MQ Q+ �ΠQ(�S)� ≔ � � i hiS i, ����� hi ∈ MQ+, S i ∈ ΠQ(�S), hi ⊥Q S i � is a unitary locality subalgebra of MQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus MQ Q+ �ΠQ(�S)� is the unitary locality MQ+-subalgebra of MQ generated by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Clearly, MQ Q+(ΠQ(�S)) is a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' As in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='6, we only need to verify that, for a, b ∈ MQ Q+(ΠQ(�S)) with a ⊥Q b, the locality product ab is well defined and lies in MQ Q+(ΠQ(�S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' To complete this, in accordance with the grading in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (9), write a = � U � i aUiS Ui = � U aU, b = � V � j bVjTVj = � V bV, where aUi, bVj ∈ MQ+, S Ui, TVj ∈ ΠQ(S) and aUi ⊥Q S Ui, bVj ⊥Q TVj, Supp(S Ui) = U, Supp(TVj) = V (with the convention that Supp(h) = 0 for h ∈ MQ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='13 gives Dep(aU) ⊂ Dep(a) and Dep(bV) ⊂ Dep(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Since a ⊥Q b means Dep(a) ⊥Q Dep(b), we have Dep(aU) ⊥Q Dep(bV), that is aU ⊥Q bV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Now let aU � 0 and bV � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (i) By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='15, we have U ⊂ Dep(a) and V ⊂ Dep(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' So U ⊥Q V and S Ui ⊥Q TVj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (ii) From Dep(aU) ⊂ Dep(a) and V ⊂ Dep(b), we obtain aU ⊥Q TVj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Similarly, bV ⊥Q S Ui;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (iii) By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='16, there exist holomorphic germs ˜aUi and ˜bVj with Dep(˜aUi) ⊂ Dep(aU), ˜aUi ⊥Q S Ui, Dep(˜bVj) ⊂ Dep(bU), ˜bVj ⊥Q TVj, such that aU = � i ˜aUiS Ui, bV = � j ˜bVjTVj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' From aU ⊥Q TVj and Dep(˜aUi) ⊂ Dep(aU), we have ˜aUi ⊥Q TVj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Likewise, ˜bVj ⊥Q S Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' In summary, ˜aUi, ˜bVj, S Ui, TVj are mutually Q-orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus aUbV = � i, j ˜aUi ˜bVjS UiTVj, is de- fined and gives an element in MQ Q+(ΠQ(�S))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Therefore, ab = � U,V aUbV is defined and lies in MQ Q+(ΠQ(�S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' □ In the subsequent examples, we implicitly fix an orthonormal basis E with respect to an inner product Q in R∞, only referring to these choices in the notation when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES 15 Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The set FCh = FCh,Q,E of Chen fractions in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='7 generates the unitary locality subalgebra MCh Q ≔ MQ Q+ � ΠQ(�FCh) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For a finite subset J of Z>0, we set zJ ≔ � i∈J zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' As considered by Speer [Sp2] (see § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2), a Feynman fraction is a simplex fraction (19) 1 � J∈J zsJ J , sJ > 0, for a finite collection J of finite subsets of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' With similar notations of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='8, the set of Feynman fractions and the locality subalgebra it generates are denoted by FFe ≔ FFe,Q,E, MFe Q ≔ MQ Q+ � ΠQ(�FFe) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Automorphism groups of simplex locality algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For a locality subalgebra A of (MQ, ⊥Q), let AutQ(A) denote the group of locality automorphisms of A, following Defini- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let S be a set of simplex fractions and let QΠQ(�S) be the unitary locality subalgebra of Q�FQ generated by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let AutQ Res(QΠQ(�S)) be the set of unitary locality algebra homomorphisms ϕ : QΠQ(�S) → QΠQ(�S) with the property that, for any fraction S in ΠQ(S), (20) ϕ (S ) = S + � i ai S i, where ai ∈ Q, S i ∈ ΠQ(S), p-ord(S i) < p-ord(S ), Supp(S i) ⊊ Supp (S ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Then AutQ Res(QΠQ(�S)) is a subgroup of AutQ(QΠQ(�S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' As a consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='15 yet to come, we have (21) AutQ Res(QΠQ(�S)) = � ϕ ∈ AutQ(QΠQ(�S)) ���� ϕ preserves the p-residue and d-residue � which justifies the notation with subscript “Res”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Denote R ≔ RS ≔ QΠQ(�S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We first prove that ϕ is one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let f = � i aiS i ∈ R, ai ∈ Q, S i ∈ ΠQ(S) ∪ {1}, be nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' If f is a constant in Q, then ϕ(f ) = f � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' If f is not a constant, we group the terms of f according to the gradation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (9): f = c0 + � U∈U fU, where U is a finite nonempty set of nonzero subspaces of R∞ and 0 � fU ∈ QΠQ(�S), for any U in U, is a sum of fractions with supporting space U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Applying ϕ yields ϕ(f ) = c0 + � U∈U � fU + � V⊊U gV � , where gV is a sum (possibly zero) of fractions with supporting space V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For a maximal element U0 in U, fU0 is the only contribution in the above sum arising in ϕ(f ) to the component with supporting space U0 in the decomposition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' So ϕ(f ) = 0 implies fU0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' It follows that ϕ(f ) � 0, which ends the proof of the injectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' To prove the surjectivity of ϕ, by the linearity of ϕ, we only need to show that every element f of ΠQ(�S) lies in the range Im ϕ of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Suppose this is not the case and let U0 � 0 be minimal among the supporting spaces of elements in ΠQ(�S)\\Imϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let f0 be one of the simplex fractions in ΠQ(�S)\\Imϕ with supporting space U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Then by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (20) we have 16 LI GUO, SYLVIE PAYCHA, AND BIN ZHANG ϕ(f0) = f0 + � V⊊U0 fV, where fV ∈ Q(ΠQ(�S)) is the sum of simplex fractions with supporting space V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The space U0 being minimal, each fV lies in the image of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Therefore, f0 = ϕ(f0) − � V⊊U0 fV also lies in the image of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This is a contradiction, showing that ϕ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We finally prove that ϕ−1 is a locality algebra homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We first show that ϕ−1 has the property in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (20), that is, for any S ∈ ΠQ(�S), we have ϕ−1(S ) = S + � i hiS i, where each S i ∈ ΠQ(�S) has smaller supporting space and p-order than those of S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Assume that this were not the case, and let S have a minimal supporting space U0 among the counterexam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Then ϕ(S ) = S + � j b jT j, where each simplex fraction T j ∈ ΠQ(�S) has it supporting space and p-order smaller than those of S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Applying ϕ−1 gives (22) S = ϕ−1(S ) + � j b jϕ−1(T j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The minimality of the supporting space of S yields ϕ−1(T j) = T j + � jk d jkT jk, where each T jk ∈ ΠQ(�S) has its supporting space and p-order smaller than those of T j and hence of S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (22) gives ϕ−1(S ) = S − � j ϕ−1(T j) = S − � j b j \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edT j + � jk d jkT jk \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 , which shows that ϕ−1(S ) has the form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This gives the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' To check that ϕ−1 is a locality map, we consider two linear combinations f, g in R = Q(ΠQ(�S)) and group the terms f = c + � U∈U fU and g = d + � V∈V gV, with c, d in Q and nonzero sums fU, gV ∈ Q(ΠQ(�S)) of fractions with supporting spaces U ∈ U and V ∈ V respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We proceed to show that f ⊥Q g implies ϕ−1(f ) ⊥Q ϕ−1(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let U be an element in U and V an element in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='13, Dep fU ⊂ Dep f and Dep gV ⊂ Dep g so that Dep f ⊥Q Dep g implies Dep fU ⊥Q Dep gV and hence fU ⊥Q gV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='14, for the linear combination fU (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' gV) of simplex fractions with the same supporting space U (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' V), we have U = Dep(fU) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' V = Dep(gV)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus fU ⊥Q gV implies U ⊥Q V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Since ϕ−1 has the property in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (20), we have Dep(ϕ−1(fU)) ⊆ Dep(fU) and Dep(ϕ−1(gV) ⊆ Dep(gV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Then ϕ−1(fU) ⊥Q ϕ−1(gV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Therefore, by the linearity of ϕ−1, we obtain ϕ−1(f ) ⊥Q ϕ−1(g), as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Finally for f, g ∈ R with f ⊥Q g, we have ϕ−1(f ) ⊥Q ϕ−1(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Hence ϕ(ϕ−1(f ) ϕ−1(g)) = ϕ(ϕ−1(f )) ϕ(ϕ−1(g)) = f g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Therefore, applying ϕ−1, we obtain ϕ−1(f ) ϕ−1(g) = ϕ−1(f g), showing that ϕ−1 is a locality homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' □ GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Locality Galois groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' In this part we consider a locality subalgebra A of (MQ, ⊥Q) containing MQ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For any locality morphism ϕ : A → A with ϕ|MQ+ = Id, we have Dep(ϕ(f )) ⊂ Dep(f ), ∀f ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For any ℓ in LQ viewed as an element of (Rk)∗ for some k ≥ 1, if ℓ ⊥Q Dep(f ), then ℓ ⊥Q f , which implies that ϕ(ℓ) ⊥Q ϕ(f ) since ϕ is a locality map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Since ℓ is in MQ+, we have ϕ(ℓ) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus ℓ ⊥Q Dep(ϕ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' So Dep(f )⊤ ⊆ Dep(ϕ(x))⊤ which yields the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' □ In the sequel, we fix a set S ⊆ F of simplex fractions and let (23) B ≔ B(S) ≔ Q(ΠQ(�S)), A ≔ A(S) ≔ MQ Q+ � ΠQ(�S) � be the unitary locality subalgebra and MQ Q+-subalgebra of MQ generated by S, defined in Propo- sitions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='7 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Define a subset of AutQ(A) by GalQ(A/MQ+) ≔ \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3ϕ ∈ AutQ(A) �������� ϕ|MQ+ = Id ϕ preserves the p-residue, d-residue and the locality subalgebra B \uf8fc\uf8f4\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8f4\uf8fe It will be called the locality Galois group of A over MQ+, thanks to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We next give a locality tensor product property of MQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let S ⊆ F and, as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (23), define B ≔ B(S) ≔ Q(ΠQ(�S)), A ≔ A(S) ≔ MQ+ � ΠQ(�S) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (i) For each Q-subspace U of R∞, let AU ≔ A ∩ MQ,U and let BU denote the linear span of simplex fractions in B with supporting space U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let MU Q+ denote the space of holomorphic germs whose dependent space is contained in U⊥Q ≔ � y ∈ L(C∞) ��� y ⊥Q u, ∀u ∈ U � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Then we have the (inner) tensor product AU = MU Q+ ⊗ BU, that is, MU Q+ and BU are linearly disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (ii) Let (V, ⊤) be a locality vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Any pair of locality linear maps ϕ : B → V and ψ : MQ+ → V uniquely extends to a locality linear map ϕ ⊗Q ψ : A → V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (i) By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='7, (24) f = � i hiS i , where S i ∈ ΠQ(�S) with Dep(S i) = U, hi is holomorphic with dependent space contained in U⊥Q and hence hi ⊥Q S i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' So AU = MU Q+BU as a product of subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' To prove the disjointness, we more generally consider a linear combination (25) � i hiS i = 0, where Dep(S i) = U, hi is holomorphic with dependent space contained in U⊥Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Suppose that {S i}i is linearly independent, but hi � 0 for all i in the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Denote V = � i Dep(hi) which is a finite-dimensional subspace of U⊥Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Then hi is defined on V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus we can choose disjoint sets 18 LI GUO, SYLVIE PAYCHA, AND BIN ZHANG of variables {zk} of U and {wℓ} of V respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' From hi � 0, there is {w0 ℓ} such that hi({w0 ℓ}) � 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (25) gives � i hi({w0 ℓ}) S i = 0, showing that {S i}i is linearly dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This gives the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (ii) By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='7, A is linearly spanned by homogeneous elements with respect to the grading by supporting space in the grading Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus A has the restricted grading A = � U⊂R∞ AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Then we just need to show that ϕ and ψ uniquely define a locality linear map (ϕ ⊗Q ψ)U : AU → V for each subspace U of R∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' By Item (i), for any linear map ϕ : BU → V and ψ : MU Q+ :→ V, there is a unique linear map (26) (ϕ ⊗Q ψ)U : AU → V, f �→ � i ψ(hi)ϕ(S i) for any element f = � i hiS i in AU, expressed in the form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Indeed, (ϕ ⊗Q ψ)U is simply the tensor product of the restriction of ψ to MU Q+ and the restriction of ϕ to BU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Taking the sum over all subspaces U of L(R∞) including U = 0, we have an extension ϕ ⊗Q ψ of ϕ and ψ to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' □ The remaining part of the section is devoted to the proof of the following theorem which extends an element of AutQ Res(B) to an element of GalQ(A/MQ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let S ⊆ F and let A and B be as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (i) Any element ϕ ∈ AutQ Res(B) (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='10) uniquely extends to an element of GalQ(A/MQ+) defined by (27) ˜ϕ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed � i hiS i \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ≔ � i hiϕ(S i) for (28) f = � i hiS i ∈ A, hi ∈ MQ+, S i ∈ ΠQ(�S) as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (ii) The subset GalQ(A/MQ+) ⊆ AutQ(A) is a subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Restricting to B gives rise to a group isomorphism GalQ(A/MQ+) � AutQ Res(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (i) Applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='14 with ψ the identity map, for any linear map ϕ : BU → BU, there is a unique linear map (29) ˜ϕ : MQ,U → MQ,U, f �→ � i hiϕ(S i) for any element f = � i hiS i in MQ,U, expressed in the form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We next show that the ˜ϕ obtained this way has the form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Let f = � i hiS i as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' By grouping the terms according to the supporting spaces of S i as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (9), we have f = � U∈U fU with fU = � j aU jS U j, where U is a set of subspaces U of R∞ for which fU � 0, and for each U ∈ U, we have S U j ∈ ΠQ(S), Dep(S U j) = Supp(S U j) = U, Dep(aU j) ⊥Q Dep(S U j) GALOIS GROUPS AND GERMS IN SEVERAL VARIABLES 19 and each term aU jS U j is one of the terms in f = � i hiS i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus fU is in MQ,U and we can apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (29) and obtain ˜ϕ(fU) = � i aU jϕ(S U j), which takes the form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Hence so is ˜ϕ(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' This is what we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The fact that ˜ϕ preserves the p-residue in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (10) and the d-residue in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (11) follows from the definition and the special form of ϕ on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For f = � i hiS i ∈ A = MQ+(ΠQ(S)) with hi ∈ MQ+, S i ∈ S, the p-residue p-res(f ) of f is of the form �′ i hi(0)S i where the sum is over simplex fractions S i in f with the highest order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' By the definition of ˜ϕ, the sum �′ i hiS i is still the part of ˜ϕ(f ) with the highest order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Therefore, p-res(˜ϕ(f )) = p-res(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The same argument, applied to the dimensions of supporting spaces of the polar germs, shows that ˜ϕ preserves the d-residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' We next check that �ϕ is a locality MQ+-algebra homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' For a, b ∈ A with a ⊥Q b, as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content='6, we can write them as a = � U � i hUiS Ui, b = � V � j gVjTVj such that hUi � 0, gVj � 0, {hUi, S Ui} ⊥ {gVj, TVj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' By the special form of ϕ, we have Dep(ϕ(S Ui)) = U, Dep(ϕ(TVi)) = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Then it follows from the definition of ˜ϕ that ˜ϕ(a) ⊥Q ˜ϕ(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Moreover (treating h0 as hUS U for U = 0 and the same for g0), ˜ϕ(ab) = ˜ϕ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed � U,V hUigVjS UiTVj \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 = � U,V hUigVjϕ(S UiTVj) = � U,V hUigVjϕ(S Ui)ϕ(TVj) = ˜ϕ(a)˜ϕ(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' By construction, the extension ϕ �→ ˜ϕ is functorial: � ϕψ = ˜ϕ ˜ψ, �idB = idA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' So for any ϕ ∈ AutQ Res(B), ˜ϕ is a linear bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' The functorial property also shows that ˜ϕ � ϕ−1 = idA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' So ˜ϕ−1 = � ϕ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus �ϕ−1 is also a locality MQ+-algebra homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Thus ˜ϕ is in GalQ(A/MQ+) for all ϕ ∈ AutQ Res(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' By � ϕ ψ = ˜ϕ ˜ψ, ˜ϕ−1 = � ϕ−1, the image of the map (30) Ψ : AutQ Res(B) → GalQ(A/MQ+) : ϕ �→ ˜ϕ, is a subgroup of AutQ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (ii) The map Ψ defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' (30) is clearly injective and its image is in GalQ(A/MQ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Now for any g ∈ GalQ(A/MQ+), since it preserves B, for any S ∈ ΠQ(S), g(S ) = � i aiS i, with ai ∈ Q, S i ∈ ΠQ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' Note that the p-residue of g(S ) equals to the p-residue of S which is just S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE0T4oBgHgl3EQfYAA8/content/2301.02300v1.pdf'} +page_content=' So we can write g(S ) = � aiS i = � p-ord(S i)=p-ord(S ) aiS i + � p-ord(S i) 0.6, that is roughly half of the range of values for +Class II sources. In order to estimate extinction-corrected spec- +tral indices (α′), we computed a correction factor of -0.31 as the +median of differences between all the α′ and α values reported +in Dunham et al. (2015). We note that this method only provides +Table 1: List of SFRs considered in this study. +SFR +R.A. +Decl. +Distance +Radius +YSOs +RNe +[deg] +[deg] +[pc] +[deg] +Ophiuchus +249.07 +-23.64 +128.0 +6.0 +450 +13 +Taurus +67.11 +26.62 +148.0 +8.0 +190 +28 +Corona Australis +287.96 +-38.11 +155.0 +4.0 +80 +6 +Lupus +240.08 +-36.56 +158.0 +8.0 +480 +10 +Chamaeleon +169.5 +-78.17 +190.0 +8.0 +148 +7 +Perseus +54.21 +31.57 +284.0 +4.0 +435 +19 +Orion +85.2 +-3.5 +420.0 +6.0 +978 +44 +Serpens +277.66 +-1.91 +495.0 +3.0 +2169 +14 +Notes. The median coordinates and distances are from Zucker et al. +(2020). The last three columns refer to the radii used to define the SFR +boundaries (see Figure A.1) and the number of RNe and YSOs found. +a rough estimate of α′ as it does not account for extinction val- +ues for individual sources. A comparison of our α and α′ values +is shown in Figure B. Following the same classification criteria +as Dunham et al. (2015), that is −1.6 ≤ α′ < −0.3, 2562 out of +4930 YSOs in our SFRs were classified as Class II sources. +The next step was to cross match our list of Class II sources +with the merged catalogue of RNe. For the specific goal of this +work, the cross-matching distance between a YSO and RN can +be empirically estimated as the length scale from where ambient +gas can be accreted onto an isolated star (Bondi-Hoyle accretion, +Bondi & Hoyle 1944; Throop & Bally 2008; Padoan et al. 2014). +The typical length scale for this interaction is given as LBH = +2GM∗/v2, where G is the gravitational constant, M∗ is the stellar +mass, and v is the stellar speed relative to the gas. For a stellar +mass of 1 M⊙ and typical stellar velocity of 1 km s−1, LBH ∼ +2000 au. The 21 late-infall Class II candidates that we found +using this threshold are listed in Appendix C. +We note that this distance threshold is a lower limit for an +interaction between clouds and YSOs because we do not ac- +count for the sizes of RNe. These show a range of physical sizes +(∼ 103–105 au) and have highly asymmetrical shapes (e.g. Con- +nelley et al. 2007). When mining the ALMA archive, we ignored +this diversity as we aimed to identify a reliable sample of YSOs +potentially interacting with RNe clouds. The implications of a +more realistic distance threshold is discussed in Section 4. +ALMA archive search: +To target higher-quality observations, +we focussed on nearby SFRs (d ≲ 200 pc), where 16 out of +the 21 Class II disks near RNe are located. For these nearby +sources, we searched for existing observations in the ALMA +Science Archive. We are primarily interested in 12CO emission +(J = 2−1 in Band 6 and J = 3−2 in Band 7) since this molecule +is expected to be a good tracer of large-scale diffuse gas struc- +tures (see e.g. Figure 1). We found a total of 66 Band 6 and 7 +observations for 16 sources in our sample, as shown in Figure 2. +Among these datasets, we found 13 observations with the +largest angular scale (LAS) corresponding to at least 1000 au to +recover the expected large-scale structures (see Kuffmeier et al. +2020). As such large-scale structures are expected to have low +column densities, and therefore faint emission, the observational +sensitivity is important. We selected a subset of five observations +with rms noise (over a channel width of 10 km s−1) smaller than +a 2.3 mJy beam−1, equal to the line sensitivity reported in the +ALMA archive for SU Aur observations (Ginski et al. 2021). +For the typical angular resolution of 0.7′′ for these observations, +the sensitivity threshold of the 2.3 mJy beam−1 corresponds to +∼ 100 mK. The sensitivity and recoverable scale thresholds are +marked as dashed-grey lines in Figure 2. +Article number, page 3 of 11 + +0.5 +1 +1.5 +MO +4 +0 +2-12900.0 +12cO J= 2 - 1 +1500.0 +Integrated Intensity +750.0 +250.0 +500 au +2.012CO(3 - 2) +1200 +2 +1050 +kms +arcsec +900 +mJy/beam l +750 +△Dec +0 +600 +450 +300 +150 +09 +1000 +12cO Integrated Intensity +7 +500 +6 +mJy beam-1 +250 +100 +0 +50 +km +-3- +25 +s +-1 +-6 +10 +50 +au +-9 +0 +9 +6 +3 +0 +-3 +-9300 +100 +50 +25 +10 +1 +100 auA&A proofs: manuscript no. main +Among these five observations, two targeted HD 142527 +(Figure 2, red circles), a well-studied binary Class II system. +These data have been published and show non-Keplerian spi- +ral structures extending to ∼ 700 au, beyond the disk radius of +∼ 200–300 au (Christiaens et al. 2014; Garg et al. 2021), as +shown in Figure 3 (panel a). Similar structures have also been +observed in optical and NIR images (Casassus et al. 2012; Hun- +ziker et al. 2021). Christiaens et al. (2014) suggested that the +innermost spiral can be explained by acoustic waves due to an +embedded companion; however, the origin of the outer spirals +is less clear, and stellar encounters and gravitational instabili- +ties have been suggested as possible causes. We note that spiral +structures have been predicted to form due to late infall (Hen- +nebelle et al. 2017; Kuffmeier et al. 2017, 2018) and a detailed +kinematic analysis can be done to distinguish among these sce- +narios, as discussed in Appendix D. HD 142527 also exhibits +inner and outer disk misalignment (Bohn et al. 2022), which can +be explained by late infall (Thies et al. 2011; Kuffmeier et al. +2021). +The other three datasets are of S CrA, HD 97048, and +Sz 68. The selected large-scale observation for Sz 68 (Project +2019.1.01135.S) does not cover any CO lines and is therefore not +discussed further. For S CrA (Project 2019.1.01792.S) and HD +97048 (Project 2015.1.00192.S), we used the standard pipeline +calibration and imaged the 12CO (2–1) data using CASA 6.4. +For both datasets, imaging was carried out using ’briggs’ weight- +ing with robust=0.5, a cell size of 0.05′′, and ’auto-multithresh’ +masking with default parameters. The resulting integrated inten- +sity (moment 0) maps and intensity-weighted velocity (moment +1) maps are shown in Figure 3 (panel b and c). Appendix F shows +corresponding channel maps for both of the datasets. +For S CrA, at least one ∼ 1000 au streamer is clearly visi- +ble (solid cyan line), north-west of the binary disks (green con- +tours), as seen in the moment 0 map (Figure 3, panel b, mid- +dle panel). This streamer seems to be redshifted with respect to +the disk emission, as seen in the corresponding moment 1 map +(see Figure 3, panel b, right panel and channels from 12.93 to +8.57 km s−1 in Figure F.1). Furthermore, there are hints of two +more streamers, denoted as dashed-cyan lines in the moment 0 +map (Figure 3, panel b, middle panel). This is the first discovery +of large-scale streamers around S CrA. As discussed in Section +2, such elongated structures can form due to the late infall of +material, but they could also be a consequence of a binary in- +teraction. Further analysis is required to ascertain the dynamical +nature of these features and will follow in a future publication. +For HD 97048, no clear streamers are visible in the moment +0 map (Figure 3, panel c, middle panel). However, significant +negative emission was observed in the central channels (Figure +F.2, channels 5.32 to 3.57 km s−1), suggesting that the source +may be surrounded by large-scale gas that may absorb or oth- +erwise obscure, through spatial filtering, kilo-au features around +the star. +These initial results suggest that at least two (HD 142527 and +S CrA) out of the three Class II sources associated with RNe, +and with good-enough ALMA data, exhibit large-scale spirals +or streamer-like structures, as are expected to form due to cloud- +disk interactions, particularly late-stage infall of material (Hen- +nebelle et al. 2017; Dullemond et al. 2019; Kuffmeier et al. 2017, +2020; Kuznetsova et al. 2020). These two sources are in addi- +tion to the already known similar sources discussed in Section +2. Other possible explanations for these large-scale structures +are discussed in Appendix E. Irrespective of the actual origin of +such features, it is promising to see that large-scale gas emission +is found around YSOs close to RNe. This may indicate that an +association with RNe can be used to look for similar structures +around a wider population of Class II disks, as discussed further +in Section 4. +Fig. 2: Largest recoverable physical scale vs line sensitivity (over +10 km s−1) for archival ALMA observations, in Band 6 and 7, of +the 16 nearby Class II YSOs associated with RNe, as discussed +in Section 3. Marker colours denote the exposure time for each +observation in minutes. The vertical dashed line denotes a re- +coverable scale of 1000 au. The horizontal dashed line denotes a +line sensitivity of 2.3 mJy beam−1 (the line sensitivity reported +in the ALMA archive for SU Aur observations by Ginski et al. +(2021)). Markers with open circles denote the observations dis- +cussed in Section 3, with red circles for HD 142527, black circle +for S CrA, and an orange circle for HD 97048. +4. Discussion +We have found 21 Class II sources associated with RNe in the +SFRs listed in Table 1, using a distance threshold of ∼ 2000 au, +equivalent to the typical Bondi-Hoyle accretion length scale. +However, this distance threshold does not account for the ob- +served range of physical extents and asymmetric shapes of RNe. +Connelley et al. (2007) provided angular sizes and distances +for RNe in their catalogue, which correspond to a mean radius +∼ 104 au. The exact distances between the centres of YSOs and +RNe that result in late infall can vary greatly for different sources +depending on their stellar (mass and velocity) and RNe proper- +ties (size and shape) and it is likely that more Class II sources in +our sample may be interacting with neighbouring material than +discussed here. +Figure A.2 plots the fraction of Class II sources and all YSOs +associated with RNe as a function of the offset distance for dif- +ferent SFRs. Considerable differences can be seen in the associ- +ation probability of YSOs with RNe for different SFRs, mostly +due to different catalogue completeness levels. The incomplete- +ness of the available RNe catalogues is a major obstacle in pro- +viding reliable statistics at the moment (see discussion in Sec. 3). +However, even with this limitation, it is clear that there are po- +tentially many Class II disks interacting with their parent cloud. +For at least four SFRs (Taurus, Lupus, Corona Aurstralis, and +Chamaeleon), ∼ 5–10% of Class II sources are close-enough +(≲ 104 au) to RNe. If the threshold is slightly increased to +≲ 4 × 104 au, ∼ 50% of Class II sources in Corona Aurstralis +would be associated with a RN, and therefore potentially accret- +ing material from the ambient cloud. +Article number, page 4 of 11 + +200 +HD 142527 +175 +S CrA +150 +HD 97048 +125 +100 +Exposure Time [min] +101 +75 +50 +25 +100 +102 +103 +Physical scale [au]A. Gupta et al.: Reflections on nebulae around young stars +HD 142527 +(a) +HD 97048 +S CrA +(b) +(c) +(Garg et al. 2021) +(Garg et al. 2021) +Fig. 3: Optical images (left), 12CO integrated intensity (moment 0) maps (middle), and 12CO intensity-weighted velocity (moment +1) maps (right) of nearby Class II sources associated with RNe, as discussed in Section 3. Panel a: HD 142527’s DSS optical image +and the ALMA 12CO (2–1) moment maps from Garg et al. (2021). Panel b: S CrA’s DSS2 (red) optical image and the ALMA 12CO +(2–1) moment maps. Solid and dashed curved-cyan lines denote prominent and potential streamer-like features, respectively. Panel +c: HD 97048’s DSS optical image and the ALMA 12CO (2–1) moment maps. For the moment maps (middle and right) in panel b +and c, only pixels with an intensity > 3σ are considered. In these panels, green contours represent continuum emission (∼ 1.3 mm, +3σ and 15σ levels) from protoplanetary disks, horizontal red lines in the bottom-left corner represent a 1000 au length scale, the +grey ellipse in the bottom-right corner represent the beam size, and black contours in moment 1 maps (right) represent moment +0 emission (starting from the error in moment 0, increased by a factor of five). Errors in moment 0 emission are 22.1 and 20.6 +mJy beam−1 km s−1 for S CrA and HD 97048, respectively. +Accounting for stellar kinematics, the number of Class II +systems that pass by RNe at some point in their lifetime could be +even greater. If an association with RNe is indeed related to late +infall, this may be an important phenomenon especially since it +can have important implications for disk evolution and planet +formation, as described in Section 1. To further test the tenta- +tive link between RNe and an interaction between disks and sur- +rounding clouds, a survey of structures and kinematics around +Class II sources with known RNe is needed, as discussed in Ap- +pendix D. Coupled with a better RNe catalogue, such a survey +will allow us to understand how frequent late infall is for Class +II sources. +5. Conclusions +In this Letter we pioneer the use of the detections of RNe close +to Class II stars to identify late-infall candidates. We find that +all of the sources with known large-scale CO structures, where +late infall is invoked as a possible explanation, also exhibit some +reflection nebulosity at OIR wavelengths. Furthermore, at least +five out of the six sources which are associated with a prominent +RNe and for which adequate ALMA observations are available +– that is, known sources AB Aur, SU Aur, and DO Tau along +with independently identified sources S CrA and HD 142527 – +exhibit some large scale structure that may be indicative of late +infall. This per se suggests that association with RNe may be +used to identify candidate Class II sources undergoing late-stage +infall of material. Finally, in nearby SFRs, the fraction of Class II +sources associated with RNe can be as large as 50%, depending +on the distance threshold, but a proper statistical analysis is still +pending improved RNe catalogues. If RNe are indeed related +to late infall, this suggests that a significant fraction of Class +II sources could be undergoing this phenomenon, with a non- +negligible impact on disk evolution and planet formation. The +catalogue of potential late accretors obtained serves as a starting +point for more systematic studies of late infall onto disks. +Acknowledgements. This work was partly funded by the Deutsche Forschungs- +gemeinschaft (DFG, German Research Foundation) - 325594231. Funded by the +European Union under the European Union’s Horizon Europe Research & In- +novation Programme 101039452 (WANDA). T.B. acknowledges funding from +the European Research Council (ERC) under the European Union’s Horizon +2020 research and innovation programme under grant agreement No 714769 +and funding by the Deutsche Forschungsgemeinschaft (DFG, German Research +Foundation) under grants 361140270, 325594231 (FOR 2634/2), and Germany’s +Excellence Strategy - EXC-2094 - 390783311. M.K. gratefully acknowledges +that this project has received funding from the European Union’s Framework +Programme for Research and Innovation Horizon 2020 (2014–2020) under the +Marie Skłodowska-Curie Grant Agreement No. 897524. This work was partly +supported by the Italian Ministero dell’Istruzione, Università e Ricerca through +the grant Progetti Premiali 2012-iALMA (CUP C52I13000140001), by the DFG +Cluster of Excellence Origins (www.origins-cluster.de). This project has re- +ceived funding from the European Union’s Horizon 2020 research and innova- +tion program under the Marie Sklodowska-Curie grant agreement No 823823 +(DUSTBUSTERS) and from the European Research Council (ERC) via the +ERC Synergy Grant ECOGAL (grant 855130). Views and opinions expressed +Article number, page 5 of 11 + +10 +-36°57'10" +Declination (J2000) +8 +Velocity [km s-1] +20 +6 +4 +30" +1000 au +2 +19h01m10s +09s +08s +RA (J2000)Integrated flux[Jybeam +km s-1] +Velocity[km s-]] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +1.5 +3.0 +4.5 +6.0 +6 +(a) +(b) +4 +2 +Dec +0 +-2 +-4 +-6-77°39'00" +6 +5 +4 +km s +Declination (J2000) +10 +3 +Intensity [Jy beam- +2 +20 +30" +1000 au +11h08m06s +03s +00s +RA (J2000)-77°39'00" +6 +Declination (J2000) +10' +5 +20 +30" +3 +1000 au +11h08m06s +03s +00s +RA (J2000)6 +5 +4 +36°57'10" +Intensity [Jy beam-1 km s- +Declination (J2000) +3 +2 +20" +30" +1000 au +19h01m10s +s60 +08s +RA (2000)A&A proofs: manuscript no. main +are however those of the author(s) only and do not necessarily reflect those of +the European Union or the European Research Council. Neither the European +Union nor the granting authority can be held responsible for them. 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F., et al. 2020, A&A, 633, A51 +1 http://www.astropy.org +Article number, page 6 of 11 + +A. Gupta et al.: Reflections on nebulae around young stars +Appendix A: Star-forming regions +Figure A.1 shows DSS optical images of all the SFRs listed in +Table 1, as discussed in Section 3. Figure A.2 shows fraction of +all YSOs (solid lines) and just Class II sources (dashed lines) +which are associated with RNe as a function of offset thresholds +used to define association, as discussed in Section 4. +Fig. A.1: DSS optical images of all the SFRs listed in Table 1. Solid-black curves denotes circular boundaries of these SFRs, as +parameterized by the "Radius" column of Table 1. Blue and purple circles represent YSOs from Marton et al. (2016) and Dunham +et al. (2015) catalogues, respectively. Red and yellow open diamonds represent RNe from Magakian (2003) and Connelley et al. +(2007) catalogues, respectively. Dashed-black curve in Chamaeleon’s map denote a circle with radius of 20◦. +Article number, page 7 of 11 + +Corona Australis +Ophiuchus +Taurus +-30° +30° +-20° +-35°. +25° +-25° +-40° +20° +-30° +-45° +16h40m +5h00m +4h40m +20m +18h40m +17h00m +20m +00m +00m +19h40m +20m +00m +Chamaeleon +Perseus +Lupus +-70° +-30° +35° +[J2000] +-35° +-75° +Decl. +30° +-40° +-80° +25° +20m +14h 12h 10h +16h40m +00m +15h40m +20m +4ho0m +3h40m +8h +20m +Orion +Serpens +5° +0° +YSOs (Marton et al. 2016) +0° +YSOs (Dunham et al. 2015) +RNe (Magakian 2003) +-5° +RNe (Connelley et al. 2007) +-5° +-10° +-10° +6h00m +5h45m +30m +15m +19h00m 18h45m +30m +15m +00m +R.A. 2000]A&A proofs: manuscript no. main +Fig. A.2: Cumulative distribution of the fraction of YSOs with distance to the nearest RNe less than the given offset, as discussed +in Section 4, for different SFRs. Solid lines represent all the YSOs and dashed lines represent only Class II sources. Vertical dotted +and dash-dotted lines denote offset values of 2000 and 10000 au, respectively. +Appendix B: Distribution of spectral indices +Figure B shows the distribution of extinction-corrected spectral +indices (α’, solid bars) and originally measured spectral indices +(α, dashed-grey line), as discussed in Section 3. The distribution +of α values are shifted to the right because foreground extinction +can artificially increase the observed infrared excess for a source. +Fig. B.1: Distribution of extinction-corrected infrared spectral indices (α’, solid bars) and measured spectral indices (α, grey-dashed +steps) for all the 4930 YSOs in SFRs, as discussed in Section 4. Blue bars denote α’ values estimated for sources exclusively from +Marton et al. (2016). Orange bars denote α’ values for sources from Dunham et al. (2015). Red vertical lines mark the range of +values for a YSO to be classified as a Class II source (−1.6 ≤ α′ < −0.3). +Article number, page 8 of 11 + +Offset [pc] +10-3 +10-2 +10-1 +100 +101 +1.0 +Ophiuchus +Taurus +Cummulative fraction of YSOs +Corona Australis +0.8 - +Lupus +Chamaeleon +0.6 +Perseus +Orion +Serpens +0.4 +0.2 +AlLYSOS +Class II YSOs +0.0 +102 +103 +104 +105 +106 +107 +Offset [au]Marton et al. (2016) +008 +Dunham et al. (2015) +Uncorrected values (α) +600 : +400: +200 +0 +3 +-2 +-1 +0 +1 +2 +3 +Corrected spectral indices (a')A. Gupta et al.: Reflections on nebulae around young stars +Appendix C: Class II sources near RNe +Table C.1 gives coordinates (first two columns), SIMBAD iden- +tifiers (third column), the SFR (fourth column), spectral indices +(fifth and sixth columns), and RNe catalogue identifiers (last two +columns) for all the Class II sources in the vicinity of RNe, as +discussed in Section 3. +Table C.1: Class II YSOs associated with RNe +R.A. [◦] +Decl. [◦] +Simbad Id. +Region +α +α’ +Magakian RNe Id. +Connelley RNe Id. +247.96698 +-24.93782 +ISO-Oph 204 +Ophiuchus +-0.17 +-0.51 +- +66 +239.17449 +-42.32318 +HD 142527 +Lupus +-0.6 +-0.91 +641 +- +236.30347 +-34.29186 +CD-33 10685 +Lupus +-0.64 +-0.95 +634 +- +237.02178 +-35.26469 +V* HN Lup +Lupus +-0.84 +-1.15 +636 +- +277.19941 +0.14439 +V* VV Ser +Serpens +-0.78 +-1.05 +766 +- +85.20028 +-8.09964 +CoKu DL Ori G1 +Orion +-0.93 +-1.24 +132 +- +167.01364 +-77.65476 +HD 97048b +Chamaeleon +-0.07 +-0.38 +533 +- +168.11282 +-76.73947 +BRAN 341D +Chamaeleon +-0.76 +-1.11 +545 +- +168.12797 +-76.73998 +V* CW Cha +Chamaeleon +-0.61 +-1.09 +545 +- +285.28588 +-36.95575 +V* S CrA B +Corona Australis +-0.8 +-1.22 +781 +- +68.13232 +24.33411 +V* FZ Tau +Taurus +-0.86 +-1.17 +74 +- +68.39192 +24.35472 +V* GI Tau +Taurus +-0.62 +-0.93 +75 +- +68.12742 +24.33257 +V* FY Tau +Taurus +-1.12 +-1.43 +74 +- +68.39412 +24.35176 +V* GK Tau +Taurus +-0.67 +-0.98 +75 +- +69.61912 +26.1804 +V* DO Tau +Taurus +-0.51 +-0.82 +78 +- +68.92066 +24.18566 +NAME CoKu Tau 3 +Taurus +-0.9 +-1.21 +76 +- +68.97004 +22.90634 +V* HP Tau +Taurus +-0.57 +-0.88 +77 +- +55.73321 +31.97828 +2MASS J03425596+3158419 +Perseus +-0.98 +-0.84 +48 +- +52.68335 +30.54634 +EM* LkHA 326 +Perseus +-0.64 +-0.89 +45 +- +52.21743 +30.75151 +EM* LkHA 325 +Perseus +-0.69 +-0.78 +41 +- +235.755 +-34.15417 +HH 185 +Lupus +-0.21 +-0.31 +- +64 +Notes. List of 21 Class II YSOs (−1.6 ≤ α′ < −0.3) associated with RNe (distance to nearest RNe ≲ 2000 au), as discussed in Section 3. α and α’ +values are measured and extinction-corrected spectral indices, respectively. Last two columns show index numbers for matched RNe in Magakian +(2003) and Connelley et al. (2007) catalogues. +Article number, page 9 of 11 + +A&A proofs: manuscript no. main +Appendix D: Required observations and analysis +In order to further test a possible link between RNe and late in- +fall, a deep uniform survey of large-scale structures is needed for +Class II sources associated with RNe, as suggested in Section 4. +Ideal observational parameters for such a survey are discussed +below. +For what concerns the angular scales, both observations (Fig- +ure 1) and simulations (e.g. Kuffmeier et al. 2020) suggest +that the infalling streamers should be roughly kilo-au scales in +length. Therefore, observations needed to study these structures +should have a large enough maximum recoverable angular scale +(≳ 1000 au), so as to not filter out large-scale emission. For the +typical distance of 150 pc to nearby SFRs, this physical scale +corresponds to the largest angular scale of ≳ 7′′. On the other +hand, spatial resolution of such observations should be roughly +≲ 100 au (≲ 0.7′′ at a distance of 150 pc), in order to resolve +the connection between large-scale structures and protoplanetary +disks. Such a resolution should also be adequate to resolve the +width of infalling streamers (Figure 1). +In terms of spectral resolution, free-fall velocity for the infall +of material can be estimated as v = √2GM∗/R, where G is the +gravitational constant, M∗ is the stellar mass, and R is the free- +fall length scale. For the typical stellar mass of ∼ 0.5M⊙ and +expected infall length scale of ∼ 1, 000 au, the free-fall veloc- +ity should be ∼ 0.95 km s−1. Assuming we see such an infalling +streamer at an intermediate inclination of 45◦, observed velocity +difference would be ∼ 0.65 km s−1. In order to resolve the veloc- +ity profile, we would need at least three independent data points, +and thus a spectral resolution of ≲ 0.2 km s−1. +The sensitivity requirements of the ideal observations can +be based on the past observations of such large-scale structures. +Among the five sources discussed in Section 2, AB Aur and SU +Aur are exceptionally bright and may not be representatives for +the overall sample. For RU Lup, the signal-to-noise ratio for the +spiral structures was sub-optimal (≲ 3) in the individual chan- +nel (see Figure 5, Huang et al. 2020), which can make it hard +to study the background dynamical processes. Thus, sensitivity +requirements of the observations can be based on observations +of GM Aur and DO Tau, and for both of which the brightness- +temperature sensitivity was ∼ 250 mK (normalised to a channel +width of 0.2 km s−1). +If large-scale structures are observed around other Class II +sources, gas kinematics can be analysed to understand the dom- +inant dynamical processes. A first step could be to check if the +material is gravitationally bound to the protostellar system. For +this the kinetic energy can be computed along the streamer, us- +ing the relative line-of-sight velocities, and compared to gravita- +tional energy, similar to the analysis done for DO Tau by Huang +et al. (2022) (see Figure 12). Furthermore, position-velocity di- +agrams, along any detected streamer, can be modelled and com- +pared to the velocity profiles expected for different kinematic +features such as rotation (v ∝ R−1, for conserved angular mo- +mentum) and infall (v ∝ R−0.5, for free fall), similar to the analy- +sis done for less evolved protostars HL Tau (Yen et al. 2019) and +Lupus 3-MMS (Thieme et al. 2022). +Another way to infer late infall could be to study gas kine- +matics together with NIR polarisation observations, as was done +for SU Aur by Ginski et al. (2021). The degree of polarisation in +such observations can be correlated to the dust scattering angles, +which are expected to depend on the three-dimensional morphol- +ogy of dust structures (e.g. Stolker et al. 2016). Studying the +morphology and gas kinematics in larger-scale (∼ 10, 000 au) +clouds can also allow us to judge the possibility of late infall +(Tang et al. 2012; Dullemond et al. 2019). +Finally, late infall can also be inferred by observing these +systems using different chemical species. Though CO has a high +surface brightness, making it ideal to detect faint structures, it +is also likely to be polluted by the emission from diffuse gas +in these clouds. For less evolved sources, infalling streamers +have also been observed in tracers such as HCO+, HC3N, HC5N, +CCS, 13CS, HNC, and H2CO (Yen et al. 2019; Pineda et al. 2020; +Murillo et al. 2022; Valdivia-Mena et al. 2022). Moreover, mate- +rial falling onto protoplanetary disks also creates shocks, which +can be observed using shock tracers such as SiO, SO, and SO2 +(e.g. Garufi et al. 2022). A dedicated chemical study of streamers +could also allow us to identify better chemical tracers for these +structures for a large-scale survey. +Appendix E: Alternative explanations for +large-scale structures +The large-scale CO structures discussed in Section 2 and 3 could +also be due to other dynamical processes besides late infall (e.g. +Huang et al. 2020). One of the other prominent causes, partic- +ularly for spiral-like structures, could be a tidal interaction of +stellar companions, as it has been observed in some other mul- +tiple systems (e.g. Rodriguez et al. 2018; Kurtovic et al. 2018; +Zapata et al. 2020). The two sources we found with large-scale +structures, HD 142527 and S CrA (Section 3), are binaries, and +thus some of the structures we observe around them (Figure 3, +panel a and c) could be due to tidal interactions between proto- +stars and surrounding gas. +Furthermore, such structures can also be created due to close +encounters by neighbouring YSOs, as predicted by several hy- +drodynamic simulations (e.g. Cuello et al. 2019; Vorobyov et al. +2020) and likely observed for a few sources (e.g. Dong et al. +2022). The role of these stellar flybys can be checked by looking +at relative distances and velocities of nearby YSOs, as was done +for SU Aur (Ginski et al. 2021, Appendix D). +Gravitational instabilities can be another possible way to +form spiral-like structures, if the disks are massive enough (e.g. +Dong et al. 2015), as generally inferred by Toomre’s Q parame- +ter (Toomre 1964). Such instabilities are expected to leave char- +acteristic ’wiggle’ signatures in the gas kinematics, which can +be used to identify them (Hall et al. 2020). Moreover, Harsono +et al. (2011) showed that such instabilities can also be triggered +by the infall of material. Irrespective of the cause of such non- +Keplerian structures, both approaches followed in Sec. 2 and in +Sec. 3 suggest that the vicinity of a RN can be an effective crite- +rion to identify Class II disks that present large-scale structures. +Appendix F: Channel maps +Figure F.1 and F.2 show channels maps of S CrA and HD 97048, +respectively, as discussed in Section 3. +Article number, page 10 of 11 + +A. Gupta et al.: Reflections on nebulae around young stars +Fig. F.1: ALMA 12CO (2–1) channel maps for S CrA archival observations (Project code: 2019.1.01792.S). Emission only from +pixels with an intensity > 2σ was considered. Grey ellipses in the bottom right corners of the maps represent the beam size. +Fig. F.2: ALMA 12CO (2–1) channel maps for HD 97048 archival observations (Project code: 2015.1.00192.S). Emission only from +pixels with an intensity > 2σ was considered. Grey ellipses in bottom right corners of the maps represent the beam size. +Article number, page 11 of 11 + +13.81 to 12.93 km / s +12.93 to 12.06 km / s +12.06 to 11.19 km / s +11.19 to 10.31 km / s +10.31 to 9.44 km / s +10.0 +0.0 +-10.0 +9.44 to 8.57 km / s +8.57 to 7.7 km / s +7.7 to 6.82 km / s +6.82 to 5.95 km / s +5.95 to 5.08 km / s +10.0 +0.0 +-10.0 +5.08 to 4.2 km / s +4.2 to 3.33 km / s +3.33 to 2.46 km / s +2.46 to 1.58 km / s +1.58 to 0.71 km / s +10.0 +0.0 +0.125 +-10.0 +0.100 +Offset in Decl. [arcsecond] +0.71 to -0.16 km / s +-0.16 to -1.04 km / s +-1.04to-1.91km/s +beam-1 +0.075 +10.0 +0.050 +Intensity +0.0 +0.025 +-10.0 +-10.0 +0.0 +10.0 +10.0 +0.0 +10.0 +-10.0 +0.0 +10.0 +Offset in R.A. [arcsecond]9.76 to 8.89 km / s +8.89 to 8.02 km / s +8.02 to 7.14 km / s +7.14 to 6.27 km / s +6.27 to 5.4 km / s +10.0 +0.0 - +-10.0 +5.4 to 4.52 km / s +4.52 to 3.65 km / s +3.65 to 2.78 km / s +2.78 to 1.9 km / s +1.9 to 1.03 km / s +10.0 +0.0 +-10.0 + 0.5 +0.4 +[arcsecond] +1.03 to 0.16 km / s +0.3 +10.0 +Offset in Decl. +0.2 +0.0 +0.1 +-10.0 +-10.0 +0.0 +10.0 +Offset in R.A. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Niels Bohrweg 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' NL-2333 CA Leiden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The Netherlands 6 Anton Pannekoek Institute for Astronomy (API),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' University of Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Science Park 904,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 1098 XH Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The Nether- lands 7 Department of Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' University of Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Türkenschanzstrasse 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 1180 Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Austria 8 Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' University of Virginia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Charlottesville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' VA 22904,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' USA 9 Max-Planck Institute for Extraterrestrial Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Gießenbachstraße 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Germany 10 Institute for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' University of Hawaii,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Honolulu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' HI 96822,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' USA 11 Academia Sinica Institute of Astronomy and Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 11F of Astro-Math Bldg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 4, Roosevelt Rd, Taipei 10617, Taiwan 12 Alma Mater Studiorum Università di Bologna, Dipartimento di Fisica e Astronomia (DIFA), Via Gobetti 93/2, I-40129, Bologna, Italy January 10, 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' While it is generally assumed that Class II sources evolve largely in isolation from their environment, many still lie close to molecular clouds and may continue to interact with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' This may result in late accretion of material onto the disk that can significantly influence disk processes and planet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' In order to systematically study late infall of gas onto disks, we identify candidate Class II sources in close vicinity to a reflection nebula (RN) that may be undergoing this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' First we targeted Class II sources with known kilo-au scale gas structures – possibly due to late infall of material – and we searched for RNe in their vicinity in optical and near-infrared images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Second, we compiled a catalogue of Class II sources associated with RNe and looked for the large-scale CO structures in archival ALMA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Using the catalogues of protostars and RNe, we also estimated the probability of Class II sources interacting with surrounding material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' All of the sources with large-scale gas structures also exhibit some reflection nebulosity in their vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Similarly, at least five Class II objects associated with a prominent RNe, and for which adequate ALMA observations are available, were found to have spirals or stream-like structures which may be due to late infall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' We report the first detection of these structures around S CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Our results suggest that a non-negligible fraction of Class II disks in nearby star-forming regions may be associated with RNe and could therefore be undergoing late accretion of gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Surveys of RNe and kilo-au scale gas structures around Class II sources will allow us to better understand the frequency and impact of late-infall phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Planets and satellites: formation, ISM: clouds, Protoplanetary disks, Stars: formation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Motivation Most stars are born in groups in giant molecular clouds through the gravitational collapse of dense molecular cores (McKee & Ostriker 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' As these protostellar systems evolve from the Class 0/I to the Class II stage, the surrounding gaseous envelope is thought to be completely dispersed and, traditionally, Class II systems are believed to evolve in general isolation to form planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' However, in reality, these systems are still in the vicinity of molecular clouds on large scales (≳ 1 pc) and may continue to dynamically interact with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Such interactions were ob- served in younger Class 0/I objects, where 1000 au scale stream- ers of molecular gas infalling onto the protostar were detected ⋆ Aashish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='Gupta@eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='org (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Pineda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Garufi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' There have been some recent serendipitous detections of ∼ 1000 au, generally stream-like, gaseous structures around Class II disks too, which are likely due to such interactions (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020, 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Late infall of material can greatly influence the physical and chemical properties of Class II disks, and thus, of the planets they form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For example, the supply of fresh material can help solve the ‘mass-budget problem’ of planet-forming disks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Mulders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2021), and explain the ob- served chemical diversity among meteorites (Nanne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Thies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2011) and Kuffmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2021) demonstrated that late infall can torque disks and explain the observed mis- alignment of some planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Approximated as Bondi- Hoyle accretion (Bondi & Hoyle 1944), late infall can explain Article number, page 1 of 11 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='02994v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='SR] 8 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' main the steep dependence of mass accretion on stellar mass (Padoan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2005, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Finally, this phenomenon may also produce disk sub-structures and instabilities as seen in vortices (Bae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Kuznetsova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2022), spiral waves (Hennebelle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Kuffmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2018), and FU Orionis outbursts (Dulle- mond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' A characterisation of the frequency and efficiency of late in- fall is therefore crucial for establishing a holistic view of the star and planet formation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Simulations suggest that a sign- post of such accretion events could be ∼ 103 au-scale arc-shaped structures, generally referred to as ‘streamers’ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Kuffmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' However, only a few have been detected, mainly because these spatial scales lie at the limits of single-dish reso- lution and they are largely filtered out in interferometric obser- vations designed to resolve protoplanetary disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' To comprehen- sively study late-stage infall, a survey of large-scale structures around Class II sources is needed and a first step in this direction would be to systematically identify suitable targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' In order to find disks that are potentially undergoing late infall, one should first identify Class II sources that are close enough to clouds to gravitationally interact with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Such clouds will scatter the protostellar light in optical and near- infrared (NIR) wavelengths and appear as reflection nebulae (RNe) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Hubble 1922).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Historically, RNe were used to iden- tify young stellar objects (YSOs) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Cohen 1980) and they were indeed one of the original defining characteristics of T Tauri stars (Joy 1945).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Recently, hydrodynamical simulations have demonstrated that kilo-au scale RNe can appear due to cloud-protostar interactions, some of which lead to late infall of material (Dullemond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' In this Letter we pioneer the use of RN detections close to Class II stars to identify late-infall candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Class II sources with large-scale CO structures Large-scale non-Keplerian gaseous structures have been de- tected around a few Class II sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' In order to test the hypoth- esis that RNe might indicate late infall, we looked for signs of nebulosity around these sources, as discussed below: AB Aur: Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2012) found four ∼ 500 au spirals around AB Aur in CO observations using the Instituto de Ra- dioastronomía Milimétrica (IRAM) 30-m telescope, Plateau de Bure interferometer (PbDI), and Submillimeter Array (SMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Analysis of the gas kinematics in these spirals, which seems to be counter-rotating with respect to the Keplerian disk, suggests that they are likely formed due to late-stage infall of gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' This source is also known to be associated with a bright arc-shaped RN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Dullemond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2019), as shown in Figure 1 (panel a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' SU Aur: Akiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2019) reported an ∼ 1000 au long tail-like streamer using the Atacama Large Millime- ter/submillimeter Array (ALMA) in CO emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Later, Gin- ski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2021) studied the morphology of its dust tails in scat- tered light using the Very Large Telescope (VLT), along with a kinematic study of CO gas, and found that the material is likely moving towards the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' A bright RN is visible in the immediate vicinity of SU Aur (Figure 1, panel b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' RU Lup: Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2020) reported at least five CO spiral arms stretching up to ∼ 1000 au around RU Lup using ALMA observations and suggested late infall as a possible explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Archival Digital Sky Survey (DSS) images show faint nebulosity just north of RU Lup, as shown in Figure 1 (panel c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' GM Aur: Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2021) found extended elongated structures around GM Aur, ∼ 1000–2000 au in length, using ALMA CO observations with a morphology and kinematics indicative of late infall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' No RNe have been found in archival DSS or Panoramic Survey Telescope and Rapid Response Sys- tem (Pan-STARRS) images, but there are elongated features in more sensitive Hubble Space Telescope (HST) NIR images, as reported in Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2003) and shown in Figure 1 (panel d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' DO Tau: Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2022) studied the kilo-au environment of DO Tau using VLT/SPHERE, HST, and ALMA observations, and found the disk to be connected to multiple ∼ 1000 au scale stream-like structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Larger-scale Herschel observations show that these structures are probably due to an interaction with the neighbouring YSO HV Tau, but late accretion of material onto DO Tau is not ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The source is associated with a promi- nent RN (Figure 1, panel e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' To summarise, the five known Class II sources that have large-scale CO structures suggestive of an interaction with sur- rounding gas have either a prominent RN or some hints of reflec- tion nebulosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Following this lead, we exploit the association of YSOs with RNe to search for candidate Class II sources under- going late-stage infall of material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Class II sources near reflection nebulae As a first step, we compiled a catalogue of RNe using the pub- lished lists in Magakian (2003) and Connelley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Magakian (2003) merged previously published RNe catalogues and presented a final list of 913 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Most of the sources in their list have been manually identified by visual inspection of DSS optical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Connelley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2007) surveyed 197 nearby, mostly Class I, protostars at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='2 µm using the Univer- sity of Hawaii 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='2 m telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' They detected 106 RNe, out of which 41 were reported as new discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The fact that ≳ 40% of RNe were not detected before suggests that the previous RNe catalogues were rather incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For example, the prominent RNe around SU Aur (see Figure 1, panel b) and HD 100546 (Ardila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2007) were not in these RNe catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Connelley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2007) estimated sizes of RNe as the square root of their area with a mean (µ) and standard deviation (σ) of ∼ 18′′ and ∼ 15′′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Using a cross-matching ra- dius of 30′′ (∼ µ + 1σ), we found ten RNe present in both of the catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' After removing duplicates, we merged the two catalogues to have a final sample of 1009 RNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' To avoid spurious detections and have a well-characterised sample of YSOs with RNe that can be further followed up on using molecular-line observations (as discussed in Appendix D), we focus on nearby well-studied star-forming regions (SFRs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The regions considered in this study are listed in Table 1 and shown in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The radii reported in the fifth column in Ta- ble 1 were used to define circular boundaries around the central coordinates listed in the second and third columns, and they are marked with black circles in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' We found 141 sources from our catalogue of 1009 RNe within these region boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The next step was to compile a catalogue of YSOs in these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For this we started with the all-sky catalogue of 133980 Class I/II sources reported in Marton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Marton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2016) analysed WISE and 2MASS photometry of these sources, using the support vector machine algorithm to identify them as YSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For a more complete sample of nearby YSOs, we also used the list of 2966 YSOs identified using Spitzer’s ’cores to disks’ and ’Gould Belt’ surveys, as reported in Dunham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Among the SFRs considered in this study, Taurus and Orion were not part of the Dunham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2015) catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Using Article number, page 2 of 11 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' : Reflections on nebulae around young stars mJy/beam km/s 150 au mJy/beam km/s mJy/beam km/s SU Aur RU Lup GM Aur AB Aur Jy/beam km/s mJy/beam km/s DO Tau (a) (b) (c) (d) (e) (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2022) (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2021) (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020) (Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2021) (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2012) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 1: Optical and NIR images (left) and 12CO integrated inten- sity moment 0 maps (right) of Class II sources with known large- scale structures, as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Panel a: AB Aur’s DSS2 optical image and the PdBI 12CO (2–1) moment 0 map from Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Panel b: SU Aur’s Pan-STARRS optical image and the ALMA 12CO (3–2) moment 0 map from Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Panel c: RU Lup’s DSS2 (red) optical image and the ALMA 12CO (2–1) moment 0 map from Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Panel d: GM Aur’s HST (NICMOS) NIR image and the ALMA 12CO (2–1) moment 0 map from Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Panel e: DO Tau’s Pan-STARRS optical image and ALMA 12CO (2–1) moment 0 map from Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Angular resolutions for the moment 0 maps are roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='5′′, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='3′′, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='3′′, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='2′′, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='7′′ for panels a to e, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' a cross-matching radius of 5′′, we found 781 common sources in both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Our final catalogue consists of 136165 YSOs of which 4930 lie within the SFR boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The spectral indices, α, or slope of the spectral energy dis- tributions in the infrared regime (∼ 2–25 µm) were given in the Dunham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2015) catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' We used their values, α′, cor- rected for foreground extinction to classify the source evolution- ary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Similarly, we determined α for sources exclusively in the Marton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2016) catalogue using the provided WISE and 2MASS (K band) photometry, and discarding those with fitting uncertainties > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='6, that is roughly half of the range of values for Class II sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' In order to estimate extinction-corrected spec- tral indices (α′), we computed a correction factor of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='31 as the median of differences between all the α′ and α values reported in Dunham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' We note that this method only provides Table 1: List of SFRs considered in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' SFR R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Distance Radius YSOs RNe [deg] [deg] [pc] [deg] Ophiuchus 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='07 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='64 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 450 13 Taurus 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='11 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='62 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 190 28 Corona Australis 287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='96 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='11 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 80 6 Lupus 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='08 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='56 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 480 10 Chamaeleon 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='5 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='17 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 148 7 Perseus 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='21 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='57 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 435 19 Orion 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='5 420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 978 44 Serpens 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='66 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='91 495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 2169 14 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The median coordinates and distances are from Zucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The last three columns refer to the radii used to define the SFR boundaries (see Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1) and the number of RNe and YSOs found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' a rough estimate of α′ as it does not account for extinction val- ues for individual sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' A comparison of our α and α′ values is shown in Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Following the same classification criteria as Dunham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2015), that is −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='6 ≤ α′ < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='3, 2562 out of 4930 YSOs in our SFRs were classified as Class II sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The next step was to cross match our list of Class II sources with the merged catalogue of RNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For the specific goal of this work, the cross-matching distance between a YSO and RN can be empirically estimated as the length scale from where ambient gas can be accreted onto an isolated star (Bondi-Hoyle accretion, Bondi & Hoyle 1944;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Throop & Bally 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Padoan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The typical length scale for this interaction is given as LBH = 2GM∗/v2, where G is the gravitational constant, M∗ is the stellar mass, and v is the stellar speed relative to the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For a stellar mass of 1 M⊙ and typical stellar velocity of 1 km s−1, LBH ∼ 2000 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The 21 late-infall Class II candidates that we found using this threshold are listed in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' We note that this distance threshold is a lower limit for an interaction between clouds and YSOs because we do not ac- count for the sizes of RNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' These show a range of physical sizes (∼ 103–105 au) and have highly asymmetrical shapes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Con- nelley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' When mining the ALMA archive, we ignored this diversity as we aimed to identify a reliable sample of YSOs potentially interacting with RNe clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The implications of a more realistic distance threshold is discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' ALMA archive search: To target higher-quality observations, we focussed on nearby SFRs (d ≲ 200 pc), where 16 out of the 21 Class II disks near RNe are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For these nearby sources, we searched for existing observations in the ALMA Science Archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' We are primarily interested in 12CO emission (J = 2−1 in Band 6 and J = 3−2 in Band 7) since this molecule is expected to be a good tracer of large-scale diffuse gas struc- tures (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' We found a total of 66 Band 6 and 7 observations for 16 sources in our sample, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Among these datasets, we found 13 observations with the largest angular scale (LAS) corresponding to at least 1000 au to recover the expected large-scale structures (see Kuffmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' As such large-scale structures are expected to have low column densities, and therefore faint emission, the observational sensitivity is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' We selected a subset of five observations with rms noise (over a channel width of 10 km s−1) smaller than a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='3 mJy beam−1, equal to the line sensitivity reported in the ALMA archive for SU Aur observations (Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For the typical angular resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='7′′ for these observations, the sensitivity threshold of the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='3 mJy beam−1 corresponds to ∼ 100 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The sensitivity and recoverable scale thresholds are marked as dashed-grey lines in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Article number, page 3 of 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='5 MO 4 0 2-12900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 12cO J= 2 - 1 1500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 Integrated Intensity 750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 500 au 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='012CO(3 - 2) 1200 2 1050 kms arcsec 900 mJy/beam l 750 △Dec 0 600 450 300 150 09 1000 12cO Integrated Intensity 7 500 6 mJy beam-1 250 100 0 50 km 3- 25 s 1 6 10 50 au 9 0 9 6 3 0 3 9300 100 50 25 10 1 100 auA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' main Among these five observations, two targeted HD 142527 (Figure 2, red circles), a well-studied binary Class II system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' These data have been published and show non-Keplerian spi- ral structures extending to ∼ 700 au, beyond the disk radius of ∼ 200–300 au (Christiaens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2021), as shown in Figure 3 (panel a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Similar structures have also been observed in optical and NIR images (Casassus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Hun- ziker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Christiaens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2014) suggested that the innermost spiral can be explained by acoustic waves due to an embedded companion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' however, the origin of the outer spirals is less clear, and stellar encounters and gravitational instabili- ties have been suggested as possible causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' We note that spiral structures have been predicted to form due to late infall (Hen- nebelle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Kuffmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2017, 2018) and a detailed kinematic analysis can be done to distinguish among these sce- narios, as discussed in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' HD 142527 also exhibits inner and outer disk misalignment (Bohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2022), which can be explained by late infall (Thies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Kuffmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The other three datasets are of S CrA, HD 97048, and Sz 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The selected large-scale observation for Sz 68 (Project 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='01135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='S) does not cover any CO lines and is therefore not discussed further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For S CrA (Project 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='01792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='S) and HD 97048 (Project 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='00192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='S), we used the standard pipeline calibration and imaged the 12CO (2–1) data using CASA 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For both datasets, imaging was carried out using ’briggs’ weight- ing with robust=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='5, a cell size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='05′′, and ’auto-multithresh’ masking with default parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The resulting integrated inten- sity (moment 0) maps and intensity-weighted velocity (moment 1) maps are shown in Figure 3 (panel b and c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Appendix F shows corresponding channel maps for both of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For S CrA, at least one ∼ 1000 au streamer is clearly visi- ble (solid cyan line), north-west of the binary disks (green con- tours), as seen in the moment 0 map (Figure 3, panel b, mid- dle panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' This streamer seems to be redshifted with respect to the disk emission, as seen in the corresponding moment 1 map (see Figure 3, panel b, right panel and channels from 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='93 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='57 km s−1 in Figure F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Furthermore, there are hints of two more streamers, denoted as dashed-cyan lines in the moment 0 map (Figure 3, panel b, middle panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' This is the first discovery of large-scale streamers around S CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' As discussed in Section 2, such elongated structures can form due to the late infall of material, but they could also be a consequence of a binary in- teraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Further analysis is required to ascertain the dynamical nature of these features and will follow in a future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For HD 97048, no clear streamers are visible in the moment 0 map (Figure 3, panel c, middle panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' However, significant negative emission was observed in the central channels (Figure F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='2, channels 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='32 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='57 km s−1), suggesting that the source may be surrounded by large-scale gas that may absorb or oth- erwise obscure, through spatial filtering, kilo-au features around the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' These initial results suggest that at least two (HD 142527 and S CrA) out of the three Class II sources associated with RNe, and with good-enough ALMA data, exhibit large-scale spirals or streamer-like structures, as are expected to form due to cloud- disk interactions, particularly late-stage infall of material (Hen- nebelle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Dullemond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Kuffmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2017, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Kuznetsova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' These two sources are in addi- tion to the already known similar sources discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Other possible explanations for these large-scale structures are discussed in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Irrespective of the actual origin of such features, it is promising to see that large-scale gas emission is found around YSOs close to RNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' This may indicate that an association with RNe can be used to look for similar structures around a wider population of Class II disks, as discussed further in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2: Largest recoverable physical scale vs line sensitivity (over 10 km s−1) for archival ALMA observations, in Band 6 and 7, of the 16 nearby Class II YSOs associated with RNe, as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Marker colours denote the exposure time for each observation in minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The vertical dashed line denotes a re- coverable scale of 1000 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The horizontal dashed line denotes a line sensitivity of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='3 mJy beam−1 (the line sensitivity reported in the ALMA archive for SU Aur observations by Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Markers with open circles denote the observations dis- cussed in Section 3, with red circles for HD 142527, black circle for S CrA, and an orange circle for HD 97048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Discussion We have found 21 Class II sources associated with RNe in the SFRs listed in Table 1, using a distance threshold of ∼ 2000 au, equivalent to the typical Bondi-Hoyle accretion length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' However, this distance threshold does not account for the ob- served range of physical extents and asymmetric shapes of RNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Connelley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2007) provided angular sizes and distances for RNe in their catalogue, which correspond to a mean radius ∼ 104 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The exact distances between the centres of YSOs and RNe that result in late infall can vary greatly for different sources depending on their stellar (mass and velocity) and RNe proper- ties (size and shape) and it is likely that more Class II sources in our sample may be interacting with neighbouring material than discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='2 plots the fraction of Class II sources and all YSOs associated with RNe as a function of the offset distance for dif- ferent SFRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Considerable differences can be seen in the associ- ation probability of YSOs with RNe for different SFRs, mostly due to different catalogue completeness levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The incomplete- ness of the available RNe catalogues is a major obstacle in pro- viding reliable statistics at the moment (see discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' However, even with this limitation, it is clear that there are po- tentially many Class II disks interacting with their parent cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For at least four SFRs (Taurus, Lupus, Corona Aurstralis, and Chamaeleon), ∼ 5–10% of Class II sources are close-enough (≲ 104 au) to RNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' If the threshold is slightly increased to ≲ 4 × 104 au, ∼ 50% of Class II sources in Corona Aurstralis would be associated with a RN, and therefore potentially accret- ing material from the ambient cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Article number, page 4 of 11 200 HD 142527 175 S CrA 150 HD 97048 125 100 Exposure Time [min] 101 75 50 25 100 102 103 Physical scale [au]A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' : Reflections on nebulae around young stars HD 142527 (a) HD 97048 S CrA (b) (c) (Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2021) (Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2021) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 3: Optical images (left), 12CO integrated intensity (moment 0) maps (middle), and 12CO intensity-weighted velocity (moment 1) maps (right) of nearby Class II sources associated with RNe, as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Panel a: HD 142527’s DSS optical image and the ALMA 12CO (2–1) moment maps from Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Panel b: S CrA’s DSS2 (red) optical image and the ALMA 12CO (2–1) moment maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Solid and dashed curved-cyan lines denote prominent and potential streamer-like features, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Panel c: HD 97048’s DSS optical image and the ALMA 12CO (2–1) moment maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For the moment maps (middle and right) in panel b and c, only pixels with an intensity > 3σ are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' In these panels, green contours represent continuum emission (∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='3 mm, 3σ and 15σ levels) from protoplanetary disks, horizontal red lines in the bottom-left corner represent a 1000 au length scale, the grey ellipse in the bottom-right corner represent the beam size, and black contours in moment 1 maps (right) represent moment 0 emission (starting from the error in moment 0, increased by a factor of five).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Errors in moment 0 emission are 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1 and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='6 mJy beam−1 km s−1 for S CrA and HD 97048, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Accounting for stellar kinematics, the number of Class II systems that pass by RNe at some point in their lifetime could be even greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' If an association with RNe is indeed related to late infall, this may be an important phenomenon especially since it can have important implications for disk evolution and planet formation, as described in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' To further test the tenta- tive link between RNe and an interaction between disks and sur- rounding clouds, a survey of structures and kinematics around Class II sources with known RNe is needed, as discussed in Ap- pendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Coupled with a better RNe catalogue, such a survey will allow us to understand how frequent late infall is for Class II sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Conclusions In this Letter we pioneer the use of the detections of RNe close to Class II stars to identify late-infall candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' We find that all of the sources with known large-scale CO structures, where late infall is invoked as a possible explanation, also exhibit some reflection nebulosity at OIR wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Furthermore, at least five out of the six sources which are associated with a prominent RNe and for which adequate ALMA observations are available – that is, known sources AB Aur, SU Aur, and DO Tau along with independently identified sources S CrA and HD 142527 – exhibit some large scale structure that may be indicative of late infall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' This per se suggests that association with RNe may be used to identify candidate Class II sources undergoing late-stage infall of material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Finally, in nearby SFRs, the fraction of Class II sources associated with RNe can be as large as 50%, depending on the distance threshold, but a proper statistical analysis is still pending improved RNe catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' If RNe are indeed related to late infall, this suggests that a significant fraction of Class II sources could be undergoing this phenomenon, with a non- negligible impact on disk evolution and planet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The catalogue of potential late accretors obtained serves as a starting point for more systematic studies of late infall onto disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' This work was partly funded by the Deutsche Forschungs- gemeinschaft (DFG, German Research Foundation) - 325594231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Funded by the European Union under the European Union’s Horizon Europe Research & In- novation Programme 101039452 (WANDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 714769 and funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grants 361140270, 325594231 (FOR 2634/2), and Germany’s Excellence Strategy - EXC-2094 - 390783311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' gratefully acknowledges that this project has received funding from the European Union’s Framework Programme for Research and Innovation Horizon 2020 (2014–2020) under the Marie Skłodowska-Curie Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 897524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' This work was partly supported by the Italian Ministero dell’Istruzione, Università e Ricerca through the grant Progetti Premiali 2012-iALMA (CUP C52I13000140001), by the DFG Cluster of Excellence Origins (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='origins-cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' This project has re- ceived funding from the European Union’s Horizon 2020 research and innova- tion program under the Marie Sklodowska-Curie grant agreement No 823823 (DUSTBUSTERS) and from the European Research Council (ERC) via the ERC Synergy Grant ECOGAL (grant 855130).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Views and opinions expressed Article number, page 5 of 11 10 36°57\'10" Declination (J2000) 8 Velocity [km s-1] 20 6 4 30" 1000 au 2 19h01m10s 09s 08s RA (J2000)Integrated flux[Jybeam km s-1] Velocity[km s-]] 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 6 (a) (b) 4 2 Dec 0 2 4 6-77°39\'00" 6 5 4 km s Declination (J2000) 10 3 Intensity [Jy beam- 2 20 30" 1000 au 11h08m06s 03s 00s RA (J2000)-77°39\'00" 6 Declination (J2000) 10\' 5 20 30" 3 1000 au 11h08m06s 03s 00s RA (J2000)6 5 4 36°57\'10" Intensity [Jy beam-1 km s- Declination (J2000) 3 2 20" 30" 1000 au 19h01m10s s60 08s RA (2000)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' main are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Neither the European Union nor the granting authority can be held responsible for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' This pa- per makes use of the following ALMA data: ADS/JAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='ALMA#2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='00192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='S and ADS/JAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='ALMA#2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='01792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' ALMA is a partnership of ESO (repre- senting its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in coop- eration with the Republic of Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' This work made use of Astropy:1 a community- developed core Python package and an ecosystem of tools and resources for astronomy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2013, 2018, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' References Akiyama, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Vorobyov, E.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Allen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Evans, Neal J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2015, ApJS, 220, 11 Garg, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Pinte, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Christiaens, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2021, MNRAS, 504, 782 Garufi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Podio, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Codella, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2022, A&A, 658, A104 Ginski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Facchini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2021, ApJ, 908, L25 Hall, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Dong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Teague, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020, ApJ, 904, 148 Harsono, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Alexander, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', & Levin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2011, MNRAS, 413, 423 Hennebelle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Lesur, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', & Fromang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2017, A&A, 599, A86 Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Andrews, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Öberg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020, ApJ, 898, 140 Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Bergin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Öberg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2021, ApJS, 257, 19 Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Ginski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Benisty, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2022, ApJ, 930, 171 Hubble, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 1922, ApJ, 56, 400 Hunziker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Schmid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Ma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2021, A&A, 648, A110 Joy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 1945, ApJ, 102, 168 Kuffmeier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Dullemond, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Reissl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', & Goicovic, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2021, A&A, 656, A161 Kuffmeier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Frimann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Jensen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', & Haugbølle, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2018, MNRAS, 475, 2642 Kuffmeier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Goicovic, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', & Dullemond, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020, A&A, 633, A3 Kuffmeier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Haugbølle, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', & Nordlund, Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2017, ApJ, 846, 7 Kurtovic, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Pérez, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Benisty, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2018, ApJ, 869, L44 Kuznetsova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Bae, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Hartmann, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', & Mac Low, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2022, ApJ, 928, 92 Kuznetsova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Hartmann, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', & Heitsch, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020, ApJ, 893, 73 Magakian, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2003, A&A, 399, 141 Manara, C.' metadata={'source': 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Paladini, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2016, MNRAS, 458, 3479 McKee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' & Ostriker, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2007, ARA&A, 45, 565 Mulders, G.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', van Dishoeck, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Hacar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Harsono, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', & Jørgensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} 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al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020, ApJ, 896, 132 Zucker, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Speagle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', Schlafly, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020, A&A, 633, A51 1 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='astropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='org Article number, page 6 of 11 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' : Reflections on nebulae around young stars Appendix A: Star-forming regions Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1 shows DSS optical images of all the SFRs listed in Table 1, as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='2 shows fraction of all YSOs (solid lines) and just Class II sources (dashed lines) which are associated with RNe as a function of offset thresholds used to define association, as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1: DSS optical images of all the SFRs listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Solid-black curves denotes circular boundaries of these SFRs, as parameterized by the "Radius" column of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Blue and purple circles represent YSOs from Marton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2016) and Dunham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2015) catalogues, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Red and yellow open diamonds represent RNe from Magakian (2003) and Connelley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2007) catalogues, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Dashed-black curve in Chamaeleon’s map denote a circle with radius of 20◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Article number, page 7 of 11 Corona Australis Ophiuchus Taurus 30° 30° 20° 35°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 25° 25° 40° 20° 30° 45° 16h40m 5h00m 4h40m 20m 18h40m 17h00m 20m 00m 00m 19h40m 20m 00m Chamaeleon Perseus Lupus 70° 30° 35° [J2000] 35° 75° Decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 30° 40° 80° 25° 20m 14h 12h 10h 16h40m 00m 15h40m 20m 4ho0m 3h40m 8h 20m Orion Serpens 5° 0° YSOs (Marton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2016) 0° YSOs (Dunham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2015) RNe (Magakian 2003) 5° RNe (Connelley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2007) 5° 10° 10° 6h00m 5h45m 30m 15m 19h00m 18h45m 30m 15m 00m R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2000]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='2: Cumulative distribution of the fraction of YSOs with distance to the nearest RNe less than the given offset, as discussed in Section 4, for different SFRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Solid lines represent all the YSOs and dashed lines represent only Class II sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Vertical dotted and dash-dotted lines denote offset values of 2000 and 10000 au, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Appendix B: Distribution of spectral indices Figure B shows the distribution of extinction-corrected spectral indices (α’, solid bars) and originally measured spectral indices (α, dashed-grey line), as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The distribution of α values are shifted to the right because foreground extinction can artificially increase the observed infrared excess for a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1: Distribution of extinction-corrected infrared spectral indices (α’, solid bars) and measured spectral indices (α, grey-dashed steps) for all the 4930 YSOs in SFRs, as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Blue bars denote α’ values estimated for sources exclusively from Marton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Orange bars denote α’ values for sources from Dunham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Red vertical lines mark the range of values for a YSO to be classified as a Class II source (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='6 ≤ α′ < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Article number, page 8 of 11 Offset [pc] 10-3 10-2 10-1 100 101 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 Ophiuchus Taurus Cummulative fraction of YSOs Corona Australis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='8 - Lupus Chamaeleon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='6 Perseus Orion Serpens 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='2 AlLYSOS Class II YSOs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 102 103 104 105 106 107 Offset [au]Marton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2016) 008 Dunham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=" (2015) Uncorrected values (α) 600 : 400: 200 0 3 2 1 0 1 2 3 Corrected spectral indices (a')A." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' : Reflections on nebulae around young stars Appendix C: Class II sources near RNe Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1 gives coordinates (first two columns), SIMBAD iden- tifiers (third column), the SFR (fourth column), spectral indices (fifth and sixth columns), and RNe catalogue identifiers (last two columns) for all the Class II sources in the vicinity of RNe, as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1: Class II YSOs associated with RNe R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' [◦] Decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' [◦] Simbad Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Region α α’ Magakian RNe Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Connelley RNe Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='96698 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='93782 ISO-Oph 204 Ophiuchus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='51 66 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='17449 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1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='43 74 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='39412 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='35176 V* GK Tau Taurus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='98 75 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='61912 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1804 V* DO Tau Taurus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='82 78 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='92066 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='18566 NAME CoKu Tau 3 Taurus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='21 76 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='97004 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='90634 V* HP Tau Taurus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='88 77 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='73321 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='97828 2MASS J03425596+3158419 Perseus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='84 48 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='68335 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='54634 EM* LkHA 326 Perseus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='89 45 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='21743 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='75151 EM* LkHA 325 Perseus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='78 41 235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='755 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='15417 HH 185 Lupus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='31 64 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' List of 21 Class II YSOs (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='6 ≤ α′ < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='3) associated with RNe (distance to nearest RNe ≲ 2000 au), as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' α and α’ values are measured and extinction-corrected spectral indices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Last two columns show index numbers for matched RNe in Magakian (2003) and Connelley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2007) catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Article number, page 9 of 11 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' main Appendix D: Required observations and analysis In order to further test a possible link between RNe and late in- fall, a deep uniform survey of large-scale structures is needed for Class II sources associated with RNe, as suggested in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Ideal observational parameters for such a survey are discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For what concerns the angular scales, both observations (Fig- ure 1) and simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Kuffmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020) suggest that the infalling streamers should be roughly kilo-au scales in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Therefore, observations needed to study these structures should have a large enough maximum recoverable angular scale (≳ 1000 au), so as to not filter out large-scale emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For the typical distance of 150 pc to nearby SFRs, this physical scale corresponds to the largest angular scale of ≳ 7′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' On the other hand, spatial resolution of such observations should be roughly ≲ 100 au (≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='7′′ at a distance of 150 pc), in order to resolve the connection between large-scale structures and protoplanetary disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Such a resolution should also be adequate to resolve the width of infalling streamers (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' In terms of spectral resolution, free-fall velocity for the infall of material can be estimated as v = √2GM∗/R, where G is the gravitational constant, M∗ is the stellar mass, and R is the free- fall length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For the typical stellar mass of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='5M⊙ and expected infall length scale of ∼ 1, 000 au, the free-fall veloc- ity should be ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='95 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Assuming we see such an infalling streamer at an intermediate inclination of 45◦, observed velocity difference would be ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='65 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' In order to resolve the veloc- ity profile, we would need at least three independent data points, and thus a spectral resolution of ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='2 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The sensitivity requirements of the ideal observations can be based on the past observations of such large-scale structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Among the five sources discussed in Section 2, AB Aur and SU Aur are exceptionally bright and may not be representatives for the overall sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For RU Lup, the signal-to-noise ratio for the spiral structures was sub-optimal (≲ 3) in the individual chan- nel (see Figure 5, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020), which can make it hard to study the background dynamical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Thus, sensitivity requirements of the observations can be based on observations of GM Aur and DO Tau, and for both of which the brightness- temperature sensitivity was ∼ 250 mK (normalised to a channel width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='2 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' If large-scale structures are observed around other Class II sources, gas kinematics can be analysed to understand the dom- inant dynamical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' A first step could be to check if the material is gravitationally bound to the protostellar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For this the kinetic energy can be computed along the streamer, us- ing the relative line-of-sight velocities, and compared to gravita- tional energy, similar to the analysis done for DO Tau by Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2022) (see Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Furthermore, position-velocity di- agrams, along any detected streamer, can be modelled and com- pared to the velocity profiles expected for different kinematic features such as rotation (v ∝ R−1, for conserved angular mo- mentum) and infall (v ∝ R−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='5, for free fall), similar to the analy- sis done for less evolved protostars HL Tau (Yen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2019) and Lupus 3-MMS (Thieme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Another way to infer late infall could be to study gas kine- matics together with NIR polarisation observations, as was done for SU Aur by Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The degree of polarisation in such observations can be correlated to the dust scattering angles, which are expected to depend on the three-dimensional morphol- ogy of dust structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Stolker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Studying the morphology and gas kinematics in larger-scale (∼ 10, 000 au) clouds can also allow us to judge the possibility of late infall (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Dullemond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Finally, late infall can also be inferred by observing these systems using different chemical species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Though CO has a high surface brightness, making it ideal to detect faint structures, it is also likely to be polluted by the emission from diffuse gas in these clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' For less evolved sources, infalling streamers have also been observed in tracers such as HCO+, HC3N, HC5N, CCS, 13CS, HNC, and H2CO (Yen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Pineda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Murillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Valdivia-Mena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Moreover, mate- rial falling onto protoplanetary disks also creates shocks, which can be observed using shock tracers such as SiO, SO, and SO2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Garufi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' A dedicated chemical study of streamers could also allow us to identify better chemical tracers for these structures for a large-scale survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Appendix E: Alternative explanations for large-scale structures The large-scale CO structures discussed in Section 2 and 3 could also be due to other dynamical processes besides late infall (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' One of the other prominent causes, partic- ularly for spiral-like structures, could be a tidal interaction of stellar companions, as it has been observed in some other mul- tiple systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Kurtovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Zapata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The two sources we found with large-scale structures, HD 142527 and S CrA (Section 3), are binaries, and thus some of the structures we observe around them (Figure 3, panel a and c) could be due to tidal interactions between proto- stars and surrounding gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Furthermore, such structures can also be created due to close encounters by neighbouring YSOs, as predicted by several hy- drodynamic simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Cuello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Vorobyov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020) and likely observed for a few sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' The role of these stellar flybys can be checked by looking at relative distances and velocities of nearby YSOs, as was done for SU Aur (Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2021, Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Gravitational instabilities can be another possible way to form spiral-like structures, if the disks are massive enough (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2015), as generally inferred by Toomre’s Q parame- ter (Toomre 1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Such instabilities are expected to leave char- acteristic ’wiggle’ signatures in the gas kinematics, which can be used to identify them (Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Moreover, Harsono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' (2011) showed that such instabilities can also be triggered by the infall of material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Irrespective of the cause of such non- Keplerian structures, both approaches followed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 2 and in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' 3 suggest that the vicinity of a RN can be an effective crite- rion to identify Class II disks that present large-scale structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Appendix F: Channel maps Figure F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1 and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='2 show channels maps of S CrA and HD 97048, respectively, as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Article number, page 10 of 11 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' : Reflections on nebulae around young stars Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1: ALMA 12CO (2–1) channel maps for S CrA archival observations (Project code: 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='01792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Emission only from pixels with an intensity > 2σ was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Grey ellipses in the bottom right corners of the maps represent the beam size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='2: ALMA 12CO (2–1) channel maps for HD 97048 archival observations (Project code: 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='00192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Emission only from pixels with an intensity > 2σ was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Grey ellipses in bottom right corners of the maps represent the beam size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' Article number, page 11 of 11 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='81 to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='93 km / s 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='93 to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='06 km / s 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='06 to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='19 km / s 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='19 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='31 km / s 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='31 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='44 km / s 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='44 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='57 km / s 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='57 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='7 km / s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='7 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='82 km / s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='82 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='95 km / s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='95 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='08 km / s 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='08 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='2 km / s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='2 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='33 km / s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='33 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='46 km / s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='46 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='58 km / s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='58 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='71 km / s 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='125 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='100 Offset in Decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content=' [arcsecond] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='71 to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='16 km / s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='16 to -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='04 km / s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='04to-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='91km/s beam-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='075 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='050 Intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='025 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} +page_content='0 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE1T4oBgHgl3EQfMwPW/content/2301.02994v1.pdf'} diff --git a/kNFPT4oBgHgl3EQf1zVk/content/tmp_files/2301.13184v1.pdf.txt b/kNFPT4oBgHgl3EQf1zVk/content/tmp_files/2301.13184v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4cffbc7d7460c74012d21dde4d99b6adf5b43e4e --- /dev/null +++ b/kNFPT4oBgHgl3EQf1zVk/content/tmp_files/2301.13184v1.pdf.txt @@ -0,0 +1,1483 @@ +Evolution of the Thermodynamic Properties of +a Coronal Mass Ejection in the Inner Corona +Jyoti Sheoran 1, Vaibhav Pant 1, Ritesh Patel 1,2, and Dipankar Banerjee 1,3,4 +1Aryabhatta Research Institute of Observational Sciences, Nainital 263002, India +2Southwest Research Institute, 1050 Walnut Street, Suite 300, Boulder, CO 80302, +USA +3Indian Institute of Astrophysics, 2nd Block Koramangala, Bangalore 560034, India +4Center of Excellence in Space Science, IISER Kolkata, Kolkata 741246, India +Correspondence*: +Jyoti Sheoran, Vaibhav Pant +jyotisheoran@aries.res.in, vaibhav.pant@aries.res.in +ABSTRACT +The thermodynamic evolution of Coronal Mass Ejections (CMEs) in the inner corona (≤ 1.5 Rsun) is +not yet completely understood. In this work, we study the evolution of thermodynamic properties of +a CME core observed in the inner corona on July 20, 2017, by combining the MLSO/K-Cor white- +light and the MLSO/CoMP Fe XIII 10747 Å line spectroscopic data. We also estimate the emission +measure weighted temperature (TEM) of the CME core by applying the Differential Emission Measure +(DEM) inversion technique on the SDO/AIA six EUV channels data and compare it with the effective +temperature (Te f f ) obtained using Fe XIII line width measurements. We find that the Te f f and TEM +of the CME core show similar variation and remain almost constant as the CME propagates from +∼1.05 to 1.35 Rsun. The temperature of the CME core is of the order of million-degree kelvin, indicating +that it is not associated with a prominence. Further, we estimate the electron density of this CME +core using K-Cor polarized brightness (pB) data and found it decreasing by a factor of ∼3.6 as the +core evolves. An interesting finding is that the temperature of the CME core remains almost constant +despite expected adiabatic cooling due to the expansion of the CME core, which suggests that the +CME core plasma must be heated as it propagates. We conclude that the expansion of this CME core +behaves more like an isothermal than an adiabatic process. +Keywords: Solar Atmosphere, Corona, Coronal Mass Ejections (CMEs), Spectroscopy, Thermodynamics +1 +INTRODUCTION +Coronal mass ejections (CMEs) are large structures of plasma and magnetic fields ejected from the solar +atmosphere into the heliosphere (Hundhausen et al. 1984; Webb and Howard 2012). CMEs are the major +drivers of space weather because they can cause interplanetary disturbances and shock waves that can lead +to the disruption of a range of technologies on Earth (Gosling 1993; Schwenn 2006; Pulkkinen 2007). +Thus, it is important to understand their evolution in the solar atmosphere. CMEs have been studied for +several decades using remote sensing and in-situ instruments. To study the evolution of CMEs, one uses +data from space and ground-based instruments such as the Sun Earth Connection Coronal and Heliospheric +Investigation (SECCHI; Howard et al. 2008), COR-1, COR-2 and Extreme ultraviolet imager (EUVI; +Wuelser et al. 2004) instruments on board Solar Terrestrial Relations Observatory (STEREO; Kaiser +2005; Vourlidas et al. 2012), the Large Angle Spectrscopic COronagraph (LASCO; Brueckner et al. 1992) +coronagraphs and Extreme-Ultraviolet Imaging Telescope (EIT; Delaboudini`ere et al. 1995) on board +the Solar and Heliospheric Observatory (SOHO; Domingo et al. 1995), the Mark IV, K-Coronagraph +(K-Cor), Coronal Multi-channel Polarimeter (CoMP; Tomczyk et al. 2008) instruments at the Mauna +1 +arXiv:2301.13184v1 [astro-ph.SR] 30 Jan 2023 + +Sheoran et al. +CME Thermodynamics in the Inner Corona +Loa Solar Observatory (MLSO; Elmore et al. 2003), Atmospheric Imaging Assembly (AIA; Lemen et al. +2012) telescopes onboard the Solar Dynamic Observatory (SDO) and the EUV Imaging Spectrometer +(EIS; Culhane et al. 2007) onboard Hinode. Most of earlier studies focused on the signature of origins of +CMEs, determination of their mass/density, dynamic evolution, and the connection between magnetic flux +ropes measured in-situ and CME’s morphology observed by the white-light coronagraphs and heliospheric +imagers. Nevertheless, from white-light data, one cannot estimate the plasma properties of CMEs, such +as plasma temperature and elemental composition. CMEs may contain a cool (∼ 104 K) chromospheric +material (prominence/filament), a hot coronal material (∼ 106 K), and flare plasma (∼ 107 K), thus, giving +rise to the emission in the broad wavelength range. Many processes occurring during CME propagation can +interchange different forms of energies (electromagnetic, kinetic, potential, thermal, etc.), causing plasma +heating or cooling (Bemporad et al. 2007). Thus, to understand these processes, it is crucial to study the +evolution of thermodynamic properties (such as density, temperature, thermal pressure, etc.) of CMEs, as it +propagates. +However, the evolution of CMEs in the inner corona (≤ 1.5 Rsun) is not yet completely understood, +primarily due to the lack of continuous observations of the inner corona. The dynamics of CMEs in the +inner corona are important because this region exhibits kinematics of CME, such as rapid expansion, +impulsive acceleration, etc. (Zhang et al. 2001, 2004; Gallagher et al. 2003; Temmer et al. 2010; Bein et al. +2011; Joshi and Srivastava 2011; Sarkar et al. 2019; Cremades et al. 2020; Majumdar et al. 2020, 2021, +2022). However, simultaneous spectroscopy and white-light imaging in the inner corona are needed to +improve our understanding of the temperature and kinematics evolution of the CMEs during their initial +phase of propagation. There are few studies using ultraviolet (UV) and extreme UV (EUV) imaging and +spectral observations, X-ray, and in situ observations where the thermodynamic properties of the CMEs +have been studied. +The spectroscopic UV observations of CMEs acquired by the Ultraviolet Coronagraph Spectrometer +(UVCS; Kohl et al. 1995) on board the SOHO has helped us to understand the evolution of CME plasma +physical parameters such as densities, temperatures, abundances in the CME core and leading front, and +their 3-D velocity structures in the corona from 1.5 to 10 Rsun. In this heliocentric distance range, the +density of the CME leading front (LF) is found to be in the range 104 - 106 cm−3 (Ciaravella et al. 2003, +2005), and the temperature has been inferred ranging from 6.0 × 103 to 2.0 × 106 K at 1.5 Rsun (Ciaravella +et al. 1997; Bemporad et al. 2007). The study by Bemporad (2022) found CME LF density of order 1 × 107 +cm−3 and peak temperature ≃ 1.9 × 106 K at 1.6 Rsun, and density ≃ 7 × 106 cm−3 and peak temperature ≃ +2.1 × 106 K at 1.9 Rsun. The bright cores of CMEs are usually believed to contain the filament/prominence +material. The densities of the CME core have been found to range from 1.4 × 106 to 7.0 × 108 cm−3 at 1.3 +Rsun, and density decreases from 1.3 × 106 to 4.0 × 107 cm−3 at 3.0 Rsun (Akmal et al. 2001; Raymond and +Ciaravella 2004). However, Bemporad (2022) found CME core density of order 1.1 × 107 cm−3 at 1.6 Rsun +and 9 × 106 cm−3 at 1.9 Rsun, and core peak temperature ≃ 2.4 × 106 K at 1.6 Rsun and ≃ 3.2 × 106 K at +1.9 Rsun. Furthermore, these studies were limited to a certain heliocentric distance and/or a particular time. +Furthermore, the differential emission measure (DEM) analysis has also been applied to diagnose the +physical properties of CMEs. Many CME eruptions have been studied using UV-EUV imagers such as the +EIT on board SOHO, the STEREO/EUVI instruments, the AIA telescopes on board SDO, and Hinode/EIS +spectroscopic observations. Long et al. 2018 studied the evolution of a coronal cavity using spectroscopic +Hinode/EIS and broadband SDO/AIA observations. The DEM inversion techniques on EUV images have +enabled us to infer the 2-D distribution of plasma temperatures and densities inside CMEs. The main +results emerge from these studies are that the CME core temperature ranges from 0.8 - 2.0 MK for CMEs +This is a provisional file, not the final typeset article +2 + +Sheoran et al. +CME Thermodynamics in the Inner Corona +associated with prominence eruption, and the core temperature was found to be ≥8.0 MK for CMEs +associated with a flux rope (Chmielewska et al. 2016). It was found that the CME core regions were heated +during the CME eruption (Cheng et al. 2012; Landi et al. 2010; Hannah and Kontar 2013), presumably +through magnetic reconnection. +In most CME thermodynamic models, the evolution of thermal energy is derived under the assumption of +adiabatic expansion of plasma material (Durand-Manterola et al. 2017). However, many studies addressing +the energy budget of CME plasma demonstrate the presence of an additional energy source responsible for +the heating of the CME plasma. The additional heating source provides thermal energy that was found +to be comparable to the total kinetic and potential energies gained by the eruptions (Akmal et al. 2001; +Ciaravella et al. 2003; Murphy et al. 2011). While in some cases, it was found to be much larger than the +total kinetic and potential energies gained by the eruptions (Lee et al. 2009; Landi et al. 2010). Also, based +on in-situ observations of ICMEs, the polytropic index (γ) of ICME plasma was suggested to be of the +order of 1.1 to 1.3 from 0.3 and 20 AU (Liu et al. 2005, 2006), implying the local heating of ICMEs plasma. +Mishra and Wang 2018 found that the polytropic index of a CME plasma decreased from 1.8 to 1.35 as the +CME moved from 5.9 to 13.9 Rsun, implying that CME first released heat to reach an adiabatic state and +then absorbed heat. However, our understanding of the evolution of thermodynamic properties of CME +during its propagation is still limited. +Despite this, we still lack an understanding of how the thermodynamic properties of CMEs change +during their propagation in the inner corona (up to ∼ 1.5 Rsun). Using UVCS spectroscopic data, the +physical parameters of CME plasma have been obtained at fixed heliocentric distances starting from 1.5 +to 10 Rsun. However, the DEM analysis using SOHO/EIT, STEREO/EUVI, & SDO/AIA, has enabled +us to estimate the physical parameters of CME in the inner corona (up to ∼ 1.5 Rsun). Nevertheless, in +none of these studies, the continuous evolution of the thermodynamic parameters of CME structures in +the inner corona has been studied. Hence, the motivation behind the work presented here. In this paper, +by combining the K-Cor white-light data and the CoMP Fe XIII 10747 Å line spectroscopic data, we +present the spectroscopic diagnostics of the temperature and density of a CME core observed in the inner +corona on July 20, 2017. We studied the continuous evolution of the thermodynamic properties, such as the +temperature and density of the CME core from ∼ 1.05 to 1.35 Rsun. In Section 2, we provide the details of +the observations. In Section 3, we describe the methods used to derive the temperature and density of this +CME core and present the main results, followed by a summary & discussion in Section 4. +2 +OBSERVATIONS +On July 20, 2017, the MLSO/K-Cor observed a CME propagating outward from the west limb starting at +around 17:00 UT. The event is observed with multiple instruments, including the K-Cor and CoMP, the +AIA onboard SDO, and the white-light coronagraph (COR-1A) of SECCHI onboard STEREO. The K-Cor +records polarized brightness (pB) images of the inner corona from 1.05 - 3 Rsun in 720 - 750 nm passband +with a cadence of 15 seconds. The K-Cor images have a pixel scale of ≈ 5.6 arcsecs pixel−1. We used +white-light level-2 pB data to estimate the electron density. Figure 1(a) shows the three-part CME in the +K-Cor FOV. +The eruption was also seen in the CoMP Fe XIII 10747 Å channel. The CoMP takes observation of the +polarization state at a few spectral locations across the profiles of three infrared lines (Fe XIII 10747 Å, +10798 Å & He I 10830 Å) using a narrow-band tunable filter. It has a field of view (FOV) of 1.05 - 1.4 Rsun, +a pixel-scale of 4.46 arcsecs pixel−1, and a typical cadence of 30 seconds. The details of the acquisition +and reduction of CoMP data are described in Tomczyk et al. 2008, and the calculation process is given by +Frontiers +3 + +Sheoran et al. +CME Thermodynamics in the Inner Corona +0 +500 +1000 +1500 +X [arcsec] +−1000 +−500 +0 +Y [arcsec] +0 +500 +1000 +1500 +−1000 +−500 +0 +(a) MLSO/K−Cor 17:04:12 UT +0 +200 +400 +600 +800 +1000 1200 +−1000 +−800 +−600 +−400 +−200 +0 +200 +0 +200 +400 +600 +800 +1000 1200 +−1000 +−800 +−600 +−400 +−200 +0 +200 +(b) MLSO/CoMP 10747Å 17:04:57 UT +0 +200 +400 +600 +800 +1000 +1200 +−800 +−600 +−400 +−200 +0 +200 +0 +200 +400 +600 +800 +1000 +1200 +−800 +−600 +−400 +−200 +0 +200 +(c) SDO/AIA 193Å 17:04:53 UT +Figure 1. (a) July 20, 2017, CME as observed by the K-Cor. All three parts of the CME are clearly visible. +The yellow rectangular box shows the ROI chosen for the analysis. (b) The CoMP 10747 Å channel +enhanced intensity image. When CoMP observation started, only the core of the CME was visible in the +CoMP FOV. The yellow box is the CoMP ROI aligned with the K-Cor ROI shown in Figure (a). (c) The +AIA 193 Å channel image processed using MGN to enhance the CME core structures. The yellow box is +the AIA ROI aligned with the CoMP ROI shown in Figure (b). (An animation of the evolution of the CME +core in the K-Cor, CoMP 10747 Å, and AIA 193 Å channel is available in the Electronic Supplementary +Material. The yellow circle in the animation represents the CoMP FOV.) +Tian et al. 2013. We used Fe XIII 10747 Å level 2 dynamic data, which contains line peak intensity, edge +enhanced line peak intensity, line-of-sight (LOS) Doppler velocity, and line width images. On this day, the +CoMP observation started at around 16:57 UT, and only the core of this CME was visible in the CoMP +FOV, as shown in Figure 1(b). This eruption was seen as a braided structure moving out in CoMP Fe XIII +10747 Å channel and caused a significant enhancement in Fe XIII line width. +This eruption was also recorded in all SDO/AIA EUV channels from around 15:00 to 18:00 UT. The +AIA takes full-disk images of the corona and transition region up to 1.3 Rsun simultaneously in seven +extreme-ultraviolet (EUV) narrow-band filters. AIA has a pixel-scale of 0.6 arcsecs pixel−1 and a cadence +of 12 seconds. Using aia prep.pro AIA images were normalized by their exposure times, rotated, and +re-scaled to ensure that each pixel is examining the same spatial location across the different wavelengths. +We also checked AIA 304 Å data-set to determine if this CME has an associated filament. No filament +could be seen in AIA 304 Å channel prior to the period of study. We used 94, 131, 171, 193, 211, and 335 +Å pass bands images, sensitive over a temperature range from 105 K to 107 K for the DEM analysis. Since +AIA 193 Å and CoMP 10747 Å images have similar temperature responses, the core of this CME as seen +in AIA 193 Å channel, is shown in Figure 1(c). The AIA 193 Å channel image has been processed using +Multiscale Gaussian Normalization (MGN: Morgan and Druckm¨uller, 2014) to enhance the CME core +structures. +The yellow rectangular box in all panels of Figure 1 corresponds to the region of interest (ROI) chosen for +the analysis. We have aligned CoMP 10747 Å ROI with respect to AIA 193 Å ROI. The yellow dashed lines +in Figure 2 (a) indicate the position of four artificial slits at the same location in both AIA and CoMP ROIs. +Each slit is approximately ten arcsec wide and the location of these slits is such that they cover almost the +entire CME core. Since K-Cor images capture white-light and CoMP images capture narrow-band emission, +we could not use intensity/brightness to align their FOVs as the features look different in white-light and +This is a provisional file, not the final typeset article +4 + +Sheoran et al. +CME Thermodynamics in the Inner Corona +narrow-band emission images. The CoMP and the K-Cor both have 1.05 Rsun occulter. We use it as a +reference to align the CoMP & the K-Cor ROIs. +This CME is also observed by the STEREO/COR-1A coronagraph. The COR-1A coronagraph offers +high-cadence (5 minutes) data with FOV from 1.4-4 Rsun, and a pixel scale of 7.5 arcsecs pixel−1. There +is a filament behind the limb, which could be seen in STEREO EUVI 304 Å channel around 11:46 UT +on July 20, 2017 which erupts around 12:46 UT, seen in COR-1A FOV ∼13:00 UT. Around 15:00 UT, a +braided structure is seen to rise in EUVI 304 Å and 195 Å FOV, which propagates out and a CME appears +(case event for this study) in COR-1 FOV at around 16:40 UT. Therefore, our case event does not have a +filament associated with it. To determine the parameters like LOS depth and volume of this CME, we used +the graduated cylindrical shell (GCS) model (Thernisien et al. 2009) to fit this CME using the K-Cor & +COR-1A vantage point observations. For GCS fitting, we have used K-Cor white light level 2 Normalizing +Radially Graded Filter (NGRF; Morgan et al. 2006) data and COR-1 level 1 data filtered using Simple +Radial Gradient Filter (SiRGraF; Patel et al. 2022). +3 +ANALYSIS AND RESULTS +3.1 +Determination of the Electron Temperature +In this work, we determined the two-dimensional plasma temperature distribution across the CME core +using two methods. First, using DEM analysis on AIA six EUV channels data, and second, using the line +broadening of Fe XIII emission line centered at 10747 Å. +3.1.1 +Temperature using the Differential Emission Measure Analysis +To infer the DEM from AIA six EUV channels data, we applied the inversion technique developed +by Cheung et al. 2015. For the inversion, we used a temperature grid spanning log T/K ∈ [5.7, 7.7] +with a grid spacing of log T = 0.1. The emission measure in each temperature grid was obtained using +aia sparse em init.pro in the SolarSoftware (SSWIDL) package. The total emission measure (EM) was +then obtained by summing the EM in all temperature bins: +EM = +n +� +j +EM j, +(1) +where EMj is the EM contained in the jth temperature bin, and n is the total number of bins (in our case +n = 21). The mean temperature can be estimated by the emission-weighted temperature, +log TEM = EM−1 +��������� +n +� +j +EM j log T j +��������� , +(2) +W2 +EM = EM−1 +��������� +n +� +j +EM j +�log T j − log TEM +�2 +��������� , +(3) +here log TEM is the EM-weighted log temperature and WEM is the effective width of the distribution in +log T space. +We created DEM maps at each pixel for the logarithmic temperature spanning from 5.7 to 7.7. The +colored boxes in the left panel of Figure 2 (b) show the selected sub-regions of AIA ROI used to reconstruct +the DEM curves. Note that we have processed AIA 193 Å channel images using the MGN technique to +Frontiers +5 + +Sheoran et al. +CME Thermodynamics in the Inner Corona +enhance the CME core structures, the MGN technique has not been used for DEM analysis. The DEM +curves for these box regions are shown in the right panel. The DEM in all regions peaked at around log +T/K ∼6.3. Figure 2 (c) show the EM maps of the AIA ROI at 2017-07-20 T 17:16:11 UT. We can see that +the EM of the core of the CME is mostly confined in log T/K ∈ [6.05, 6.65]. Then using Equation (2) we +created EM-weighted log temperature (log TEM) maps of the AIA ROI. We obtained the intensity and log +TEM values along the AIA slits (shown in the right panel of Figure 2 (a)) and took their averaged over the +width of each slit to ensure a good signal-to-noise ratio. To illustrate the full evolution of the core of the +CME, we constructed space-time maps for all four slits. The left panel of Figure 3 shows the AIA 193 Å +intensity space-time maps, and the right panel shows the log TEM space-time maps for slit 2 and 3. We +visually inspected the AIA 193 Å intensity space-time map of slit 2 and tracked the eruption along this slit +as shown by the blue dashed curve in the top left panel of Figure 3, which we fitted using the cubic spline +interpolation method. Using these height-time values, we fitted the spline curve on log TEM space-time +map of slit 2 as shown in the top right panel of Figure 3 by blue dashed curve and obtained the values +of log TEM and WEM (WEM is calculated using Equation (3)) along this curve. The same procedure is +repeated for the remaining slits. Note that the spline fitted curve has approx 24 arcsec spatial extent at a +particular time. +(c) +(a) +(b) +Figure 2. (a) The left panel shows the AIA ROI (shown by yellow box in Figure 1 (c)), and right panel +shows the CoMP ROI (shown by yellow box in Figure 1 (b)). The yellow dashed lines in both panels show +the locations of four co-spatial slits chosen in the two ROIs. (b) The left panel shows AIA 193 Å ROI +image at 2017-07-20 T 17:16:17 UT. The colored boxes show the selected sub-regions used to reconstruct +the DEM curves. The DEM curves for these box regions are shown in the right panel. The DEM in all +regions peaked at around log T/K ∼ 6.3. (c) EM maps of the AIA ROI at 2017-07-20 T 17:16:11 UT. The +color coding indicates the total EM contained within a log temperature range indicated in the bottom left +corner of each panel. (An animation is available in the Electronic Supplementary Material.) +This is a provisional file, not the final typeset article +6 + +0 +-100 +Y [arcsec] +-200 +-300 +193A T17:00:05 UT +1050 1100 1150 1200 +X [arcsec] +1.0 +0.5 +0.0 +5.5 6.0 6.5 7.0 7.5 8.0 +Log T/K0 +0 +-100 +-100 +Y [arcsec] +EE +[arcsec] +Y +-200 +-200 +-300 +-300 +10747A 17:04:57 UT +193A 17:04:53 UT +1050 1100 1150 1200 +1050 1100 1150 1200 +X [arcsec] +X[arcsec]0 +0 +-100 +-100 +Y [arcsec] +Y [arcsec] +-200 +-200 +-300 +-300 +10747A 17:16:24 UT +193A 17:16:17 UT +1050110011501200 +1050110011501200 +X [arcsec] +X [arcsec]0 +-100 +[arcsec] +Y +-200 +-300 +193AT17:16:17U +1050110011501200 +X[arcsec] +2.0F +1.5 +1.0 +0.5 +0.0 +5.56.06.57.07.58.0 +LogT/K2017-07-20 T 17:16:11 UT +EM in Igt=[5.75,6.05] +EM in Igt=[6.05,6.35] +EM in Igt=[6.35,6.65] +27.5 +Log Emissian Measure [cm +27.0 +26.5 +26.0 +25.5 +EM in lgt=[6.65,6.95] +EM in Igt=[6.95.7.25] +EM in Igt=[7.25,7.55]Sheoran et al. +CME Thermodynamics in the Inner Corona +AIA 193Å, Slit 2 +15:00 +15:30 +16:00 +16:30 +17:00 +17:30 +18:00 +2017−07−20, Time [UT] +1050 +1100 +1150 +1200 +X [arcsec] +15:00 +15:30 +16:00 +16:30 +17:00 +17:30 +18:00 +1050 +1100 +1150 +1200 +AIA, Slit 2 +15:00 +15:30 +16:00 +16:30 +17:00 +17:30 +18:00 +2017-07-20, Time [UT] +1050 +1100 +1150 +1200 +X [arcsec] +15:00 +15:30 +16:00 +16:30 +17:00 +17:30 +18:00 +1050 +1100 +1150 +1200 +6.20 +6.25 +6.30 +6.35 +6.40 +6.45 +6.50 +log TEM /K +AIA 193Å, Slit 3 +15:00 +15:30 +16:00 +16:30 +17:00 +17:30 +18:00 +2017−07−20, Time [UT] +1050 +1100 +1150 +1200 +X [arcsec] +15:00 +15:30 +16:00 +16:30 +17:00 +17:30 +18:00 +1050 +1100 +1150 +1200 +AIA, Slit 3 +15:00 +15:30 +16:00 +16:30 +17:00 +17:30 +18:00 +2017-07-20, Time [UT] +1050 +1100 +1150 +1200 +X [arcsec] +15:00 +15:30 +16:00 +16:30 +17:00 +17:30 +18:00 +1050 +1100 +1150 +1200 +6.20 +6.25 +6.30 +6.35 +6.40 +6.45 +6.50 +log TEM /K +Figure 3. The left panel shows AIA 193 Å intensity space-time maps, and the right panel shows the +EM-weighted log temperature space-time maps for all slits. We visually traced the eruption along each slit +intensity space-time map shown by the blue dashed curve in the left panel, which is fitted using the cubic +spline interpolation method. The same curve is overplotted over log TEM space-time map of the respective +slit. +The calculation of EM-weighted temperature is based on the assumption that the plasma is isothermal +along the LOS. However, the left panel of Figure 2 (b) demonstrates that the emission measure has a spread +over a range of temperatures. To investigate the temporal evolution of the CME core emission in different +temperature bins, we produce space-time maps of log EM for slit 3. Figure 4 shows temporal variation +in log EM along slit 3 for different temperature bins. The space-time maps show that the CME core rises +gradually, and at ∼17:30 UT, the CME core moves out of the AIA FOV. The CME core appears in most of +the temperature bins indicating that the CME core is multi-thermal. However, the EM of the CME core +is mainly confined in log T/K ∈ [6.05, 6.65]. The blue dashed curve in all panels of Figure 4 is the same +spline curve as in the bottom panel of Figure 3. +log T = [5.75, 6.05] +1050 +1100 +1150 +1200 +X [arcsec] +1050 +1100 +1150 +1200 +25.0 +25.2 +25.4 +25.6 +25.8 +26.0 +log EM [cm-5] +log T = [6.05, 6.35] +1050 +1100 +1150 +1200 +X [arcsec] +1050 +1100 +1150 +1200 +25.0 +25.5 +26.0 +26.5 +27.0 +log EM [cm-5] +log T = [6.35, 6.65] +1050 +1100 +1150 +1200 +X [arcsec] +1050 +1100 +1150 +1200 +25.0 +25.5 +26.0 +26.5 +27.0 +log EM [cm-5] +log T = [6.65, 6.95] +1050 +1100 +1150 +1200 +X [arcsec] +1050 +1100 +1150 +1200 +25.0 +25.2 +25.4 +25.6 +25.8 +26.0 +log EM [cm-5] +log T = [6.95, 7.25] +15:00 +15:30 +16:00 +16:30 +17:00 +17:30 +18:00 +2017-07-20, Time [UT] +1050 +1100 +1150 +1200 +X [arcsec] +15:00 +15:30 +16:00 +16:30 +17:00 +17:30 +18:00 +1050 +1100 +1150 +1200 +24.0 +24.5 +25.0 +25.5 +26.0 +log EM [cm-5] +log T = [7.25, 7.55] +15:00 +15:30 +16:00 +16:30 +17:00 +17:30 +18:00 +2017-07-20, Time [UT] +1050 +1100 +1150 +1200 +X [arcsec] +15:00 +15:30 +16:00 +16:30 +17:00 +17:30 +18:00 +1050 +1100 +1150 +1200 +23.0 +23.5 +24.0 +24.5 +25.0 +25.5 +log EM [cm-5] +Figure 4. The temporal variation in log EM along slit 3 for different temperature bins. The blue dashed +curve in all panels of is the same spline curve as in the bottom panel of Figure 3. +Frontiers +7 + +Sheoran et al. +CME Thermodynamics in the Inner Corona +3.1.2 +Effective temperature from Fe XIII Line Width +Using the Fe XIII line width and considering that the broadening of the line may occur due to the thermal +motions of ions, the non-thermal motions in the corona, the instrumental broadening, the expansion of +CME as it moves outward, and the additional turbulence created by the CME propagation in the corona, the +CME plasma temperature can be obtained. In order to calculate the non-thermal line width (NTLW) in the +corona, we used the previous day, i.e., July 19, 2017, CoMP Fe XIII 10747 Å line width data when CME is +not present. We subtract the thermal width (21 km s−1, corresponding to the peak formation temperature of +Fe XIII of ∼1.6 MK) and instrumental width (21 km s−1) calculated by Morton et al. 2015 from the line +width obtained for each pixel (McIntosh and De Pontieu 2012). To obtain the radial profile of NTLW, we +took a median value of NTLW of pixels lying along 40 degrees (to ensure a good signal-to-noise ratio) +about the equator at each radius. Figure 5 shows the variation of the NTLW with height above the solar +surface. We found that the NTLW did not vary significantly with height. Therefore, for our calculations, +we used the mean value of NTLW, 19.32 km s−1. We subtract non-thermal width (NTLW) and instrument +0 +50 +100 +150 +200 +250 +Height above photoshphere [Mm] +0 +10 +20 +30 +40 +NTLW [km s -1] +2017-07-19 T 22:33:32 UT + = 19.32 km s -1 +Figure 5. The variation of NTLW with height above the solar surface. +width in quadrature from the total line width; the residual width is Doppler/thermal width (i.e., width due +to thermal motions of ions). Furthermore, the plasma temperature can be estimated using the following +Equation: +T = 1 +2 +m +kB +v2 +1/e, +(4) +where m is the mass of the Fe XIII ion, kB is the Boltzmann constant, and v1/e is the velocity derived +from the Doppler half-width, ∆λ1/e. Note that we did not take into account the broadening of the line +due to the expansion of the CME core and the additional turbulence created by the propagation of the +CME. Hence Equation 4 provides the upper limit to the plasma temperature. We refer to it as effective +temperature, Te f f . +We selected an ROI from the CoMP (Figure 1 (b)) image same as AIA 193 ROI and placed four artificial +slits co-spatial with AIA slits (Figure 2 (a)). We averaged the Fe XIII line enhanced intensity and line width +over the width of the slits and created the space-time maps of line enhanced intensity and total width as +shown in the left and right panels of Figure 6, respectively. Then, using the height-time values along the +spline fitted curve on AIA 193 slit 2 intensity space-time map, we fitted the Fe XIII enhanced intensity and +line width space-time maps of slit 2, which is shown by the blue dashed curve in the top panel of Figure +6. Since the CoMP has a larger FOV than AIA, we visually tracked the eruption in CoMP FOV outside +the AIA FOV shown by the cyan curve in Figure 6. The line width values were obtained along the blue +and the cyan curves. Moreover, to calculate the Doppler/thermal width values, the NTLW (19 km s−1) and +This is a provisional file, not the final typeset article +8 + +Sheoran et al. +CME Thermodynamics in the Inner Corona +instrumental width (21 km s−1) were subtracted from the total line width values. The effective temperature +was calculated using Equation (4). This procedure is repeated for the remaining slits also. +CoMP, Slit 2 (Enhanced Intensity) +16:57 +17:13 +17:30 +17:45 +18:03 +18:19 + 2017-07-20, Time [UT] +1050 +1100 +1150 +1200 +1250 +X [arcsec] +16:57 +17:13 +17:30 +17:45 +18:03 +18:19 +1050 +1100 +1150 +1200 +1250 +CoMP, Slit 2 (Line Width) +16:57 +17:13 +17:30 +17:45 +18:03 +18:19 + 2017-07-20, Time [UT] +1050 +1100 +1150 +1200 +1250 +X [arcsec] +16:57 +17:13 +17:30 +17:45 +18:03 +18:19 +1050 +1100 +1150 +1200 +1250 +30 +34 +39 +44 +Line Width [km s -1] +CoMP, Slit 3 (Enhanced Intensity) +16:57 +17:13 +17:30 +17:45 +18:03 +18:19 + 2017-07-20, Time [UT] +1050 +1100 +1150 +1200 +1250 +X [arcsec] +16:57 +17:13 +17:30 +17:45 +18:03 +18:19 +1050 +1100 +1150 +1200 +1250 +CoMP, Slit 3 (Line Width) +16:57 +17:13 +17:30 +17:45 +18:03 +18:19 + 2017-07-20, Time [UT] +1050 +1100 +1150 +1200 +1250 +X [arcsec] +16:57 +17:13 +17:30 +17:45 +18:03 +18:19 +1050 +1100 +1150 +1200 +1250 +30 +34 +39 +44 +Line Width [km s -1] +Figure 6. The left panel shows the CoMP Fe XIII 10747 Å enhanced-intensity space-time maps, and the +right panel shows the Fe XIII line width space-time maps. The blue dashed curve in each slit space-time +map is fitted using the height-time values obtained from spline fitting AIA 193 Å intensity space-time map +of the respective slit. The eruption is further tracked in the CoMP FOV outside the AIA FOV and is shown +by the cyan curve. Black stripes represent gaps in data. +In addition to this, expansion of the CME core and additional turbulence created by the CME propagation +will also contribute to the non-thermal broadening. To model the expansion of the CME core, we used +the graduated cylindrical shell (GCS) model developed by Thernisien et al. 2009; Thernisien 2011 and +followed the fitting procedure described in Majumdar et al. 2022. We used the GCS ice-cream cone model +to fit the core of the CME. Figure 7 shows the variation of the LOS nominal depth of the CME core plasma +(LCME) with time and heliocentric distance of the CME core front. We found that the LCME of the CME +core increases almost two times as the core evolves during this period. The expansion velocity of the CME +core is found to be approx. 50.76 km/s, which is larger than the total line width. Since the features look +different in the white-light & IR emission band ( Fe XIII 10747 Å), we conclude that we may not be +modelling the same feature using the GCS model fit, which we are tracking in IR emission. Therefore, the +expansion factor for these two wavelength regimes would be different, which makes it difficult to apply the +expansion factor obtained from the GCS model to account for the broadening of the Fe XIII 10747 Å line +due to the expansion of the CME core. +Figure 8 shows the evolution of log temperature of the CME core with height and time. The green +diamond symbols show the variation of the EM-weighted log temperature (log TEM) averaged over the +spatial direction along the blue curve shown in Figure 3. The shaded grey region shows the effective +width of the distribution in log T space, which is obtained by adding WEM and the standard deviation for +this spline-fitted curve. We found that within this region of uncertainty, the EM-weighted temperature of +the core of the CME remains almost constant as CME evolves, and the mean value of log TEM/K of the +core of the CME is found to be in the range of 6.28 - 6.36. The blue cross symbols in the Figure 8 show +the variation of log effective temperature averaged over the spatial direction along the blue and the cyan +curves shown in Figure 6, and the red bars are one sigma deviation in log Te f f values. The log effective +Frontiers +9 + +Sheoran et al. +CME Thermodynamics in the Inner Corona +temperature of the core of the CME has a similar variation to the EM-weighted log temperature and has +mean values in the range of 5.97 - 6.57. + 7211 + 7696 + 8667 + 9638 + 10201 10501 10801 +Time (t) since 14:59:59 UT [seconds] +0.2 +0.3 +0.4 +0.5 +0.6 +LOS width, LCME [Rsun] + 1.64 1.66 1.69 + 1.71 + 1.78 + 1.80 + 1.87 + 1.94 1.97 + 2.06 + 2.10 +Heliocentric height [Rsun] +KCOR + COR1 +Linear fit +LCME = -0.279 + (7.29e-05)*t +Figure 7. The variation of LOS width (LCME) of the CME core with time or heliocentric height. +4000 +5000 +6000 +7000 +8000 +9000 +Time since 2017-07-20 14:59:59 [seconds] +5.0 +5.5 +6.0 +6.5 +7.0 +Log10T + 795.1 + 796.4 + 798.4 + 801.6 806.0 + 815.5 828.4 852.8 908.5 + Height along the slit 2 [Mm] +log TEM +log Teff +4000 +5000 +6000 +7000 +8000 +9000 +Time since 2017-07-20 14:59:59 [seconds] +5.0 +5.5 +6.0 +6.5 +7.0 +Log10T + 793.3 + 800.9 + 808.2 + 817.0 + 824.9 + 838.6 + 868.9 + Height along the slit 3 [Mm] +log TEM +log Teff +Figure 8. The evolution of log temperature of the CME core with height and time. The green diamond +symbols show the variation of the EM-weighted log temperature (log TEM) averaged over the spatial +direction along the blue curve shown in Figure 3. The grey shaded region is the effective width of the +distribution in log T space. The blue cross symbols show the variation of log effective temperature averaged +over the spatial direction along the blue and the cyan curves shown in Figure 6, and the red bars are one +sigma deviation in log Tef f values. +3.2 +Estimation of the Electron Density +The electron density can be derived using the CoMP Fe XIII 10747 & 10798 Å density-sensitive line +pair. Since there were no good signal-to-noise ratio data frames available during the period of CME under +consideration, we could not use this line pair for density calculation. Therefore, we use K-Cor polarized +brightness data to derive the electron density. +The polarized brightness observed by the white-light coronagraphs primarily depends on the column +electron density (LOS integration of the electron density). Transient phenomena, such as CMEs, cause +the intensity (hence, density) enhancements in a sequence of coronagraph images. A suitable pre-event +image is subtracted from the frames containing the CME to calculate the CME density. Since no pre-event +frame was available for the CME event on July 20, 2017, we chose a frame on July 21, 2017, at around +02:21:59 UT, much later after the CME had passed the K-Cor FOV, and subtracted this image from the +CME frames. As a result, the background F-corona and static K-corona are removed, leaving us with the +brightness changes caused by the CME. +The excess column electron density Ne (due to the CME) can be estimated by taking the ratio of the +excess observed pB (pBobs) to the polarized brightness of a single electron (pBe) assumed to lie on the +This is a provisional file, not the final typeset article +10 + +Sheoran et al. +CME Thermodynamics in the Inner Corona +plane of sky (POS) (Colaninno and Vourlidas 2009). The POS assumption is valid for July 20, 2017, +CME as indicated by very small LOS Doppler velocities values in CoMP Fe XIII 10747 Å data. The pBe +is computed analytically from the scattering geometry using the equations given in Billings 1966. We +used eltheory.pro in the SolarSoftware (SSWIDL) package to calculate the pBe, and the limb darkening +coefficient used in this routine was calculated using the Equation (5) of Hestroffer and Magnan 1998 for +wavelength = 735 nm (for the K-Cor). The CME electron density ne, can be obtained by dividing the +derived column density Ne to the LOS nominal depth of the CME plasma LCME, ne = Ne/LCME (Bemporad +2022). +We determined the column electron density for the entire K-Cor ROI (the yellow box in Figure 1(a)) +and chose four slits co-spatial with AIA and CoMP slits. The resulting 2-D maps of the column density +(Ne) for slit 2 & 3 are shown in Figure 9. The green region in all panels shows the evolution of the core +of the CME with time along the respective slit. The blue and cyan curves are the same as described in +section 3.1.2. We obtained the values of Ne along the spline fitted curves and divided it by LCME to obtain +the CME core electron density values (ne). The variation of the CME core electron density (ne) along the +spline fitted curves for slit 2 & 3 is shown in Figure 10 by blue cross symbols. The dark grey bars show one +sigma uncertainty in the electron density. We found that the CME core’s density falls by a factor of ∼3.6 as +the core evolves. The density values of the CME core are in the range (5.85- 20.85)× 107 cm−3. We also +derived the volume of the CME core using the GCS model fitted parameters. The volume of the CME core +increases approx. six times while the core evolves in the K-Cor FOV. Hence, the decrease in CME core +density is consistent with the volume increase of the CME core. Similar results were also obtained for slits +1 & 4. +K-Cor, Slit 2 +17:00 +17:10 +17:20 +17:30 +17:40 +17:50 +18:01 +2017-07-20, Time [UT] +1050 +1100 +1150 +1200 +1250 +X [arcsec] +17:00 +17:10 +17:20 +17:30 +17:40 +17:50 +18:01 +1050 +1100 +1150 +1200 +1250 +0.0 +1.0 +2.0 +3.0 +4.0 +Ne [1017 cm-2] +K-Cor, Slit 3 +17:00 +17:10 +17:20 +17:30 +17:40 +17:50 +18:01 +2017-07-20, Time [UT] +1050 +1100 +1150 +1200 +1250 +X [arcsec] +17:00 +17:10 +17:20 +17:30 +17:40 +17:50 +18:01 +1050 +1100 +1150 +1200 +1250 +0.0 +1.0 +2.0 +3.0 +4.0 +Ne [1017 cm-2] +Figure 9. The column electron density space-time maps. The blue and the cyan curves are the same as in +Figure 6. +7500 +8000 +8500 +9000 +9500 +Time since 2017-07-20 14:59:59 [seconds] +0 +5 +10 +15 +20 +25 +ne [107 cm-3] + 810.0 812.3 815.2 + 823.7 + 837.1 + 862.3 + 908.5 + Height along the slit 2 [Mm] +7500 +8000 +8500 +9000 +9500 +Time since 2017-07-20 14:59:59 [seconds] +0 +5 +10 +15 +20 +25 +ne [107 cm-3] + 820.7 823.2 825.9 + 831.9 + 841.1 + 864.9 + 907.1 + Height along the slit 3 [Mm] +Figure 10. The evolution of electron density (ne) of the CME core with height and time. The blue cross +symbols show the variation of ne averaged over the spatial direction along the blue and the cyan curves +shown in Figure 9, and the dark grey bars are one sigma deviation in ne values. +Frontiers +11 + +Sheoran et al. +CME Thermodynamics in the Inner Corona +4 +SUMMARY AND DISCUSSIONS +In this work, we carried out the spectroscopic diagnostics in the inner corona (1.05 - 1.35 Rsun) to derive +the thermodynamic property of a CME by combining the CoMP Fe XIII 10747 Å line width data and the +K-Cor polarized brightness (pB) data. We studied the evolution of a CME core’s thermodynamic properties +that occurred on July 20, 2017. We obtained the effective temperature of the CME core using the line +broadening of the Fe XIII emission line centered at 10747 Å. We have also applied the DEM inversion +technique on AIA six EUV channels data to determine the EM - weighted temperature of the CME core +and compare it with the effective temperature obtained using Fe XIII line width. The column density of the +CME core is derived using the K-Cor pB intensity. To obtain the LOS depth (LCME) and volume of the +CME core, we used the graduated cylindrical shell (GCS) model to fit this CME core using the two (K-Cor +& COR-1A) vantage point observations. We obtained the electron density of the CME core by dividing the +CME core column density by LCME of the CME core plasma. +We find that within one sigma error, the EM-weighted temperature of the CME core remains almost +constant as CME evolves, and the mean log TEM/K of the CME core is found to be in the range 6.28 - 6.36. +The effective temperature of the CME core also has a similar variation to the EM- weighted temperature, +and mean log Tef f/K has values in the range of 5.97 - 6.57. The non-thermal motions in the corona +contribute significantly to the broadening of a spectral line (McIntosh and De Pontieu 2012; Brooks and +Warren 2016; Pant et al. 2019). In earlier studies, the plasma temperatures have also been obtained using the +width of spectral lines (e.g., H I Ly-α and Ly-β, C III, etc.) observed in the CMEs (Ciaravella et al. 2000; +Heinzel et al. 2016). Nevertheless, these studies did not take into account the contribution of non-thermal +motions in the width of the spectral line. Hence, the plasma temperature limit provided by these studies +is not quite accurate. Therefore, it is required to subtract NTLW from the total width of a spectral line +(Heinzel et al. 2016). In our analysis, we have calculated NTLW and subtracted this from the total width of +the Fe XIII 10747 Å line. The consequence is that the CME core’s effective and EM-weighted temperatures +have similar values within the error bars. Thus, we can conclude that the effective temperature derived from +line width by taking into account the non-thermal in the corona and instrumental broadenings is a good +measure of CME plasma temperature. The million-degree kelvin temperature of the CME core indicates +that the core of this CME is not associated with a prominence material. +We find that the CME core’s electron density falls by a factor of ∼3.6 as the core evolves and has values +in the range (5.85- 20.85)× 107 cm−3. As pointed out by Bemporad et al. 2018, the uncertainty in the +density determination using pB is mainly due to the POS assumption (the assumption that all the plasma is +located on the POS) and due to the uncertainty in the depth of a CME structure along the LOS (LCME). +However, the POS assumption is valid for July 20, 2017, CME as indicated by very small LOS Doppler +velocities values in CoMP Fe XIII 10747 Å data, and we get a better estimate of LCME by performing GCS +model fit to the CME core. +We find that the temperature of the CME core remains almost constant despite expected adiabatic cooling +due to the expansion of the CME core, which suggests that the CME core plasma must be heated as +it propagates. In previous studies based on in-situ observations of ICMEs, the polytropic index (γ) of +ICME plasma is also found to be close to unity implying the isothermal expansion of ICME plasma. γ of +ICME plasma was suggested to be of the order of 1.1 to 1.3 from 0.3 and 20 au (Liu et al. 2005, 2006), +indicating local heating of ICME plasma. Furthermore, in few MHD models of CME have also used γ +close to unity. Gibson and Low 1998 constructed a theoretical MHD model describing the ejection of +a 3-D CME out of the solar corona by making an assumption γ ∼ 4/3. Odstrcil et al. 2002 have used γ += 1.05 in a coronal 2-D MHD model to simulate the disruption of a sheared helmet streamer launching +This is a provisional file, not the final typeset article +12 + +Sheoran et al. +CME Thermodynamics in the Inner Corona +a CME. Chen 1996 & Krall et al. 2000 have used γ = 1.2 in their theoretical treatment describing the +initiation, propagation, and driving mechanisms of ICMEs. Thus, we can infer that the expansion of July +20, 2017, CME core behaves more like an isothermal than an adiabatic process during its evolution in the +inner corona from 1.05 - 1.35 Rsun. It may also be possible that the thermal force is the internal driver of +CME expansion, as highlighted in the study by Mishra and Wang 2018. The presence of plasma heating +processes occurring during the CME expansion is also reported in many studies in the literature (Akmal +et al. 2001; Ciaravella et al. 2003; Lee et al. 2009; Landi et al. 2010; Murphy et al. 2011; Bemporad 2022). +The candidate heating mechanisms are briefly discussed in studies by Kahler and Reames 1991; Kumar and +Rust 1996; Murphy et al. 2011; however, there is no widely accepted heating mechanism. Moreover, the +above conclusion that the CME core is not a filament/prominence and the CME core is continually heated +during its early expansion has an important inference on the initiation mechanism of CMEs. Currently, the +existing theories of CME initiation fall into two categories: one is based on the ideal MHD instability of +magnetic flux rope ( T¨or¨ok and Kliem 2005; Kliem and T¨or¨ok 2006; Fan and Gibson 2007; Aulanier et al. +2010; Kliem et al. 2014; Amari et al. 2018), which often contains a filament, and the other is based on +magnetic reconnection (Antiochos et al. 1999; Moore et al. 2001; Wyper et al. 2017; Jiang et al. 2021b,a), +which does not require a pre-existing flux rope or filament. In the reconnection model, the core of the CME +is formed by reconnection, and thus it is continually heated by the reconnection. On the other hand, in the +ideal instability-based model, the CME core is the pre-existing flux rope; thus, no heating can be provided +during the ideal expansion. Hence, July 20, 2017, CME event supports the reconnection model. +Our work demonstrates the potential of the CoMP and the K-Cor and future multi-channel coronagraphs +upgraded-CoMP (UCoMP) at MLSO and Visible Emission Line Coronagraph (VELC: Prasad et al., +2017; Banerjee et al., 2017) onboard Aditya-L1 to study the thermodynamic evolution of CMEs in the +inner corona. The UCoMP has the capability to perform simultaneous 2-D imaging and high-resolution +spectroscopy in the inner corona. It has a FOV of 1.03 - 1.95 Rsun and a spatial resolution of 6 arcsecs +(3 arcsecs/pixel). The UCoMP has seven coronal emission lines (FeXIV 530.3 nm, FeX 637.4 nm, ArXI +691.8 nm, FeXV 706.2, FeXI 789.4 nm, FeXIII 1074.7 nm, and 1079.8 nm) covering a wide range of +temperature (log Tef f ∼ 5.80 - 6.63). A similar analysis can be applied to the observations of CMEs that +will be acquired simultaneously by MLSO/UCoMP and MLSO/K-Cor. In addition, FeX 637.4 nm and +FeXI 789.4 nm are temperature-sensitive lines, and FeXIII 1074.7 & 1079.8 nm are density-sensitive lines. +The ratio of these lines, together with the help of atomic spectral line databases (Dere et al. 1997), it is +possible to infer the 2-D distribution of plasma temperatures and densities inside the CMEs. This study is a +testing bed for VELC/Aditya-L1, which will perform both spectroscopy and imaging of the CMEs in the +inner corona in three visible (one continuum centered at 500 nm and two emission lines; FeXIV 530.3 nm +& FeXI 789.2 nm ) and one infrared (FeXIII 1074.7 nm) passbands (Patel et al. 2021). +CONFLICT OF INTEREST STATEMENT +The authors declare that the research was conducted in the absence of any commercial or financial +relationships that could be construed as a potential conflict of interest. +AUTHOR CONTRIBUTIONS +JS led the analysis and carried out the image processing and spectroscopic based investigations. VP and RP +planned the analysis and identified the case for analysis. VP and DB assisted in the interpretation of the +results. JS prepared the manuscript. All authors took part in the discussion. +Frontiers +13 + +Sheoran et al. +CME Thermodynamics in the Inner Corona +FUNDING +JS is supported by funds of the Council of Scientific & Industrial Research (CSIR), India, under file no. +09/0948(12550)/2021-EMR-I. +ACKNOWLEDGMENTS +We would like to thank ARIES for providing the computational facilities. Courtesy of the Mauna Loa Solar +Observatory, operated by the High Altitude Observatory, as part of the National Center for Atmospheric +Research (NCAR). NCAR is supported by the National Science Foundation. The SECCHI data used here +were produced by an international consortium of the Naval Research Laboratory (USA), Lockheed Martin +Solar and Astrophysics Lab (USA), NASA Goddard Space Flight Center (USA), Rutherford Appleton +Laboratory (UK), University of Birmingham (UK), Max-Planck-Institut for Solar System Research +(Germany), Centre Spatiale de Li`ege (Belgium), Institut d’Optique Th´eorique et Appliqu´ee (France), +Institut d’Astrophysique Spatiale (France). We also acknowledge SDO team to make AIA data available. +DATA AVAILABILITY STATEMENT +The MLSO/CoMP, MLSO/K-Cor, SDO/AIA, and STEREO/SECCHI data sets analyzed for this study can +be found in their respective data archives under the open data policy. The data sets generated in this study +can be made available upon request. +REFERENCES +Akmal, A., Raymond, J. C., Vourlidas, A., Thompson, B., Ciaravella, A., Ko, Y. K., et al. (2001). SOHO +Observations of a Coronal Mass Ejection. Astrophys. 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J. 604, 420–432. doi:10.1086/381725 +Frontiers +19 + diff --git a/kNFPT4oBgHgl3EQf1zVk/content/tmp_files/load_file.txt b/kNFPT4oBgHgl3EQf1zVk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..72825d33e9de96be23d55c426d3e9f534996dfe5 --- /dev/null +++ b/kNFPT4oBgHgl3EQf1zVk/content/tmp_files/load_file.txt @@ -0,0 +1,1799 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf,len=1798 +page_content='Evolution of the Thermodynamic Properties of a Coronal Mass Ejection in the Inner Corona Jyoti Sheoran 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Vaibhav Pant 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Ritesh Patel 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' and Dipankar Banerjee 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='4 1Aryabhatta Research Institute of Observational Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Nainital 263002,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' India 2Southwest Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 1050 Walnut Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Suite 300,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CO 80302,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' USA 3Indian Institute of Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2nd Block Koramangala,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Bangalore 560034,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' India 4Center of Excellence in Space Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' IISER Kolkata,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Kolkata 741246,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' India Correspondence*: Jyoti Sheoran,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Vaibhav Pant jyotisheoran@aries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='in, vaibhav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='pant@aries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='in ABSTRACT The thermodynamic evolution of Coronal Mass Ejections (CMEs) in the inner corona (≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 Rsun) is not yet completely understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' In this work, we study the evolution of thermodynamic properties of a CME core observed in the inner corona on July 20, 2017, by combining the MLSO/K-Cor white- light and the MLSO/CoMP Fe XIII 10747 Å line spectroscopic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We also estimate the emission measure weighted temperature (TEM) of the CME core by applying the Differential Emission Measure (DEM) inversion technique on the SDO/AIA six EUV channels data and compare it with the effective temperature (Te f f ) obtained using Fe XIII line width measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We find that the Te f f and TEM of the CME core show similar variation and remain almost constant as the CME propagates from ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='05 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='35 Rsun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The temperature of the CME core is of the order of million-degree kelvin, indicating that it is not associated with a prominence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Further, we estimate the electron density of this CME core using K-Cor polarized brightness (pB) data and found it decreasing by a factor of ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='6 as the core evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' An interesting finding is that the temperature of the CME core remains almost constant despite expected adiabatic cooling due to the expansion of the CME core, which suggests that the CME core plasma must be heated as it propagates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We conclude that the expansion of this CME core behaves more like an isothermal than an adiabatic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Keywords: Solar Atmosphere, Corona, Coronal Mass Ejections (CMEs), Spectroscopy, Thermodynamics 1 INTRODUCTION Coronal mass ejections (CMEs) are large structures of plasma and magnetic fields ejected from the solar atmosphere into the heliosphere (Hundhausen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Webb and Howard 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CMEs are the major drivers of space weather because they can cause interplanetary disturbances and shock waves that can lead to the disruption of a range of technologies on Earth (Gosling 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Schwenn 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Pulkkinen 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Thus, it is important to understand their evolution in the solar atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CMEs have been studied for several decades using remote sensing and in-situ instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' To study the evolution of CMEs, one uses data from space and ground-based instruments such as the Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2008), COR-1, COR-2 and Extreme ultraviolet imager (EUVI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Wuelser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2004) instruments on board Solar Terrestrial Relations Observatory (STEREO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Kaiser 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Vourlidas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2012), the Large Angle Spectrscopic COronagraph (LASCO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Brueckner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 1992) coronagraphs and Extreme-Ultraviolet Imaging Telescope (EIT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Delaboudini`ere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 1995) on board the Solar and Heliospheric Observatory (SOHO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Domingo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 1995), the Mark IV, K-Coronagraph (K-Cor), Coronal Multi-channel Polarimeter (CoMP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Tomczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2008) instruments at the Mauna 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='13184v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='SR] 30 Jan 2023 Sheoran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CME Thermodynamics in the Inner Corona Loa Solar Observatory (MLSO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Elmore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2003), Atmospheric Imaging Assembly (AIA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Lemen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2012) telescopes onboard the Solar Dynamic Observatory (SDO) and the EUV Imaging Spectrometer (EIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Culhane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2007) onboard Hinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Most of earlier studies focused on the signature of origins of CMEs, determination of their mass/density, dynamic evolution, and the connection between magnetic flux ropes measured in-situ and CME’s morphology observed by the white-light coronagraphs and heliospheric imagers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Nevertheless, from white-light data, one cannot estimate the plasma properties of CMEs, such as plasma temperature and elemental composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CMEs may contain a cool (∼ 104 K) chromospheric material (prominence/filament), a hot coronal material (∼ 106 K), and flare plasma (∼ 107 K), thus, giving rise to the emission in the broad wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Many processes occurring during CME propagation can interchange different forms of energies (electromagnetic, kinetic, potential, thermal, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' ), causing plasma heating or cooling (Bemporad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Thus, to understand these processes, it is crucial to study the evolution of thermodynamic properties (such as density, temperature, thermal pressure, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=') of CMEs, as it propagates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' However, the evolution of CMEs in the inner corona (≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 Rsun) is not yet completely understood, primarily due to the lack of continuous observations of the inner corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The dynamics of CMEs in the inner corona are important because this region exhibits kinematics of CME, such as rapid expansion, impulsive acceleration, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2001, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Gallagher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Temmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Bein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Joshi and Srivastava 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Sarkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Cremades et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Majumdar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2020, 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' However, simultaneous spectroscopy and white-light imaging in the inner corona are needed to improve our understanding of the temperature and kinematics evolution of the CMEs during their initial phase of propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' There are few studies using ultraviolet (UV) and extreme UV (EUV) imaging and spectral observations, X-ray, and in situ observations where the thermodynamic properties of the CMEs have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The spectroscopic UV observations of CMEs acquired by the Ultraviolet Coronagraph Spectrometer (UVCS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Kohl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 1995) on board the SOHO has helped us to understand the evolution of CME plasma physical parameters such as densities, temperatures, abundances in the CME core and leading front, and their 3-D velocity structures in the corona from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 to 10 Rsun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' In this heliocentric distance range, the density of the CME leading front (LF) is found to be in the range 104 - 106 cm−3 (Ciaravella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2003, 2005), and the temperature has been inferred ranging from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 × 103 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 × 106 K at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 Rsun (Ciaravella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Bemporad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The study by Bemporad (2022) found CME LF density of order 1 × 107 cm−3 and peak temperature ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='9 × 106 K at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='6 Rsun, and density ≃ 7 × 106 cm−3 and peak temperature ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1 × 106 K at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='9 Rsun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The bright cores of CMEs are usually believed to contain the filament/prominence material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The densities of the CME core have been found to range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='4 × 106 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 × 108 cm−3 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='3 Rsun, and density decreases from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='3 × 106 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 × 107 cm−3 at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 Rsun (Akmal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Raymond and Ciaravella 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' However, Bemporad (2022) found CME core density of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1 × 107 cm−3 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='6 Rsun and 9 × 106 cm−3 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='9 Rsun, and core peak temperature ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='4 × 106 K at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='6 Rsun and ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='2 × 106 K at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='9 Rsun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Furthermore, these studies were limited to a certain heliocentric distance and/or a particular time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Furthermore, the differential emission measure (DEM) analysis has also been applied to diagnose the physical properties of CMEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Many CME eruptions have been studied using UV-EUV imagers such as the EIT on board SOHO, the STEREO/EUVI instruments, the AIA telescopes on board SDO, and Hinode/EIS spectroscopic observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2018 studied the evolution of a coronal cavity using spectroscopic Hinode/EIS and broadband SDO/AIA observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The DEM inversion techniques on EUV images have enabled us to infer the 2-D distribution of plasma temperatures and densities inside CMEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The main results emerge from these studies are that the CME core temperature ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='8 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 MK for CMEs This is a provisional file, not the final typeset article 2 Sheoran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CME Thermodynamics in the Inner Corona associated with prominence eruption, and the core temperature was found to be ≥8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 MK for CMEs associated with a flux rope (Chmielewska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' It was found that the CME core regions were heated during the CME eruption (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Landi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Hannah and Kontar 2013), presumably through magnetic reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' In most CME thermodynamic models, the evolution of thermal energy is derived under the assumption of adiabatic expansion of plasma material (Durand-Manterola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' However, many studies addressing the energy budget of CME plasma demonstrate the presence of an additional energy source responsible for the heating of the CME plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The additional heating source provides thermal energy that was found to be comparable to the total kinetic and potential energies gained by the eruptions (Akmal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Ciaravella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' While in some cases, it was found to be much larger than the total kinetic and potential energies gained by the eruptions (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Landi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Also, based on in-situ observations of ICMEs, the polytropic index (γ) of ICME plasma was suggested to be of the order of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='3 from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='3 and 20 AU (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2005, 2006), implying the local heating of ICMEs plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Mishra and Wang 2018 found that the polytropic index of a CME plasma decreased from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='8 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='35 as the CME moved from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='9 to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='9 Rsun, implying that CME first released heat to reach an adiabatic state and then absorbed heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' However, our understanding of the evolution of thermodynamic properties of CME during its propagation is still limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Despite this, we still lack an understanding of how the thermodynamic properties of CMEs change during their propagation in the inner corona (up to ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 Rsun).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Using UVCS spectroscopic data, the physical parameters of CME plasma have been obtained at fixed heliocentric distances starting from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 to 10 Rsun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' However, the DEM analysis using SOHO/EIT, STEREO/EUVI, & SDO/AIA, has enabled us to estimate the physical parameters of CME in the inner corona (up to ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 Rsun).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Nevertheless, in none of these studies, the continuous evolution of the thermodynamic parameters of CME structures in the inner corona has been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Hence, the motivation behind the work presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' In this paper, by combining the K-Cor white-light data and the CoMP Fe XIII 10747 Å line spectroscopic data, we present the spectroscopic diagnostics of the temperature and density of a CME core observed in the inner corona on July 20, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We studied the continuous evolution of the thermodynamic properties, such as the temperature and density of the CME core from ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='05 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='35 Rsun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' In Section 2, we provide the details of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' In Section 3, we describe the methods used to derive the temperature and density of this CME core and present the main results, followed by a summary & discussion in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2 OBSERVATIONS On July 20, 2017, the MLSO/K-Cor observed a CME propagating outward from the west limb starting at around 17:00 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The event is observed with multiple instruments, including the K-Cor and CoMP, the AIA onboard SDO, and the white-light coronagraph (COR-1A) of SECCHI onboard STEREO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The K-Cor records polarized brightness (pB) images of the inner corona from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='05 - 3 Rsun in 720 - 750 nm passband with a cadence of 15 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The K-Cor images have a pixel scale of ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='6 arcsecs pixel−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We used white-light level-2 pB data to estimate the electron density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Figure 1(a) shows the three-part CME in the K-Cor FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The eruption was also seen in the CoMP Fe XIII 10747 Å channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The CoMP takes observation of the polarization state at a few spectral locations across the profiles of three infrared lines (Fe XIII 10747 Å, 10798 Å & He I 10830 Å) using a narrow-band tunable filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' It has a field of view (FOV) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='05 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='4 Rsun, a pixel-scale of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='46 arcsecs pixel−1, and a typical cadence of 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The details of the acquisition and reduction of CoMP data are described in Tomczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2008, and the calculation process is given by Frontiers 3 Sheoran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='CME Thermodynamics in the Inner Corona ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='X [arcsec] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='−1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='−500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='Y [arcsec] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='−1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='−500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='(a) MLSO/K−Cor 17:04:12 UT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='−400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='−200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='(b) MLSO/CoMP 10747Å 17:04:57 UT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='200 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='−200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='(c) SDO/AIA 193Å 17:04:53 UT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' (a) July 20, 2017, CME as observed by the K-Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' All three parts of the CME are clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The yellow rectangular box shows the ROI chosen for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' (b) The CoMP 10747 Å channel enhanced intensity image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' When CoMP observation started, only the core of the CME was visible in the CoMP FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The yellow box is the CoMP ROI aligned with the K-Cor ROI shown in Figure (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' (c) The AIA 193 Å channel image processed using MGN to enhance the CME core structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The yellow box is the AIA ROI aligned with the CoMP ROI shown in Figure (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' (An animation of the evolution of the CME core in the K-Cor, CoMP 10747 Å, and AIA 193 Å channel is available in the Electronic Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The yellow circle in the animation represents the CoMP FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=') Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We used Fe XIII 10747 Å level 2 dynamic data, which contains line peak intensity, edge enhanced line peak intensity, line-of-sight (LOS) Doppler velocity, and line width images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' On this day, the CoMP observation started at around 16:57 UT, and only the core of this CME was visible in the CoMP FOV, as shown in Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' This eruption was seen as a braided structure moving out in CoMP Fe XIII 10747 Å channel and caused a significant enhancement in Fe XIII line width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' This eruption was also recorded in all SDO/AIA EUV channels from around 15:00 to 18:00 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The AIA takes full-disk images of the corona and transition region up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='3 Rsun simultaneously in seven extreme-ultraviolet (EUV) narrow-band filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' AIA has a pixel-scale of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='6 arcsecs pixel−1 and a cadence of 12 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Using aia prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='pro AIA images were normalized by their exposure times, rotated, and re-scaled to ensure that each pixel is examining the same spatial location across the different wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We also checked AIA 304 Å data-set to determine if this CME has an associated filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' No filament could be seen in AIA 304 Å channel prior to the period of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We used 94, 131, 171, 193, 211, and 335 Å pass bands images, sensitive over a temperature range from 105 K to 107 K for the DEM analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Since AIA 193 Å and CoMP 10747 Å images have similar temperature responses, the core of this CME as seen in AIA 193 Å channel, is shown in Figure 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The AIA 193 Å channel image has been processed using Multiscale Gaussian Normalization (MGN: Morgan and Druckm¨uller, 2014) to enhance the CME core structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The yellow rectangular box in all panels of Figure 1 corresponds to the region of interest (ROI) chosen for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We have aligned CoMP 10747 Å ROI with respect to AIA 193 Å ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The yellow dashed lines in Figure 2 (a) indicate the position of four artificial slits at the same location in both AIA and CoMP ROIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Each slit is approximately ten arcsec wide and the location of these slits is such that they cover almost the entire CME core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Since K-Cor images capture white-light and CoMP images capture narrow-band emission, we could not use intensity/brightness to align their FOVs as the features look different in white-light and This is a provisional file, not the final typeset article 4 Sheoran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CME Thermodynamics in the Inner Corona narrow-band emission images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The CoMP and the K-Cor both have 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='05 Rsun occulter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We use it as a reference to align the CoMP & the K-Cor ROIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' This CME is also observed by the STEREO/COR-1A coronagraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The COR-1A coronagraph offers high-cadence (5 minutes) data with FOV from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='4-4 Rsun, and a pixel scale of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 arcsecs pixel−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' There is a filament behind the limb, which could be seen in STEREO EUVI 304 Å channel around 11:46 UT on July 20, 2017 which erupts around 12:46 UT, seen in COR-1A FOV ∼13:00 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Around 15:00 UT, a braided structure is seen to rise in EUVI 304 Å and 195 Å FOV, which propagates out and a CME appears (case event for this study) in COR-1 FOV at around 16:40 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Therefore, our case event does not have a filament associated with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' To determine the parameters like LOS depth and volume of this CME, we used the graduated cylindrical shell (GCS) model (Thernisien et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2009) to fit this CME using the K-Cor & COR-1A vantage point observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' For GCS fitting, we have used K-Cor white light level 2 Normalizing Radially Graded Filter (NGRF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Morgan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2006) data and COR-1 level 1 data filtered using Simple Radial Gradient Filter (SiRGraF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 3 ANALYSIS AND RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1 Determination of the Electron Temperature In this work, we determined the two-dimensional plasma temperature distribution across the CME core using two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' First, using DEM analysis on AIA six EUV channels data, and second, using the line broadening of Fe XIII emission line centered at 10747 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1 Temperature using the Differential Emission Measure Analysis To infer the DEM from AIA six EUV channels data, we applied the inversion technique developed by Cheung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' For the inversion, we used a temperature grid spanning log T/K ∈ [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='7, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='7] with a grid spacing of log T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The emission measure in each temperature grid was obtained using aia sparse em init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='pro in the SolarSoftware (SSWIDL) package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The total emission measure (EM) was then obtained by summing the EM in all temperature bins: EM = n � j EM j, (1) where EMj is the EM contained in the jth temperature bin, and n is the total number of bins (in our case n = 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The mean temperature can be estimated by the emission-weighted temperature, log TEM = EM−1 ��������� n � j EM j log T j ��������� , (2) W2 EM = EM−1 ��������� n � j EM j �log T j − log TEM �2 ��������� , (3) here log TEM is the EM-weighted log temperature and WEM is the effective width of the distribution in log T space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We created DEM maps at each pixel for the logarithmic temperature spanning from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='7 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The colored boxes in the left panel of Figure 2 (b) show the selected sub-regions of AIA ROI used to reconstruct the DEM curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Note that we have processed AIA 193 Å channel images using the MGN technique to Frontiers 5 Sheoran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CME Thermodynamics in the Inner Corona enhance the CME core structures, the MGN technique has not been used for DEM analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The DEM curves for these box regions are shown in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The DEM in all regions peaked at around log T/K ∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Figure 2 (c) show the EM maps of the AIA ROI at 2017-07-20 T 17:16:11 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We can see that the EM of the core of the CME is mostly confined in log T/K ∈ [6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='05, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Then using Equation (2) we created EM-weighted log temperature (log TEM) maps of the AIA ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We obtained the intensity and log TEM values along the AIA slits (shown in the right panel of Figure 2 (a)) and took their averaged over the width of each slit to ensure a good signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' To illustrate the full evolution of the core of the CME, we constructed space-time maps for all four slits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The left panel of Figure 3 shows the AIA 193 Å intensity space-time maps, and the right panel shows the log TEM space-time maps for slit 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We visually inspected the AIA 193 Å intensity space-time map of slit 2 and tracked the eruption along this slit as shown by the blue dashed curve in the top left panel of Figure 3, which we fitted using the cubic spline interpolation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Using these height-time values, we fitted the spline curve on log TEM space-time map of slit 2 as shown in the top right panel of Figure 3 by blue dashed curve and obtained the values of log TEM and WEM (WEM is calculated using Equation (3)) along this curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The same procedure is repeated for the remaining slits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Note that the spline fitted curve has approx 24 arcsec spatial extent at a particular time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' (c) (a) (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' (a) The left panel shows the AIA ROI (shown by yellow box in Figure 1 (c)), and right panel shows the CoMP ROI (shown by yellow box in Figure 1 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The yellow dashed lines in both panels show the locations of four co-spatial slits chosen in the two ROIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' (b) The left panel shows AIA 193 Å ROI image at 2017-07-20 T 17:16:17 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The colored boxes show the selected sub-regions used to reconstruct the DEM curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The DEM curves for these box regions are shown in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The DEM in all regions peaked at around log T/K ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' (c) EM maps of the AIA ROI at 2017-07-20 T 17:16:11 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The color coding indicates the total EM contained within a log temperature range indicated in the bottom left corner of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' (An animation is available in the Electronic Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=') This is a provisional file, not the final typeset article 6 0 100 Y [arcsec] 200 300 193A T17:00:05 UT 1050 1100 1150 1200 X [arcsec] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 Log T/K0 0 100 100 Y [arcsec] EE [arcsec] Y 200 200 300 300 10747A 17:04:57 UT 193A 17:04:53 UT 1050 1100 1150 1200 1050 1100 1150 1200 X [arcsec] X[arcsec]0 0 100 100 Y [arcsec] Y [arcsec] 200 200 300 300 10747A 17:16:24 UT 193A 17:16:17 UT 1050110011501200 1050110011501200 X [arcsec] X [arcsec]0 100 [arcsec] Y 200 300 193AT17:16:17U 1050110011501200 X[arcsec] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 LogT/K2017-07-20 T 17:16:11 UT EM in Igt=[5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='75,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='05] EM in Igt=[6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='05,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='35] EM in Igt=[6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='35,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='65] 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 Log Emissian Measure [cm 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 EM in lgt=[6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='65,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='95] EM in Igt=[6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='25] EM in Igt=[7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='25,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='55]Sheoran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CME Thermodynamics in the Inner Corona AIA 193Å, Slit 2 15:00 15:30 16:00 16:30 17:00 17:30 18:00 2017−07−20, Time [UT] 1050 1100 1150 1200 X [arcsec] 15:00 15:30 16:00 16:30 17:00 17:30 18:00 1050 1100 1150 1200 AIA, Slit 2 15:00 15:30 16:00 16:30 17:00 17:30 18:00 2017-07-20, Time [UT] 1050 1100 1150 1200 X [arcsec] 15:00 15:30 16:00 16:30 17:00 17:30 18:00 1050 1100 1150 1200 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='30 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='35 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='40 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='45 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='50 log TEM /K AIA 193Å, Slit 3 15:00 15:30 16:00 16:30 17:00 17:30 18:00 2017−07−20, Time [UT] 1050 1100 1150 1200 X [arcsec] 15:00 15:30 16:00 16:30 17:00 17:30 18:00 1050 1100 1150 1200 AIA, Slit 3 15:00 15:30 16:00 16:30 17:00 17:30 18:00 2017-07-20, Time [UT] 1050 1100 1150 1200 X [arcsec] 15:00 15:30 16:00 16:30 17:00 17:30 18:00 1050 1100 1150 1200 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='30 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='35 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='40 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='45 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='50 log TEM /K Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The left panel shows AIA 193 Å intensity space-time maps, and the right panel shows the EM-weighted log temperature space-time maps for all slits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We visually traced the eruption along each slit intensity space-time map shown by the blue dashed curve in the left panel, which is fitted using the cubic spline interpolation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The same curve is overplotted over log TEM space-time map of the respective slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The calculation of EM-weighted temperature is based on the assumption that the plasma is isothermal along the LOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' However, the left panel of Figure 2 (b) demonstrates that the emission measure has a spread over a range of temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' To investigate the temporal evolution of the CME core emission in different temperature bins, we produce space-time maps of log EM for slit 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Figure 4 shows temporal variation in log EM along slit 3 for different temperature bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The space-time maps show that the CME core rises gradually, and at ∼17:30 UT, the CME core moves out of the AIA FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The CME core appears in most of the temperature bins indicating that the CME core is multi-thermal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' However, the EM of the CME core is mainly confined in log T/K ∈ [6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='05, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The blue dashed curve in all panels of Figure 4 is the same spline curve as in the bottom panel of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' log T = [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='75, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='05] 1050 1100 1150 1200 X [arcsec] 1050 1100 1150 1200 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='4 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='8 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 log EM [cm-5] log T = [6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='05, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='35] 1050 1100 1150 1200 X [arcsec] 1050 1100 1150 1200 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 log EM [cm-5] log T = [6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='35, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='65] 1050 1100 1150 1200 X [arcsec] 1050 1100 1150 1200 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 log EM [cm-5] log T = [6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='65, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='95] 1050 1100 1150 1200 X [arcsec] 1050 1100 1150 1200 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='4 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='8 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 log EM [cm-5] log T = [6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='95, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='25] 15:00 15:30 16:00 16:30 17:00 17:30 18:00 2017-07-20, Time [UT] 1050 1100 1150 1200 X [arcsec] 15:00 15:30 16:00 16:30 17:00 17:30 18:00 1050 1100 1150 1200 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 log EM [cm-5] log T = [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='25, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='55] 15:00 15:30 16:00 16:30 17:00 17:30 18:00 2017-07-20, Time [UT] 1050 1100 1150 1200 X [arcsec] 15:00 15:30 16:00 16:30 17:00 17:30 18:00 1050 1100 1150 1200 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 log EM [cm-5] Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The temporal variation in log EM along slit 3 for different temperature bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The blue dashed curve in all panels of is the same spline curve as in the bottom panel of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Frontiers 7 Sheoran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CME Thermodynamics in the Inner Corona 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='2 Effective temperature from Fe XIII Line Width Using the Fe XIII line width and considering that the broadening of the line may occur due to the thermal motions of ions, the non-thermal motions in the corona, the instrumental broadening, the expansion of CME as it moves outward, and the additional turbulence created by the CME propagation in the corona, the CME plasma temperature can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' In order to calculate the non-thermal line width (NTLW) in the corona, we used the previous day, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=', July 19, 2017, CoMP Fe XIII 10747 Å line width data when CME is not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We subtract the thermal width (21 km s−1, corresponding to the peak formation temperature of Fe XIII of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='6 MK) and instrumental width (21 km s−1) calculated by Morton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2015 from the line width obtained for each pixel (McIntosh and De Pontieu 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' To obtain the radial profile of NTLW, we took a median value of NTLW of pixels lying along 40 degrees (to ensure a good signal-to-noise ratio) about the equator at each radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Figure 5 shows the variation of the NTLW with height above the solar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We found that the NTLW did not vary significantly with height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Therefore, for our calculations, we used the mean value of NTLW, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='32 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We subtract non-thermal width (NTLW) and instrument 0 50 100 150 200 250 Height above photoshphere [Mm] 0 10 20 30 40 NTLW [km s -1] 2017-07-19 T 22:33:32 UT = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='32 km s -1 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The variation of NTLW with height above the solar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' width in quadrature from the total line width;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' the residual width is Doppler/thermal width (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=', width due to thermal motions of ions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Furthermore, the plasma temperature can be estimated using the following Equation: T = 1 2 m kB v2 1/e, (4) where m is the mass of the Fe XIII ion, kB is the Boltzmann constant, and v1/e is the velocity derived from the Doppler half-width, ∆λ1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Note that we did not take into account the broadening of the line due to the expansion of the CME core and the additional turbulence created by the propagation of the CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Hence Equation 4 provides the upper limit to the plasma temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We refer to it as effective temperature, Te f f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We selected an ROI from the CoMP (Figure 1 (b)) image same as AIA 193 ROI and placed four artificial slits co-spatial with AIA slits (Figure 2 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We averaged the Fe XIII line enhanced intensity and line width over the width of the slits and created the space-time maps of line enhanced intensity and total width as shown in the left and right panels of Figure 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Then, using the height-time values along the spline fitted curve on AIA 193 slit 2 intensity space-time map, we fitted the Fe XIII enhanced intensity and line width space-time maps of slit 2, which is shown by the blue dashed curve in the top panel of Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Since the CoMP has a larger FOV than AIA, we visually tracked the eruption in CoMP FOV outside the AIA FOV shown by the cyan curve in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The line width values were obtained along the blue and the cyan curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Moreover, to calculate the Doppler/thermal width values, the NTLW (19 km s−1) and This is a provisional file, not the final typeset article 8 Sheoran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CME Thermodynamics in the Inner Corona instrumental width (21 km s−1) were subtracted from the total line width values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The effective temperature was calculated using Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' This procedure is repeated for the remaining slits also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CoMP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Slit 2 (Enhanced Intensity) 16:57 17:13 17:30 17:45 18:03 18:19 2017-07-20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Time [UT] 1050 1100 1150 1200 1250 X [arcsec] 16:57 17:13 17:30 17:45 18:03 18:19 1050 1100 1150 1200 1250 CoMP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Slit 2 (Line Width) 16:57 17:13 17:30 17:45 18:03 18:19 2017-07-20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Time [UT] 1050 1100 1150 1200 1250 X [arcsec] 16:57 17:13 17:30 17:45 18:03 18:19 1050 1100 1150 1200 1250 30 34 39 44 Line Width [km s -1] CoMP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Slit 3 (Enhanced Intensity) 16:57 17:13 17:30 17:45 18:03 18:19 2017-07-20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Time [UT] 1050 1100 1150 1200 1250 X [arcsec] 16:57 17:13 17:30 17:45 18:03 18:19 1050 1100 1150 1200 1250 CoMP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Slit 3 (Line Width) 16:57 17:13 17:30 17:45 18:03 18:19 2017-07-20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Time [UT] 1050 1100 1150 1200 1250 X [arcsec] 16:57 17:13 17:30 17:45 18:03 18:19 1050 1100 1150 1200 1250 30 34 39 44 Line Width [km s -1] Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The left panel shows the CoMP Fe XIII 10747 Å enhanced-intensity space-time maps, and the right panel shows the Fe XIII line width space-time maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The blue dashed curve in each slit space-time map is fitted using the height-time values obtained from spline fitting AIA 193 Å intensity space-time map of the respective slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The eruption is further tracked in the CoMP FOV outside the AIA FOV and is shown by the cyan curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Black stripes represent gaps in data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' In addition to this, expansion of the CME core and additional turbulence created by the CME propagation will also contribute to the non-thermal broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' To model the expansion of the CME core, we used the graduated cylindrical shell (GCS) model developed by Thernisien et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Thernisien 2011 and followed the fitting procedure described in Majumdar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We used the GCS ice-cream cone model to fit the core of the CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Figure 7 shows the variation of the LOS nominal depth of the CME core plasma (LCME) with time and heliocentric distance of the CME core front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We found that the LCME of the CME core increases almost two times as the core evolves during this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The expansion velocity of the CME core is found to be approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='76 km/s, which is larger than the total line width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Since the features look different in the white-light & IR emission band ( Fe XIII 10747 Å), we conclude that we may not be modelling the same feature using the GCS model fit, which we are tracking in IR emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Therefore, the expansion factor for these two wavelength regimes would be different, which makes it difficult to apply the expansion factor obtained from the GCS model to account for the broadening of the Fe XIII 10747 Å line due to the expansion of the CME core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Figure 8 shows the evolution of log temperature of the CME core with height and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The green diamond symbols show the variation of the EM-weighted log temperature (log TEM) averaged over the spatial direction along the blue curve shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The shaded grey region shows the effective width of the distribution in log T space, which is obtained by adding WEM and the standard deviation for this spline-fitted curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We found that within this region of uncertainty, the EM-weighted temperature of the core of the CME remains almost constant as CME evolves, and the mean value of log TEM/K of the core of the CME is found to be in the range of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='28 - 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The blue cross symbols in the Figure 8 show the variation of log effective temperature averaged over the spatial direction along the blue and the cyan curves shown in Figure 6, and the red bars are one sigma deviation in log Te f f values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The log effective Frontiers 9 Sheoran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CME Thermodynamics in the Inner Corona temperature of the core of the CME has a similar variation to the EM-weighted log temperature and has mean values in the range of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='97 - 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 7211 7696 8667 9638 10201 10501 10801 Time (t) since 14:59:59 UT [seconds] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='6 LOS width, LCME [Rsun] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='66 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='69 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='10 Heliocentric height [Rsun] KCOR + COR1 Linear fit LCME = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='279 + (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='29e-05)*t Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The variation of LOS width (LCME) of the CME core with time or heliocentric height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 4000 5000 6000 7000 8000 9000 Time since 2017-07-20 14:59:59 [seconds] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 Log10T 795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1 796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='4 798.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='4 801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='6 806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='4 852.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='8 908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 Height along the slit 2 [Mm] log TEM log Teff 4000 5000 6000 7000 8000 9000 Time since 2017-07-20 14:59:59 [seconds] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 Log10T 793.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='3 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='9 808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='2 817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='9 838.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='6 868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='9 Height along the slit 3 [Mm] log TEM log Teff Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The evolution of log temperature of the CME core with height and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The green diamond symbols show the variation of the EM-weighted log temperature (log TEM) averaged over the spatial direction along the blue curve shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The grey shaded region is the effective width of the distribution in log T space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The blue cross symbols show the variation of log effective temperature averaged over the spatial direction along the blue and the cyan curves shown in Figure 6, and the red bars are one sigma deviation in log Tef f values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='2 Estimation of the Electron Density The electron density can be derived using the CoMP Fe XIII 10747 & 10798 Å density-sensitive line pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Since there were no good signal-to-noise ratio data frames available during the period of CME under consideration, we could not use this line pair for density calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Therefore, we use K-Cor polarized brightness data to derive the electron density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The polarized brightness observed by the white-light coronagraphs primarily depends on the column electron density (LOS integration of the electron density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Transient phenomena, such as CMEs, cause the intensity (hence, density) enhancements in a sequence of coronagraph images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' A suitable pre-event image is subtracted from the frames containing the CME to calculate the CME density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Since no pre-event frame was available for the CME event on July 20, 2017, we chose a frame on July 21, 2017, at around 02:21:59 UT, much later after the CME had passed the K-Cor FOV, and subtracted this image from the CME frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' As a result, the background F-corona and static K-corona are removed, leaving us with the brightness changes caused by the CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The excess column electron density Ne (due to the CME) can be estimated by taking the ratio of the excess observed pB (pBobs) to the polarized brightness of a single electron (pBe) assumed to lie on the This is a provisional file, not the final typeset article 10 Sheoran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CME Thermodynamics in the Inner Corona plane of sky (POS) (Colaninno and Vourlidas 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The POS assumption is valid for July 20, 2017, CME as indicated by very small LOS Doppler velocities values in CoMP Fe XIII 10747 Å data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The pBe is computed analytically from the scattering geometry using the equations given in Billings 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We used eltheory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='pro in the SolarSoftware (SSWIDL) package to calculate the pBe, and the limb darkening coefficient used in this routine was calculated using the Equation (5) of Hestroffer and Magnan 1998 for wavelength = 735 nm (for the K-Cor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The CME electron density ne, can be obtained by dividing the derived column density Ne to the LOS nominal depth of the CME plasma LCME, ne = Ne/LCME (Bemporad 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We determined the column electron density for the entire K-Cor ROI (the yellow box in Figure 1(a)) and chose four slits co-spatial with AIA and CoMP slits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The resulting 2-D maps of the column density (Ne) for slit 2 & 3 are shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The green region in all panels shows the evolution of the core of the CME with time along the respective slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The blue and cyan curves are the same as described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We obtained the values of Ne along the spline fitted curves and divided it by LCME to obtain the CME core electron density values (ne).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The variation of the CME core electron density (ne) along the spline fitted curves for slit 2 & 3 is shown in Figure 10 by blue cross symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The dark grey bars show one sigma uncertainty in the electron density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We found that the CME core’s density falls by a factor of ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='6 as the core evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The density values of the CME core are in the range (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='85- 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='85)× 107 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We also derived the volume of the CME core using the GCS model fitted parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The volume of the CME core increases approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' six times while the core evolves in the K-Cor FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Hence, the decrease in CME core density is consistent with the volume increase of the CME core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Similar results were also obtained for slits 1 & 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' K-Cor, Slit 2 17:00 17:10 17:20 17:30 17:40 17:50 18:01 2017-07-20, Time [UT] 1050 1100 1150 1200 1250 X [arcsec] 17:00 17:10 17:20 17:30 17:40 17:50 18:01 1050 1100 1150 1200 1250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 Ne [1017 cm-2] K-Cor, Slit 3 17:00 17:10 17:20 17:30 17:40 17:50 18:01 2017-07-20, Time [UT] 1050 1100 1150 1200 1250 X [arcsec] 17:00 17:10 17:20 17:30 17:40 17:50 18:01 1050 1100 1150 1200 1250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 Ne [1017 cm-2] Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The column electron density space-time maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The blue and the cyan curves are the same as in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 7500 8000 8500 9000 9500 Time since 2017-07-20 14:59:59 [seconds] 0 5 10 15 20 25 ne [107 cm-3] 810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='0 812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='3 815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='2 823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='7 837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1 862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='3 908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='5 Height along the slit 2 [Mm] 7500 8000 8500 9000 9500 Time since 2017-07-20 14:59:59 [seconds] 0 5 10 15 20 25 ne [107 cm-3] 820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='7 823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='2 825.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='9 831.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='9 841.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1 864.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='9 907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1 Height along the slit 3 [Mm] Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The evolution of electron density (ne) of the CME core with height and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The blue cross symbols show the variation of ne averaged over the spatial direction along the blue and the cyan curves shown in Figure 9, and the dark grey bars are one sigma deviation in ne values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Frontiers 11 Sheoran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CME Thermodynamics in the Inner Corona 4 SUMMARY AND DISCUSSIONS In this work, we carried out the spectroscopic diagnostics in the inner corona (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='05 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='35 Rsun) to derive the thermodynamic property of a CME by combining the CoMP Fe XIII 10747 Å line width data and the K-Cor polarized brightness (pB) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We studied the evolution of a CME core’s thermodynamic properties that occurred on July 20, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We obtained the effective temperature of the CME core using the line broadening of the Fe XIII emission line centered at 10747 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We have also applied the DEM inversion technique on AIA six EUV channels data to determine the EM - weighted temperature of the CME core and compare it with the effective temperature obtained using Fe XIII line width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The column density of the CME core is derived using the K-Cor pB intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' To obtain the LOS depth (LCME) and volume of the CME core, we used the graduated cylindrical shell (GCS) model to fit this CME core using the two (K-Cor & COR-1A) vantage point observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We obtained the electron density of the CME core by dividing the CME core column density by LCME of the CME core plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We find that within one sigma error, the EM-weighted temperature of the CME core remains almost constant as CME evolves, and the mean log TEM/K of the CME core is found to be in the range 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='28 - 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The effective temperature of the CME core also has a similar variation to the EM- weighted temperature, and mean log Tef f/K has values in the range of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='97 - 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The non-thermal motions in the corona contribute significantly to the broadening of a spectral line (McIntosh and De Pontieu 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Brooks and Warren 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Pant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' In earlier studies, the plasma temperatures have also been obtained using the width of spectral lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=', H I Ly-α and Ly-β, C III, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=') observed in the CMEs (Ciaravella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Heinzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Nevertheless, these studies did not take into account the contribution of non-thermal motions in the width of the spectral line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Hence, the plasma temperature limit provided by these studies is not quite accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Therefore, it is required to subtract NTLW from the total width of a spectral line (Heinzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' In our analysis, we have calculated NTLW and subtracted this from the total width of the Fe XIII 10747 Å line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The consequence is that the CME core’s effective and EM-weighted temperatures have similar values within the error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Thus, we can conclude that the effective temperature derived from line width by taking into account the non-thermal in the corona and instrumental broadenings is a good measure of CME plasma temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The million-degree kelvin temperature of the CME core indicates that the core of this CME is not associated with a prominence material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We find that the CME core’s electron density falls by a factor of ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='6 as the core evolves and has values in the range (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='85- 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='85)× 107 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' As pointed out by Bemporad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2018, the uncertainty in the density determination using pB is mainly due to the POS assumption (the assumption that all the plasma is located on the POS) and due to the uncertainty in the depth of a CME structure along the LOS (LCME).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' However, the POS assumption is valid for July 20, 2017, CME as indicated by very small LOS Doppler velocities values in CoMP Fe XIII 10747 Å data, and we get a better estimate of LCME by performing GCS model fit to the CME core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We find that the temperature of the CME core remains almost constant despite expected adiabatic cooling due to the expansion of the CME core, which suggests that the CME core plasma must be heated as it propagates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' In previous studies based on in-situ observations of ICMEs, the polytropic index (γ) of ICME plasma is also found to be close to unity implying the isothermal expansion of ICME plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' γ of ICME plasma was suggested to be of the order of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='3 from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='3 and 20 au (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2005, 2006), indicating local heating of ICME plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Furthermore, in few MHD models of CME have also used γ close to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Gibson and Low 1998 constructed a theoretical MHD model describing the ejection of a 3-D CME out of the solar corona by making an assumption γ ∼ 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Odstrcil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2002 have used γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='05 in a coronal 2-D MHD model to simulate the disruption of a sheared helmet streamer launching This is a provisional file, not the final typeset article 12 Sheoran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CME Thermodynamics in the Inner Corona a CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Chen 1996 & Krall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2000 have used γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='2 in their theoretical treatment describing the initiation, propagation, and driving mechanisms of ICMEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Thus, we can infer that the expansion of July 20, 2017, CME core behaves more like an isothermal than an adiabatic process during its evolution in the inner corona from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='05 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='35 Rsun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' It may also be possible that the thermal force is the internal driver of CME expansion, as highlighted in the study by Mishra and Wang 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The presence of plasma heating processes occurring during the CME expansion is also reported in many studies in the literature (Akmal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Ciaravella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Landi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Bemporad 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The candidate heating mechanisms are briefly discussed in studies by Kahler and Reames 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Kumar and Rust 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' however, there is no widely accepted heating mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Moreover, the above conclusion that the CME core is not a filament/prominence and the CME core is continually heated during its early expansion has an important inference on the initiation mechanism of CMEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Currently, the existing theories of CME initiation fall into two categories: one is based on the ideal MHD instability of magnetic flux rope ( T¨or¨ok and Kliem 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Kliem and T¨or¨ok 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Fan and Gibson 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Aulanier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Kliem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Amari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2018), which often contains a filament, and the other is based on magnetic reconnection (Antiochos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Wyper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2021b,a), which does not require a pre-existing flux rope or filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' In the reconnection model, the core of the CME is formed by reconnection, and thus it is continually heated by the reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' On the other hand, in the ideal instability-based model, the CME core is the pre-existing flux rope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' thus, no heating can be provided during the ideal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Hence, July 20, 2017, CME event supports the reconnection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Our work demonstrates the potential of the CoMP and the K-Cor and future multi-channel coronagraphs upgraded-CoMP (UCoMP) at MLSO and Visible Emission Line Coronagraph (VELC: Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=', 2017) onboard Aditya-L1 to study the thermodynamic evolution of CMEs in the inner corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The UCoMP has the capability to perform simultaneous 2-D imaging and high-resolution spectroscopy in the inner corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' It has a FOV of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='03 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='95 Rsun and a spatial resolution of 6 arcsecs (3 arcsecs/pixel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The UCoMP has seven coronal emission lines (FeXIV 530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='3 nm, FeX 637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='4 nm, ArXI 691.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='8 nm, FeXV 706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='2, FeXI 789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='4 nm, FeXIII 1074.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='7 nm, and 1079.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='8 nm) covering a wide range of temperature (log Tef f ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='80 - 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' A similar analysis can be applied to the observations of CMEs that will be acquired simultaneously by MLSO/UCoMP and MLSO/K-Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' In addition, FeX 637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='4 nm and FeXI 789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='4 nm are temperature-sensitive lines, and FeXIII 1074.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='7 & 1079.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='8 nm are density-sensitive lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The ratio of these lines, together with the help of atomic spectral line databases (Dere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 1997), it is possible to infer the 2-D distribution of plasma temperatures and densities inside the CMEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' This study is a testing bed for VELC/Aditya-L1, which will perform both spectroscopy and imaging of the CMEs in the inner corona in three visible (one continuum centered at 500 nm and two emission lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' FeXIV 530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='3 nm & FeXI 789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='2 nm ) and one infrared (FeXIII 1074.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='7 nm) passbands (Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CONFLICT OF INTEREST STATEMENT The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' AUTHOR CONTRIBUTIONS JS led the analysis and carried out the image processing and spectroscopic based investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' VP and RP planned the analysis and identified the case for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' VP and DB assisted in the interpretation of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' JS prepared the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' All authors took part in the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Frontiers 13 Sheoran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' CME Thermodynamics in the Inner Corona FUNDING JS is supported by funds of the Council of Scientific & Industrial Research (CSIR), India, under file no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' 09/0948(12550)/2021-EMR-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' ACKNOWLEDGMENTS We would like to thank ARIES for providing the computational facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' Courtesy of the Mauna Loa Solar Observatory, operated by the High Altitude Observatory, as part of the National Center for Atmospheric Research (NCAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' NCAR is supported by the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The SECCHI data used here were produced by an international consortium of the Naval Research Laboratory (USA), Lockheed Martin Solar and Astrophysics Lab (USA), NASA Goddard Space Flight Center (USA), Rutherford Appleton Laboratory (UK), University of Birmingham (UK), Max-Planck-Institut for Solar System Research (Germany), Centre Spatiale de Li`ege (Belgium), Institut d’Optique Th´eorique et Appliqu´ee (France), Institut d’Astrophysique Spatiale (France).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' We also acknowledge SDO team to make AIA data available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' DATA AVAILABILITY STATEMENT The MLSO/CoMP, MLSO/K-Cor, SDO/AIA, and STEREO/SECCHI data sets analyzed for this study can be found in their respective data archives under the open data policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' The data sets generated in this study can be made available upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' REFERENCES Akmal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=', Raymond, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=', Vourlidas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=', Thompson, B.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} +page_content='1086/381725 Frontiers 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFPT4oBgHgl3EQf1zVk/content/2301.13184v1.pdf'} diff --git a/oNAzT4oBgHgl3EQfqf3Z/content/tmp_files/2301.01631v1.pdf.txt b/oNAzT4oBgHgl3EQfqf3Z/content/tmp_files/2301.01631v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fab4b2c9672a417266704171c20df87cb2e4f9bd --- /dev/null +++ b/oNAzT4oBgHgl3EQfqf3Z/content/tmp_files/2301.01631v1.pdf.txt @@ -0,0 +1,1259 @@ +arXiv:2301.01631v1 [cs.DS] 4 Jan 2023 +R´enyi-Ulam Games and Online Computation with Imperfect +Advice +Spyros Angelopoulos1 and Shahin Kamali2 +1CNRS and LIP6-Sorbonne University, Paris, France. spyros.angelopoulos@lip6.fr +2York University, Canada. kamalis@yorku.ca +Abstract +We study the nascent setting of online computation with imperfect advice, in which the online algo- +rithm is enhanced by some prediction encoded in the form of a possibly erroneous binary string. The +algorithm is oblivious to the advice error, but defines a desired tolerance, namely an upper bound on the +number of erroneous advice bits it can tolerate. This is a model that generalizes the untrusted advice +model [Angelopoulos et al. ITCS 2020], in which the performance of the algorithm is only evaluated at +the extreme values of error (namely, if the advice has either no errors, or if it is generated adversarially). +In this work, we establish connections between games with a lying responder, also known as R´enyi- +Ulam games, and the design and analysis of online algorithms with imperfect advice. Specifically, we +demonstrate how to obtain upper and lower bounds on the competitive ratio for well-studied online +problems such as time-series search, online bidding, and fractional knapsack. Our techniques provide +the first lower bounds for online problems in this model. We also highlight and exploit connections +between competitive analysis with imperfect advice and fault-tolerance in multiprocessor systems. Last, +we show how to waive the dependence on the tolerance parameter, by means of resource augmentation +and robustification. +Keywords Online computation, noisy queries, R´enyi-Ulam games, beyond worst-case analysis, fault- +tolerant algorithms. +1 +Introduction +Online computation, and competitive analysis, in particular, have served as the definitive framework for the +theoretical analysis of algorithms in a state of uncertainty. While the early, standard definition of online +computation [37] assumes that the algorithm has no knowledge in regards to the request sequence, in practi- +cal situations the algorithm may indeed have certain limited, but possibly inaccurate such information (e.g., +some lookahead, or historical information on typical sequences). Hence the need for more nuanced models +that capture the power and limitations of online algorithms enhanced with external information. +One such approach, within Theoretical Computer Science, is the framework of advice complexity; +see [19, 11, 21], the survey [12] and the book [25]. In the advice-complexity model (and in particular, +the tape model [10]), the online algorithm receives a string that encodes information concerning the request +sequence, and which can help improve its performance. The objective is to quantify the trade-offs between +the size of advice (in terms of number of bits), and the competitive ratio of the algorithm. This model places +stringent requirements: the advice is assumed to be error free, and may be provided by an omnipotent oracle. +Thus, as noted in [34], this model is mostly of theoretical significance. +1 + +A different, and more practical approach, studies the effect of predictions towards improving the com- +petitive ratio. In this model, the online algorithm is enhanced with some imperfect information concerning +the request sequence, without restrictions on its size. One is interested in algorithms whose performance +degrades gently as function of the prediction error, and specifically perform well if the prediction is error +free (what is called the consistency of the algorithm), but also remain robust under any possible error (what +is called the robustness of the algorithm). This line of research was initiated with the works [31] and [35], +and a very large number of online problems have been studied under this model (see, e.g., the survey [34] +and the online collection [1]). +A combination of the advice complexity and prediction models is the untrusted advice model, introduced +in [7]. Here, some of the advice bits may be erroneous, and the algorithm’s performance is evaluated at two +extreme situations, in regards to the advice error. At the one extreme, the advice is error-free, whereas at the +other extreme, the advice is generated by a (malicious) adversary who aims to maximize the performance +degradation of the algorithm. Using the terminology of algorithms with predictions, these two competitive +ratios are called consistency and robustness, respectively. The objective is to identify algorithms that are +Pareto-efficient, and ideally Pareto-optimal, for these two extreme measures. Several online problems have +been studied recently within this framework of Pareto-optimality (both within the untrusted advice and the +predictions models); see, e.g., [38, 28, 26, 5, 8]. +1.1 +Online computation with imperfect advice +In this work, we focus on a nascent model in which the advice can be imperfect. The starting observation is +that the Pareto-based framework of untrusted advice only focuses on extreme competitive ratios, namely the +consistency and the robustness. A more general issue is to evaluate the performance of an online algorithm +as function of the advice error. Given an advice string of size k, we denote by η ≤ k the number of erroneous +bits. The objective is then to study the power and limitations of online algorithms within this setting, i.e., +from the point of view of both upper and lower bounds on the competitive ratio. +Naturally, the algorithm does not know the exact advice error ahead of time. Instead, the algorithm +defines an appliction-specific parameter H ≤ k which determines the desired tolerance to errors, or, equiv- +alently, an anticipated upper bound on the advice error. This parameter appears very often in the analysis of +games with a lying responder such as R´enyi-Ulam games [36], which are of interest to our work, as we will +discuss shortly. It is also further motivated by recent works in learning-enhanced online algorithms with +weak predictions, in which the prediction is an upper bound of some pertinent parameter of the input (see +e.g., online knapsack with frequency predictions [23], where the prediction is an upper bound on the size +of items that appear online). Our objective is to quantify the tradeoffs between advice size, tolerance and +competitive ratio, both from the point of upper and lower bounds. +A different interpretation of the imperfect advice model treats each advice bit as a response to a binary +query concerning the input. Hence, one may think of a k-bit advice string as a prediction elicited in the form +of k binary queries, not all of which may receive correct responses. Queries are known to help improve the +performance of approximation algorithms in ML applications; see, e.g, clustering with noisy queries [33], +in which a query asks whether two points should belong in the same cluster, and where each query receives +a correct response with probability p that is known to the algorithm. In this work, we study the power, +but also the limitations of online algorithms with noisy queries. However, unlike [33], we do not rely on +any probabilistic assumptions concerning the query responses. To our knowledge, the imperfect advice +model (in particular, its binary query-based interpretation) has only been applied to the problems of contract +scheduling [8], and time-series search [9], and solely from the point of view of upper bounds. +2 + +1.2 +Contribution +We establish connections between games with a lying responder and the design and analysis of online +algorithms with imperfect advice. Namely, we show how to leverage results from the analysis of R´enyi- +Ulam games, and obtain both positive and negative results on the competitive analysis. We apply these tools +towards well-studied problems such as time-series search, online bidding, and online fractional knapsack. +Our results improve the known upper bounds for these problems, where such results were already known, +but also provide the first lower bounds on the competitive ratio of online problems in this setting. +More precisely, we begin as a warm-up with the time-series search problem in Section 3, which illus- +trates how these techniques can help us improve upon the results of [9]; we also show how to evaluate the +competitive ratios, using approximations based on the binary entropy function. In Section 4, we study a +more complex application, namely the online bidding problem, first studied in [7] in the context of untrusted +advice. Here, the crucial part is establishing near-optimal lower bounds. We achieve this by formulating a +multi-processor version of online bidding in l ≤ 2k processors, in which a certain number of processors may +be faulty; we then relate the competitive ratio of this problem to the imperfect advice setting, by relating +fault-tolerance in the processor level, to the inherent error in R´enyi-Ulam games. In Section 5 we study +the online fractional knapsack problem. Here, we present an algorithm whose competitive ratio converges +to 1 at a rate exponential to k, as long as H < k/2 . We also present a near-matching lower bound that +shows that our algorithm is close-to-optimal. For the upper bound, the crux is to use to allocate queries so +as to approximate two appropriately defined parameters of the instance. For the lower bound, we use an +information theoretic argument. Specifically, we show a reduction from R´enyi-Ulam games: if there existed +an algorithm of competitive ratio better than a certain value, one could play the game beyond the theoretical +performance bound, which is a contradiction. +As explained above, the parameter H expresses the algorithm’s desired tolerance to errors, and is thus +application-specific. In Section 6 we argue that the results are useful even in settings in which the pre- +cise tolerance is not known ahead of time. We accomplish this in two different ways: First, by resource- +augmentation arguments, i.e., by comparing the performance of an algorithm with perfect (error-free) advice +of size k to that of an algorithm with l > k advice bits but potentially very high advice error. Second, by +robustifying the algorithm, namely by requiring that the algorithm performs well even if the error happens +to exceed the tolerance parameter. +The techniques we develop can be applicable to other online problems. Specifically, our approach +to the online bidding problem defines the following general framework: For upper bounds, one would +aim to define a collection of “candidate” algorithms that are closely ranked in terms of their worst-case +performance. Then the advice can be used so as to select a suitable candidate from this collection that is +close to the best-possible. For lower bounds, one would aim to show that in any collection of candidate +algorithms, the erroneous queries may have to always return a solution sufficiently far, in terms of “rank”, +from the best one; then one needs to relate the concept of “rank” to performance, from a lower-bound point +of view. This last part highlights connections between an online problem with imperfect advice, and its fault- +tolerant version in a parallel system (with no advice). On the other hand, our approach to the time-series and +fractional knapsack problems illustrate another general technique: For upper bounds, one should identify +some important parameters of the problem, then allocate the queries appropriately so as to approximate them +in the presence of response errors. For lower bounds, information-theoretic arguments should establish a +reduction from a R´enyi-Ulam game to the online problem. +3 + +2 +Games with a lying responder +We review some core results related to games with a lying responder which will be in the heart of the +analysis of online problems with imperfect advice. We are interested, in particular, in [36], which studied +games between a questioner and a responder, related to an unknown value x drawn from a domain D. The +questioner may ask general queries of the form “is x in S”, where S is some subset of D, and which are +called subset queries. The upper bounds of [36] hold even if the questioner asks much simpler queries, +namely comparison queries of the form “is x at most a”, for some given a. Both the upper and lower bounds +in [36] are expressed in terms of partial sums of binomial coefficients. Formally, we define: +��N +m +�� +:= +m +� +j=0 +�N +j +� +, for m ≤ N. +We are interested, in particular, in the following game played over a continuous space: +CONTINIOUSSEARCH(k, H) +game: In this game, x is a real number with x ∈ D = (0, 1], and the +questioner asks k queries, at most H of which may receive erroneous responses. The objective of the +questioner is to find an interval Ix such that x ∈ Ix and |Ix| is minimized. +Lemma 1. +[36] Any questioner’s strategy for CONTINIOUSSEARCH(k, H) with H ≤ k/2 is such that +|Ix| ≥ +�� +k +H +�� +/2k. Moreover, for H ≤ k/2, there is a strategy, named C-WEIGHTING, that uses comparison +queries and outputs an interval IW,x with |IW,x| ≤ +�� +k−H +H +�� +/2k−H. +The following game will be useful in our analysis of online time-series and fractional knapsack. +FIND(k, H) game: In this game, given k and H ≤ k/2, and D = {1, . . . , m}, the objective of to find an +unknown x ∈ D, using k queries, up to H of which may be answered incorrectly. +The proof of the following theorem is direct from Lemma 1: +Theorem 2. The largest positive integer µ(k, H) such that a questioner can identify any number x ∈ +{1, 2, . . . , µ(k, H)} in the FIND(k, H) game is such that 2k−H/ +�� +k−H +H +�� +≤ µ(k, H) ≤ 2k/ +�� +k +H +�� +. +We define two further games that will be of interest to our analysis. The first is related to searching in +cyclic permutations, and will be useful in the upper-bound analysis of online bidding. +MINCYCLIC(n, k, H) game: Given an array A[0 . . . n − 1] whose elements are an unknown cyclic per- +mutation of {0, . . . , n − 1}, the objective is to use k queries, at most H ≤ k/2 of which can be erroneous, +so as to output an index of the array whose element is as small as possible. +Theorem 3 (Appendix). There is a questioner’s strategy for MINCYCLIC(n, k, H) based on k comparison +queries that outputs an index j such that A[j] ≤ ⌈n +��k−H +H +�� +/2k−H⌉, for all H ≤ k/2. +Last, we define a game that is related to searching in general permutations, and it will be useful in +establishing lower bounds on the competitiveness of online bidding. +SEARCH(n, k, H) game: Given an array, A[0, . . . , n − 1] whose elements are an unknown permutation of +{0, . . . , n − 1}, the objective is to use k queries, at most H of which can be erroneous, so as to output an +index of the array whose element is as small as possible. +Theorem 4 (Appendix). For any questioner’s strategy for the SEARCH(n, k, H) game, there is a respon- +der’s strategy such that if e is the element of A that is returned, then A[e] ≥ ⌊n +�� k +H +�� +/2k⌋. +4 + +3 +A warm-up: Online time-series search +The online (time series) search problem formulates a simple, yet fundamental setting in decision-making +under uncertainty. In this problem, a player must sell an indivisible asset within a certain time horizon, e.g., +within a certain number of days d, that is unknown to the player. On each day i, a price pi is revealed, +and the player has two choices: either accept the price, and gain a profit pi (at which point the game ends), +or reject the price (at which point the game continues to day i + 1). If the player has not accepted a price +by day d, then it accepts by default the last price pd. The competitive ratio of the player’s algorithm is +the worst-case ratio, over all price sequences, of the maximum price in the sequence divided by the price +accepted by the player. +The problem was introduced and studied in [20] that gave a simple, deterministic algorithm that achieves +a competitive ratio equal to +� +M/m, where M, m are upper and lower bounds on the maximum and mini- +mum price in the sequence, respectively, and which are assumed to be known to the algorithm. This bound +is optimal for deterministic algorithms. Time-series search is a basic paradigm in online financial optimiza- +tion, and several variants and generalizations have been studied [18, 30, 39, 17]; see also the survey [27]. +The problem has also been used as a case study for evaluating several performance measures of online +algorithms, including measures alternative to competitive analysis [13, 2]. +Time-series search was recently studied under the imperfect advice framework in [9], who showed +an upper bound of (M/m)22H−k/2 on the competitive ratio with k-bit advice and tolerance H, under the +assumption that H ≤ k/4. Note that no upper bound is known for H ∈ (k/4, k/2]. If the advice is error- +free, i.e., in the advice-complexity model, then a tight bound on the competitive ratio equal to (M/m) +1 +2k+1 +is due to [17]. +We show the following result, as an application of the FIND(k, H) game discussed in Section 2. +Theorem 5. Consider the online time series search problem, with imperfect advice of size k and tolerance +H ≤ k/2. There is an algorithm that uses k comparison queries, and that has competitive ratio at most +(M/m) +1 +U+1, where U = ⌊2k−H/ +�� +k−H +H +�� +⌋, for any H ≤ k/2. In contrast, every (deterministic) algorithm +based on k subset queries has competitive ratio less than (M/m) +1 +L+1, where L = ⌈2k/ +�� +k +H +�� +⌉. +Proof. We first show the upper bound. Let a1, . . . aU, r be defined such that r = a1 +m = a2 +a1 = . . . = +aU +aU−1 = +M +aU , hence r = (M/m)1/(U+1). The algorithm uses k comparison queries so as to find the best reservation +price, in the set {ai}U +i=1, i.e., the threshold p above which the algorithm will always accept a price in the +sequence. This follows from Theorem 2, since U ≤ 2k−H/ +�� +k−H +H +�� +. The algorithm then uses p as its +reservation price, namely it accepts the first price in the request sequence that is at least as large as p. From +the definition of the set {ai}U +i=1, it easily follows that this algorithm has competitive ratio at most r, which +completes the proof of the upper bound. +We now show the lower bound. By way of contradiction, suppose that there is an algorithm A for time- +series search with k-bit imperfect advice, and of competitive ratio less than C = (M/m) +1 +L+1. We will show +that A could then be used in the FIND(k, H) game so as to identify, using k queries, an unknown value in +{1, . . . , L + 1}, which is a contradiction to the upper bound of Theorem 2. +To arrive at the contradiction, define a1, . . . , aL such that r′ = a1 +m = a2 +a1 = . . . = +aL +aL−1 = M +aL , hence +r′ = (M/m) +1 +L+1 = C. Consider a game between the online algorithm A and the adversary, in which +the request sequences consist of prices in {m, a1, . . . , aL, M}. More precisely, consider the set of request +sequences of the form σi = m, a1, . . . , ai, m, for all i ∈ [1, L + 1], where aL+1 is defined to be equal to +5 + +M. In σi, A must accept price ai (the last request in the sequence) to be strictly less than C-competitive. +Equivalently, A uses k queries with at most H errors, and finds ai in the set {aj}L+1 +j=1 , which contradicts +Theorem 2. +3.1 +Comparison of the bounds +In order to compare the upper and lower bounds of Theorem 5, we need to be able to evaluate the partial sum +of binomial coefficients. Since this partial sum does not have a closed form, we will rely on the following +useful approximation from [32]. Let H denote the binary entropy function. Then +2NH( m +N ) +� +8m(1 − m +N ) ≤ +��N +m +�� +≤ 2NH( m +N ), +for 0 < m < N/2. +(1) +We will also use the following property of the binary entropy function +4p(1 − p) ≤ H(p) ≤ (4p(1 − p))1/ ln 4, for all p ∈ (0, 1). +(2) +We first show that the algorithm of Theorem 5 improves upon the one of [9]. First, note that [9] assumes +that H ≤ k/4, whereas Theorem 5 applies to all H ≤ k/2. Furthermore, we improve on the competitive +ratio for all values of H and k. For this, it suffices to show that +�� +k−H +H +�� +/2k−H < 22H−k/2, which, from (1) +holds if 2(k−H)(H( +H +k−H )−1) < 22H−k/2, or equivalently (k − H)(H( +H +k−H ) − 1) < 2H − k/2. Let τ be such +that τ = H/k (hence τ ≤ 1/2), then the latter is equivalent to showing that H( +τ +1−τ ) < 1+2τ +2−2τ . Using (2), it +suffices to show that +(4τ(1 − 2τ) +(1 − τ)2 )1/ ln 4 < 1 + 2τ +2 − 2τ , +which holds for all τ ≤ 1/2. +Next, we investigate how close the upper and lower bounds of Theorem 5 are to each other. Recall that +the bounds are of the form (M/m)1/(U+1), and (M/m)1/(L+1). Using (1), and ignoring for simplicity the +floors and ceilings, we obtain that +U ≥ 2k(1−τ)(1−H( +τ +1−τ )) and L ≤ +� +8kτ(1 − τ)2k(1−H(τ)). +The above inequalities, along with (2) show that the upper and lower bounds are very close to each other, +since for any fixed value of τ, we have that U ≥ 2Θ(k) and L ≤ 2Θ(k). +4 +Online bidding +Online bidding was introduced in [16] as a canonical problem for formalizing doubling-based strategies in +online and offline optimization problems, such as searching for a target on the line, minimum latency, and +hierarchical clustering. In this problem, a player wants to guess a hidden, unknown real value u ≥ 1. To +this end, the player defines an (infinite) sequence X = (xi) of positive, increasing bids, which is called its +strategy. The cost of discovering the hidden value u using the strategy X, denoted by c(X, u), is defined to +be equal to �ju +i=1 xi, where ju is such that xju−1 < u ≤ xju. Hence one naturally defines the competitive +ratio of the bidder’s strategy X as Cr(X) = supu +c(X,u) +u +. +In the standard version of the problem, i.e, assuming no advice, the doubling strategy xi = 2i achieves +optimal competitive ratio equal to 4. Online bidding was also studied under the untrusted advice model +in [7], which gave bounds on the consistency/robustness tradeoffs. The problem is also related to contract +scheduling, studied in [8], see also the discussion in Section 4.1.3. +6 + +4.1 +Online bidding with imperfect advice +4.1.1 +Upper bound +The idea behind the upper bound is as follows. We will consider bidding sequences from a space of 2k +geometrically-increasing sequences (see Definition 6). In the ideal situation of perfect advice, the k advice +bits could be used to identify the best strategy in this space. In the presence of advice errors, we will show +how to exploit the cyclic structure of this space, in conjunction with our upper bound for the MINCYCLIC +game (Theorem 3), so as to find a strategy that is not too far from the optimal. +We first define the space of geometrically-increasing bidding sequences. +Definition 6. For given b > 1, and l ∈ N+ define Xb,l as the set of bidding sequences {X0, . . . Xl−1}, in +which Xi = (bi+jl)∞ +j=0, for all i ∈ [0, l − 1]. +From the definition of Xb,l, it is easy to see that for any potential target u, there is a cyclic permutation π +of {0, . . . l − 1} which determines an ordering of the strategies in Xb,l in terms of their performance. More +precisely, suppose that Xπ(0) is the best sequence that discovers u at least cost, say C. Then Xπ(i) discovers +u at cost at most biC. This property can help us show the following upper bound: +Theorem 7 (Appendix). There is a bidding strategy based on k comparison queries of competitive ratio at +most 1+U +2k +� +1 + +2k +1+U +�1+ 1+U +2k , where U = ⌈2H ��k−H +H +�� +⌉. +4.1.2 +Lower bound +The idea behind the lower bound is as follows. With k advice bits, the best one can do is choose the best +strategy from a set X that consists of at most 2k strategies. Note that if the advice were error-free, |X| could +be as large as 2k; however, in the presence of errors, the algorithm may choose to narrow |X|. +Our approach combines two ideas. The first idea uses the abstraction of the SEARCH(n, k, H) game, +and the lower bound of Theorem 4. This result will allow us to place a lower bound on the rank of the +chosen strategy, where the best strategy has rank 0. The second idea is to define a measure that relates how +much worse a strategy of rank j in X has to be relative to the best strategy in X. We will accomplish this by +appealing to the concepts of parallelism and fault tolerance. +More precisely, given integers p, and φ, with φ < p, we define the fault-tolerant parallel bidding +problem, denoted by FPB(p, φ), as follows. The player is allowed to run, in parallel, p bidding strategies; +however, φ of these strategies can be faulty, in that they never discover the target; e.g., we can think of a +fault strategy as one in which the player abruptly stops submitting bids, at some point in time, akin to a +“byzantine” failure. The cost of discovering a target u is then defined as the minimum cost at which one of +the p − φ non-faulty strategies discovers the target, noting that the faults are dictated by an adversary that +aims to maximize this cost. The competitive ratio is defined accordingly. +The next theorem is the main technical result for FPB(p, φ), which gives a lower bound on the competi- +tive ratio of any strategy for this problem, as a function of the parameters p, φ and α ¯ +X. Here, ¯X is defined +as the sorted sequence of all bids in the p-parallel strategy X, in non-decreasing order. Moreover, given a +sequence X of positive reals, we define αX to be equal to lim supi→∞ x1/i +i +. +Theorem 8 (Appendix). Every p-parallel strategy X for FPB(p, φ) has competitive ratio Cr(X) ≥ +αp+1+φ +¯ +X +αp +¯ +X−1 . +7 + +Proof sketch. We use properties of p-parallel strategies so as to show that any such strategy satisfies Cr(X) ≥ +supq +�q+φ+1 +i=0 +¯xi +�q−(p−1) +i=q +¯xi . We then use Gal’s functional theorem [22] to obtain the result. We refer to Appendix for +many technical details. +We now show how to obtain a lower bound for the problem by combining the above ideas. We emphasize +a subtle point: unlike error-free advice of size k, where one should always choose the best strategy out of a +collection of exactly 2k strategies, it is conceivable that, in the presence of errors, this collection could very +well be of size l < 2k. This is because, as l decreases, so does the effect of errors on the competitive ratio. +In other words, we need to establish the result for all values l ≤ 2k, and not only for l = 2k. +Theorem 9. For every bidding sequence X and k subset queries in the imperfect advice model, we have +Cr(X) ≥ 1 +L(1 + L)1+1/L, where L = 2k/ +�� k +H +�� +. +Proof. Every bidding strategy will use the query responses so as to select a strategy from a set X = +{X0, . . . , Xl−1} of candidate sequences, for some l ≤ 2k. For a given target value u, there is an ordering +of the l sequences in X such that Xπ(i) has no worse competitive ratio than Xπ(i+1), namely the permuta- +tion orders the sequences in decreasing order of performance. From Theorem 4, it follows that the strategy +will choose a sequence Xj such that π(j) ≥ ⌊l +�� k +H +�� +/2k⌋. The competitive ratio of the selected sequence +is at least the competitive ratio of the l-parallel strategy defined by X, in which up to φl = ⌊l +�� k +H +�� +/2k⌋ +sequences may be faulty. From Theorem 8, +Cr(X) ≥ +αl+1+φl +¯ +X +αl¯ +X − 1 , +with φl = ⌊l +�� k +H +�� +/2k⌋. +(3) +We now consider two cases. Suppose first that l < L. In this case, case φl = 0, and therefore (3) implies that +Cr(X) ≥ αl+1 +¯ +X /(αl¯ +X − 1), which is minimized for α ¯ +X = (l + 1)1/l > 1, therefore Cr(X) ≥ 1 +l (l + 1)1+1/l. +This function is decreasing in l, and since l < L we have Cr(X) ≥ 1 +L(1 + L)1+1/L. Next, suppose that l ∈ +[L, 2k]. In this case, (3) gives Cr(X) ≥ +αl(1+1/L) +¯ +X +αl +¯ +X−1 . The above expression is minimized for α ¯ +X = (1 + L)1/l, +and by substitution we obtain again Cr(X) ≥ 1 +L(1 + L)1+1/L. +4.1.3 +Comparison of the bounds +In the Appendix we prove that the ratio between the two bounds is approximately +log UB +LB ≤ +� +8kτ(1 − τ)k(1 − τ)(1 − H( +τ +1−τ )) +2k(1−τ)(1−H( +τ +1−τ )) +− k(1 − H(τ)) +2k(1−H(τ)) , +where τ = H/k. We infer that as k increases, and for any fixed value of τ, the upper and lower bounds +become very close to each other. +5 +Online fractional knapsack +In the online fractional knapsack problem, the request sequence consists of items, where item i has a value +vi ∈ R+ and a size si ∈ (0, 1]. The online algorithm has a knapsack of unit capacity, and when considering +item i, it can accept irrevocably a fraction fi ∈ (0, 1] of the item, subject to capacity constraints. More +precisely, the algorithm aims to maximize � +i +(fi · vi) subject to � +i +(fi · si) ≤ 1. +8 + +Let di = vi/si denote the density of item i. While the offline version of the problem admits a simple, +optimal solution via a greedy algorithm (that sorts all items by non-decreasing order of density, and accepts +items in this order until the knapsack is full), the online version is more challenging. Suppose that di ∈ +[L, U], for L, U known to the algorithm. [14, 15] gave matching O(log(U/L)) and Ω(log(U/L)) upper and +lower bounds on the competitive ratio of the problem, respectively, and [40] showed an optimal bound of +ln(U/L) + 1 for deterministic algorithms. Online fractional knapsack has applications in sponsored search +auctions, and online ad allocation, and has been studied in several other settings, e.g., [3, 24]. In this section, +we study this problem in the imperfect advice setting. +5.1 +Upper bound +As in all previous work, we assume that the density of all items is in [L, U] for known values of L and +U. Let d∗ denote the smallest density of an item included at a positive fraction in the optimal solution. +That is, the optimal algorithm OPT accepts a fraction 1 of items with density larger than d∗, and fills the +remaining space with a fraction of items of density d∗. Unfortunately, knowing d∗ (even its exact value) is +not sufficient for an online algorithm to be anywhere as efficient as OPT. For example, an algorithm that +accepts a fraction 1 of items of density larger than d∗ has unbounded competitive ratio in sequences that +consist only of items of density d∗. Similarly, an algorithm that accepts a fraction 1 of items with density +at least d∗ has unbounded competitive ratio in sequences in which items of density d∗ appear early in the +sequence, and items of greater density later in the sequence. However, if we denote by c∗ ∈ (0, 1) the +fraction of the knapsack in the optimal solution that is either empty, or occupied with items of density d∗, +then knowing the exact value of both d∗ and c∗ suffices to achieve optimality. Our approach will then aim +to use k comparison queries so as to approximate the values of c∗ and d∗, then use these approximations to +choose fractional items. +5.1.1 +Algorithm and analysis +We describe the online algorithm. We first define two types of partitions, related to the parameters d∗ and +c∗. In what concerns d∗, partition the interval [L, U] into s sub-intervals I1, . . . , Is such that Ii = [di−1, di), +for s that will be specified later. We also set L = d0, U = ds. The values di are defined so that: β = d1 +d0 = +d2 +d1 = . . . = +ds +ds−1 . Thus, we have β = (U/L)1/s and di = L · βi, and note that d∗ ∈ Ix for some x ∈ [1, s]. +In what concerns the parameter c∗, we partition the interval [0, 1] into m sub-intervals I′ +1, . . . , I′ +m such +that I′ +i = [ci−1, ci); we have c0 = 0 and cm = 1. The value of m will be determined later; the values ci are +defined so that c1 = c2 − c1 +β = c3 − c2 +β = . . . = cm − cm−1 +β +. +It readily follows that for i ≥ 1, we have ci = βm+i−1−βm+i−2 +βm−1 +. In particular, c1 = βm−βm−1 +βm−1 +, and +1 +1−c1 = +βm−1 +βm−1−1. Note also that c∗ ∈ I′ +y for some y ∈ [1, m]. +Provided that s · m ≤ ⌊2k−H/ +�� +k−H +H +�� +⌋, Theorem 2 shows that the algorithm can use k comparison +queries so as to identify both x and y. Given these values, the algorithm reserves, in its knapsack, a capacity +c = cy−1 for items with density in the range Ix = [dx−1, dx), to which we refer as critical items. The +algorithm uses the remaining capacity of 1 − c for items of density larger than dx, to which we refer as +heavy items. The algorithm accepts a fraction 1 of all critical items, as long as the capacity c reserved for +them allows. Similarly, the algorithm accepts a fraction 1 of heavy items and places them in their dedicated +space of the knapsack. Given that c∗ ∈ Iy, we have 1 − c > 1 − c∗; that is, the reserved capacity for heavy +items is at least equal to the total size of these items. In other words, the algorithm can afford to accept all +heavy items. The algorithm rejects all items of density smaller than dx−1. +9 + +Theorem 10 (Appendix). For any H ≤ k/2, the above algorithm has competitive ratio at most +min +s,m∈N +fm(β) +where +β = (U/L)1/s, and fm(β) = +βm − 1 +βm−1 − 1 +subject to s · m ≤ ⌊2k−H/ +��k − H +H +�� +⌋. +5.2 +Lower bound +We will show a lower bound C(k, H) on the competitive ratio of any algorithm with imperfect advice. For +the sake of contradiction, suppose there is an algorithm A of competitive ratio better than C(k, H). Our +proof is based on a reduction from the FIND(k, H) game. Specifically, we prove that, based on A, we obtain +a questioner’s strategy for FIND(k, H) which can find a value z ∈ {1, . . . , p}, with p = ⌈2k/ +�� +k +H +�� +⌉ + 1, +which contradicts Theorem 2. +We give the intuition behind the proof. Let s and m be any two positive integers such that s · m ≤ p and +s · (m + 1) > p. Define β = (U/L)1/s, and di = U · βi, for i ∈ [1, s]. Given a pair (x, y) of integers, where +x ∈ {1, . . . , s} and y ∈ {1, . . . m + 1}, define the sequence +σx,y = ((d1, 1), (d2, 1), . . . , (dx−1, 1), (dx, cy), +where (di, j) indicates a subsequence of j/ǫ items, each of which has size ǫ and density di, and where ǫ +is infinitesimally small. cy ∈ [0, 1] is defined appropriately in the proof. For this sequence, OPT(σx,y) = +(1−cy)dx−1 +cydx. There are s·(m+1) > p such sequences, and σx,y is a prefix sequence of σx,y+1, and +σx,m is a prefix sequence of σx+1,1. In the proof, we consider request sequences of this form, and we show +that if A is C(k, H)-competitive, its decisions can help find any given z ∈ {1, . . . , p}, which contradicts +Theorem 2. We refer to Appendix for the technical details. +Theorem 11 (Appendix). For the fractional knapsack problem, where items densities are in [L, U], no +deterministic algorithm with k subset queries, out of which H ≤ k/2 may have erroneous responses, can +achieve a competitive ratio better than +C(k, H) = min +s,m∈N +gm(β) where +β = (U/L)1/s, gm(β) = (β2 − β + 1 +2β + 1 +)1/(m+1) +subject to s · m ≤ ⌈2k/ +�� k +H +�� +⌉ + 1. +Comparison of the bounds +Let τ = H/k. Since +βm−1 +βm−1−1 ≤ β, using (1), the upper bound of Theorem 10 +is at most (U/L)q, where q ≤ 1/2k(1−τ)(1−H( +τ +1−τ )). Furthermore, since β2−β+1 +2β+1 +≥ β +3 (for all β ≥ 3), the +lower bound of Theorem 11 is at least (U/L)q′(1/3)q′, where q′ ≥ 1/(2 +� +8kτ(1 − τ)2k(1−H(τ)) + 1), for +all U/L ≥ 3. For simplicity, we omitted the floors and ceilings. +6 +Waiving the assumption of the tolerance parameter +In the imperfect advice setting we studied so far, the algorithm defines an application-specific tolerance +parameter that measures its desired tolerance to errors (or equivalently, an anticipated upper bound on the +10 + +error). This parameter is in a sense required, since the analysis of R´enyi-Ulam games in [36] involves +the extreme value of error (i.e., H) instead of the instance-specific error value (i.e., η). Nevertheless, in +this section, we discuss how to mitigate the need for pre-determining a tolerance parameter. We propose +two different approaches, based on resource-augmentation, and robustification, which we discuss in what +follows. We use the time-series search and online bidding problems as illustration, even though our approach +may carry through in other online problems, at the expense of more complex calculations. +6.1 +Resource augmentation +In this setting, we compare an oblivious online algorithm A with l advice bits and no information on the error +bound, to an online algorithm B that has k ideal (i.e. error-free) advice bits. Specifically, we are interested +in finding the smallest l ≥ k (as function of k) for which algorithm A is at least as good as algorithm B, +regardless of the error in the advice of A. +The following theorem shows that O(1)-factor resource augmentation suffices to obtain an oblivious +algorithm that is at least as efficient as any algorithm that operates in the ideal setting of error-free advice, +and even if a fraction 1/3 − c of the advice bits may be erroneous, for any constant c. +Theorem 12 (Appendix). Consider the time-series and the online bidding problems. For all sufficiently +large k, and any c ∈ (0, 1/3), there is an oblivious online algorithm A with advice of size l, whose compet- +itive ratio is at least as good as that of any online algorithm B with k bits of perfect (i.e. error-free) advice, +where l = +1 +( 2 +3+c)(1−H( +1 +3 −c +2 +3 +c )) +k + 1, for any error η ≤ (1/3 − c)l in the advice of A. +6.2 +Robustification +In this setting, we augment the imperfect advice framework by requiring not only that the algorithm min- +imizes the competitive ratio assuming that the advice error is at most the tolerance H, but also that its +competitive ratio does not exceed a robustness requirement r, for some specified r, if the error exceeds H +(and in particular, if the advice is adversarially generated). We call such online algorithms r-robust. Thus, +this model can be seen as an extension of both the imperfect advice and the untrusted advice model of [7]. +For the time-series problem, we obtain the following result, which generalizes Theorem 5. In particular, +note that Theorem 5 is a special case of Theorem 13 for ρ = 1. +Theorem 13 (Appendix). Consider the online time series search problem, with imperfect advice of size k, +tolerance H ≤ k/2, and robustness r = (M/m)ρ, where ρ ∈ (1/2, 1]. There is an r-robust algorithm that +uses k comparison queries, and has competitive ratio at most (M/m) +2ρ−1 +U+1 , where U = ⌊2k−H/ +�� +k−H +H +�� +⌋, +for any H ≤ k/2. Moreover, every (deterministic) algorithm based on k subset queries has competitive +ratio better than (M/m) +2ρ−1 +L+1 , where L = ⌈2k/ +�� +k−H +H +�� +⌉. +The analysis of r-robust algorithms for online bidding is more challenging, in particular in what concerns +the impossibility results. We give an overview of the approach. For the upper bound, we can follow an +analysis along the lines of Theorem 7, however, each bidding sequence in the collection Xb,2k must be +individually r-robust. This is easy to enforce, and it requires that b much be such that b2/(b − 1) ≤ r. +The lower bound is more subtle: the proof follows the lines of Theorem 9, but uses the fact that if all the l +sequences in X0, . . . , Xl−1 must be r-robust, then α2¯ +X/(α ¯ +X − 1) ≤ r. We obtain the following: +11 + +Theorem 14 (Appendix). 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Budget constrained bidding in keyword +auctions and online knapsack problems. In Proceedings of the International Workshop on Internet and +Network Economics (WINE), pages 566–576. Springer, 2008. +A +Appendix +A.1 +Omitted material of Section 2 +Proof of Theorem 3. We will reduce MINCYCLIC(n, k, H) to the following game that was studied in [36]: +14 + +IDENTIFY(m, H) game: In this game, x is an integer in {0, 1, . . . , m − 1} for some known m, and the +objective is to identify x with as few queries as possible, if up to H queries may be answered incorrectly. +We will use the following result in the analysis: +Lemma 15. [36] The number Q(m, H) of queries required to identify x in an instance of IDENTIFY(m, H) +is such that +min{k′|2k′ ≥ m · +�� k′ +H +�� +} ≤ Q(m, H) ≤ min{k|2k−H ≥ m +��k − H +H +�� +}. +Given an instance of MINCYCLIC(n, k, H), we create an instance of IDENTIFY(m, H) , with m = +2k−H/ +��k−H +H +�� +(note that H is the same for both instances). Partition the interval [0, . . . , n − 1] into m +disjoint subintervals, each of length at most ⌈n/m⌉. Let x being the index in [0, . . . , n − 1] for which +A[x] = 0, and let Ix denote the interval that contains x. A query q of the WEIGHTING strategy of [36] that +asks “is Ix ≤ b for some b ∈ {0, m − 1}?” translates to query µ(x) in the MINCYCLIC(n, k, H) instance +that asks “is x ≤ f(b)?”, where f(b) is the largest value in the interval Ib. Note that the answer to q is ‘yes’ +if and only if the answer to µ(q) is ‘yes’. The response to µ(q) is then given to WEIGHTING, which updates +its state and proceeds with the next query. For m defined as above, we have that 2k−H ≥ m · +��k−H +H +�� +and +using WEIGHTING, we can find Ix using k queries. Subsequently, we return the largest integer f(x) in Ix. +Given that x is in Ix and the length of the intervals is at least ⌈n/m⌉, we conclude that the returned index j +is such that A[j] =≤ ⌈n/m⌉ = ⌈n +��k−H +H +�� +/2k−H⌉. +Proof of Theorem 4. Consider the following game that is defined as the CONTINIOUSSEARCH(k, H) game, +with the only difference that the goal is to guess a value as close to some r ∈ [0, n), for some fixed n (for +the purpose of the proof, we can think of n as sufficiently large). After receiving the responses to the +k queries, the questioner returns a number r′ ∈ [0, n), and the objective is to minimize |r′ − r|. We +will show a reduction from this game that will help us establish the lower bound. Suppose, by way of +contradiction, that there exists a strategy, say ALG for SEARCH(n, k, H) that returns an element e with +π(e) < ⌊n +�� k +H +�� +/2k⌋ − 1. We devise a strategy for the questioner in the continuous game based on ALG. +Given values of r, k and H that define an instance G of the continuous game over the continuous interval +I = [0, n), create an instance S of SEARCH(n, k, H) on a space A of n elements, with the same values of +k and H. Consider a bijective mapping β that maps an element of rank i in A (i ∈ {0, . . . , n − 1}) to an +interval β(i) = [i, i+1) in I. Similarly, define a bijective mapping µ between queries asked for G and those +asked for S. Any range [i, j] of indices that is a part of a subset query q asked for S is mapped to an interval +[i, j + 1) in the query µ(q) asked for G. Let r denote the searched value in G and let x denote an index of +A such that r belongs to β(x). To search for r, we consider queries that ALG asks for S and for any such +query q, we ask µ(q) for G. The response to µ(q) is then given to ALG so that it can update its state and ask +its next query. +Recall that we supposed that ALG outputs an element e such that π(e) < ⌊n +�� k +H +�� +/2k⌋ − 1, and +that β(π(e)) = [π(e), π(e) + 1). As an output for G, we return r′ = π(e) + 1 as the answer for G. +Note that there are exactly e + 1 intervals from the range of β that lie between r and r′ in I. That is +|r − r′| < ⌊n +�� k +H +�� +/2k − 1⌋ + 1 ≤ n( +�� k +H +�� +/2k. This, however, contradicts Lemma 1. +A.2 +Omitted material of Section 4 +Proof of Theorem 7. We apply the algorithm of Theorem 3 on the set of indices of all sequences in Xb,2k +with n = 2k, where b > 1 will be chosen later. The output is the index of a strategy in Xb,2k which is +15 + +ranked at most U among the sequences in Xb,2k From the definition of Xb,2k, and in particular its cyclic +property, this means that the selected strategy discovers the target with cost at most bU times larger than +the best strategy in Xb,2k. We infer that the competitive ratio of the chosen strategy is at most b2k+1+U +b2k −1 . +This expression is minimized for b = (2k+U+1 +U+1 +)1/2k, from which we obtain that the competitive ratio of our +strategy is at most +1 + U +2k +� +1 + +2k +1 + U +�1+ 1+U +2k +. +Proof of Theorem 8. Consider a p-parallel strategy X, defined by p bidding strategies X0, . . . , Xp−1, each +run on a dedicated processor. Let xj,i denote bid i in Xj; we say that xj,i precedes bid xj,i′ in j if i < i′. +We define the prefix cost of a bid in Xj as the sum of the values of all bids that precede that bid in Xj. For +j ∈ [0, . . . , p−1], we denote by uX(c, j) as the value of the largest bid in Xj, such that the sum of the prefix +cost of that bid and the value of that bid do not exceed c. We also define by uX,φ(c) at the (φ + 1)-largest +quantity in the set {uX(c, j)}p−1 +j=0. +Consider an arbitrary indexing of all bids in X, i.e., the i-th bid is such that it is the m-th bid in Xj, +for some m, j. We will represent this bid as a pair of the form (Ci, Di), where Ci is the cost of all bids +that precede bid i in the sequence to which it belongs, and Di is the bid itself (i.e., its value). Note that +this representation ignores the specific sequence to which the bid is assigned, since this is not important for +the purposes of the proof, as we will see. Given a bid represented as (Ci, Di) we define di to be equal to +uX,φ(Ci + Di): we call this value the (φ + 1)-largest bid relative to Di, in X. +Recall that ¯X denotes the sequence of all bid values in X, in non-decreasing order. Hence, each bid in +X is mapped via its length to an element of this sequence (breaking ties arbitrarily). +Fix a bid i0 of the form bi0 = (Ci0, Di0), and suppose, without loss of generality, that bi0 belongs to +sequence X0. Let c = Ci0 + Di0. For all m ∈ [1, p − 1], let bim = (Cim, Dim) denote the largest bid in Xm +for which the sum of its prefix cost and the value of its bid are at most c. For every m ∈ [0, p − 1], define +Im as the set of indices in N such that i ∈ Im if and only if a bid of value xi is such that its prefix cost plus +xi does not exceed c. From the definition of the competitive ratio we have that +Cr(X) ≥ +� +i∈Im xi +dim +, +for all m ∈ [0, p − 1]. +Therefore, +Cr(X) ≥ +max +0≤m≤p−1 +� +i∈Im xi +dim +, +and using the property max{a/b, c/d} ≥ a+b +c+d, for all a, b, c, d > 0, we obtain that +Cr(X) ≥ +�p−1 +m=0 +� +i∈Im xi +�p−1 +m=0 dim +. +(4) +Next, we will bound the numerator of the fraction in (4) from below, and its denominator from above. We +begin with a useful observation: we can assume, without loss of generality, that for cost c (defined earlier), +no bid of value di0 or smaller has prefix cost larger than c minus the value of the bid in question. This +follows from the definition of di0: if such a bid existed, then one could simply “remove” this bid from X, +16 + +and obtain a p-parallel sequence of no worse competitive ratio (in other words, such a bid is useless, and +one can derive a sequence of no larger competitive ratio than X that does not contain it). +Using the above observation, it follows that the numerator in (4) includes, as summands, all bids of +value at most di0, as well as at least φ + 1 bids that are at least as large as di0 (φ of those bids are from the +definition of di0, and the additional one is bid bi0). Let q denote an index such that di0 = ¯xq, then we have +that +p−1 +� +m=0 +� +i∈Im +xi ≥ +q+φ+1 +� +i=0 +¯xi. +We now show how to upper-bound the denominator, using the monotonicity implied in the definition of the +(φ + 1)-largest value relative to a given bid value, and the definition of the bids bi0, . . . , bip−1. Specifically, +for every bid bim, with m ∈ [1, p − 1], we have that dim ≤ di0. It thus follows that +p−1 +� +m=0 +dim ≤ +q−(p−1) +� +i=q +¯xi. +Combining the two bounds, it follows that +Cr(X) ≥ +sup +0≤q<∞ +�q+φ+1 +i=0 +¯xi +�q−(p−1) +i=q +¯xi +. +In the last step of the proof, we will use a result from search theory, namely Gal’s functional theorem, +stated below: +Theorem 16 (Gal [22]). Let q be a positive integer, and X = (xi)∞ +i=0 a sequence of positive numbers with +supn≥0 xn+1/xn < ∞ and αX > 0. Suppose that Fi is a sequence of functionals that satisfy the following +properties: +(1) Fi(X) depends only on x0, x1, . . . xi+q, +(2) Fi(X) is continuous in every variable, for all positive sequences X, +(3) Fi(aX) = Fi(X), for all a > 0, +(4) Fi(X + Y ) ≤ max(Fi(X), Fi(Y )), for all positive sequences X, Y , and +(5) Fi+j(X) ≥ Fi(X+j), for all j ≥ 1, where X+j = (xj, xj+1, . . .). +Then +sup +0≤k<∞ +Fk(X) ≥ +sup +0≤k<∞ +Fk(GαX), +where Ga is defined as the geometric sequence (ai)∞ +i=0. +Define now the functional Fq( ¯X) = +�q+φ+1 +i=0 +¯xi +�q−(p−1) +i=q +¯xi , for every q. The functional satisfies the conditions +(1)-(5) of Theorem 16 (see Example 7.3 in [4]). By applying Gal’s Theorem, it follows that +Cr(X) ≥ +sup +0≤q<∞ +�q+φ+1 +i=0 +αi¯ +X +�q−(p−1) +i=q +αi¯ +X +. +If α ¯ +X ≤ 1, then it is easy to show that the above expression shows that Cr(X) = ∞; see, e.g. [29]. +Otherwise, i.e., if α ¯ +X > 1, after some simple calculations we arrive at the desired result. +17 + +Comparison between the upper and the lower bounds. We compare the upper and lower bounds of Sec- +tions 4.1.1 and 4.1.2. Define f as the function f(x) = 1 +x(1 + x)1+1/x, and note that f is decreasing in +x, with f(1) = 4, and limx→∞ f(x) = 1. Then, the upper bound of Theorem 7 is equal to f(2k/(U + 1)), +whereas the lower bound of Theorem 9 is equal to f(L), where U, L are defined in the statements of the +corresponding theorems. +For every y > x we have +f(y) +f(x) = +1 +y(1 + y)1+1/y +1 +x(1 + x)1+1/x ≤ (1 + y)1/y +(1 + x)1/x . +Moreover, using some more elementary calculus, +log f(y) +f(x) ≤ log (1 + y)1/y +(1 + x)1/x ≤ log y1/y +x1/x = 1 +y log y − 1 +x log x. +We will use the above inequality to compare f(2k/(U + 1)) to f(L). To simplify the calculations, +we will assume that the upper bound is f(2k/U), since the additive “one” in the numerator has virtually +no effect on the competitive ratio as k becomes large. For the same reasons, we ignore the ceiling in the +expression of U. Using the approximation of the partial sum of binomial coefficients of (1), and defining +τ = k/H, we obtain that the ratio UB/LB of the upper and lower bounds satisfies +log UB +LB ≤ +� +8kτ(1 − τ)k(1 − τ)(1 − H( +τ +1−τ )) +2k(1−τ)(1−H( +τ +1−τ )) +− k(1 − H(τ)) +2k(1−H(τ)) , +A.3 +Omitted material of Section 5 +Proof of Theorem 10. Note that a fraction 1 of heavy items is accepted by both the online algorithm and +OPT. Therefore, the contribution of heavy items to the profits of the algorithm and OPT are the same, say +∆; we have δ ≥ (1 − c∗)dx ≥ (1 − cy)dx; this is because c∗ ∈ [cy−1, cy) and all heavy items have density +larger than dx. The algorithm fills the reserved space of size c with critical items, which are of density at +least dx−1, while OPT fills a space of c∗ < cy with critical items, which are of density at most dx. Thus, we +have +Cr(A) ≤ +∆ + cy · dx +∆ + cy−1 · dx−1 +≤ +(1 − cy)dx + cy · dx +(1 − cy)dx + cy−1 · dx−1 += +1 +1 − (cy − cy−1/β) = +1 +1 − c1 += +βm − 1 +βm−1 − 1. +Proof of Theorem 11. +By way of contradiction, suppose there is an algorithm A with a better competitive +ratio than C(k, H). We will show that A could then be used in the FIND(k, H) game with k queries so as to +identify an unknown value in {1, . . . , p}, which contradicts the upper bound of Theorem 2. +Fix the values of (s, m′) that minimize gm(β) subject to s.m′ ≤ p, and let m = m′+1. Define p = s·m; +we have p > ⌈2k/ +�� +k +H +�� +⌉, otherwise, the pair (s, m) results in a smaller value for gm(β) (note that gm(β) +18 + +is a decreasing function of m). Suppose we want to identify an unknown value z ∈ {1, . . . , s · m}; this is +equivalent to finding a pair (x, y) with x ∈ {1, . . . s} and y ∈ {1, . . . , m}. For i ∈ {1, . . . , s}, let di = L·βi. +Moreover, for j ∈ {0, . . . , m}, define cj such that the following hold: +c0 = min{1/β, 1 − 1/β}, +cm = max{1/β, 1 − 1/β}, +r = c1 + β(1 − c1) +c0 + β(1 − c0) = c2 + β(1 − c2) +c1 + β(1 − c1) = . . . = ci+1 + β(1 − ci+1) +ci + β(1 − ci) += . . . =cm−1 + β(1 − cm−1) +cm + β(1 − cm) +. +It can be verified that r = (β2−β+1 +2β+1 )1/m. For any x ∈ {1, . . . , s} and y ∈ {1, . . . , m}, create an input +sequence as +σx,y = (1, d0), (1, d1), (1, d2), . . . , (1, dx−1), (cy, dx). +Here (a, b) indicates a sequence of items, all of infinitesimal small size ǫ and density b, and total size a. The +optimal solution fills a capacity cy with the item of density dx, and the remaining capacity of 1 − cy with +the item of density dx+1. We have +OPT(σx,y) = (1 − cy) · dx−1 + cy · dx. +In what follows, we describe how an algorithm A with a competitive ratio better than C(k, H) can be +used to correctly find unknown values x ∈ {1, . . . s} and y ∈ {1, . . . , m}. Suppose the next item has +density dα. Let wα denote the total size of items of density ≤ dα−1, and suppose wα ∈ [1 − cq, 1 − cq−1) +for some q ∈ {[1, . . . , m}, that is, 1 − wα ∈ (cq−1, cq]. Define ∆α = cq+1 − cq. When the empty space +in the knapsack of A becomes less than ∆α, the algorithm “guesses” x = α and y = q. In what follows, +we show that A makes these guesses at some point and the guesses made by A are correct. For the sake +of contradiction, suppose A does not make a guess, or at least one of its guesses is incorrect. We show +that the competitive ratio of A will be larger than C(k, H). Suppose wx ∈ [1 − cq, 1 − cq−1) for some +q ∈ {1, . . . , m}, that is, 1 − w ∈ (cq−1, cq]. There are four possibilities to consider: +A does not make a guess: Since A does not make a guess, the empty space in the knapsack is at least +∆x = cq+1 − cq. Also, the total size of items of density ≤ dx−1 is at least 1 − cq. Therefore, the +contribution of items of density dx to the value of the knapsack is at most cq · dx, and the contribution +of other items is at most (1 − cq − ∆x)dx−1 = (1 − cq+1)dx−1. Therefore, the total value of value of +the knapsack of A is at most (1 − cq+1)dx−1 + cqdx ≤ (1 − cy+1)dx−1 + cydx. We can write: +Cr(A) ≥ +(1 − cy)dx−1 + cydx +(1 − cy+1)dx−1 + cydx +> r. +Wrong guess for y: Suppose the algorithm stops but makes the wrong guess for y, that is, q ̸= y. +First, suppose q ≤ y − 1, that is, A reserves too little space for items of density dx. The total size of +items of density ≤ dx−1 is at least 1 − cq. Therefore, the total size of items of density dx is at most +cq. The final value of the knapsack is thus at most (1 − cq)dx−1 + cqdx ≤ (1 − cy−1)dx−1 + cy−1dx. +We can write: +Cr(A) ≥ +(1 − cy)dx−1 + cydx +(1 − cy−1)dx−1 + cy−1dx += r. +Next, suppose p ≥ y + 1, that is, too much space is reserved for items of density dx and some +of this space stays empty. The total size of items of density at most dx−1 in the knapsack will be +19 + +at most 1 − cq+1, and the total size of items of density dx is at most 1 − cq. Therefore, we have +A(σx,y) ≤ (1 − cq+1)dx−1 + cqdx ≤ (1 − cy+1)dx−1 + cydx. We can write +Cr(A) ≥ +(1 − cy)dx−1 + cydx +(1 − cy+1)dx−1 + cydx +> r. +Wrong guess for x: Suppose p = y but α < x (note that α cannot be larger than x). The value of +the knapsack is maximized when the algorithm fills a capacity cy with items of density α and the rest +with items of density dα−1, that is, A(σx,y) ≤ (1 − cy)dα−1 + βcydα ≤ (1 − cy)dx−2 + βcydx−1. +We can write +Cr(A) ≥ +(1 − cy)dx−1 + cydx +(1 − cy)dx−2 + βcydx−1 += β > r. +To summarize, as long as Cr(A) < gm(β), one can use A to guess both values of x and y correctly, that +is, it can identify z ∈ {1, . . . , p} with k queries. This, however, contradicts the lower bound of Theorem 2. +Therefore, the initial assumption about the competitive ratio of A does not hold, and we conclude that +Cr(A) ≥ gm(β). +A.4 +Omitted material of Section 6 +Proof of Theorem 12. Consider the algorithms (upper bounds) of Theorems 5 and 7, with l imperfect advice +bits. The advice error η is, from the assumption at most (1/3 − c)l, thus at most a fraction equal to 1/3 − c +of the advice bits may be erroneous. Then, from the discussion in Sections 3.1 and 4.1.3, it follows that +these two algorithms have better competitive ratio (for the corresponding problem) than any algorithm with +k bits of advice (irrespectively of the latter’s advice error), as long as +l(1 − (1/3 − c))(1 − H( +1/3 − c +1 − (1/3 − c)) > k, +for all sufficiently large k, which yields the result. +Proof of Theorem 13. Define p1 and p2 to be such that M/p1 = (M/m)ρ, and p2/m = (M/m)ρ, re- +spectively, and note that m ≤ p1 ≤ p2 ≤ M. Then an algorithm for time-series search is r-robust if +and only if it sets its reservation price equal to some p ∈ [p1, p2]. The proof proceeds along the lines +of the proof of Theorem 5; instead of using the queries to find a suitable reservation price in [m, M], we +search instead for a reservation price in [p1, p2]. In particular, note that by definition of p1, p2, we have that +p2/p1 = (M/m)2ρ−1. +Proof of Theorem 14. For the upper bound, the proof is similar to that of Theorem 7. The only difference is +that b must be optimized under the condition that each strategy in Xb,2k must be individually r-robust. We +know that a geometric strategy for online bidding with base b has competitive ratio at most b2/(b−1). Since +each strategy is geometric with base b2k, it follows that as a long as b2k+1/(b2k − 1) ≤ r, every strategy in +Xb,2k is r-robust, hence the strategy chosen by the imperfect advice as well. +For the lower bound, we appeal to the following property shown in [6]: If all l bidding sequences in the +collection X0, . . . , Xl−1 are r-robust, then it must be that α2¯ +X/(α ¯ +X − 1) ≤ r, where recall that ¯X is the +sequence of the union of all bids in the l strategies, sorted in non-decreasing order. Then the proof follows +along the lines of the proof of Theorem 9, with the observation that the competitive ratio is minimized if +l = 2k. +20 + diff --git a/oNAzT4oBgHgl3EQfqf3Z/content/tmp_files/load_file.txt b/oNAzT4oBgHgl3EQfqf3Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d7b1432571f3f5fc3d988348265834bde33b6952 --- /dev/null +++ b/oNAzT4oBgHgl3EQfqf3Z/content/tmp_files/load_file.txt @@ -0,0 +1,851 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf,len=850 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='01631v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='DS] 4 Jan 2023 R´enyi-Ulam Games and Online Computation with Imperfect Advice Spyros Angelopoulos1 and Shahin Kamali2 1CNRS and LIP6-Sorbonne University, Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' spyros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='angelopoulos@lip6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='fr 2York University, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' kamalis@yorku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='ca Abstract We study the nascent setting of online computation with imperfect advice, in which the online algo- rithm is enhanced by some prediction encoded in the form of a possibly erroneous binary string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The algorithm is oblivious to the advice error, but defines a desired tolerance, namely an upper bound on the number of erroneous advice bits it can tolerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This is a model that generalizes the untrusted advice model [Angelopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' ITCS 2020], in which the performance of the algorithm is only evaluated at the extreme values of error (namely, if the advice has either no errors, or if it is generated adversarially).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In this work, we establish connections between games with a lying responder, also known as R´enyi- Ulam games, and the design and analysis of online algorithms with imperfect advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Specifically, we demonstrate how to obtain upper and lower bounds on the competitive ratio for well-studied online problems such as time-series search, online bidding, and fractional knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Our techniques provide the first lower bounds for online problems in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We also highlight and exploit connections between competitive analysis with imperfect advice and fault-tolerance in multiprocessor systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Last, we show how to waive the dependence on the tolerance parameter, by means of resource augmentation and robustification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Keywords Online computation, noisy queries, R´enyi-Ulam games, beyond worst-case analysis, fault- tolerant algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 1 Introduction Online computation, and competitive analysis, in particular, have served as the definitive framework for the theoretical analysis of algorithms in a state of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' While the early, standard definition of online computation [37] assumes that the algorithm has no knowledge in regards to the request sequence, in practi- cal situations the algorithm may indeed have certain limited, but possibly inaccurate such information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', some lookahead, or historical information on typical sequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Hence the need for more nuanced models that capture the power and limitations of online algorithms enhanced with external information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' One such approach, within Theoretical Computer Science, is the framework of advice complexity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' see [19, 11, 21], the survey [12] and the book [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In the advice-complexity model (and in particular, the tape model [10]), the online algorithm receives a string that encodes information concerning the request sequence, and which can help improve its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The objective is to quantify the trade-offs between the size of advice (in terms of number of bits), and the competitive ratio of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This model places stringent requirements: the advice is assumed to be error free, and may be provided by an omnipotent oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Thus, as noted in [34], this model is mostly of theoretical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 1 A different, and more practical approach, studies the effect of predictions towards improving the com- petitive ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In this model, the online algorithm is enhanced with some imperfect information concerning the request sequence, without restrictions on its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' One is interested in algorithms whose performance degrades gently as function of the prediction error, and specifically perform well if the prediction is error free (what is called the consistency of the algorithm), but also remain robust under any possible error (what is called the robustness of the algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This line of research was initiated with the works [31] and [35], and a very large number of online problems have been studied under this model (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', the survey [34] and the online collection [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' A combination of the advice complexity and prediction models is the untrusted advice model, introduced in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Here, some of the advice bits may be erroneous, and the algorithm’s performance is evaluated at two extreme situations, in regards to the advice error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' At the one extreme, the advice is error-free, whereas at the other extreme, the advice is generated by a (malicious) adversary who aims to maximize the performance degradation of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Using the terminology of algorithms with predictions, these two competitive ratios are called consistency and robustness, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The objective is to identify algorithms that are Pareto-efficient, and ideally Pareto-optimal, for these two extreme measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Several online problems have been studied recently within this framework of Pareto-optimality (both within the untrusted advice and the predictions models);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', [38, 28, 26, 5, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1 Online computation with imperfect advice In this work, we focus on a nascent model in which the advice can be imperfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The starting observation is that the Pareto-based framework of untrusted advice only focuses on extreme competitive ratios, namely the consistency and the robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' A more general issue is to evaluate the performance of an online algorithm as function of the advice error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Given an advice string of size k, we denote by η ≤ k the number of erroneous bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The objective is then to study the power and limitations of online algorithms within this setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', from the point of view of both upper and lower bounds on the competitive ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Naturally, the algorithm does not know the exact advice error ahead of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Instead, the algorithm defines an appliction-specific parameter H ≤ k which determines the desired tolerance to errors, or, equiv- alently, an anticipated upper bound on the advice error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This parameter appears very often in the analysis of games with a lying responder such as R´enyi-Ulam games [36], which are of interest to our work, as we will discuss shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' It is also further motivated by recent works in learning-enhanced online algorithms with weak predictions, in which the prediction is an upper bound of some pertinent parameter of the input (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', online knapsack with frequency predictions [23], where the prediction is an upper bound on the size of items that appear online).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Our objective is to quantify the tradeoffs between advice size, tolerance and competitive ratio, both from the point of upper and lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' A different interpretation of the imperfect advice model treats each advice bit as a response to a binary query concerning the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Hence, one may think of a k-bit advice string as a prediction elicited in the form of k binary queries, not all of which may receive correct responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Queries are known to help improve the performance of approximation algorithms in ML applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='g, clustering with noisy queries [33], in which a query asks whether two points should belong in the same cluster, and where each query receives a correct response with probability p that is known to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In this work, we study the power, but also the limitations of online algorithms with noisy queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' However, unlike [33], we do not rely on any probabilistic assumptions concerning the query responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' To our knowledge, the imperfect advice model (in particular, its binary query-based interpretation) has only been applied to the problems of contract scheduling [8], and time-series search [9], and solely from the point of view of upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='2 Contribution We establish connections between games with a lying responder and the design and analysis of online algorithms with imperfect advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Namely, we show how to leverage results from the analysis of R´enyi- Ulam games, and obtain both positive and negative results on the competitive analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We apply these tools towards well-studied problems such as time-series search, online bidding, and online fractional knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Our results improve the known upper bounds for these problems, where such results were already known, but also provide the first lower bounds on the competitive ratio of online problems in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' More precisely, we begin as a warm-up with the time-series search problem in Section 3, which illus- trates how these techniques can help us improve upon the results of [9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' we also show how to evaluate the competitive ratios, using approximations based on the binary entropy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In Section 4, we study a more complex application, namely the online bidding problem, first studied in [7] in the context of untrusted advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Here, the crucial part is establishing near-optimal lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We achieve this by formulating a multi-processor version of online bidding in l ≤ 2k processors, in which a certain number of processors may be faulty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' we then relate the competitive ratio of this problem to the imperfect advice setting, by relating fault-tolerance in the processor level, to the inherent error in R´enyi-Ulam games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In Section 5 we study the online fractional knapsack problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Here, we present an algorithm whose competitive ratio converges to 1 at a rate exponential to k, as long as H < k/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We also present a near-matching lower bound that shows that our algorithm is close-to-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For the upper bound, the crux is to use to allocate queries so as to approximate two appropriately defined parameters of the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For the lower bound, we use an information theoretic argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Specifically, we show a reduction from R´enyi-Ulam games: if there existed an algorithm of competitive ratio better than a certain value, one could play the game beyond the theoretical performance bound, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' As explained above, the parameter H expresses the algorithm’s desired tolerance to errors, and is thus application-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In Section 6 we argue that the results are useful even in settings in which the pre- cise tolerance is not known ahead of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We accomplish this in two different ways: First, by resource- augmentation arguments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', by comparing the performance of an algorithm with perfect (error-free) advice of size k to that of an algorithm with l > k advice bits but potentially very high advice error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Second, by robustifying the algorithm, namely by requiring that the algorithm performs well even if the error happens to exceed the tolerance parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The techniques we develop can be applicable to other online problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Specifically, our approach to the online bidding problem defines the following general framework: For upper bounds, one would aim to define a collection of “candidate” algorithms that are closely ranked in terms of their worst-case performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Then the advice can be used so as to select a suitable candidate from this collection that is close to the best-possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For lower bounds, one would aim to show that in any collection of candidate algorithms, the erroneous queries may have to always return a solution sufficiently far, in terms of “rank”, from the best one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' then one needs to relate the concept of “rank” to performance, from a lower-bound point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This last part highlights connections between an online problem with imperfect advice, and its fault- tolerant version in a parallel system (with no advice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' On the other hand, our approach to the time-series and fractional knapsack problems illustrate another general technique: For upper bounds, one should identify some important parameters of the problem, then allocate the queries appropriately so as to approximate them in the presence of response errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For lower bounds, information-theoretic arguments should establish a reduction from a R´enyi-Ulam game to the online problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 3 2 Games with a lying responder We review some core results related to games with a lying responder which will be in the heart of the analysis of online problems with imperfect advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We are interested, in particular, in [36], which studied games between a questioner and a responder, related to an unknown value x drawn from a domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The questioner may ask general queries of the form “is x in S”, where S is some subset of D, and which are called subset queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The upper bounds of [36] hold even if the questioner asks much simpler queries, namely comparison queries of the form “is x at most a”, for some given a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Both the upper and lower bounds in [36] are expressed in terms of partial sums of binomial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Formally, we define: ��N m �� := m � j=0 �N j � , for m ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We are interested, in particular, in the following game played over a continuous space: CONTINIOUSSEARCH(k, H) game: In this game, x is a real number with x ∈ D = (0, 1], and the questioner asks k queries, at most H of which may receive erroneous responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The objective of the questioner is to find an interval Ix such that x ∈ Ix and |Ix| is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' [36] Any questioner’s strategy for CONTINIOUSSEARCH(k, H) with H ≤ k/2 is such that |Ix| ≥ �� k H �� /2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Moreover, for H ≤ k/2, there is a strategy, named C-WEIGHTING, that uses comparison queries and outputs an interval IW,x with |IW,x| ≤ �� k−H H �� /2k−H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The following game will be useful in our analysis of online time-series and fractional knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' FIND(k, H) game: In this game, given k and H ≤ k/2, and D = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , m}, the objective of to find an unknown x ∈ D, using k queries, up to H of which may be answered incorrectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The proof of the following theorem is direct from Lemma 1: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The largest positive integer µ(k, H) such that a questioner can identify any number x ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , µ(k, H)} in the FIND(k, H) game is such that 2k−H/ �� k−H H �� ≤ µ(k, H) ≤ 2k/ �� k H �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We define two further games that will be of interest to our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The first is related to searching in cyclic permutations, and will be useful in the upper-bound analysis of online bidding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' MINCYCLIC(n, k, H) game: Given an array A[0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' n − 1] whose elements are an unknown cyclic per- mutation of {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , n − 1}, the objective is to use k queries, at most H ≤ k/2 of which can be erroneous, so as to output an index of the array whose element is as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Theorem 3 (Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' There is a questioner’s strategy for MINCYCLIC(n, k, H) based on k comparison queries that outputs an index j such that A[j] ≤ ⌈n ��k−H H �� /2k−H⌉, for all H ≤ k/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Last, we define a game that is related to searching in general permutations, and it will be useful in establishing lower bounds on the competitiveness of online bidding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' SEARCH(n, k, H) game: Given an array, A[0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , n − 1] whose elements are an unknown permutation of {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , n − 1}, the objective is to use k queries, at most H of which can be erroneous, so as to output an index of the array whose element is as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Theorem 4 (Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For any questioner’s strategy for the SEARCH(n, k, H) game, there is a respon- der’s strategy such that if e is the element of A that is returned, then A[e] ≥ ⌊n �� k H �� /2k⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 4 3 A warm-up: Online time-series search The online (time series) search problem formulates a simple, yet fundamental setting in decision-making under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In this problem, a player must sell an indivisible asset within a certain time horizon, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', within a certain number of days d, that is unknown to the player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' On each day i, a price pi is revealed, and the player has two choices: either accept the price, and gain a profit pi (at which point the game ends), or reject the price (at which point the game continues to day i + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' If the player has not accepted a price by day d, then it accepts by default the last price pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The competitive ratio of the player’s algorithm is the worst-case ratio, over all price sequences, of the maximum price in the sequence divided by the price accepted by the player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The problem was introduced and studied in [20] that gave a simple, deterministic algorithm that achieves a competitive ratio equal to � M/m, where M, m are upper and lower bounds on the maximum and mini- mum price in the sequence, respectively, and which are assumed to be known to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This bound is optimal for deterministic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Time-series search is a basic paradigm in online financial optimiza- tion, and several variants and generalizations have been studied [18, 30, 39, 17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' see also the survey [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The problem has also been used as a case study for evaluating several performance measures of online algorithms, including measures alternative to competitive analysis [13, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Time-series search was recently studied under the imperfect advice framework in [9], who showed an upper bound of (M/m)22H−k/2 on the competitive ratio with k-bit advice and tolerance H, under the assumption that H ≤ k/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Note that no upper bound is known for H ∈ (k/4, k/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' If the advice is error- free, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', in the advice-complexity model, then a tight bound on the competitive ratio equal to (M/m) 1 2k+1 is due to [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We show the following result, as an application of the FIND(k, H) game discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Consider the online time series search problem, with imperfect advice of size k and tolerance H ≤ k/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' There is an algorithm that uses k comparison queries, and that has competitive ratio at most (M/m) 1 U+1, where U = ⌊2k−H/ �� k−H H �� ⌋, for any H ≤ k/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In contrast, every (deterministic) algorithm based on k subset queries has competitive ratio less than (M/m) 1 L+1, where L = ⌈2k/ �� k H �� ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We first show the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' aU, r be defined such that r = a1 m = a2 a1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' = aU aU−1 = M aU , hence r = (M/m)1/(U+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The algorithm uses k comparison queries so as to find the best reservation price, in the set {ai}U i=1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', the threshold p above which the algorithm will always accept a price in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This follows from Theorem 2, since U ≤ 2k−H/ �� k−H H �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The algorithm then uses p as its reservation price, namely it accepts the first price in the request sequence that is at least as large as p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' From the definition of the set {ai}U i=1, it easily follows that this algorithm has competitive ratio at most r, which completes the proof of the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We now show the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' By way of contradiction, suppose that there is an algorithm A for time- series search with k-bit imperfect advice, and of competitive ratio less than C = (M/m) 1 L+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We will show that A could then be used in the FIND(k, H) game so as to identify, using k queries, an unknown value in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , L + 1}, which is a contradiction to the upper bound of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' To arrive at the contradiction, define a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , aL such that r′ = a1 m = a2 a1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' = aL aL−1 = M aL , hence r′ = (M/m) 1 L+1 = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Consider a game between the online algorithm A and the adversary, in which the request sequences consist of prices in {m, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , aL, M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' More precisely, consider the set of request sequences of the form σi = m, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , ai, m, for all i ∈ [1, L + 1], where aL+1 is defined to be equal to 5 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In σi, A must accept price ai (the last request in the sequence) to be strictly less than C-competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Equivalently, A uses k queries with at most H errors, and finds ai in the set {aj}L+1 j=1 , which contradicts Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1 Comparison of the bounds In order to compare the upper and lower bounds of Theorem 5, we need to be able to evaluate the partial sum of binomial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Since this partial sum does not have a closed form, we will rely on the following useful approximation from [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Let H denote the binary entropy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Then 2NH( m N ) � 8m(1 − m N ) ≤ ��N m �� ≤ 2NH( m N ), for 0 < m < N/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' (1) We will also use the following property of the binary entropy function 4p(1 − p) ≤ H(p) ≤ (4p(1 − p))1/ ln 4, for all p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' (2) We first show that the algorithm of Theorem 5 improves upon the one of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' First, note that [9] assumes that H ≤ k/4, whereas Theorem 5 applies to all H ≤ k/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Furthermore, we improve on the competitive ratio for all values of H and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For this, it suffices to show that �� k−H H �� /2k−H < 22H−k/2, which, from (1) holds if 2(k−H)(H( H k−H )−1) < 22H−k/2, or equivalently (k − H)(H( H k−H ) − 1) < 2H − k/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Let τ be such that τ = H/k (hence τ ≤ 1/2), then the latter is equivalent to showing that H( τ 1−τ ) < 1+2τ 2−2τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Using (2), it suffices to show that (4τ(1 − 2τ) (1 − τ)2 )1/ ln 4 < 1 + 2τ 2 − 2τ , which holds for all τ ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Next, we investigate how close the upper and lower bounds of Theorem 5 are to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Recall that the bounds are of the form (M/m)1/(U+1), and (M/m)1/(L+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Using (1), and ignoring for simplicity the floors and ceilings, we obtain that U ≥ 2k(1−τ)(1−H( τ 1−τ )) and L ≤ � 8kτ(1 − τ)2k(1−H(τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The above inequalities, along with (2) show that the upper and lower bounds are very close to each other, since for any fixed value of τ, we have that U ≥ 2Θ(k) and L ≤ 2Θ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 4 Online bidding Online bidding was introduced in [16] as a canonical problem for formalizing doubling-based strategies in online and offline optimization problems, such as searching for a target on the line, minimum latency, and hierarchical clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In this problem, a player wants to guess a hidden, unknown real value u ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' To this end, the player defines an (infinite) sequence X = (xi) of positive, increasing bids, which is called its strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The cost of discovering the hidden value u using the strategy X, denoted by c(X, u), is defined to be equal to �ju i=1 xi, where ju is such that xju−1 < u ≤ xju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Hence one naturally defines the competitive ratio of the bidder’s strategy X as Cr(X) = supu c(X,u) u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In the standard version of the problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='e, assuming no advice, the doubling strategy xi = 2i achieves optimal competitive ratio equal to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Online bidding was also studied under the untrusted advice model in [7], which gave bounds on the consistency/robustness tradeoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The problem is also related to contract scheduling, studied in [8], see also the discussion in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1 Online bidding with imperfect advice 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1 Upper bound The idea behind the upper bound is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We will consider bidding sequences from a space of 2k geometrically-increasing sequences (see Definition 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In the ideal situation of perfect advice, the k advice bits could be used to identify the best strategy in this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In the presence of advice errors, we will show how to exploit the cyclic structure of this space, in conjunction with our upper bound for the MINCYCLIC game (Theorem 3), so as to find a strategy that is not too far from the optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We first define the space of geometrically-increasing bidding sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For given b > 1, and l ∈ N+ define Xb,l as the set of bidding sequences {X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Xl−1}, in which Xi = (bi+jl)∞ j=0, for all i ∈ [0, l − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' From the definition of Xb,l, it is easy to see that for any potential target u, there is a cyclic permutation π of {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' l − 1} which determines an ordering of the strategies in Xb,l in terms of their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' More precisely, suppose that Xπ(0) is the best sequence that discovers u at least cost, say C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Then Xπ(i) discovers u at cost at most biC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This property can help us show the following upper bound: Theorem 7 (Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' There is a bidding strategy based on k comparison queries of competitive ratio at most 1+U 2k � 1 + 2k 1+U �1+ 1+U 2k , where U = ⌈2H ��k−H H �� ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='2 Lower bound The idea behind the lower bound is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' With k advice bits, the best one can do is choose the best strategy from a set X that consists of at most 2k strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Note that if the advice were error-free, |X| could be as large as 2k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' however, in the presence of errors, the algorithm may choose to narrow |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Our approach combines two ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The first idea uses the abstraction of the SEARCH(n, k, H) game, and the lower bound of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This result will allow us to place a lower bound on the rank of the chosen strategy, where the best strategy has rank 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The second idea is to define a measure that relates how much worse a strategy of rank j in X has to be relative to the best strategy in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We will accomplish this by appealing to the concepts of parallelism and fault tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' More precisely, given integers p, and φ, with φ < p, we define the fault-tolerant parallel bidding problem, denoted by FPB(p, φ), as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The player is allowed to run, in parallel, p bidding strategies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' however, φ of these strategies can be faulty, in that they never discover the target;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', we can think of a fault strategy as one in which the player abruptly stops submitting bids, at some point in time, akin to a “byzantine” failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The cost of discovering a target u is then defined as the minimum cost at which one of the p − φ non-faulty strategies discovers the target, noting that the faults are dictated by an adversary that aims to maximize this cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The competitive ratio is defined accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The next theorem is the main technical result for FPB(p, φ), which gives a lower bound on the competi- tive ratio of any strategy for this problem, as a function of the parameters p, φ and α ¯ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Here, ¯X is defined as the sorted sequence of all bids in the p-parallel strategy X, in non-decreasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Moreover, given a sequence X of positive reals, we define αX to be equal to lim supi→∞ x1/i i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Theorem 8 (Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Every p-parallel strategy X for FPB(p, φ) has competitive ratio Cr(X) ≥ αp+1+φ ¯ X αp ¯ X−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 7 Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We use properties of p-parallel strategies so as to show that any such strategy satisfies Cr(X) ≥ supq �q+φ+1 i=0 ¯xi �q−(p−1) i=q ¯xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We then use Gal’s functional theorem [22] to obtain the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We refer to Appendix for many technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We now show how to obtain a lower bound for the problem by combining the above ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We emphasize a subtle point: unlike error-free advice of size k, where one should always choose the best strategy out of a collection of exactly 2k strategies, it is conceivable that, in the presence of errors, this collection could very well be of size l < 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This is because, as l decreases, so does the effect of errors on the competitive ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In other words, we need to establish the result for all values l ≤ 2k, and not only for l = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For every bidding sequence X and k subset queries in the imperfect advice model, we have Cr(X) ≥ 1 L(1 + L)1+1/L, where L = 2k/ �� k H �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Every bidding strategy will use the query responses so as to select a strategy from a set X = {X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , Xl−1} of candidate sequences, for some l ≤ 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For a given target value u, there is an ordering of the l sequences in X such that Xπ(i) has no worse competitive ratio than Xπ(i+1), namely the permuta- tion orders the sequences in decreasing order of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' From Theorem 4, it follows that the strategy will choose a sequence Xj such that π(j) ≥ ⌊l �� k H �� /2k⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The competitive ratio of the selected sequence is at least the competitive ratio of the l-parallel strategy defined by X, in which up to φl = ⌊l �� k H �� /2k⌋ sequences may be faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' From Theorem 8, Cr(X) ≥ αl+1+φl ¯ X αl¯ X − 1 , with φl = ⌊l �� k H �� /2k⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' (3) We now consider two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Suppose first that l < L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In this case, case φl = 0, and therefore (3) implies that Cr(X) ≥ αl+1 ¯ X /(αl¯ X − 1), which is minimized for α ¯ X = (l + 1)1/l > 1, therefore Cr(X) ≥ 1 l (l + 1)1+1/l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This function is decreasing in l, and since l < L we have Cr(X) ≥ 1 L(1 + L)1+1/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Next, suppose that l ∈ [L, 2k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In this case, (3) gives Cr(X) ≥ αl(1+1/L) ¯ X αl ¯ X−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The above expression is minimized for α ¯ X = (1 + L)1/l, and by substitution we obtain again Cr(X) ≥ 1 L(1 + L)1+1/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='3 Comparison of the bounds In the Appendix we prove that the ratio between the two bounds is approximately log UB LB ≤ � 8kτ(1 − τ)k(1 − τ)(1 − H( τ 1−τ )) 2k(1−τ)(1−H( τ 1−τ )) − k(1 − H(τ)) 2k(1−H(τ)) , where τ = H/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We infer that as k increases, and for any fixed value of τ, the upper and lower bounds become very close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 5 Online fractional knapsack In the online fractional knapsack problem, the request sequence consists of items, where item i has a value vi ∈ R+ and a size si ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The online algorithm has a knapsack of unit capacity, and when considering item i, it can accept irrevocably a fraction fi ∈ (0, 1] of the item, subject to capacity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' More precisely, the algorithm aims to maximize � i (fi · vi) subject to � i (fi · si) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 8 Let di = vi/si denote the density of item i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' While the offline version of the problem admits a simple, optimal solution via a greedy algorithm (that sorts all items by non-decreasing order of density, and accepts items in this order until the knapsack is full), the online version is more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Suppose that di ∈ [L, U], for L, U known to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' [14, 15] gave matching O(log(U/L)) and Ω(log(U/L)) upper and lower bounds on the competitive ratio of the problem, respectively, and [40] showed an optimal bound of ln(U/L) + 1 for deterministic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Online fractional knapsack has applications in sponsored search auctions, and online ad allocation, and has been studied in several other settings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', [3, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In this section, we study this problem in the imperfect advice setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1 Upper bound As in all previous work, we assume that the density of all items is in [L, U] for known values of L and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Let d∗ denote the smallest density of an item included at a positive fraction in the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' That is, the optimal algorithm OPT accepts a fraction 1 of items with density larger than d∗, and fills the remaining space with a fraction of items of density d∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Unfortunately, knowing d∗ (even its exact value) is not sufficient for an online algorithm to be anywhere as efficient as OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For example, an algorithm that accepts a fraction 1 of items of density larger than d∗ has unbounded competitive ratio in sequences that consist only of items of density d∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Similarly, an algorithm that accepts a fraction 1 of items with density at least d∗ has unbounded competitive ratio in sequences in which items of density d∗ appear early in the sequence, and items of greater density later in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' However, if we denote by c∗ ∈ (0, 1) the fraction of the knapsack in the optimal solution that is either empty, or occupied with items of density d∗, then knowing the exact value of both d∗ and c∗ suffices to achieve optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Our approach will then aim to use k comparison queries so as to approximate the values of c∗ and d∗, then use these approximations to choose fractional items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1 Algorithm and analysis We describe the online algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We first define two types of partitions, related to the parameters d∗ and c∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In what concerns d∗, partition the interval [L, U] into s sub-intervals I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , Is such that Ii = [di−1, di), for s that will be specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We also set L = d0, U = ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The values di are defined so that: β = d1 d0 = d2 d1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' = ds ds−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Thus, we have β = (U/L)1/s and di = L · βi, and note that d∗ ∈ Ix for some x ∈ [1, s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In what concerns the parameter c∗, we partition the interval [0, 1] into m sub-intervals I′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , I′ m such that I′ i = [ci−1, ci);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' we have c0 = 0 and cm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The value of m will be determined later;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' the values ci are defined so that c1 = c2 − c1 β = c3 − c2 β = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' = cm − cm−1 β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' It readily follows that for i ≥ 1, we have ci = βm+i−1−βm+i−2 βm−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In particular, c1 = βm−βm−1 βm−1 , and 1 1−c1 = βm−1 βm−1−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Note also that c∗ ∈ I′ y for some y ∈ [1, m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Provided that s · m ≤ ⌊2k−H/ �� k−H H �� ⌋, Theorem 2 shows that the algorithm can use k comparison queries so as to identify both x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Given these values, the algorithm reserves, in its knapsack, a capacity c = cy−1 for items with density in the range Ix = [dx−1, dx), to which we refer as critical items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The algorithm uses the remaining capacity of 1 − c for items of density larger than dx, to which we refer as heavy items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The algorithm accepts a fraction 1 of all critical items, as long as the capacity c reserved for them allows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Similarly, the algorithm accepts a fraction 1 of heavy items and places them in their dedicated space of the knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Given that c∗ ∈ Iy, we have 1 − c > 1 − c∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' that is, the reserved capacity for heavy items is at least equal to the total size of these items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In other words, the algorithm can afford to accept all heavy items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The algorithm rejects all items of density smaller than dx−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 9 Theorem 10 (Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For any H ≤ k/2, the above algorithm has competitive ratio at most min s,m∈N fm(β) where β = (U/L)1/s, and fm(β) = βm − 1 βm−1 − 1 subject to s · m ≤ ⌊2k−H/ ��k − H H �� ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='2 Lower bound We will show a lower bound C(k, H) on the competitive ratio of any algorithm with imperfect advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For the sake of contradiction, suppose there is an algorithm A of competitive ratio better than C(k, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Our proof is based on a reduction from the FIND(k, H) game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Specifically, we prove that, based on A, we obtain a questioner’s strategy for FIND(k, H) which can find a value z ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , p}, with p = ⌈2k/ �� k H �� ⌉ + 1, which contradicts Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We give the intuition behind the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Let s and m be any two positive integers such that s · m ≤ p and s · (m + 1) > p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Define β = (U/L)1/s, and di = U · βi, for i ∈ [1, s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Given a pair (x, y) of integers, where x ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , s} and y ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' m + 1}, define the sequence σx,y = ((d1, 1), (d2, 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , (dx−1, 1), (dx, cy), where (di, j) indicates a subsequence of j/ǫ items, each of which has size ǫ and density di, and where ǫ is infinitesimally small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' cy ∈ [0, 1] is defined appropriately in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For this sequence, OPT(σx,y) = (1−cy)dx−1 +cydx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' There are s·(m+1) > p such sequences, and σx,y is a prefix sequence of σx,y+1, and σx,m is a prefix sequence of σx+1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In the proof, we consider request sequences of this form, and we show that if A is C(k, H)-competitive, its decisions can help find any given z ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , p}, which contradicts Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We refer to Appendix for the technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Theorem 11 (Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For the fractional knapsack problem, where items densities are in [L, U], no deterministic algorithm with k subset queries, out of which H ≤ k/2 may have erroneous responses, can achieve a competitive ratio better than C(k, H) = min s,m∈N gm(β) where β = (U/L)1/s, gm(β) = (β2 − β + 1 2β + 1 )1/(m+1) subject to s · m ≤ ⌈2k/ �� k H �� ⌉ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Comparison of the bounds Let τ = H/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Since βm−1 βm−1−1 ≤ β, using (1), the upper bound of Theorem 10 is at most (U/L)q, where q ≤ 1/2k(1−τ)(1−H( τ 1−τ )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Furthermore, since β2−β+1 2β+1 ≥ β 3 (for all β ≥ 3), the lower bound of Theorem 11 is at least (U/L)q′(1/3)q′, where q′ ≥ 1/(2 � 8kτ(1 − τ)2k(1−H(τ)) + 1), for all U/L ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For simplicity, we omitted the floors and ceilings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 6 Waiving the assumption of the tolerance parameter In the imperfect advice setting we studied so far, the algorithm defines an application-specific tolerance parameter that measures its desired tolerance to errors (or equivalently, an anticipated upper bound on the 10 error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This parameter is in a sense required, since the analysis of R´enyi-Ulam games in [36] involves the extreme value of error (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', H) instead of the instance-specific error value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Nevertheless, in this section, we discuss how to mitigate the need for pre-determining a tolerance parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We propose two different approaches, based on resource-augmentation, and robustification, which we discuss in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We use the time-series search and online bidding problems as illustration, even though our approach may carry through in other online problems, at the expense of more complex calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1 Resource augmentation In this setting, we compare an oblivious online algorithm A with l advice bits and no information on the error bound, to an online algorithm B that has k ideal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' error-free) advice bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Specifically, we are interested in finding the smallest l ≥ k (as function of k) for which algorithm A is at least as good as algorithm B, regardless of the error in the advice of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The following theorem shows that O(1)-factor resource augmentation suffices to obtain an oblivious algorithm that is at least as efficient as any algorithm that operates in the ideal setting of error-free advice, and even if a fraction 1/3 − c of the advice bits may be erroneous, for any constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Theorem 12 (Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Consider the time-series and the online bidding problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For all sufficiently large k, and any c ∈ (0, 1/3), there is an oblivious online algorithm A with advice of size l, whose compet- itive ratio is at least as good as that of any online algorithm B with k bits of perfect (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' error-free) advice, where l = 1 ( 2 3+c)(1−H( 1 3 −c 2 3 +c )) k + 1, for any error η ≤ (1/3 − c)l in the advice of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='2 Robustification In this setting, we augment the imperfect advice framework by requiring not only that the algorithm min- imizes the competitive ratio assuming that the advice error is at most the tolerance H, but also that its competitive ratio does not exceed a robustness requirement r, for some specified r, if the error exceeds H (and in particular, if the advice is adversarially generated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We call such online algorithms r-robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Thus, this model can be seen as an extension of both the imperfect advice and the untrusted advice model of [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For the time-series problem, we obtain the following result, which generalizes Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In particular, note that Theorem 5 is a special case of Theorem 13 for ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Theorem 13 (Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Consider the online time series search problem, with imperfect advice of size k, tolerance H ≤ k/2, and robustness r = (M/m)ρ, where ρ ∈ (1/2, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' There is an r-robust algorithm that uses k comparison queries, and has competitive ratio at most (M/m) 2ρ−1 U+1 , where U = ⌊2k−H/ �� k−H H �� ⌋, for any H ≤ k/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Moreover, every (deterministic) algorithm based on k subset queries has competitive ratio better than (M/m) 2ρ−1 L+1 , where L = ⌈2k/ �� k−H H �� ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The analysis of r-robust algorithms for online bidding is more challenging, in particular in what concerns the impossibility results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We give an overview of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For the upper bound, we can follow an analysis along the lines of Theorem 7, however, each bidding sequence in the collection Xb,2k must be individually r-robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This is easy to enforce, and it requires that b much be such that b2/(b − 1) ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The lower bound is more subtle: the proof follows the lines of Theorem 9, but uses the fact that if all the l sequences in X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , Xl−1 must be r-robust, then α2¯ X/(α ¯ X − 1) ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We obtain the following: 11 Theorem 14 (Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For every r ≥ 4 there is an r-robust bidding strategy with k-bit imperfect advice that has competitive ratio at most min b>1 b2k+U+1 b2k − 1 , subject to b2k+1/(b2k − 1) ≤ r, and where U = ⌈2H ��k − H H �� ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Furthermore, every r-robust bidding strategy with k-bit imperfect advice has competitive ratio at least min α>1 α2k+L+1 α2k − 1 subject to α2k/(αk − 1) ≤ r, and where L = ⌊ �� k H �� ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' References [1] Repository of works on algorithms with predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 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Xu, Wenming Zhang, and Feifeng Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Optimal algorithms for the online time series search problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Theoretical Computer Science, 412(3):192–197, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' [40] Yunhong Zhou, Deeparnab Chakrabarty, and Rajan Lukose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Budget constrained bidding in keyword auctions and online knapsack problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In Proceedings of the International Workshop on Internet and Network Economics (WINE), pages 566–576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Springer, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' A Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1 Omitted material of Section 2 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We will reduce MINCYCLIC(n, k, H) to the following game that was studied in [36]: 14 IDENTIFY(m, H) game: In this game, x is an integer in {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , m − 1} for some known m, and the objective is to identify x with as few queries as possible, if up to H queries may be answered incorrectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We will use the following result in the analysis: Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' [36] The number Q(m, H) of queries required to identify x in an instance of IDENTIFY(m, H) is such that min{k′|2k′ ≥ m · �� k′ H �� } ≤ Q(m, H) ≤ min{k|2k−H ≥ m ��k − H H �� }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Given an instance of MINCYCLIC(n, k, H), we create an instance of IDENTIFY(m, H) , with m = 2k−H/ ��k−H H �� (note that H is the same for both instances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Partition the interval [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , n − 1] into m disjoint subintervals, each of length at most ⌈n/m⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Let x being the index in [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , n − 1] for which A[x] = 0, and let Ix denote the interval that contains x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' A query q of the WEIGHTING strategy of [36] that asks “is Ix ≤ b for some b ∈ {0, m − 1}?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' translates to query µ(x) in the MINCYCLIC(n, k, H) instance that asks “is x ≤ f(b)?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', where f(b) is the largest value in the interval Ib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Note that the answer to q is ‘yes’ if and only if the answer to µ(q) is ‘yes’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The response to µ(q) is then given to WEIGHTING, which updates its state and proceeds with the next query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For m defined as above, we have that 2k−H ≥ m · ��k−H H �� and using WEIGHTING, we can find Ix using k queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Subsequently, we return the largest integer f(x) in Ix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Given that x is in Ix and the length of the intervals is at least ⌈n/m⌉, we conclude that the returned index j is such that A[j] =≤ ⌈n/m⌉ = ⌈n ��k−H H �� /2k−H⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Consider the following game that is defined as the CONTINIOUSSEARCH(k, H) game, with the only difference that the goal is to guess a value as close to some r ∈ [0, n), for some fixed n (for the purpose of the proof, we can think of n as sufficiently large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' After receiving the responses to the k queries, the questioner returns a number r′ ∈ [0, n), and the objective is to minimize |r′ − r|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We will show a reduction from this game that will help us establish the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Suppose, by way of contradiction, that there exists a strategy, say ALG for SEARCH(n, k, H) that returns an element e with π(e) < ⌊n �� k H �� /2k⌋ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We devise a strategy for the questioner in the continuous game based on ALG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Given values of r, k and H that define an instance G of the continuous game over the continuous interval I = [0, n), create an instance S of SEARCH(n, k, H) on a space A of n elements, with the same values of k and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Consider a bijective mapping β that maps an element of rank i in A (i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , n − 1}) to an interval β(i) = [i, i+1) in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Similarly, define a bijective mapping µ between queries asked for G and those asked for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Any range [i, j] of indices that is a part of a subset query q asked for S is mapped to an interval [i, j + 1) in the query µ(q) asked for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Let r denote the searched value in G and let x denote an index of A such that r belongs to β(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' To search for r, we consider queries that ALG asks for S and for any such query q, we ask µ(q) for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The response to µ(q) is then given to ALG so that it can update its state and ask its next query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Recall that we supposed that ALG outputs an element e such that π(e) < ⌊n �� k H �� /2k⌋ − 1, and that β(π(e)) = [π(e), π(e) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' As an output for G, we return r′ = π(e) + 1 as the answer for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Note that there are exactly e + 1 intervals from the range of β that lie between r and r′ in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' That is |r − r′| < ⌊n �� k H �� /2k − 1⌋ + 1 ≤ n( �� k H �� /2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This, however, contradicts Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='2 Omitted material of Section 4 Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We apply the algorithm of Theorem 3 on the set of indices of all sequences in Xb,2k with n = 2k, where b > 1 will be chosen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The output is the index of a strategy in Xb,2k which is 15 ranked at most U among the sequences in Xb,2k From the definition of Xb,2k, and in particular its cyclic property, this means that the selected strategy discovers the target with cost at most bU times larger than the best strategy in Xb,2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We infer that the competitive ratio of the chosen strategy is at most b2k+1+U b2k −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This expression is minimized for b = (2k+U+1 U+1 )1/2k, from which we obtain that the competitive ratio of our strategy is at most 1 + U 2k � 1 + 2k 1 + U �1+ 1+U 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Consider a p-parallel strategy X, defined by p bidding strategies X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , Xp−1, each run on a dedicated processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Let xj,i denote bid i in Xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' we say that xj,i precedes bid xj,i′ in j if i < i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We define the prefix cost of a bid in Xj as the sum of the values of all bids that precede that bid in Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For j ∈ [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , p−1], we denote by uX(c, j) as the value of the largest bid in Xj, such that the sum of the prefix cost of that bid and the value of that bid do not exceed c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We also define by uX,φ(c) at the (φ + 1)-largest quantity in the set {uX(c, j)}p−1 j=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Consider an arbitrary indexing of all bids in X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', the i-th bid is such that it is the m-th bid in Xj, for some m, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We will represent this bid as a pair of the form (Ci, Di), where Ci is the cost of all bids that precede bid i in the sequence to which it belongs, and Di is the bid itself (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', its value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Note that this representation ignores the specific sequence to which the bid is assigned, since this is not important for the purposes of the proof, as we will see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Given a bid represented as (Ci, Di) we define di to be equal to uX,φ(Ci + Di): we call this value the (φ + 1)-largest bid relative to Di, in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Recall that ¯X denotes the sequence of all bid values in X, in non-decreasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Hence, each bid in X is mapped via its length to an element of this sequence (breaking ties arbitrarily).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Fix a bid i0 of the form bi0 = (Ci0, Di0), and suppose, without loss of generality, that bi0 belongs to sequence X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Let c = Ci0 + Di0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For all m ∈ [1, p − 1], let bim = (Cim, Dim) denote the largest bid in Xm for which the sum of its prefix cost and the value of its bid are at most c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For every m ∈ [0, p − 1], define Im as the set of indices in N such that i ∈ Im if and only if a bid of value xi is such that its prefix cost plus xi does not exceed c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' From the definition of the competitive ratio we have that Cr(X) ≥ � i∈Im xi dim , for all m ∈ [0, p − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Therefore, Cr(X) ≥ max 0≤m≤p−1 � i∈Im xi dim , and using the property max{a/b, c/d} ≥ a+b c+d, for all a, b, c, d > 0, we obtain that Cr(X) ≥ �p−1 m=0 � i∈Im xi �p−1 m=0 dim .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' (4) Next, we will bound the numerator of the fraction in (4) from below, and its denominator from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We begin with a useful observation: we can assume, without loss of generality, that for cost c (defined earlier), no bid of value di0 or smaller has prefix cost larger than c minus the value of the bid in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This follows from the definition of di0: if such a bid existed, then one could simply “remove” this bid from X, 16 and obtain a p-parallel sequence of no worse competitive ratio (in other words, such a bid is useless, and one can derive a sequence of no larger competitive ratio than X that does not contain it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Using the above observation, it follows that the numerator in (4) includes, as summands, all bids of value at most di0, as well as at least φ + 1 bids that are at least as large as di0 (φ of those bids are from the definition of di0, and the additional one is bid bi0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Let q denote an index such that di0 = ¯xq, then we have that p−1 � m=0 � i∈Im xi ≥ q+φ+1 � i=0 ¯xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We now show how to upper-bound the denominator, using the monotonicity implied in the definition of the (φ + 1)-largest value relative to a given bid value, and the definition of the bids bi0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , bip−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Specifically, for every bid bim, with m ∈ [1, p − 1], we have that dim ≤ di0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' It thus follows that p−1 � m=0 dim ≤ q−(p−1) � i=q ¯xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Combining the two bounds, it follows that Cr(X) ≥ sup 0≤q<∞ �q+φ+1 i=0 ¯xi �q−(p−1) i=q ¯xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In the last step of the proof, we will use a result from search theory, namely Gal’s functional theorem, stated below: Theorem 16 (Gal [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Let q be a positive integer, and X = (xi)∞ i=0 a sequence of positive numbers with supn≥0 xn+1/xn < ∞ and αX > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Suppose that Fi is a sequence of functionals that satisfy the following properties: (1) Fi(X) depends only on x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' xi+q, (2) Fi(X) is continuous in every variable, for all positive sequences X, (3) Fi(aX) = Fi(X), for all a > 0, (4) Fi(X + Y ) ≤ max(Fi(X), Fi(Y )), for all positive sequences X, Y , and (5) Fi+j(X) ≥ Fi(X+j), for all j ≥ 1, where X+j = (xj, xj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Then sup 0≤k<∞ Fk(X) ≥ sup 0≤k<∞ Fk(GαX), where Ga is defined as the geometric sequence (ai)∞ i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Define now the functional Fq( ¯X) = �q+φ+1 i=0 ¯xi �q−(p−1) i=q ¯xi , for every q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The functional satisfies the conditions (1)-(5) of Theorem 16 (see Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='3 in [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' By applying Gal’s Theorem, it follows that Cr(X) ≥ sup 0≤q<∞ �q+φ+1 i=0 αi¯ X �q−(p−1) i=q αi¯ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' If α ¯ X ≤ 1, then it is easy to show that the above expression shows that Cr(X) = ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Otherwise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=', if α ¯ X > 1, after some simple calculations we arrive at the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 17 Comparison between the upper and the lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We compare the upper and lower bounds of Sec- tions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Define f as the function f(x) = 1 x(1 + x)1+1/x, and note that f is decreasing in x, with f(1) = 4, and limx→∞ f(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Then, the upper bound of Theorem 7 is equal to f(2k/(U + 1)), whereas the lower bound of Theorem 9 is equal to f(L), where U, L are defined in the statements of the corresponding theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For every y > x we have f(y) f(x) = 1 y(1 + y)1+1/y 1 x(1 + x)1+1/x ≤ (1 + y)1/y (1 + x)1/x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Moreover, using some more elementary calculus, log f(y) f(x) ≤ log (1 + y)1/y (1 + x)1/x ≤ log y1/y x1/x = 1 y log y − 1 x log x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We will use the above inequality to compare f(2k/(U + 1)) to f(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' To simplify the calculations, we will assume that the upper bound is f(2k/U), since the additive “one” in the numerator has virtually no effect on the competitive ratio as k becomes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For the same reasons, we ignore the ceiling in the expression of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Using the approximation of the partial sum of binomial coefficients of (1), and defining τ = k/H, we obtain that the ratio UB/LB of the upper and lower bounds satisfies log UB LB ≤ � 8kτ(1 − τ)k(1 − τ)(1 − H( τ 1−τ )) 2k(1−τ)(1−H( τ 1−τ )) − k(1 − H(τ)) 2k(1−H(τ)) , A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='3 Omitted material of Section 5 Proof of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Note that a fraction 1 of heavy items is accepted by both the online algorithm and OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Therefore, the contribution of heavy items to the profits of the algorithm and OPT are the same, say ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' we have δ ≥ (1 − c∗)dx ≥ (1 − cy)dx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' this is because c∗ ∈ [cy−1, cy) and all heavy items have density larger than dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The algorithm fills the reserved space of size c with critical items, which are of density at least dx−1, while OPT fills a space of c∗ < cy with critical items, which are of density at most dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Thus, we have Cr(A) ≤ ∆ + cy · dx ∆ + cy−1 · dx−1 ≤ (1 − cy)dx + cy · dx (1 − cy)dx + cy−1 · dx−1 = 1 1 − (cy − cy−1/β) = 1 1 − c1 = βm − 1 βm−1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Proof of Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' By way of contradiction, suppose there is an algorithm A with a better competitive ratio than C(k, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We will show that A could then be used in the FIND(k, H) game with k queries so as to identify an unknown value in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , p}, which contradicts the upper bound of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Fix the values of (s, m′) that minimize gm(β) subject to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='m′ ≤ p, and let m = m′+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Define p = s·m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' we have p > ⌈2k/ �� k H �� ⌉, otherwise, the pair (s, m) results in a smaller value for gm(β) (note that gm(β) 18 is a decreasing function of m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Suppose we want to identify an unknown value z ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , s · m};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' this is equivalent to finding a pair (x, y) with x ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' s} and y ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , s}, let di = L·βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Moreover, for j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , m}, define cj such that the following hold: c0 = min{1/β, 1 − 1/β}, cm = max{1/β, 1 − 1/β}, r = c1 + β(1 − c1) c0 + β(1 − c0) = c2 + β(1 − c2) c1 + β(1 − c1) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' = ci+1 + β(1 − ci+1) ci + β(1 − ci) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' =cm−1 + β(1 − cm−1) cm + β(1 − cm) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' It can be verified that r = (β2−β+1 2β+1 )1/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For any x ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , s} and y ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , m}, create an input sequence as σx,y = (1, d0), (1, d1), (1, d2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , (1, dx−1), (cy, dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Here (a, b) indicates a sequence of items, all of infinitesimal small size ǫ and density b, and total size a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The optimal solution fills a capacity cy with the item of density dx, and the remaining capacity of 1 − cy with the item of density dx+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We have OPT(σx,y) = (1 − cy) · dx−1 + cy · dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In what follows, we describe how an algorithm A with a competitive ratio better than C(k, H) can be used to correctly find unknown values x ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' s} and y ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Suppose the next item has density dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Let wα denote the total size of items of density ≤ dα−1, and suppose wα ∈ [1 − cq, 1 − cq−1) for some q ∈ {[1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , m}, that is, 1 − wα ∈ (cq−1, cq].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Define ∆α = cq+1 − cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' When the empty space in the knapsack of A becomes less than ∆α, the algorithm “guesses” x = α and y = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In what follows, we show that A makes these guesses at some point and the guesses made by A are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For the sake of contradiction, suppose A does not make a guess, or at least one of its guesses is incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We show that the competitive ratio of A will be larger than C(k, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Suppose wx ∈ [1 − cq, 1 − cq−1) for some q ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , m}, that is, 1 − w ∈ (cq−1, cq].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' There are four possibilities to consider: A does not make a guess: Since A does not make a guess, the empty space in the knapsack is at least ∆x = cq+1 − cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Also, the total size of items of density ≤ dx−1 is at least 1 − cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Therefore, the contribution of items of density dx to the value of the knapsack is at most cq · dx, and the contribution of other items is at most (1 − cq − ∆x)dx−1 = (1 − cq+1)dx−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Therefore, the total value of value of the knapsack of A is at most (1 − cq+1)dx−1 + cqdx ≤ (1 − cy+1)dx−1 + cydx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We can write: Cr(A) ≥ (1 − cy)dx−1 + cydx (1 − cy+1)dx−1 + cydx > r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Wrong guess for y: Suppose the algorithm stops but makes the wrong guess for y, that is, q ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' First, suppose q ≤ y − 1, that is, A reserves too little space for items of density dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The total size of items of density ≤ dx−1 is at least 1 − cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Therefore, the total size of items of density dx is at most cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The final value of the knapsack is thus at most (1 − cq)dx−1 + cqdx ≤ (1 − cy−1)dx−1 + cy−1dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We can write: Cr(A) ≥ (1 − cy)dx−1 + cydx (1 − cy−1)dx−1 + cy−1dx = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Next, suppose p ≥ y + 1, that is, too much space is reserved for items of density dx and some of this space stays empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The total size of items of density at most dx−1 in the knapsack will be 19 at most 1 − cq+1, and the total size of items of density dx is at most 1 − cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Therefore, we have A(σx,y) ≤ (1 − cq+1)dx−1 + cqdx ≤ (1 − cy+1)dx−1 + cydx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We can write Cr(A) ≥ (1 − cy)dx−1 + cydx (1 − cy+1)dx−1 + cydx > r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Wrong guess for x: Suppose p = y but α < x (note that α cannot be larger than x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The value of the knapsack is maximized when the algorithm fills a capacity cy with items of density α and the rest with items of density dα−1, that is, A(σx,y) ≤ (1 − cy)dα−1 + βcydα ≤ (1 − cy)dx−2 + βcydx−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We can write Cr(A) ≥ (1 − cy)dx−1 + cydx (1 − cy)dx−2 + βcydx−1 = β > r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' To summarize, as long as Cr(A) < gm(β), one can use A to guess both values of x and y correctly, that is, it can identify z ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , p} with k queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' This, however, contradicts the lower bound of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Therefore, the initial assumption about the competitive ratio of A does not hold, and we conclude that Cr(A) ≥ gm(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='4 Omitted material of Section 6 Proof of Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Consider the algorithms (upper bounds) of Theorems 5 and 7, with l imperfect advice bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The advice error η is, from the assumption at most (1/3 − c)l, thus at most a fraction equal to 1/3 − c of the advice bits may be erroneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Then, from the discussion in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content='3, it follows that these two algorithms have better competitive ratio (for the corresponding problem) than any algorithm with k bits of advice (irrespectively of the latter’s advice error), as long as l(1 − (1/3 − c))(1 − H( 1/3 − c 1 − (1/3 − c)) > k, for all sufficiently large k, which yields the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Proof of Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Define p1 and p2 to be such that M/p1 = (M/m)ρ, and p2/m = (M/m)ρ, re- spectively, and note that m ≤ p1 ≤ p2 ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Then an algorithm for time-series search is r-robust if and only if it sets its reservation price equal to some p ∈ [p1, p2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The proof proceeds along the lines of the proof of Theorem 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' instead of using the queries to find a suitable reservation price in [m, M], we search instead for a reservation price in [p1, p2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' In particular, note that by definition of p1, p2, we have that p2/p1 = (M/m)2ρ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Proof of Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For the upper bound, the proof is similar to that of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' The only difference is that b must be optimized under the condition that each strategy in Xb,2k must be individually r-robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' We know that a geometric strategy for online bidding with base b has competitive ratio at most b2/(b−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Since each strategy is geometric with base b2k, it follows that as a long as b2k+1/(b2k − 1) ≤ r, every strategy in Xb,2k is r-robust, hence the strategy chosen by the imperfect advice as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' For the lower bound, we appeal to the following property shown in [6]: If all l bidding sequences in the collection X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' , Xl−1 are r-robust, then it must be that α2¯ X/(α ¯ X − 1) ≤ r, where recall that ¯X is the sequence of the union of all bids in the l strategies, sorted in non-decreasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' Then the proof follows along the lines of the proof of Theorem 9, with the observation that the competitive ratio is minimized if l = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} +page_content=' 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAzT4oBgHgl3EQfqf3Z/content/2301.01631v1.pdf'} diff --git a/oNE5T4oBgHgl3EQfkA8i/content/tmp_files/2301.05659v1.pdf.txt b/oNE5T4oBgHgl3EQfkA8i/content/tmp_files/2301.05659v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d2cbb617d7909c605a4f3ccb0ef43af2b9424bd0 --- /dev/null +++ b/oNE5T4oBgHgl3EQfkA8i/content/tmp_files/2301.05659v1.pdf.txt @@ -0,0 +1,1031 @@ +From stage to page: language independent +bootstrap measures of distinctiveness in +fictional speech +Artjoms ˇSel¸a +Institute of Polish Language, +Polish Academy of Sciences +and University of Tartu +artjoms.sela@ijp.pan.pl +Ben Nagy +Institute of Polish Language, +Polish Academy of Sciences +benjamin.nagy@ijp.pan.pl +Joanna Byszuk +Institute of Polish Language, +Polish Academy of Sciences +joanna.byszuk@ijp.pan.pl +Laura Hern´andez-Lorenzo +University of Seville +lhernandez1@us.es +Botond Szemes +Research Centre for the Humanities +Institute for Literary Studies Budapest +szemes.botond@abtk.hu +Maciej Eder +Institute of Polish Language, +Polish Academy of Sciences +maciej.eder@ijp.pan.pl +Abstract +Stylometry is mostly applied to authorial style. More recently, +researchers have begun investigating the style of characters, +finding that, although there is detectable stylistic variation, +the variation remains within authorial bounds. In this article, +we address the stylistic distinctiveness of characters in drama. +Our primary contribution is methodological; we introduce and +evaluate two non-parametric methods to produce a summary +statistic for character distinctiveness that can be usefully ap- +plied and compared across languages and times. This is a sig- +nificant advance—previous approaches have either been based +on pairwise similarities (which cannot be easily compared) or +indirect methods that attempt to infer distinctiveness using +classification accuracy. +Our first method is based on boot- +strap distances between 3-gram probability distributions, the +second (reminiscent of ‘unmasking’ techniques) on word key- +ness curves. Both methods are validated and explored by ap- +plying them to a reasonably large corpus (a subset of DraCor): +1 +arXiv:2301.05659v1 [cs.CL] 13 Jan 2023 + +we analyse 3301 characters drawn from 2324 works, covering +five centuries and four languages (French, German, Russian, +and the works of Shakespeare). +Both methods appear use- +ful; the 3-gram method is statistically more powerful but the +word keyness method offers rich interpretability. Both meth- +ods are able to capture phonological differences such as accent +or dialect, as well as broad differences in topic and lexical +richness. Based on exploratory analysis, we find that smaller +characters tend to be more distinctive, and that women are +cross-linguistically more distinctive than men, with this latter +finding carefully interrogated using multiple regression. This +greater distinctiveness stems from a historical tendency for +female characters to be restricted to an ‘internal narrative do- +main’ covering mainly direct discourse and family/romantic +themes. It is hoped that direct, comparable statistical mea- +sures will form a basis for more sophisticated future studies, +and advances in theory. +1. Introduction +Since Vladimir Propp’s work, structural narratology has approached fictional characters +mainly through their role or function—by what they do, or what is done to them (Eder, +Jannidis, and Schneider 2010). This character typology relied on recurring functions +in the narrative (lover, villain, victim, detective, etc.) and the same perspective was +often adopted in computational research, where characters in novels were modelled on +the basis of narrative passages rather than dialogue (Bamman, Underwood, and Smith +2014; Bonch-Osmolovskaya and Skorinkin 2017; Underwood, Bamman, and Lee 2018; +Stammbach, Antoniak, and Ash 2022). +In dramatic texts, however, the dominant device for characterisation is an utterance. +While the script usually contains some stage directions, the specifics of characterisation +and style of performance are not determined by the text itself, but developed by a spe- +cific theatre, director or a troupe. Over the course of history, many plays were written +for specific theatre stages, and it was common practice to write characters for specific +actors (Fischer-Lichte 2002). Of course, this kind of ‘outsourced characterisation’ was +supported by dramatic conventions and formulas. Viewers’ expectations could be shaped +without a single word being uttered on stage, just by a character wearing a costume, +operating a puppet, or changing a dell’arte stock mask. At the same time, the things +characters say and how they say them are the main textual source of information about +them. It is reasonable to assume that dramatists make significant efforts to create lin- +guistic distinctions between princes and paupers, lovers and schemers, aristocrats and +merchants. Tragic monologue is written differently to a comedic exchange between ser- +vants. Some previous computational works treat linguistic distinctiveness of characters +from the perspective of this stylistic continuum (Vishnubhotla, Hammond, and Hirst +2019), noting that it can be influenced by genre, character gender, or their social and +professional dispositions. +2 + +A parallel narratological tradition, tied to Bakhtin’s ideas of heteroglossia, focuses not +on abstract character roles, but on the words characters say (Bronwen 2012; Culpeper +2001; Sternberg 1982). +The modern novelistic space of dialogic exchange, ‘educated +conversation’ (Moretti 2011) and the clash of styles in reported discourse become central +here. +Available stylometric research on fictional speech and micro-stylistic variation +suggests that characters within a text are often distinguishable by their local linguistic +patterns without obscuring the global authorial trace (Burrows 1987; Hoover 2017). +As put by Burrows and Craig: ‘Characters speak in measurably different ways, but +the authorial contrasts transcend this differentiation. The diversity of styles within an +author always remains within bounds’ (Burrows and Craig 2012, 307–8). +Conceptually and methodologically, the majority of previous works examined not the +distinctiveness of characters, but their (pairwise) similarity. Similarity measures are +meaningful in pairwise contexts, but cannot be analysed and compared as individual +summary statistics. Since Burrows’ seminal study of speech patterns in Jane Austen’s +characters (Burrows 1987), these approaches focused on calculating similarity within +a collection of characters: how different is character X from character Y, and each of +them from character Z. Burrows measured the correlation between characters’ usage of 30 +most frequent words (technically, he fit a linear regression for two sets of log-frequencies); +later, similarity was most often inferred through clustering based on pairwise distance +calculations (Hoover 2017; Reeve 2015; Craig and Greatley-Hirsch 2017). Sometimes +linguistic similarity served as a basis for arguing functional similarity, as well. A recent +study that linked Bakhtin’s dialogism and the stylistic diversity of characters’ speech +(Vishnubhotla, Hammond, and Hirst 2019), proposed the analysis of distinctiveness +rather than similarity using supervised classification. +Instead of using a network of +pairwise relationships, the authors asked how well a classifier can recognise character +X as being written by author A. Classification accuracy in this scenario becomes an +explicit summary statistic for distinctiveness that can be assigned to a character (or, +in an aggregated manner, to a play or an author). However, the supervised approach, +proposed by Vishnubhotla et al., is data hungry: it suffers from extreme class imbalance, +an abundance of short samples (most characters speak only a little) and is dependent +on language-specific feature construction procedures. +By contrast, this paper will present a simple, non-parametric measure of character dis- +tinctiveness that is based on bootstrapped probability distributions representing a char- +acter and all others present in a given play: an approach largely informed by authorship +verification techniques. This measure is language-independent and relies only on the +context of a single work, which, in turn, minimises problems of language variation, +authorial signal and chronological change in a comparative setting. Individual distinc- +tiveness scores can be then tested against other measures and metadata categories in a +hypothesis-driven manner, not only across languages, but also across genres (e.g. novel +vs. drama). Do comedies tend to employ more distinct characters? Does distinctiveness +increase (authors get better), or decrease (social and linguistic homogenisation occurs) +over time? Is there a difference between the distinctiveness of fictional women and men? +If so is it the direct result of perceived gender differences, or is it constructed by imagined +3 + +differences in social and professional status? +Lacking good descriptive metadata on the dramatic characters, this paper will not answer +above mentioned questions in any satisfying way. Instead we focus on presenting and +justifying the measure of distinctiveness and exploring several factors that might shape +the final scores (like the year of composition, character gender and characters’ sample +size). +2. Materials +Total +Characters +Unique +Unique +Total +Total +Corpus +Characters +Analysed +3-grams +Words +3-grams +Words +French +15462 +1744 +9896 +79994 +29.79 m +5.47 m +German +14010 +1182 +14341 +150956 +24.80 m +4.31 m +Russian +3707 +248 +12542 +71217 +4.05 m +0.72 m +Shakespeare +1431 +127 +5921 +19595 +2.16 m +0.43 m +Table 1.: A summary of the corpus. All word and 3-gram counts are for the filtered +corpus (characters that speak at least 2000 words) only. +As the beginning of our exploration of cross-linguistic variation, we examined four dra- +matic corpora from DraCor (Fischer et al. 2019): Shakespeare, French, German, and +Russian. DraCor is a project that gathers dramatic corpora in various languages, pri- +marily European, encoded in TEI-XML. With 15 corpora available so far, including the +Shakespeare corpus available both in English and German, DraCor facilitates large scale +analysis of dramatic conventions across language traditions, and offers a wide variety of +useful metadata, at the level of both plays and characters. While the analysis of all Dra- +Cor corpora would be possible with the methods we developed, for the purpose of this +preliminary study we focused on the languages and dramatic traditions well known to +the members of our team, eventually selecting the full corpora for Shakespeare, French, +German, and Russian: a total of 2324 texts, the majority of which come from French +and German. The corpus is summarised in Table 1. +3. Methods +3.1. General Approach and Definitions +Our understanding of character distinctiveness is largely informed by ‘authorship ver- +ification’ approaches, which centre around verifying that a text is written by a target +author. This problem is more general than ‘authorship attribution’ that tries to identify +the nearest stylistic neighbour for a text (Halvani, Winter, and Graner 2019). Instead, +authorship verification asks about the relative magnitude of similarity: is a target text +more similar to same-author samples, or different-author samples? With this in mind, +4 + +Figure 1.: Character distinctiveness, per corpus, versus % Dialogue. Women are shown +smaller, in orange, men (and undefined) larger and in blue. GAM (Generalised +Additive Model) trendlines are superimposed in the same colours. Baseline +data (GAM trend for distinctiveness of character vs self) is shown as a dashed +line. +we define a character’s ‘distinctiveness’ as the degree to which the style of their speech +differs from that of other characters. We understand ‘style’ here instrumentally, as a +deviation from an unobserved average language (Herrmann, Dalen-Oskam, and Sch¨och +2015) and do not introduce aggressive feature filtering, allowing both ‘grammatical’ and +‘thematic’ signal to contribute to the final measures. We anchor our distinctiveness mea- +sure in the context of the specific text in which a character appears. In theory, the frame +of reference could be all plays from one author, or all plays from the same period, or even +5 + +Distinctiveness vs Percent Dialogue +French +German +0.20 - +0.15 - +0.10 +rence) +differe +0.05 +3 +Russian +Shakespeare +istinctiveness +0.20 - +三 +Character +0.15 - +0.10 - +0.05 +20 +40 +60 +20 +40 +60 +Proportion of total dialogue (%)Figure 2.: Character distinctiveness, per corpus, versus year composed (DraCor data). +Women are shown smaller, in orange, men (and undefined) larger and in blue. +GAM (Generalised Additive Model) trendlines are superimposed in the same +colours. Baseline data (GAM trend for distinctiveness of character vs self) is +shown as a dashed line. +some external corpus—however all of these would greatly complicate any comparative +study. +3.2. Bootstrap 3-gram Distinctiveness +Based on our definition of distinctiveness above, we considered a character’s style to be +an idiolect sampled from a frequency distribution of character 3-grams. As a natural +6 + +French +German +0.20 +0.15 +0.10 +difference) +0.05 + distribution +1600 +1700 +1800 +1900 +1800 +Russian +Shakespeare +isti +0.20 +Character +0.15 +0.10 +0.05 +750 +1850 +1900 +1595 +Normalized Year Composed(a) % Dialogue +(b) Distinctiveness +(c) Vocab size (3-grams) +Figure 3.: An analysis, per corpus, of the distribution of various features by gender. +Distributions are estimated, with the median shown as a solid line. Actual +points are shown as rug plots with outliers ‘o’ plotted for points outside 3Q ++ 2×IQR. +language distribution, this was expected to be generally Zipfian, a family of heavy-tailed +distributions, so non-parametric methods were seen to be important. +3-grams were +preferred to words for a number of reasons: first, they capture sub-word information +which means they will reflect general sonic preferences (so they can capture things like +accent) and, particularly in inflected languages, also reflect some grammatical style; +second, as a practical matter, they effectively expand the sample data, since a string +of text produces approximately one 3-gram per character. This increased sample size +should reduce the variance of the statistics. Finally, the number of unique 3-grams in +a language is considerably smaller than the number of words, so the frequency data is +less sparse, which again is expected to increase robustness. To now operationalise the +distinctiveness, as defined, we used standard bootstrap methods to measure the median +energy distance (Sz´ekely and Rizzo 2013) with bootstrap confidence intervals between +the two distributions (character 3-gram frequencies vs ‘other’ 3-gram frequencies). The +energy distance is one of a family of related metrics that are commonly used to measure +difference between probability distributions. +Some limitations and choices were required. As mentioned, we measured distinctiveness +only within the context of a single work (even for authors with multiple works). To +expand beyond single works would produce very mismatched sample sizes, since some +authors were prolific and some produced just one play; even with non-parametric meth- +7 + +% Dialogue by Gender +French +German +Russian +Shakespeare +60 +8 +8000 +oaooooao oll +C +00 +68 +% Dialogue +20 +0Distinctiveness by Gender +French +German +Russian +Shakespeare +0.2 +0000 +Distinctiveness (Bootstrap Median) +0.1 +0.0 Vocab Size by Gender +French +German +Russian +Shakespeare +5000 +- +4000 +Vocabulary Size (Unique 3-grams) +3000 +2000 +1000ods, hugely mismatched sample sizes are problematic. +Further, the plays span four +languages and roughly five centuries, making the ‘distant’ context seem ridiculous. As +well as the selected distinctiveness statistic (median energy distance) we also recorded +a ‘baseline’ distinctiveness, being each character’s distance from themselves. The theo- +retical baseline is, of course, zero, but the sample baselines will not be, so this gives us +an idea of the inherent variance of the samples. Finally, when selecting characters to +examine, we chose a minimum size of 2000 words. Sample sizes are somewhat arbitrary, +and are matters of debate (Eder 2015, 2017), but this seemed a reasonable, or perhaps +even slightly aggressive, lower bound. +3.3. Area under keywords +Our second, supplementary approach was informed by ‘unmasking’ techniques, often em- +ployed in stylometric research (Koppel and Schler 2004; Kestemont et al. 2016; Plech´aˇc +and ˇSel¸a 2021). Unmasking refers to a range of methods that share one goal: to measure +and compare the depth of the differences between two sets of texts. For example, an +author might write both high fantasy fiction and historical novels: a classifier would +have little difficulty distinguishing one genre from another by simply using superficial +features (e.g. ‘dragons’, ‘magic’, ‘elves’). However, by assumption, if these most distinc- +tive features are removed, the classifier will have more trouble determining which text +came from which pool, because the texts share one deep similarity—a common authorial +style. Conversely, if we compare books by two different fiction writers, these texts will +also have superficial differences. However, while removing more and more distinctive +features, the classifier should remain confident in distinguishing the authors from each +other, because the texts do not share an authorial style that is deeply rooted in common +linguistic elements and distributed over many features. By comparing the speed with +which the rates of accuracy decay we can approach authorship verification problems, i.e. +how plausible is that this text belongs to author A? +We applied the same thinking to fictional characters, as opposed to authors: the dis- +tinctiveness of a character may rely on a small number of catch-phrases (‘Gadzooks!’ or +‘Cowabunga!’), or it may be driven by non-stylistic, referential factors (Mary, speaking +to John is not likely to use word ‘Mary’, but likely to use word ‘John’, and vice-versa). +On the other hand, there are characters whose speech systematically differs from the +neutral language: such as when the author imitates dialects, slang, regionalism, speech +and phonetic idiosyncrasies. In the former case, an imaginary classifier should quickly +lose accuracy (since John and Mary speak quite similarly), but in the latter case the +removal of a small number of features would not be enough to disrupt classification. +In our case, it was impractical to use ‘standard’, supervised (i.e. classifier-based) un- +masking because individual characters, as samples, were simply too small. Instead we +used word keyness—a character’s relative preference for a word in the context of a given +drama—to calculate an alternative distinctiveness score together with a bag of easily +interpretable features per character. First, we use weighted log-odds (Monroe, Colaresi, +8 + +and Quinn 2008) to calculate keywords for a character relative to the speech pool of the +rest of the cast; second, we represented each character by their 100 words with highest +keyness, arranged by rank; finally, we measured the area under this curve, which we +interpret as distinctiveness—characters with just a few key words will exhibit less area +under the keyness curve. By comparing these final areas, we can compare the amount +of difference each character has in relation to all other speech in the play. In a similar +manner to the bootstrapped approach, we upsample each character’s word pool to match +the size of the rest of the words in the play to minimise, as much as possible, the effect +of sample size. +4. Results +Overall, the distinctiveness energy statistic appears useful. +The baseline (character +vs self) is quite stable cross-linguistically, although it is slightly higher for characters +with a very large share of dialogue (Fig. 1). Note also that the distinctiveness statistic +appears roughly Gaussian (see Appendix B for more discussion) and its range is relatively +consistent between languages (peaking at roughly 0.20), although this consistency does +not apply at the level of authors. The obvious issue is that there is a strong negative +correlation between character size and distinctiveness, but this is not only a limitation +of the method—lead characters naturally set the dominant style of a text (and, possibly, +inherit more of the ‘true’ authorial voice). Importantly, distinctiveness does not increase +with the number of speakers in a play. The method works best when there are reasonable +sample sizes for both the examined character and the ‘other’ class. This is illustrated +by the ‘U’ curve visible in the French corpus in Figure 1 as the examined characters’ +dialogue share passes 50%. As hoped, the energy-distance method does appear to capture +characters who are written with distinctive idiolects, representing things like foreign +accents or social class. For a discussion of this see Section 5. +As seen in Figure 2, there is no clear correlation between the date of composition and +character distinctiveness which suggests that language change does not disturb the mea- +sure. The finding that seems clear is that women are written differently to men. Female +characters are generally more distinctive in all corpora (Fig. 3b), although this is not +visible using the keyness AUC measure—leading us to conclude that the keyness measure +has lower power. This difference in the distinctiveness of female characters can partly +be explained by the fact that they tend to have smaller parts (Fig 3a), and smaller char- +acters in general are more distinctive (Fig. 1), but that is not the whole story. Female +parts have more restricted 3-gram vocabularies (Fig. 3c), suggesting that they are also +restricted in their semantic fields. This becomes clearer when the relative frequencies of +their (word) vocabularies are examined. As well as the stereotypical tendencies (women +say ‘love’, men say ‘sword’), the female characters, cross-linguistically, seem to be less +likely to reference the ‘external world’ of the drama. As seen in Appendix A, relatively +more frequent words for women are dominated by personal pronouns representing ‘I’, +‘me’, ‘you’, etc. or words relating to family. The male lists are dominated by indicative +9 + +articles and political terms (‘law’, ‘noble’, ‘king’, etc.). +The higher distinctiveness of female characters is further supported by a formal linear +model: we fit a Bayesian multiple regression where distinctiveness was conditioned on +both gender and size (characters’ percentage of total dialogue). A direct gender effect is +present in all corpora, as expected from Figure 3a, but, when we account for variation +among authors, the effect may be less pronounced than it appears (for analysis and more +detailed discussed, the posterior estimates are described in Appendix B). Our finding +interlocks with the observation by Underwood, Bamman, and Lee (2018) that female +characters found in English 18–20th century fiction displayed high distinctiveness due +to the particular way they were narrated, suggesting a pervasive authorial mentality. +5. Discussion +The measures of stylistic character distinctiveness that were proposed in this paper ap- +pear to be effective in capturing a degree to which characters stand out from others. The +most distinctive characters, by both of our metrics, often have systematically different +speech, in the form of dialects, regionalisms or class markers. For example, Shakespeare’s +Captain Fluellen (Henry V ) is Welsh, and his accent is written for comedic effect. The +systematic replacements b→p and d→t make him the most distinctive Shakespearean +character according to both the 3-gram and word measures: +Fluellen +Your grandfather of famous memory, an’t please your +majesty, and your great-uncle Edward the Plack +Prince of Wales, as I have read in the chronicles, +fought a most prave pattle here in France. +King Henry V +They did, Fluellen. +Fluellen +Your majesty says very true: if your majesties is +remembered of it, the Welshmen did good service in a +garden where leeks did grow, wearing leeks in their +Monmouth caps; which, your majesty know, to this +hour is an honourable badge of the service; and I do +believe your majesty takes no scorn to wear the leek +upon Saint Tavy’s day. +Regional differences also contribute to high distinctiveness in the German corpus. For +example, Emerike, written by Johanna von Weißenthurn, uses -ey instead of -ei (zwey, +bey, Freylich) which is a form indicative of pre-standardised Southern German spelling. +John, in Hauptmann’s Die Ratten, speaks Plattdeutsch, a variant heavily influenced by +Dutch, e.g. ‘Det hat er jesacht, det ick noch ma hin m¨ußte und janz jenau anjeben’. +In the French corpus, the most distinctive character by keyness is Gareau, from Le P´edant +Jou´e (Cyrano de Bergerac), who speaks a ‘patois’ or rural dialect. In his critical edition, +Fr´ed´eric Lach`evre comments on this distinct idiolect when Gareau is first introduced +10 + +(Cyrano de Bergerac 1921, 25): +Cyrano a fabriqu´e de toutes pi`eces le patois de Gareau. Le manuscrit de la +BN donne un langage tout diff´erent que celui imprim´e en 1654, la prono- +ciation des mots n’est pas tout `a fait la mˆeme. Nous avons naturellement +maintenu pour Gareau le texte de 1654 publi´e par Cyrano lui-mˆeme. +Cyrano created the patois of Gareau from scratch. The manuscript of the +[Biblioth`eque Nationale] offers quite a different language to the one printed +in 1654, the pronunciation of the words is not quite the same. We have nat- +urally maintained for Gareau the text of 1654 published by Cyrano himself. +The most distinctive Russian characters come from Ostrovskii, who gave the main stage +to Muscovite merchants and their families with their vernacular, non-aristocratic lan- +guage. Tolstoy’s Nikita (high on both the 3-gram and keyness lists) from The Power of +Darkness has heavily stylised speech suggestive of Western or Southern Russian dialects, +e.g. featuring a word-initial [w]. +It must be borne in mind, however, that dialects or accents do not automatically cause +high distinctiveness—what is being detected is the difference in speech patterns. +In +a text where everyone speaks Welsh, an English character would score highly on dis- +tinctiveness, and vice versa. Cross-linguistic inference must also account for systematic +language differences: the lexical and morphological features of the various languages lead +naturally to different probability distributions for both words and N-grams (although +the exact nature of those differences is too complex to grapple with here). Word-based +distinctiveness measures permit easier interpretation, but appear less (statistically) pow- +erful. In addition, word-based measures operate in much higher dimensions, with all the +usual problems that entails (sparsity, the ‘curse of dimensionality’, etc. +See, for ex- +ample Moisl (2011)). Finally, word-based measures naturally invite lemmatisation for +highly inflected languages (like Russian and German), which might cause problems for +future work dealing with languages that are non-standard, historical, or otherwise less +well-resourced. +We have noted that our distinctiveness measure has a strong negative correlation to the +size of the character. This relationship should not be understood as a simple artefact that +renders our measurement useless. Distinctive speech is always a construct, a subset of +linguistic and stylistic reality. If a minor character has just a few lines about gallows and +graves—like Shakespeare’s gravedigger—we will never know more about their language. +However, Hamlet is not only about gallows and graves; if we imagine bootstrapping +the gravedigger’s speech, it would be endlessly populated by these few words: we don’t +know how the gravedigger would speak when ruling a country, or murdering their uncle. +From this perspective, a protagonist is more likely to represent lexical and stylistic norm, +while minor characters will sample the Other in their ethnic, dialectal, or professional +distinctiveness. +Despite the few limitations, we hope that these measures of character distinctiveness will +support improved theories about style, characterisation and history. The most important +11 + +question to be asked concerns the source(s) of this representational distinctiveness that +authors instil in their characters. To even begin to address this issue, we need much +richer annotation for characters: their social class, profession, region of origins, age. +Determining the drivers of distinctiveness will not be easy. Even to carefully verify the +effect of character gender was quite complicated. We know that part of the effect comes +from size: women are more likely to be minor characters. However, it is reasonable +to assume that gender difference can also be confounded by genre (e.g. in comedies +there are more women playing larger roles) and social class (rural people speak more in +comedies). There is also the effect of time: changing the relative dynamics of character +sizes (Algee-Hewitt 2017), improving the representation of women as dramatists and +altering the depiction of social class—all of which complicates the analysis even further. +However, having a clear summary measure for a character’s stylistic distinctiveness may +help us to refine our theories about the speech of fictional characters, leading in turn to +better causal models. +6. Availability of Data and Code +The details of our approach, including data acquisition and preprocessing, are published +in a Zenodo repository, allowing for full replication of all reported results: https://doi. +org/10.5281/zenodo.7383687. +Acknowledgements +AˇS, JB, LHL and ME were funded by the “Large-Scale Text Analysis and Methodolog- +ical Foundations of Computational Stylistics” project (SONATA-BIS +2017/26/E/HS2/01019). 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Relatively-More-Frequent Words +French +German +Shakespeare +Female +Male +Female +Male +Female +Male +vous +diable +ach +der +husband +the +´epoux +la +o +die +you +of +m`ere +ami +du +teufel +alas +this +amant +les +vater +und +love +sir +mari +parbleu +mutter +ein +husbands +and +tante +maˆıtre +er +des +me +we +h´elas +morbleu +mich +in +romeo +king +coeur +des +liebe +den +lysander +our +rivale +amis +mama +kerl +willow +their +ne +morgu´e +papa +kaiser +pisanio +duke +malheureuse +serviteur +nein +ihr +sister +three +quil +belle +dat +euch +nerissa +her +mon +vin +mein +auf +yours +to +me +un +herz +dem +o +whom +maman +heureux +gemahl +wir +pray +lordship +fr`ere +leur +gott +k¨onig +mother +in +fˆach´ee +rome +geliebter +sache +nurse +stand +p`ere +peuple +kind +also +i +noble +oblig´ee +boire +ihn +hm +woman +ha +sˆure +soldats +lieber +majest¨at +malvolio +dog +dorante +peste +nicht +oder +prithee +certain +il +prˆet +nich +volk +my +kate +soeur +rival +sie +euer +orlando +master +amour +d´e +mann +das +boyet +sword +lui +s´enat +weh +unter +do +follow +heureuse +¸ca +dir +im +false +soldiers +que +messieurs +dich +zum +ring +his +pleurs +coquin +mellefont +freund +emilia +caesar +cruel +gens +ja +krieg +refuse +us +lamour +du +ihm +durch +troilus +law +chevalier +allons +angst +h¨olle +pilgrim +friends +seule +beaut´e +freundin +zu +windsor +york +aim´ee +au +wat +gnaden +would +money +lingrat +lhomme +doch +wein +rosalind +pompey +val`ere +par +mamachen +heer +such +england +aime +oblig´e +so +mit +weep +present +aimer +l´e +mir +b¨urger +faith +warwick +maime +bon +fritz +jeder +suit +great +hans +cents +arme +herren +am +heads +ch`ere +bˆaton +gurli +rom +diana +ready +ingrat +quatre +lieb +land +never +business +Table 2.: 40 most relatively-more-frequent words (Weighted Log-Odds) for the French, +German and Shakespearean corpora. + +B. Bayesian Regression Models: effect of gender on +distinctiveness +Is the perceived gender effect ‘real’? In technical terms, what is the direct influence of +character gender (G) on distinctiveness scores (D) across traditions (T), conditioned on +the share of dialogue they have (S)? To answer this, we fit a Bayesian multilevel multiple +regression with group-level estimates for individual plays (P). We chose to model at the +level of plays both because our D statistic is tied to the context of a single play, and +Figure 4.: Character distinctiveness, predicted from posterior, estimate of grand mean +(no group-level effects), 6000 draws. Predictions are made for a counterfactual +”median” character role, who has 20.9% of dialogue share. Predictions are +presented at natural scale. + +Posterior predictions, estimates of global grand mean +French +German +400 - +300 - +200 - +100 - +-0 +nsity +Den +Russian +Shakespeare +400 +300 - +200 - +100 - +-0 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 0.04 +0.06 +0.08 +0.10 +0.12 +0.1 +DistinctivenessFigure 5.: Posterior predictions for gender, marginal of individual plays. Errorbars show +.95 CI. Empirical data is plotted in colour, 5 extreme cases (>0.3) are filtered +out. Predictions are presented at natural scale. +because character features coming from the same play are not independent (e.g. there +cannot be two characters with 60% of the dialogue). Modelling this way also significantly +improved predictions. Gender is allowed to interact by corpus, yielding a single, cross- +linguistic model that makes compatible predictions for different traditions. +In brms +formula syntax: +log(D) ∼ G * T + T*(S + I(S^2)) + (1|P) +Based on sample observation, we used a Gaussian prior for log-transformed D scores. We +could have also fitted the original values, but D scores have extreme outliers that extend + +Posterior predictions, marginal of plays +French +German +0.3 +0.2 +0.1 +Distinctiveness +0.0 +Russian +Shakespeare +0.3 +0.2 +0.1 +0.0 +FEMALE +MALE +FEMALE +MALEthe tail: the model has much easier time with sampling and chain convergence on a log- +transformed domain. We chose a quadratic term for S, because the relationship between +D and S is U-shaped. Importantly, ‘unknown’ gender entities are filtered, because often +(but not always) this is not data that is missing, but entries that are incompatible with a +binary classification:1 primarily collective or compound entities (people, choirs, soldiers). +It would have been possible to use standard strategies, like imputation, to ‘repair’ the +data, but that approach would be incorrect. +Posterior estimates for distinctiveness by gender are shown in Figure 4. Based on the +figure, we can be most confident about the difference in German and least confident in +Shakespeare (few characters and, specifically, few women with large dialogue shares). +The differences in means, however, appear consistent. As calculated from the posterior: +in French, female characters are more distinctive by only .009 (± .003); in German, by +.017 (± .003); in Russian by .023 (± .009, the widest CI); and in Shakespeare by .012 +(± .008). +To understand the full extent of variation across different plays, it is useful to look at the +marginal posterior means of the plays (Fig. 5). Here, the difference in distinctiveness +between genders remains visible, but there is a better estimation of the global uncertainty +and variation across different texts. Note that the confidence intervals in Fig. 5 are +asymmetric (wider on the upper arm), having been transformed from symmetric intervals +on a log domain. +1. In modern terms, it is vexing to be forced to reduce characters to a gender binary, but since gender +non-conforming characters are virtually unrepresented in this predominantly historical corpus, the point +is moot. + diff --git a/oNE5T4oBgHgl3EQfkA8i/content/tmp_files/load_file.txt b/oNE5T4oBgHgl3EQfkA8i/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..adfc66216f08d55ac2ac6247424274085a89de59 --- /dev/null +++ b/oNE5T4oBgHgl3EQfkA8i/content/tmp_files/load_file.txt @@ -0,0 +1,912 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf,len=911 +page_content='From stage to page: language independent bootstrap measures of distinctiveness in fictional speech Artjoms ˇSel¸a Institute of Polish Language, Polish Academy of Sciences and University of Tartu artjoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='sela@ijp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='pan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='pl Ben Nagy Institute of Polish Language, Polish Academy of Sciences benjamin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='nagy@ijp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='pan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='pl Joanna Byszuk Institute of Polish Language, Polish Academy of Sciences joanna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='byszuk@ijp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='pan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='pl Laura Hern´andez-Lorenzo University of Seville lhernandez1@us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='es Botond Szemes Research Centre for the Humanities Institute for Literary Studies Budapest szemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='botond@abtk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='hu Maciej Eder Institute of Polish Language, Polish Academy of Sciences maciej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='eder@ijp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='pan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='pl Abstract Stylometry is mostly applied to authorial style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' More recently, researchers have begun investigating the style of characters, finding that, although there is detectable stylistic variation, the variation remains within authorial bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' In this article, we address the stylistic distinctiveness of characters in drama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Our primary contribution is methodological;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' we introduce and evaluate two non-parametric methods to produce a summary statistic for character distinctiveness that can be usefully ap- plied and compared across languages and times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' This is a sig- nificant advance—previous approaches have either been based on pairwise similarities (which cannot be easily compared) or indirect methods that attempt to infer distinctiveness using classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Our first method is based on boot- strap distances between 3-gram probability distributions, the second (reminiscent of ‘unmasking’ techniques) on word key- ness curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Both methods are validated and explored by ap- plying them to a reasonably large corpus (a subset of DraCor): 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='05659v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='CL] 13 Jan 2023 we analyse 3301 characters drawn from 2324 works, covering five centuries and four languages (French, German, Russian, and the works of Shakespeare).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Both methods appear use- ful;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' the 3-gram method is statistically more powerful but the word keyness method offers rich interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Both meth- ods are able to capture phonological differences such as accent or dialect, as well as broad differences in topic and lexical richness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Based on exploratory analysis, we find that smaller characters tend to be more distinctive, and that women are cross-linguistically more distinctive than men, with this latter finding carefully interrogated using multiple regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' This greater distinctiveness stems from a historical tendency for female characters to be restricted to an ‘internal narrative do- main’ covering mainly direct discourse and family/romantic themes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' It is hoped that direct, comparable statistical mea- sures will form a basis for more sophisticated future studies, and advances in theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Introduction Since Vladimir Propp’s work, structural narratology has approached fictional characters mainly through their role or function—by what they do, or what is done to them (Eder, Jannidis, and Schneider 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' This character typology relied on recurring functions in the narrative (lover, villain, victim, detective, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=') and the same perspective was often adopted in computational research, where characters in novels were modelled on the basis of narrative passages rather than dialogue (Bamman, Underwood, and Smith 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Bonch-Osmolovskaya and Skorinkin 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Underwood, Bamman, and Lee 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Stammbach, Antoniak, and Ash 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' In dramatic texts, however, the dominant device for characterisation is an utterance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' While the script usually contains some stage directions, the specifics of characterisation and style of performance are not determined by the text itself, but developed by a spe- cific theatre, director or a troupe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Over the course of history, many plays were written for specific theatre stages, and it was common practice to write characters for specific actors (Fischer-Lichte 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Of course, this kind of ‘outsourced characterisation’ was supported by dramatic conventions and formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Viewers’ expectations could be shaped without a single word being uttered on stage, just by a character wearing a costume, operating a puppet, or changing a dell’arte stock mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' At the same time, the things characters say and how they say them are the main textual source of information about them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' It is reasonable to assume that dramatists make significant efforts to create lin- guistic distinctions between princes and paupers, lovers and schemers, aristocrats and merchants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Tragic monologue is written differently to a comedic exchange between ser- vants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Some previous computational works treat linguistic distinctiveness of characters from the perspective of this stylistic continuum (Vishnubhotla, Hammond, and Hirst 2019), noting that it can be influenced by genre, character gender, or their social and professional dispositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 2 A parallel narratological tradition, tied to Bakhtin’s ideas of heteroglossia, focuses not on abstract character roles, but on the words characters say (Bronwen 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Culpeper 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Sternberg 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The modern novelistic space of dialogic exchange, ‘educated conversation’ (Moretti 2011) and the clash of styles in reported discourse become central here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Available stylometric research on fictional speech and micro-stylistic variation suggests that characters within a text are often distinguishable by their local linguistic patterns without obscuring the global authorial trace (Burrows 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Hoover 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' As put by Burrows and Craig: ‘Characters speak in measurably different ways, but the authorial contrasts transcend this differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The diversity of styles within an author always remains within bounds’ (Burrows and Craig 2012, 307–8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Conceptually and methodologically, the majority of previous works examined not the distinctiveness of characters, but their (pairwise) similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Similarity measures are meaningful in pairwise contexts, but cannot be analysed and compared as individual summary statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Since Burrows’ seminal study of speech patterns in Jane Austen’s characters (Burrows 1987), these approaches focused on calculating similarity within a collection of characters: how different is character X from character Y, and each of them from character Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Burrows measured the correlation between characters’ usage of 30 most frequent words (technically, he fit a linear regression for two sets of log-frequencies);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' later, similarity was most often inferred through clustering based on pairwise distance calculations (Hoover 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Reeve 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Craig and Greatley-Hirsch 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Sometimes linguistic similarity served as a basis for arguing functional similarity, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' A recent study that linked Bakhtin’s dialogism and the stylistic diversity of characters’ speech (Vishnubhotla, Hammond, and Hirst 2019), proposed the analysis of distinctiveness rather than similarity using supervised classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Instead of using a network of pairwise relationships, the authors asked how well a classifier can recognise character X as being written by author A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Classification accuracy in this scenario becomes an explicit summary statistic for distinctiveness that can be assigned to a character (or, in an aggregated manner, to a play or an author).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' However, the supervised approach, proposed by Vishnubhotla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=', is data hungry: it suffers from extreme class imbalance, an abundance of short samples (most characters speak only a little) and is dependent on language-specific feature construction procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' By contrast, this paper will present a simple, non-parametric measure of character dis- tinctiveness that is based on bootstrapped probability distributions representing a char- acter and all others present in a given play: an approach largely informed by authorship verification techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' This measure is language-independent and relies only on the context of a single work, which, in turn, minimises problems of language variation, authorial signal and chronological change in a comparative setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Individual distinc- tiveness scores can be then tested against other measures and metadata categories in a hypothesis-driven manner, not only across languages, but also across genres (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' novel vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' drama).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Do comedies tend to employ more distinct characters?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Does distinctiveness increase (authors get better), or decrease (social and linguistic homogenisation occurs) over time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Is there a difference between the distinctiveness of fictional women and men?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' If so is it the direct result of perceived gender differences, or is it constructed by imagined 3 differences in social and professional status?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Lacking good descriptive metadata on the dramatic characters, this paper will not answer above mentioned questions in any satisfying way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Instead we focus on presenting and justifying the measure of distinctiveness and exploring several factors that might shape the final scores (like the year of composition, character gender and characters’ sample size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Materials Total Characters Unique Unique Total Total Corpus Characters Analysed 3-grams Words 3-grams Words French 15462 1744 9896 79994 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='79 m 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='47 m German 14010 1182 14341 150956 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='80 m 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='31 m Russian 3707 248 12542 71217 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='05 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='72 m Shakespeare 1431 127 5921 19595 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='16 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='43 m Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=': A summary of the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' All word and 3-gram counts are for the filtered corpus (characters that speak at least 2000 words) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' As the beginning of our exploration of cross-linguistic variation, we examined four dra- matic corpora from DraCor (Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 2019): Shakespeare, French, German, and Russian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' DraCor is a project that gathers dramatic corpora in various languages, pri- marily European, encoded in TEI-XML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' With 15 corpora available so far, including the Shakespeare corpus available both in English and German, DraCor facilitates large scale analysis of dramatic conventions across language traditions, and offers a wide variety of useful metadata, at the level of both plays and characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' While the analysis of all Dra- Cor corpora would be possible with the methods we developed, for the purpose of this preliminary study we focused on the languages and dramatic traditions well known to the members of our team, eventually selecting the full corpora for Shakespeare, French, German, and Russian: a total of 2324 texts, the majority of which come from French and German.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The corpus is summarised in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' General Approach and Definitions Our understanding of character distinctiveness is largely informed by ‘authorship ver- ification’ approaches, which centre around verifying that a text is written by a target author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' This problem is more general than ‘authorship attribution’ that tries to identify the nearest stylistic neighbour for a text (Halvani, Winter, and Graner 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Instead, authorship verification asks about the relative magnitude of similarity: is a target text more similar to same-author samples, or different-author samples?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' With this in mind, 4 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=': Character distinctiveness, per corpus, versus % Dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Women are shown smaller, in orange, men (and undefined) larger and in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' GAM (Generalised Additive Model) trendlines are superimposed in the same colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Baseline data (GAM trend for distinctiveness of character vs self) is shown as a dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' we define a character’s ‘distinctiveness’ as the degree to which the style of their speech differs from that of other characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' We understand ‘style’ here instrumentally, as a deviation from an unobserved average language (Herrmann, Dalen-Oskam, and Sch¨och 2015) and do not introduce aggressive feature filtering, allowing both ‘grammatical’ and ‘thematic’ signal to contribute to the final measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' We anchor our distinctiveness mea- sure in the context of the specific text in which a character appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' In theory, the frame of reference could be all plays from one author, or all plays from the same period, or even 5 Distinctiveness vs Percent Dialogue French German 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='15 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='10 rence) differe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='05 3 Russian Shakespeare istinctiveness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='20 - 三 Character 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='15 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='10 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='05 20 40 60 20 40 60 Proportion of total dialogue (%)Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=': Character distinctiveness, per corpus, versus year composed (DraCor data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Women are shown smaller, in orange, men (and undefined) larger and in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' GAM (Generalised Additive Model) trendlines are superimposed in the same colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Baseline data (GAM trend for distinctiveness of character vs self) is shown as a dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' some external corpus—however all of these would greatly complicate any comparative study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Bootstrap 3-gram Distinctiveness Based on our definition of distinctiveness above, we considered a character’s style to be an idiolect sampled from a frequency distribution of character 3-grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' As a natural 6 French German 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='10 difference) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='05 distribution 1600 1700 1800 1900 1800 Russian Shakespeare isti 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='20 Character 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='05 750 1850 1900 1595 Normalized Year Composed(a) % Dialogue (b) Distinctiveness (c) Vocab size (3-grams) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=': An analysis, per corpus, of the distribution of various features by gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Distributions are estimated, with the median shown as a solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Actual points are shown as rug plots with outliers ‘o’ plotted for points outside 3Q + 2×IQR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' language distribution, this was expected to be generally Zipfian, a family of heavy-tailed distributions, so non-parametric methods were seen to be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 3-grams were preferred to words for a number of reasons: first, they capture sub-word information which means they will reflect general sonic preferences (so they can capture things like accent) and, particularly in inflected languages, also reflect some grammatical style;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' second, as a practical matter, they effectively expand the sample data, since a string of text produces approximately one 3-gram per character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' This increased sample size should reduce the variance of the statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Finally, the number of unique 3-grams in a language is considerably smaller than the number of words, so the frequency data is less sparse, which again is expected to increase robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' To now operationalise the distinctiveness, as defined, we used standard bootstrap methods to measure the median energy distance (Sz´ekely and Rizzo 2013) with bootstrap confidence intervals between the two distributions (character 3-gram frequencies vs ‘other’ 3-gram frequencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The energy distance is one of a family of related metrics that are commonly used to measure difference between probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Some limitations and choices were required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' As mentioned, we measured distinctiveness only within the context of a single work (even for authors with multiple works).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' To expand beyond single works would produce very mismatched sample sizes, since some authors were prolific and some produced just one play;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' even with non-parametric meth- 7 % Dialogue by Gender French German Russian Shakespeare 60 8 8000 oaooooao oll C 00 68 % Dialogue 20 0Distinctiveness by Gender French German Russian Shakespeare 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='2 0000 Distinctiveness (Bootstrap Median) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='0 Vocab Size by Gender French German Russian Shakespeare 5000 4000 Vocabulary Size (Unique 3-grams) 3000 2000 1000ods, hugely mismatched sample sizes are problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Further, the plays span four languages and roughly five centuries, making the ‘distant’ context seem ridiculous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' As well as the selected distinctiveness statistic (median energy distance) we also recorded a ‘baseline’ distinctiveness, being each character’s distance from themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The theo- retical baseline is, of course, zero, but the sample baselines will not be, so this gives us an idea of the inherent variance of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Finally, when selecting characters to examine, we chose a minimum size of 2000 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Sample sizes are somewhat arbitrary, and are matters of debate (Eder 2015, 2017), but this seemed a reasonable, or perhaps even slightly aggressive, lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Area under keywords Our second, supplementary approach was informed by ‘unmasking’ techniques, often em- ployed in stylometric research (Koppel and Schler 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Kestemont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Plech´aˇc and ˇSel¸a 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Unmasking refers to a range of methods that share one goal: to measure and compare the depth of the differences between two sets of texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' For example, an author might write both high fantasy fiction and historical novels: a classifier would have little difficulty distinguishing one genre from another by simply using superficial features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' ‘dragons’, ‘magic’, ‘elves’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' However, by assumption, if these most distinc- tive features are removed, the classifier will have more trouble determining which text came from which pool, because the texts share one deep similarity—a common authorial style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Conversely, if we compare books by two different fiction writers, these texts will also have superficial differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' However, while removing more and more distinctive features, the classifier should remain confident in distinguishing the authors from each other, because the texts do not share an authorial style that is deeply rooted in common linguistic elements and distributed over many features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' By comparing the speed with which the rates of accuracy decay we can approach authorship verification problems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' how plausible is that this text belongs to author A?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' We applied the same thinking to fictional characters, as opposed to authors: the dis- tinctiveness of a character may rely on a small number of catch-phrases (‘Gadzooks!’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' or ‘Cowabunga!’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='), or it may be driven by non-stylistic, referential factors (Mary, speaking to John is not likely to use word ‘Mary’, but likely to use word ‘John’, and vice-versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' On the other hand, there are characters whose speech systematically differs from the neutral language: such as when the author imitates dialects, slang, regionalism, speech and phonetic idiosyncrasies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' In the former case, an imaginary classifier should quickly lose accuracy (since John and Mary speak quite similarly), but in the latter case the removal of a small number of features would not be enough to disrupt classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' In our case, it was impractical to use ‘standard’, supervised (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' classifier-based) un- masking because individual characters, as samples, were simply too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Instead we used word keyness—a character’s relative preference for a word in the context of a given drama—to calculate an alternative distinctiveness score together with a bag of easily interpretable features per character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' First, we use weighted log-odds (Monroe, Colaresi, 8 and Quinn 2008) to calculate keywords for a character relative to the speech pool of the rest of the cast;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' second, we represented each character by their 100 words with highest keyness, arranged by rank;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' finally, we measured the area under this curve, which we interpret as distinctiveness—characters with just a few key words will exhibit less area under the keyness curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' By comparing these final areas, we can compare the amount of difference each character has in relation to all other speech in the play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' In a similar manner to the bootstrapped approach, we upsample each character’s word pool to match the size of the rest of the words in the play to minimise, as much as possible, the effect of sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Results Overall, the distinctiveness energy statistic appears useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The baseline (character vs self) is quite stable cross-linguistically, although it is slightly higher for characters with a very large share of dialogue (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Note also that the distinctiveness statistic appears roughly Gaussian (see Appendix B for more discussion) and its range is relatively consistent between languages (peaking at roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='20), although this consistency does not apply at the level of authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The obvious issue is that there is a strong negative correlation between character size and distinctiveness, but this is not only a limitation of the method—lead characters naturally set the dominant style of a text (and, possibly, inherit more of the ‘true’ authorial voice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Importantly, distinctiveness does not increase with the number of speakers in a play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The method works best when there are reasonable sample sizes for both the examined character and the ‘other’ class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' This is illustrated by the ‘U’ curve visible in the French corpus in Figure 1 as the examined characters’ dialogue share passes 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' As hoped, the energy-distance method does appear to capture characters who are written with distinctive idiolects, representing things like foreign accents or social class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' For a discussion of this see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' As seen in Figure 2, there is no clear correlation between the date of composition and character distinctiveness which suggests that language change does not disturb the mea- sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The finding that seems clear is that women are written differently to men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Female characters are generally more distinctive in all corpora (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 3b), although this is not visible using the keyness AUC measure—leading us to conclude that the keyness measure has lower power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' This difference in the distinctiveness of female characters can partly be explained by the fact that they tend to have smaller parts (Fig 3a), and smaller char- acters in general are more distinctive (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 1), but that is not the whole story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Female parts have more restricted 3-gram vocabularies (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 3c), suggesting that they are also restricted in their semantic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' This becomes clearer when the relative frequencies of their (word) vocabularies are examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' As well as the stereotypical tendencies (women say ‘love’, men say ‘sword’), the female characters, cross-linguistically, seem to be less likely to reference the ‘external world’ of the drama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' As seen in Appendix A, relatively more frequent words for women are dominated by personal pronouns representing ‘I’, ‘me’, ‘you’, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' or words relating to family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The male lists are dominated by indicative 9 articles and political terms (‘law’, ‘noble’, ‘king’, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The higher distinctiveness of female characters is further supported by a formal linear model: we fit a Bayesian multiple regression where distinctiveness was conditioned on both gender and size (characters’ percentage of total dialogue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' A direct gender effect is present in all corpora, as expected from Figure 3a, but, when we account for variation among authors, the effect may be less pronounced than it appears (for analysis and more detailed discussed, the posterior estimates are described in Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Our finding interlocks with the observation by Underwood, Bamman, and Lee (2018) that female characters found in English 18–20th century fiction displayed high distinctiveness due to the particular way they were narrated, suggesting a pervasive authorial mentality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Discussion The measures of stylistic character distinctiveness that were proposed in this paper ap- pear to be effective in capturing a degree to which characters stand out from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The most distinctive characters, by both of our metrics, often have systematically different speech, in the form of dialects, regionalisms or class markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' For example, Shakespeare’s Captain Fluellen (Henry V ) is Welsh, and his accent is written for comedic effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The systematic replacements b→p and d→t make him the most distinctive Shakespearean character according to both the 3-gram and word measures: Fluellen Your grandfather of famous memory, an’t please your majesty, and your great-uncle Edward the Plack Prince of Wales, as I have read in the chronicles, fought a most prave pattle here in France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' King Henry V They did, Fluellen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Fluellen Your majesty says very true: if your majesties is remembered of it, the Welshmen did good service in a garden where leeks did grow, wearing leeks in their Monmouth caps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' which, your majesty know, to this hour is an honourable badge of the service;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' and I do believe your majesty takes no scorn to wear the leek upon Saint Tavy’s day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Regional differences also contribute to high distinctiveness in the German corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' For example, Emerike, written by Johanna von Weißenthurn, uses -ey instead of -ei (zwey, bey, Freylich) which is a form indicative of pre-standardised Southern German spelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' John, in Hauptmann’s Die Ratten, speaks Plattdeutsch, a variant heavily influenced by Dutch, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' ‘Det hat er jesacht, det ick noch ma hin m¨ußte und janz jenau anjeben’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' In the French corpus, the most distinctive character by keyness is Gareau, from Le P´edant Jou´e (Cyrano de Bergerac), who speaks a ‘patois’ or rural dialect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' In his critical edition, Fr´ed´eric Lach`evre comments on this distinct idiolect when Gareau is first introduced 10 (Cyrano de Bergerac 1921, 25): Cyrano a fabriqu´e de toutes pi`eces le patois de Gareau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Le manuscrit de la BN donne un langage tout diff´erent que celui imprim´e en 1654, la prono- ciation des mots n’est pas tout `a fait la mˆeme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Nous avons naturellement maintenu pour Gareau le texte de 1654 publi´e par Cyrano lui-mˆeme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Cyrano created the patois of Gareau from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The manuscript of the [Biblioth`eque Nationale] offers quite a different language to the one printed in 1654, the pronunciation of the words is not quite the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' We have nat- urally maintained for Gareau the text of 1654 published by Cyrano himself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The most distinctive Russian characters come from Ostrovskii, who gave the main stage to Muscovite merchants and their families with their vernacular, non-aristocratic lan- guage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Tolstoy’s Nikita (high on both the 3-gram and keyness lists) from The Power of Darkness has heavily stylised speech suggestive of Western or Southern Russian dialects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' featuring a word-initial [w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' It must be borne in mind, however, that dialects or accents do not automatically cause high distinctiveness—what is being detected is the difference in speech patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' In a text where everyone speaks Welsh, an English character would score highly on dis- tinctiveness, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Cross-linguistic inference must also account for systematic language differences: the lexical and morphological features of the various languages lead naturally to different probability distributions for both words and N-grams (although the exact nature of those differences is too complex to grapple with here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Word-based distinctiveness measures permit easier interpretation, but appear less (statistically) pow- erful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' In addition, word-based measures operate in much higher dimensions, with all the usual problems that entails (sparsity, the ‘curse of dimensionality’, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' See, for ex- ample Moisl (2011)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Finally, word-based measures naturally invite lemmatisation for highly inflected languages (like Russian and German), which might cause problems for future work dealing with languages that are non-standard, historical, or otherwise less well-resourced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' We have noted that our distinctiveness measure has a strong negative correlation to the size of the character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' This relationship should not be understood as a simple artefact that renders our measurement useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Distinctive speech is always a construct, a subset of linguistic and stylistic reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' If a minor character has just a few lines about gallows and graves—like Shakespeare’s gravedigger—we will never know more about their language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' However, Hamlet is not only about gallows and graves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' if we imagine bootstrapping the gravedigger’s speech, it would be endlessly populated by these few words: we don’t know how the gravedigger would speak when ruling a country, or murdering their uncle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' From this perspective, a protagonist is more likely to represent lexical and stylistic norm, while minor characters will sample the Other in their ethnic, dialectal, or professional distinctiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Despite the few limitations, we hope that these measures of character distinctiveness will support improved theories about style, characterisation and history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The most important 11 question to be asked concerns the source(s) of this representational distinctiveness that authors instil in their characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' To even begin to address this issue, we need much richer annotation for characters: their social class, profession, region of origins, age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Determining the drivers of distinctiveness will not be easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Even to carefully verify the effect of character gender was quite complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' We know that part of the effect comes from size: women are more likely to be minor characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' However, it is reasonable to assume that gender difference can also be confounded by genre (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' in comedies there are more women playing larger roles) and social class (rural people speak more in comedies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' There is also the effect of time: changing the relative dynamics of character sizes (Algee-Hewitt 2017), improving the representation of women as dramatists and altering the depiction of social class—all of which complicates the analysis even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' However, having a clear summary measure for a character’s stylistic distinctiveness may help us to refine our theories about the speech of fictional characters, leading in turn to better causal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Availability of Data and Code The details of our approach, including data acquisition and preprocessing, are published in a Zenodo repository, allowing for full replication of all reported results: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='7383687.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='quatre ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='lieb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='land ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='never ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='business ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=': 40 most relatively-more-frequent words (Weighted Log-Odds) for the French, German and Shakespearean corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Bayesian Regression Models: effect of gender on distinctiveness Is the perceived gender effect ‘real’?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' In technical terms, what is the direct influence of character gender (G) on distinctiveness scores (D) across traditions (T), conditioned on the share of dialogue they have (S)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' To answer this, we fit a Bayesian multilevel multiple regression with group-level estimates for individual plays (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' We chose to model at the level of plays both because our D statistic is tied to the context of a single play, and Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=': Character distinctiveness, predicted from posterior, estimate of grand mean (no group-level effects), 6000 draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Predictions are made for a counterfactual ”median” character role, who has 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='9% of dialogue share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Predictions are presented at natural scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Posterior predictions, estimates of global grand mean French German 400 - 300 - 200 - 100 - 0 nsity Den Russian Shakespeare 400 300 - 200 - 100 - 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='1 DistinctivenessFigure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=': Posterior predictions for gender, marginal of individual plays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Errorbars show .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='95 CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Empirical data is plotted in colour, 5 extreme cases (>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='3) are filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Predictions are presented at natural scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' because character features coming from the same play are not independent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' there cannot be two characters with 60% of the dialogue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Modelling this way also significantly improved predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Gender is allowed to interact by corpus, yielding a single, cross- linguistic model that makes compatible predictions for different traditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' In brms formula syntax: log(D) ∼ G * T + T*(S + I(S^2)) + (1|P) Based on sample observation, we used a Gaussian prior for log-transformed D scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' We could have also fitted the original values, but D scores have extreme outliers that extend Posterior predictions, marginal of plays French German 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='1 Distinctiveness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='0 Russian Shakespeare 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='0 FEMALE MALE FEMALE MALEthe tail: the model has much easier time with sampling and chain convergence on a log- transformed domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' We chose a quadratic term for S, because the relationship between D and S is U-shaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Importantly, ‘unknown’ gender entities are filtered, because often (but not always) this is not data that is missing, but entries that are incompatible with a binary classification:1 primarily collective or compound entities (people, choirs, soldiers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' It would have been possible to use standard strategies, like imputation, to ‘repair’ the data, but that approach would be incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Posterior estimates for distinctiveness by gender are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' Based on the figure, we can be most confident about the difference in German and least confident in Shakespeare (few characters and, specifically, few women with large dialogue shares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' The differences in means, however, appear consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' As calculated from the posterior: in French, female characters are more distinctive by only .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='009 (± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' in German, by .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='017 (± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' in Russian by .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='023 (± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='009, the widest CI);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' and in Shakespeare by .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='012 (± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content='008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' To understand the full extent of variation across different plays, it is useful to look at the marginal posterior means of the plays (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} +page_content=' 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In modern terms, it is vexing to be forced to reduce characters to a gender binary, but since gender non-conforming characters are virtually unrepresented in this predominantly historical corpus, the point is moot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE5T4oBgHgl3EQfkA8i/content/2301.05659v1.pdf'} diff --git a/oNFPT4oBgHgl3EQfKjR1/content/tmp_files/2301.13019v1.pdf.txt b/oNFPT4oBgHgl3EQfKjR1/content/tmp_files/2301.13019v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d96af7f9c562e030811b2eb5a6c1f25e2160db7f --- /dev/null +++ b/oNFPT4oBgHgl3EQfKjR1/content/tmp_files/2301.13019v1.pdf.txt @@ -0,0 +1,467 @@ +Winning Solution of Real Robot Challenge III +Qiang Wang, Robert McCarthy, David Cordova Bulens, and Stephen J. Redmond +October 2022 +Abstract +This report introduces our winning solution of the real-robot phase of the Real Robot Challenge (RRC) 2022. +The goal of this year’s challenge is to solve dexterous manipulation tasks with offline reinforcement learning (RL) +or imitation learning. To this end, participants are provided with datasets containing dozens of hours of robotic +data. For each task an expert1 dataset and a mixed dataset are provided. In our experiments, when learning from +the expert datasets, we find standard Behavioral Cloning (BC) outperforms state-of-the-art offline RL algorithms. +When learning from the mixed datasets, BC performs poorly, as expected, while surprisingly offline RL performs +suboptimally, failing to match the average performance of the baseline model used for collecting the datasets. To +remedy this, motivated by the strong performance of BC on the expert datasets we elect to use a semi-supervised +classification technique to filter the subset of expert data out from the mixed datasets, and subsequently perform +BC on this extracted subset of data. To further improve results, in all settings we use a simple data augmentation +method that exploits the geometric symmetry of the RRC physical robotic environment. Our submitted BC policies +each surpass the mean return2 of their respective raw datasets, and the policies trained on the filtered mixed datasets +come close to matching the performances of those trained on the expert datasets. +1 +Real Robot Challenge 2022 +Data-driven learning methods are promising for dexterous robotic manipulation tasks - they can learn complex, skillful +manipulation strategies from scratch, and indeed have begun to outperform more traditional control methods in some +settings (CITE). However, learning manipulation policies from real robots usually involves costly and time-consuming +data collection. However, these issues can potentially be mitigated by making use of available pre-collected data. The +RRC 2022 seeks to encourage the development of offline algorithms that can make efficient use of such real-world data, +and thus improve the performance of our learning methods when applied in practical real-world scenarios. +(a) The real TriFinger robot used in the RRC 2021. +(b) The top view of the robot arena. +Figure 1: The TriFinger robot[21]. Three identical robotic fingers are equally spaced around the circular arena. The +colored cube is the target object that must be moved. +In the RRC 2022, participant are provided with dozens of hours of TriFinger[21] robotic data. Datasets are provided +for two robotic tasks, push and lift. In the push task, the cube must be moved to target positions on the arena floor. +In the more challenging lift task, the cube must be lifted and maintained at a target position and orientation. For each +task, two separate datasets are provided; one collected by an expert policy (the expert dataset), the other collected by +a number of policies with various skill levels (mixed dataset). Thus, there are four separate datasets. Participants must +submit a different policy for each respective dataset. Only learning-based approaches are permitted, and datasets cannot +be combined in the training process. Access is also provided to a cluster of real TriFinger robots (see Figure1(a)) to +allow participants to evaluate their trained policies; however, data from the evaluation, either performed in simulation +or on the real robot, is not allowed to be used to further refine a policy. For more details, please see the official website +of RRC3. We now describe our approach to the challenge. +1For convenience, we name the datasets collected by expert policies as the expert dataset and the data contained within it as expert data. +For the dataset collected by mixed policies, we name it the mixed dataset and the data contained within it mixed data. +2The cumulative reward acquired in each episode. +3https://real-robot-challenge.com/protocol +1 +arXiv:2301.13019v1 [cs.RO] 30 Jan 2023 + +Table 1: Real-robot evaluated score comparison of algorithms trained from both expert and mixed dataset of the lift +task, where the Baseline refers to the mean episodic return in the dataset. +BC +BCQ +PLAS +IQL +Ours +Baseline +Lift exp +928.91 +646.18 +911.91 +789.12 +1129.70 +1064.00 +Lift mix +489.25 +432.75 +711.28 +550.23 +998.82 +851.00 +2 +Methodology +2.1 +Our approach +In our experiments we found that standard BC outperforms a number of state-of-the-art offline reinforcement learning +algorithms4 on the lift expert dataset. Specifically, the offline reinforcement learning algorithms we tested included +Batch-Constrained Q-learning (BCQ)[17], Policy in Latent Action Space (PLAS)[18], and Implicit Q-Learning (IQL)[19]. +The evaluated results of these algorithms on the lift task are compared in Table 1. +Now, when experimenting on the mixed dataset, we find both standard BC and offline RL to perform relatively +poorly, failing to reach the mean returns of the dataset in all cases. BC’s poor performance is to be expected – the +mixed dataset presents two challenges for BC: +1. It contains a large number of less successful episodes (see Figure 2); imitating the behaviour in these episodes +will hurt performance. +2. It contains data collected by multiple policies; i.e., the data when sorted by summed reward over the episode +is multi-modal (see Figure 2). Unless accounted for explicitly, a naive BC approach may perform poorly on a +multi-modal dataset[16], even if the data is of high quality, as it is trying to find a single policy that can replicate +the behaviour of two or more very different policies. +Offline RL should in theory be capable of learning expert behaviour from such a mixed dataset, and indeed the +tested algorithms outperform BC, however, performance is still suboptimal. +Data filtering: +Motivated by the strong performance of BC on the expert dataset, we attempt to improve results +on the mixed dataset by splitting the mixed dataset into subsets, where each subset contains the data from a single +policy, and we subsequently apply BC to the subset of data came from the best-performing policy. Note, we find the +approach of naively splitting the dataset into two subsets based on reward returns (for example, from where the red +arrow points in Figure 2(b)) and applying BC to the subset with higher rewards leads to poor performance. Here, the +subset sorted by summed reward is still multimodal, containing a number of higher reward episodes collected by the +weaker policies. Many of these episodes collected by weaker policies can be further filtered out by increasing the reward +return threshold (like Eq. 1), ensuring that almost all of the filtered data is collected by the higher performing policy. +However, we find this new subset is then too small to train a BC model with good performance. Therefore, we propose +a semi-supervised data filtering approach which can effectively extract the data collected by the best performing policy +(expert policy) from the mixed dataset. +(a) Reward distribution of the lift expert dataset. +(b) Reward distribution of the lift mixed dataset. +Figure 2: The histogram of the accumulated reward over an episode, calculating by summing all rewards of each +episode together. The expert dataset consists mostly of successful episodes, in which the cube was moved to the target. +The mixed dataset appears to consist two distinct peaks, therefore, we assume that the mixed dataset is collected +jointly by at least two policies, at least one of which is an expert. +4We primarily make use of algorithms implemented in the d3rlpy library[15] +2 + +200 +175 +150 +Amount +125 +100 +75 +50 +25 +0 +200 +400 +600 +800 +1000 +1200 +1400 +Summedreward overanepisode120- +100 +80 +Amount +60 +40 +20 +0 +0 +200 +400 +600 +800 +1000 +1200 +1400 +Summed reward overan episodeMethod summary: +In summary, for our final submissions, we use BC-based approaches in all cases. On the mixed +datasets we use our data filtering approach to extract the expert data and use it to train the BC policy. As a supervised +learning algorithm, the BC is affected by the compound error led by the covariate shift[16]. The root cause is that the +trained BC model encounters many unseen data during deployment. One solution is to increase the training dataset +size; hence, to further enhance performances we also introduce a data augmentation method that exploits the symmetry +of the robot environment using simple rotational transforms, which can increase the amount of the training data by +three times. We describe these methods in more detail in the next section. +2.2 +Expert data filter +A data filter (i.e., a binary classifier that assigns a label to an epoch) is trained and applied to the mixed dataset to +extract the data collected by the/an expert policy. Figure 3 shows the filter training process, which includes: 1) First +filtering the raw dataset such that only a small subset of the highest return episodes are retained (see section 2.2.2) to +form a superb dataset, which is small, but, we expect, mostly consists of data collected by the expert policy; 2) Use the +superb dataset to pre-train the filter classifier; 3) Use the trained (or pre-trained) filter classifier to relabel the entire +dataset, and then use the relabelled dataset to retrain the filter classifier again; 4) Repeat step 3 until the filtered +dataset’s membership does not change or vary, within some reasonable tolerance. +Figure 3: Filter training flow chart, where the blue arrows indicate the pre-training process and the red arrows +indicate one iteration of the process of training the filter. +2.2.1 +Structure of the filter +Figure 4 shows the filter’s structure, which is a classifier that outputs the binary prediction indicating whether the +input data was collected by using an expert policy, or not. The significant dimensional gap between the observation and +the action may result in data bias; hence the dimension of the observation vector is reduced before concatenating with +the action vector. The classifier operates on vectors from a single time-step; then, a post-processing step is performed +by aggregating the classifier-applied label across all time-steps in one episode to get a final label. This is done by +summing the binary labels across all time-steps and then applying a threshold, where the threshold is selected by +observing the performance of BC trained from the filtered subset. +Figure 4: The structure of the filter. The encoder and predictor are fully-connected neural networks; the architecture +is shown in Appendix A.1. +2.2.2 +Generate the training data +The expert and non-expert policy have significantly different summed reward when the episode has a high difficulty. +For this reason, the superb dataset was formed using the criterion presented in Eq. 1: +ExpertEpisode = +� +True, if +(E(R[tstart]) ≤ thlow) AND (E(R[tend]) ≥ thhigh) +False, otherwise +(1) +3 + +Raw +Filter n +- +dataset +- +Filtered +Eq. 1 +Binary +BC +Classifier +subset +Superb +- +dataset +-Binary Classifiel +Observation +Expert +Encoder +Predictor +OR +Action +Non-expertwhere R is the episodic reward5, a set of values, with each value for the reward of each time-step. And the expectation +operator E is implemented by taking the sample mean. tstart is the set of time indices addressing reward value for the +first five time-steps in the episode. tend is the set of time indices addressing reward value for the last 150 time-steps +in the episode. thlow value is 0.33 for both tasks, and thhigh value for the lift task is 0.98, and for the push task is +0.96. The first constraint (to the left of the logical AND in Eq. 1) makes only the relatively challenging episodes +be chosen by taking those with lower average rewards over the initial five time-steps, which removes episodes where +the cube started at or near the target. The second constraint (to the right of the logical AND in Eq. 1) is set to +choose episodes with excellent behavioural quality by taking those having high average rewards over the final 150 +time-steps. The second constraint is based on the assumption that if rewards are high throughout the end of episode, +then the agents has performed the task well. Hence, by combining both constraints, we only keep difficult episodes +where the agent manages to successfully completes them; it is highly possible to make the vast majority of the filtered +episodes collected by the expert. The setting of the above constants(tstart, tend, thlow, thhigh) performs the best over +our attempts. In the beginning, we roughly set reasonable values for the above constants, enabling the classifier’s +training on the superb dataset to converge. Afterwards, we fine-tune these values based on observing the loss during +the classifier’s training(smaller converged loss and smoother loss curve means the threshold is better) and the evaluated +score of BC trained from the final filtered subset. +We name the data from either the superb and filtered subset as labelled EXPERT; that is, coming from an expert +policy. We propose two ways to generate NON-EXPERT-labelled training data, which we consider to not come from an +expert policy, shown in Figure 5. 1) Combine the observations of EXPERT samples with the actions in the part of data +that is not filtered. 2) Combine the observations of EXPERT samples with randomly-generated action data. Only the +actions with a relatively larger difference from the EXPERT actions can be deemed as negative; this reduces the overall +similarity between NON-EXPERT and EXPERT training samples and thus encourages the learning of the filter. +Figure 5: The illustration of generating the EXPERT and NON-EXPERT samples for training the filter. The filtered +subset involves the data filtered by both Eq. 1 and the trained classifier filter. The Non-filtered subset is the raw +dataset that excludes the Filtered subset. +2.2.3 +Filter process +The filter is a binary classifier, which takes the SOFTMAX function as the last-layer activation function. One of +SOFTMAX’s outputs estimates the probability that the episode is expert-collected. In the filtering phase, we feed the +raw dataset into the filter network to obtain the estimated probabilities of being expert of each time-step and sum +them in each episode. This summed probability value is called the confidence that one episode is expert-collected. +When the confidence of one episode is greater than a specific threshold thconf, it is deemed a EXPERT episode. +The selection of the thconf value may lead to results with a significant difference, which is determined by both +reasonable hypotheses and observing the evaluated score of BC trained from the final filtered subset. Our experiment +found that the output confidence is large over most episodes at the beginning of the iterations6, with most of the +episodes in the lift mixed dataset having confidence in the range 1100-1500(max possible confidence is 1500)7, and +that of the push mixed dataset is 650-750 (max possible confidence is 750)8. This is led by the high similarity between +the EXPERT and NON-EXPERT samples since they share the same observations in the filtered subset, as seen in +Figure 5. Hence, we set a relatively large thconf at the beginning of the iterations, which of the lift mixed dataset is +1420 and the push mixed dataset is 730. At the end of the iterations, we found that the confidence of a big part of +episodes becomes smaller; we reasonably guess this is because the filter classifier becomes more robust, hence clarifying +the NON-EXPERT more clearly. Therefore, to filter the final subset for training the BC, we set the thconf to a +relatively smaller value making more episodes can be chosen, which of the lift mixed dataset is 1390 and the push +mixed dataset is 500. Same as above, this settings of thconf performs the best over our attempts. In the beginning, we +roughly set reasonable values through assumption and then we fine-tune these values based on observing the evaluated +score of BC trained from the final filtered subset. +5The method for calculating the reward of each time-step in the dataset was designed by the organizer and can be found in the RRC +documentation: https://webdav.tuebingen.mpg.de/real-robot-challenge/2022/docs/tasks.html#rewards-and-success +6The classifier filter is repeatedly trained by the latest filtered subset mentioned above at the beginning of 2.1. We name each repeat +process iteration. +7The episodic length of lift task is 1500 +8The episodic length of push task is 750 +4 + +EXPERT +Raw dataset +Observation +Action +Observations +Observations +NON-EXPERT +Actions +Actions +Non-filtered +Observation +Action +Filtered subset +I Non-filtered subset +Random +Observation +Action2.3 +Data augmentation through rotational transformation +The top view of the robotic arena is shown in Figure 1(b). Three fingers of the robot are placed evenly around the +center of the rounded working platform with an angle difference between the nearby two fingers of 120◦. Since the +structure of each finger is identical, theoretically, the correctness of the data, including the states of the object and +fingers, remains unchanged after rotating clockwise (blue arrows in Figure1(b)) or counterclockwise (red arrows in +Figure 1(b)) around the central point of the arena by 120◦. We divided observations into robot and object states9 and +conducted the following mathematical calculations for augmentation: +Robot state +The state data of the robot is augmented by cycling the state data around the fingers: +F ingerState(θ) = F ingerState(θ + k · 120◦) +(2) +where θ ∈ (0◦, 120◦, 240◦) and k ∈ (0, 1, 2). For example, the state data of the robot will be rotated 120◦clockwise +when k is 1. +Object state +The x-y Cartesian coordinates of the object state is rotationally transformed by: +�Objx′ +Objy′ +� += +�cos(k · 120◦) +−sin(k · 120◦) +sin(k · 120◦) +cos(k · 120◦) +� +· +�Objx +Objy +� +(3) +where Objx and Objy are the xy coordinates values of the object and k ∈ (0, 1, 2), The z coordinate of the object +stays the same because the rotation transformation is conducted on the horizontal plane (the arena floor). +The data after rotational transformation is concatenated with the original data to form a large augmented dataset. +2.4 +Train & tune +The precision of the above augmentation approach only stands in the theoretical robotic system, which is not possible +in the real world since the physical parameters between each finger are slightly different. Hence, we first trained a +BC model with the large augmented dataset, which can alleviate the issue led by the compound error and acquire a +more general policy. Then the trained model is tuned on the actual data(before augmentation) at a lower learning rate, +making the final deployed BC model a better fit for the real-robot data distribution. +The neural network’s architecture of both BC and filter, and training parameters are given in the AppendixA.1. +Because the RRC dataset is large, our approach for processing the mixed datasets requires a machine with 16 GB +RAM, whereas for the expert datasets we require 32 GB. Our experiments of mixed dataset ran on a PC with an Intel +I7-10875H CPU(2.30 GHz × 16, 16 GB RAM) and an NVIDIA 2060 GPU (6 GB RAM). Moreover, our experiments +of the expert datasets ran on Sonic high-performance cluster at University College Dublin, where the machines are +configured with 2 Intel Xeon Gold 6152 CPUs(2.1 GHz × 22, 64 GB RAM). +Table 2: Real-robot evaluated scores of our approach and standard BC, where the Train&Tune is our full final +approach +Push expert +Push mixed +Lift expert +Lift mixed +Standard BC +628.12 +497.26 +928.65 +469.25 +Train&Tune +662.30 +636.49 +1129.70 +998.82 +Baseline +660.00 +429.00 +1064.00 +851.00 +3 +Results and Discussion +On expert datasets, our train & tune method effectively improves the BC models’ performance (see Table 2), demon- +strating the effectiveness of our augmentation approach. Our approach can be extended to tasks with specific geometry +properties, such as symmetry, and it effectively increases the data utilization efficiency. However, due to the theory- +to-real gap, the data inferred from mathematics may shift from the actual robotic data distribution, resulting in a +mismatch between the trained policy by the augmented dataset and the robot. Hence the policy should be tuned on +non-augmented data before deployment to make the trained model fit the actual robotic environment. +The BC models trained jointly by our filter and train & tune approach on the mixed dataset acquire the mean +return close to that in the expert dataset in the evaluation stage. They outperform the baselines, which demonstrates +the effectiveness of our proposed filter method. +9More details about the observation space see https://webdav.tuebingen.mpg.de/real-robot-challenge/2022/docs/tasks.html +5 + +4 +Conclusion +Our approach dramatically improves the performance of BC on both expert and mixed datasets, which exceeds the +baselines in all tasks and has reasonable generalization capability. In our future work, we will integrate the filter’s +iterative training process with the controller policy’s training process to avoid the complex iterative process of filtering. +References +[1] Mnih, V., Kavukcuoglu, K., Silver, D., Graves, A., Antonoglou, I., Wierstra, D. and Riedmiller, M., 2013. Playing +Atari with deep reinforcement learning. arXiv preprint arXiv:1312.5602. +[2] Lillicrap, T.P., Hunt, J.J., Pritzel, A., Heess, N., Erez, T., Tassa, Y., Silver, D. and Wierstra, D., 2015. Continuous +control with deep reinforcement learning. arXiv preprint arXiv:1509.02971. +[3] Schulman, J., Wolski, F., Dhariwal, P., Radford, A. and Klimov, O., 2017. Proximal policy optimization algorithms. +arXiv preprint arXiv:1707.06347. +[4] Haarnoja, T., Zhou, A., Abbeel, P. and Levine, S., 2018, July. Soft actor-critic: Off-policy maximum entropy deep +reinforcement learning with a stochastic actor. 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Offline reinforcement learning with implicit Q-learning. arXiv preprint +arXiv:2110.06169. +[20] Kingma, D.P. and Ba, J., 2014. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980. +[21] Wüthrich, M., Widmaier, F., Grimminger, F., Akpo, J., Joshi, S., Agrawal, V., Hammoud, B., Khadiv, M., +Bogdanovic, M., Berenz, V. and Viereck, J., 2020. Trifinger: An open-source robot for learning dexterity. arXiv +preprint arXiv:2008.03596. +6 + +A +Neural network +We implemented the BC by referencing d3rlpy[15]. Our implemtations of both BC and filter will be open-sourced on +GitHub:https://github.com/wq13552463699/Real-Robot-Challenge-2022.git after RRC2022. +A.1 +Architecture +Figure 6: Behaviour Cloning model’s neural network architecture +(a) Filter encoder’s neural network architecture +(b) Filter predictor’s neural network architecture +Figure 7: The neural network architectures of two separated components of the filter +A.2 +Hyper-parameter +A.2.1 +BC +Train +The training of BC in our approach includes train & tune. In the Train stage, we used Adam[20] for learning +the neural network parameters with the learning of 10−3, where the weight decay was not included. The Train process +lasted 50 epochs with a batch size of 1024. The hyperparameter of the Tune stage is the same as Train, except the +learning rate is 8 × 10−4. +A.2.2 +Filter +The filter training for the push mixed dataset lasts 3 iteration, while the lift mixed dataset is 4. For each iteration, we +used Adam[20] for learning the neural network parameters with the learning of 10−3, where the weight decay was not +included. The training process lasted 50 epochs with a batch size of 1024. +7 + +ReLu + BatchNorm +ReLu + BatchNorm +Softmax +Action+18 +128 +128 +2ReLu + BatchNorm +ReLu + BatchNorm +ReLu + BatchNorm +ReLu + BatchNorm +Observation +Tanh +Action +512 +512 +256 +128 ReLu + BatchNorm + ReLu + BatchNorm +ReLu + BatchNorm +ReLu + BatchNorm +Observation +2 +2 +256 +8 +1 +2 ++ +5 +5 +2 +1 \ No newline at end of file diff --git a/oNFPT4oBgHgl3EQfKjR1/content/tmp_files/load_file.txt b/oNFPT4oBgHgl3EQfKjR1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7df8c5320a412ace35a1e160415b519bb043795d --- /dev/null +++ b/oNFPT4oBgHgl3EQfKjR1/content/tmp_files/load_file.txt @@ -0,0 +1,420 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf,len=419 +page_content='Winning Solution of Real Robot Challenge III Qiang Wang, Robert McCarthy, David Cordova Bulens, and Stephen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Redmond October 2022 Abstract This report introduces our winning solution of the real-robot phase of the Real Robot Challenge (RRC) 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The goal of this year’s challenge is to solve dexterous manipulation tasks with offline reinforcement learning (RL) or imitation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' To this end, participants are provided with datasets containing dozens of hours of robotic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' For each task an expert1 dataset and a mixed dataset are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' In our experiments, when learning from the expert datasets, we find standard Behavioral Cloning (BC) outperforms state-of-the-art offline RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' When learning from the mixed datasets, BC performs poorly, as expected, while surprisingly offline RL performs suboptimally, failing to match the average performance of the baseline model used for collecting the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' To remedy this, motivated by the strong performance of BC on the expert datasets we elect to use a semi-supervised classification technique to filter the subset of expert data out from the mixed datasets, and subsequently perform BC on this extracted subset of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' To further improve results, in all settings we use a simple data augmentation method that exploits the geometric symmetry of the RRC physical robotic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Our submitted BC policies each surpass the mean return2 of their respective raw datasets, and the policies trained on the filtered mixed datasets come close to matching the performances of those trained on the expert datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 1 Real Robot Challenge 2022 Data-driven learning methods are promising for dexterous robotic manipulation tasks - they can learn complex, skillful manipulation strategies from scratch, and indeed have begun to outperform more traditional control methods in some settings (CITE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' However, learning manipulation policies from real robots usually involves costly and time-consuming data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' However, these issues can potentially be mitigated by making use of available pre-collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The RRC 2022 seeks to encourage the development of offline algorithms that can make efficient use of such real-world data, and thus improve the performance of our learning methods when applied in practical real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' (a) The real TriFinger robot used in the RRC 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' (b) The top view of the robot arena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Figure 1: The TriFinger robot[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Three identical robotic fingers are equally spaced around the circular arena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The colored cube is the target object that must be moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' In the RRC 2022, participant are provided with dozens of hours of TriFinger[21] robotic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Datasets are provided for two robotic tasks, push and lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' In the push task, the cube must be moved to target positions on the arena floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' In the more challenging lift task, the cube must be lifted and maintained at a target position and orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' For each task, two separate datasets are provided;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' one collected by an expert policy (the expert dataset), the other collected by a number of policies with various skill levels (mixed dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Thus, there are four separate datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Participants must submit a different policy for each respective dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Only learning-based approaches are permitted, and datasets cannot be combined in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Access is also provided to a cluster of real TriFinger robots (see Figure1(a)) to allow participants to evaluate their trained policies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' however, data from the evaluation, either performed in simulation or on the real robot, is not allowed to be used to further refine a policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' For more details, please see the official website of RRC3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' We now describe our approach to the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 1For convenience, we name the datasets collected by expert policies as the expert dataset and the data contained within it as expert data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' For the dataset collected by mixed policies, we name it the mixed dataset and the data contained within it mixed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 2The cumulative reward acquired in each episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 3https://real-robot-challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='com/protocol 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='13019v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='RO] 30 Jan 2023 Table 1: Real-robot evaluated score comparison of algorithms trained from both expert and mixed dataset of the lift task, where the Baseline refers to the mean episodic return in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' BC BCQ PLAS IQL Ours Baseline Lift exp 928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='91 646.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='18 911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='91 789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='12 1129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='70 1064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='00 Lift mix 489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='25 432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='75 711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='28 550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='23 998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='82 851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='00 2 Methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='1 Our approach In our experiments we found that standard BC outperforms a number of state-of-the-art offline reinforcement learning algorithms4 on the lift expert dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Specifically, the offline reinforcement learning algorithms we tested included Batch-Constrained Q-learning (BCQ)[17], Policy in Latent Action Space (PLAS)[18], and Implicit Q-Learning (IQL)[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The evaluated results of these algorithms on the lift task are compared in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Now, when experimenting on the mixed dataset, we find both standard BC and offline RL to perform relatively poorly, failing to reach the mean returns of the dataset in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' BC’s poor performance is to be expected – the mixed dataset presents two challenges for BC: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' It contains a large number of less successful episodes (see Figure 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' imitating the behaviour in these episodes will hurt performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' It contains data collected by multiple policies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=', the data when sorted by summed reward over the episode is multi-modal (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Unless accounted for explicitly, a naive BC approach may perform poorly on a multi-modal dataset[16], even if the data is of high quality, as it is trying to find a single policy that can replicate the behaviour of two or more very different policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Offline RL should in theory be capable of learning expert behaviour from such a mixed dataset, and indeed the tested algorithms outperform BC, however, performance is still suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Data filtering: Motivated by the strong performance of BC on the expert dataset, we attempt to improve results on the mixed dataset by splitting the mixed dataset into subsets, where each subset contains the data from a single policy, and we subsequently apply BC to the subset of data came from the best-performing policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Note, we find the approach of naively splitting the dataset into two subsets based on reward returns (for example, from where the red arrow points in Figure 2(b)) and applying BC to the subset with higher rewards leads to poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Here, the subset sorted by summed reward is still multimodal, containing a number of higher reward episodes collected by the weaker policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Many of these episodes collected by weaker policies can be further filtered out by increasing the reward return threshold (like Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 1), ensuring that almost all of the filtered data is collected by the higher performing policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' However, we find this new subset is then too small to train a BC model with good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Therefore, we propose a semi-supervised data filtering approach which can effectively extract the data collected by the best performing policy (expert policy) from the mixed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' (a) Reward distribution of the lift expert dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' (b) Reward distribution of the lift mixed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Figure 2: The histogram of the accumulated reward over an episode, calculating by summing all rewards of each episode together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The expert dataset consists mostly of successful episodes, in which the cube was moved to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The mixed dataset appears to consist two distinct peaks, therefore, we assume that the mixed dataset is collected jointly by at least two policies, at least one of which is an expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 4We primarily make use of algorithms implemented in the d3rlpy library[15] 2 200 175 150 Amount 125 100 75 50 25 0 200 400 600 800 1000 1200 1400 Summedreward overanepisode120- 100 80 Amount 60 40 20 0 0 200 400 600 800 1000 1200 1400 Summed reward overan episodeMethod summary: In summary, for our final submissions, we use BC-based approaches in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' On the mixed datasets we use our data filtering approach to extract the expert data and use it to train the BC policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' As a supervised learning algorithm, the BC is affected by the compound error led by the covariate shift[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The root cause is that the trained BC model encounters many unseen data during deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' One solution is to increase the training dataset size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' hence, to further enhance performances we also introduce a data augmentation method that exploits the symmetry of the robot environment using simple rotational transforms, which can increase the amount of the training data by three times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' We describe these methods in more detail in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='2 Expert data filter A data filter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=', a binary classifier that assigns a label to an epoch) is trained and applied to the mixed dataset to extract the data collected by the/an expert policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Figure 3 shows the filter training process, which includes: 1) First filtering the raw dataset such that only a small subset of the highest return episodes are retained (see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='2) to form a superb dataset, which is small, but, we expect, mostly consists of data collected by the expert policy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 2) Use the superb dataset to pre-train the filter classifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 3) Use the trained (or pre-trained) filter classifier to relabel the entire dataset, and then use the relabelled dataset to retrain the filter classifier again;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 4) Repeat step 3 until the filtered dataset’s membership does not change or vary, within some reasonable tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Figure 3: Filter training flow chart, where the blue arrows indicate the pre-training process and the red arrows indicate one iteration of the process of training the filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='1 Structure of the filter Figure 4 shows the filter’s structure, which is a classifier that outputs the binary prediction indicating whether the input data was collected by using an expert policy, or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The significant dimensional gap between the observation and the action may result in data bias;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' hence the dimension of the observation vector is reduced before concatenating with the action vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The classifier operates on vectors from a single time-step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' then, a post-processing step is performed by aggregating the classifier-applied label across all time-steps in one episode to get a final label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' This is done by summing the binary labels across all time-steps and then applying a threshold, where the threshold is selected by observing the performance of BC trained from the filtered subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Figure 4: The structure of the filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The encoder and predictor are fully-connected neural networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' the architecture is shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='2 Generate the training data The expert and non-expert policy have significantly different summed reward when the episode has a high difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' For this reason, the superb dataset was formed using the criterion presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 1: ExpertEpisode = � True, if (E(R[tstart]) ≤ thlow) AND (E(R[tend]) ≥ thhigh) False, otherwise (1) 3 Raw Filter n dataset Filtered Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 1 Binary BC Classifier subset Superb dataset Binary Classifiel Observation Expert Encoder Predictor OR Action Non-expertwhere R is the episodic reward5, a set of values, with each value for the reward of each time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' And the expectation operator E is implemented by taking the sample mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' tstart is the set of time indices addressing reward value for the first five time-steps in the episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' tend is the set of time indices addressing reward value for the last 150 time-steps in the episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' thlow value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='33 for both tasks, and thhigh value for the lift task is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='98, and for the push task is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The first constraint (to the left of the logical AND in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 1) makes only the relatively challenging episodes be chosen by taking those with lower average rewards over the initial five time-steps, which removes episodes where the cube started at or near the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The second constraint (to the right of the logical AND in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 1) is set to choose episodes with excellent behavioural quality by taking those having high average rewards over the final 150 time-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The second constraint is based on the assumption that if rewards are high throughout the end of episode, then the agents has performed the task well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Hence, by combining both constraints, we only keep difficult episodes where the agent manages to successfully completes them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' it is highly possible to make the vast majority of the filtered episodes collected by the expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The setting of the above constants(tstart, tend, thlow, thhigh) performs the best over our attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' In the beginning, we roughly set reasonable values for the above constants, enabling the classifier’s training on the superb dataset to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Afterwards, we fine-tune these values based on observing the loss during the classifier’s training(smaller converged loss and smoother loss curve means the threshold is better) and the evaluated score of BC trained from the final filtered subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' We name the data from either the superb and filtered subset as labelled EXPERT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' that is, coming from an expert policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' We propose two ways to generate NON-EXPERT-labelled training data, which we consider to not come from an expert policy, shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 1) Combine the observations of EXPERT samples with the actions in the part of data that is not filtered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 2) Combine the observations of EXPERT samples with randomly-generated action data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Only the actions with a relatively larger difference from the EXPERT actions can be deemed as negative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' this reduces the overall similarity between NON-EXPERT and EXPERT training samples and thus encourages the learning of the filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Figure 5: The illustration of generating the EXPERT and NON-EXPERT samples for training the filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The filtered subset involves the data filtered by both Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 1 and the trained classifier filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The Non-filtered subset is the raw dataset that excludes the Filtered subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='3 Filter process The filter is a binary classifier, which takes the SOFTMAX function as the last-layer activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' One of SOFTMAX’s outputs estimates the probability that the episode is expert-collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' In the filtering phase, we feed the raw dataset into the filter network to obtain the estimated probabilities of being expert of each time-step and sum them in each episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' This summed probability value is called the confidence that one episode is expert-collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' When the confidence of one episode is greater than a specific threshold thconf, it is deemed a EXPERT episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The selection of the thconf value may lead to results with a significant difference, which is determined by both reasonable hypotheses and observing the evaluated score of BC trained from the final filtered subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Our experiment found that the output confidence is large over most episodes at the beginning of the iterations6, with most of the episodes in the lift mixed dataset having confidence in the range 1100-1500(max possible confidence is 1500)7, and that of the push mixed dataset is 650-750 (max possible confidence is 750)8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' This is led by the high similarity between the EXPERT and NON-EXPERT samples since they share the same observations in the filtered subset, as seen in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Hence, we set a relatively large thconf at the beginning of the iterations, which of the lift mixed dataset is 1420 and the push mixed dataset is 730.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' At the end of the iterations, we found that the confidence of a big part of episodes becomes smaller;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' we reasonably guess this is because the filter classifier becomes more robust, hence clarifying the NON-EXPERT more clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Therefore, to filter the final subset for training the BC, we set the thconf to a relatively smaller value making more episodes can be chosen, which of the lift mixed dataset is 1390 and the push mixed dataset is 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Same as above, this settings of thconf performs the best over our attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' In the beginning, we roughly set reasonable values through assumption and then we fine-tune these values based on observing the evaluated score of BC trained from the final filtered subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 5The method for calculating the reward of each time-step in the dataset was designed by the organizer and can be found in the RRC documentation: https://webdav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='tuebingen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='de/real-robot-challenge/2022/docs/tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='html#rewards-and-success 6The classifier filter is repeatedly trained by the latest filtered subset mentioned above at the beginning of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' We name each repeat process iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 7The episodic length of lift task is 1500 8The episodic length of push task is 750 4 EXPERT Raw dataset Observation Action Observations Observations NON-EXPERT Actions Actions Non-filtered Observation Action Filtered subset I Non-filtered subset Random Observation Action2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='3 Data augmentation through rotational transformation The top view of the robotic arena is shown in Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Three fingers of the robot are placed evenly around the center of the rounded working platform with an angle difference between the nearby two fingers of 120◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Since the structure of each finger is identical, theoretically, the correctness of the data, including the states of the object and fingers, remains unchanged after rotating clockwise (blue arrows in Figure1(b)) or counterclockwise (red arrows in Figure 1(b)) around the central point of the arena by 120◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' We divided observations into robot and object states9 and conducted the following mathematical calculations for augmentation: Robot state The state data of the robot is augmented by cycling the state data around the fingers: F ingerState(θ) = F ingerState(θ + k · 120◦) (2) where θ ∈ (0◦, 120◦, 240◦) and k ∈ (0, 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' For example, the state data of the robot will be rotated 120◦clockwise when k is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Object state The x-y Cartesian coordinates of the object state is rotationally transformed by: �Objx′ Objy′ � = �cos(k · 120◦) −sin(k · 120◦) sin(k · 120◦) cos(k · 120◦) � �Objx Objy � (3) where Objx and Objy are the xy coordinates values of the object and k ∈ (0, 1, 2), The z coordinate of the object stays the same because the rotation transformation is conducted on the horizontal plane (the arena floor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The data after rotational transformation is concatenated with the original data to form a large augmented dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='4 Train & tune The precision of the above augmentation approach only stands in the theoretical robotic system, which is not possible in the real world since the physical parameters between each finger are slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Hence, we first trained a BC model with the large augmented dataset, which can alleviate the issue led by the compound error and acquire a more general policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Then the trained model is tuned on the actual data(before augmentation) at a lower learning rate, making the final deployed BC model a better fit for the real-robot data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The neural network’s architecture of both BC and filter, and training parameters are given in the AppendixA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Because the RRC dataset is large, our approach for processing the mixed datasets requires a machine with 16 GB RAM, whereas for the expert datasets we require 32 GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Our experiments of mixed dataset ran on a PC with an Intel I7-10875H CPU(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='30 GHz × 16, 16 GB RAM) and an NVIDIA 2060 GPU (6 GB RAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Moreover, our experiments of the expert datasets ran on Sonic high-performance cluster at University College Dublin, where the machines are configured with 2 Intel Xeon Gold 6152 CPUs(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='1 GHz × 22, 64 GB RAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Table 2: Real-robot evaluated scores of our approach and standard BC, where the Train&Tune is our full final approach Push expert Push mixed Lift expert Lift mixed Standard BC 628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='12 497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='26 928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='65 469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='25 Train&Tune 662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='30 636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='49 1129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='70 998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='82 Baseline 660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='00 429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='00 1064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='00 851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='00 3 Results and Discussion On expert datasets, our train & tune method effectively improves the BC models’ performance (see Table 2), demon- strating the effectiveness of our augmentation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Our approach can be extended to tasks with specific geometry properties, such as symmetry, and it effectively increases the data utilization efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' However, due to the theory- to-real gap, the data inferred from mathematics may shift from the actual robotic data distribution, resulting in a mismatch between the trained policy by the augmented dataset and the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Hence the policy should be tuned on non-augmented data before deployment to make the trained model fit the actual robotic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The BC models trained jointly by our filter and train & tune approach on the mixed dataset acquire the mean return close to that in the expert dataset in the evaluation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' They outperform the baselines, which demonstrates the effectiveness of our proposed filter method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 9More details about the observation space see https://webdav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='tuebingen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='de/real-robot-challenge/2022/docs/tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='html 5 4 Conclusion Our approach dramatically improves the performance of BC on both expert and mixed datasets, which exceeds the baselines in all tasks and has reasonable generalization capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' In our future work, we will integrate the filter’s iterative training process with the controller policy’s training process to avoid the complex iterative process of filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' References [1] Mnih, V.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=', Akpo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=', Joshi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=', Agrawal, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=', Hammoud, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=', Khadiv, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=', Bogdanovic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=', Berenz, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' and Viereck, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Trifinger: An open-source robot for learning dexterity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' arXiv preprint arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='03596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 6 A Neural network We implemented the BC by referencing d3rlpy[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' Our implemtations of both BC and filter will be open-sourced on GitHub:https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='com/wq13552463699/Real-Robot-Challenge-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='git after RRC2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='1 Architecture Figure 6: Behaviour Cloning model’s neural network architecture (a) Filter encoder’s neural network architecture (b) Filter predictor’s neural network architecture Figure 7: The neural network architectures of two separated components of the filter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='2 Hyper-parameter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='1 BC Train The training of BC in our approach includes train & tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' In the Train stage, we used Adam[20] for learning the neural network parameters with the learning of 10−3, where the weight decay was not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The Train process lasted 50 epochs with a batch size of 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The hyperparameter of the Tune stage is the same as Train, except the learning rate is 8 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content='2 Filter The filter training for the push mixed dataset lasts 3 iteration, while the lift mixed dataset is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' For each iteration, we used Adam[20] for learning the neural network parameters with the learning of 10−3, where the weight decay was not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' The training process lasted 50 epochs with a batch size of 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} +page_content=' 7 ReLu + BatchNorm ReLu + BatchNorm Softmax Action+18 128 128 2ReLu + BatchNorm ReLu + BatchNorm ReLu + BatchNorm ReLu + BatchNorm Observation Tanh Action 512 512 256 128 ReLu + BatchNorm ReLu + BatchNorm ReLu + BatchNorm ReLu + BatchNorm Observation 2 2 256 8 1 2 + 5 5 2 1' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFPT4oBgHgl3EQfKjR1/content/2301.13019v1.pdf'} diff --git a/p9E2T4oBgHgl3EQfKgZi/content/tmp_files/2301.03703v1.pdf.txt b/p9E2T4oBgHgl3EQfKgZi/content/tmp_files/2301.03703v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..36e925ce1a72ead2dc5aa7d3e5153d83d800d8cf --- /dev/null +++ b/p9E2T4oBgHgl3EQfKgZi/content/tmp_files/2301.03703v1.pdf.txt @@ -0,0 +1,984 @@ +On the Susceptibility and Robustness of Time Series +Models through Adversarial Attack and Defense +Asadullah Hill Galib +Computer Science and Engineering +Michigan State University +galibasa@msu.edu +Bidhan Bashyal +Computer Science and Engineering +Michigan State University +bashyalb@msu.edu +Abstract +Under adversarial attacks, time series regression and classification are vulnerable. +Adversarial defense, on the other hand, can make the models more resilient. It is +important to evaluate how vulnerable different time series models are to attacks +and how well they recover using defense. The sensitivity to various attacks and +the robustness using the defense of several time series models are investigated in +this study. Experiments are run on seven time series models with three adversarial +attacks and one adversarial defense. According to the findings, all models, particu- +larly GRU and RNN, appear to be vulnerable. LSTM and GRU also have better +defense recovery. FGSM exceeds the competitors in terms of attacks. PGD attacks +are more difficult to recover from than other sorts of attacks. +1 +Introduction +Time series analysis is a vital problem in data mining with many applications including finance, +weather forecasting, industrial maintenance, and many others. Deep Learning (DL) methods have +shown great success in analyzing time series data [13]. One downside of DL methods is that they +can be fooled easily by generating adversarial attacks. Adversarial attacks in deep learning have +been vastly studied for image recognition and classification problem. Similarly, it is very important +to explore adversarial attacks in the time-series analysis as time series analysis uses various DL +algorithms. Designing a defense mechanism against these attacks is even essential for robustness. +From an adversarial standpoint, there has not been much research into DL models for time series +regression and classification. +DL models for time series are also susceptible under adversarial attacks [10, 11, 12, 9, 8, 7]. Most +of this research concentrate on performing adversarial attacks for either regression or classification +problems. Defense mechanism against the adversarial attacks for time series is also explored [9]. +Mostly, this research focuses on different adversarial attacks and defenses rather than the model +itself. But it’s also worth considering how vulnerable different time series models are to attacks +and how effectively they can recover utilizing defense. This research examines the susceptibility +and robustness of various time series models to various attacks and defenses. To the best of our +knowledge, no previous research has explored this. Our contribution and the things we are addressing +are summarized in Table 1. +In this paper, we use 7 different DL time series models, such as Long -Short Term Memory(LSTM), +Stacked-LSTM (LSTM with more than one layer), Gated Recurrent Unit (GRU), Recurrent Neural +Network (RNN), Convolutional Neural Network (CNN), CNN-LSTM and ConvLSTM to train for +regression and classification problems for time series analysis. Different adversarial mechanisms +such as Fast Gradient Sign Method (FGSM), Basic Iterative Method (BIM), and Projected Gradient +Descent (PGD) are used to attack the models. Additionally, we also implemented adversarial training +as a defense mechanism to tackle the attacks. We evaluate the performance of each adversarial attack +Preprint. Under review. +arXiv:2301.03703v1 [cs.LG] 9 Jan 2023 + +References +Regression +Classification +Models +Adversarial Attack +Adversarial +Defense +LSTM +GRU +RNN +CNN +CNN-LSTM +ConvLSTM +ResNet +Other +FGSM +BIM +PGD +Other +Fawaz et al., 2019 [10] +✓ +✓ +✓ +✓ +Harford et al., 2020 [11] +✓ +✓ +✓ +Mode et al., 2020 [12] +✓ +✓ +✓ +✓ +✓ +✓ +Siddiqui et al., 2020 [9] +✓ +✓ +✓ +✓ +✓ +✓ +Wu et al., 2022[8] +✓ +✓ +✓ +✓ +✓ +✓ +Rathore et al., 2020 [7] +✓ +✓ +✓ +✓ +Our Study +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +Table 1: Contribution of this Study +over the 7 different models in 6 different datasets (5 regression and 1 classification). Results suggest +all models are vulnerable, especially GRU and RNN. Also, LSTM and GRU show better recovery +through the defense. FGSM outperforms others in terms of attacks. Recovering from a PGD attack is +harder than others. +The rest of the paper is organized as follows. Section 2 briefly discusses the related works, Section +3 introduces the problem statement. Section 4 covers in detail the models we used for training, +adversarial attacks, and defense mechanism. Section 5 describes data sets, experimental setup, and +results. Section 6 provides discussions on the results. Section 7 briefly provides the individual +contribution. Section 8 concludes this paper. +2 +Related Works +Several studies investigate adversarial attacks and defenses for time series problems. [10] proposes +to use adversarial attack techniques such as (FGSM and BIM) to reduce model performance when +classifying instances at test time. As a time series classifier, it employs ResNet. The classifier is +subject to adversarial attacks, according to its findings. It does not take into account PGD attacks +and only attacks one classifier model. It also focuses solely on time series classification rather than +regression. [11] studies black-box and white-box attacks on multivariate time series. Adversarial +Transformation Network (ATN) and Gradient Adversarial Transformation Network (GATN) are used +to create adversarial samples. It makes use of 18 datasets to attack 1-Nearest Neighbor Dynamic Time +Warping (1-NN DTW) and a Fully Convolutional Network (FCN). On all 18 datasets, both models +were vulnerable to attack. It does not consider time series regression and typical attacks (FGSM, +BIM, PGD). Using time-series data, [9] performs extensive benchmarking of well-proven adversarial +defensive approaches. It uses 2 white-box attacks (FGSM and PGD) and 3 black-box attacks (Noise +Attack, Boundary Attack, and Simple Black-box Attack (SIMB). It evaluates robustness using +multiple adversarial defenses (adversarial training, TRADES, Feature Denoising). It, on the other +hand, ignores the regression problem and does not assess the robustness of different models, instead of +analyzing multiple defenses with a single classifier. FGSM and BIM attacks are used by another study +[7] to undertake untargeted, targeted, and universal adversarial attacks on time series classification +problems. It does not, however, make any adversarial defenses and focuses solely on the classification +problem. +[12] presents the concept of adversarial attacks (FGSM and BIM) on various deep learning models +(CNN, LSTM, and GRU) for multivariate linear regression. Although it introduces the concept of +adversarial attacks on regression tasks, it does not take account into the defense against the attacks. +The results showed that DL regression models are vulnerable to adversarial attacks. On LSTM, CNN, +RNN, and Multi-Head Attention Network, [8] examines two different attacks: FGSM and Adversarial +attack with importance measurement (AAIM) (MHANet). However, it exclusively concentrates on +the regression problem and makes no adversarial defenses. +None of the studies take both regression and categorization into account. Also, none of them +investigate how vulnerable various models are to attacks and how well they can recover using defense. +3 +Problem Statement +Previous works focus only on the adversarial attacks only for both regression and the classification +problems in the series data. We have implemented the adversarial attacks and defense mechanism +against attacks on both the classification and regression problems in the time series data. Adversarial +attacks refer to finding adversarial examples for well-trained models. For a classification task, we +2 + +use f(x; θ) to denote a model that maps an input to a discrete label set with k classes. Given a +perturbation size ϵ, the adversary tries to find a perturbation δ that maximizes the loss. Thus, the +adversarial counterpart xi of x can be expressed as: xi = x + δ. Some of the common attacks for +generating adversarial attacks are FGSM, BIM, and PGD. +We designed a defense against these attacks using adversarial training for adversarial robustness. +In general, Adversarial robustness is the model’s performance on test data with adversarial attacks. +Similar attacks techniques can also be used for adversarial training by expanding it to the min-max +optimization problem. Min-max optimization problem consists of two parts: inner maximization +which can be single or multiple steps (depending on the attack) is used to generate maximum +perturbation and outer minimization is a single step stochastic gradient descent to minimize the loss +of a model. +4 +Methodology +Various DL algorithms are trained for regression and classification problems for time series analysis. +These models are attacked by generating perturbations on the test data. We implemented adversarial +training as a defense mechanism to tackle the attacks. The subsections below introduce the models +used for training, methods for generating adversarial attacks, and adversarial training mechanisms. +Figure 1 depicts the overview of this study. +Figure 1: Overview of this study +4.1 +Models +For the time series regression and classification, we used the following 7 models: LSTM (Long +short-term memory), Stacked LSTM, GRU (Gated recurrent unit), RNN (Recurrent neural network), +CNN (Convolutional neural network), CNN-LSTM, and ConvLSTM. The models are widely used for +time series problems due to their capability of capturing sequential patterns. For sequential data, RNN, +LSTM, and GRU are the go-to models. CNN is often used for time-series data as its convolutional +capability is useful for modeling time-series patterns. CNN-LSTM and ConvLSTM combine the +CNN and LSTM architectures, the latter of which is especially suitable for two-dimensional data. All +the models are evaluated on time series classification and regression. Adversarial attacks and defense +are employed in these models for analyzing their vulnerability and robustness. +4.2 +Adversarial Attacks +We have employed three well-known gradient-based attacks. Gradient-based attacks employ a +perturbation for the input time series by altering the back-propagation process slightly. They use +the gradients with respect to the input given the output for perturbation. All of these attacks are +performed on the models for time series classification and regression. +3 + +Time Series +Regression +Classification +Stacked +Models +LSTM +GRU +RNN +CNN +CNN-LSTM +ConvLSTM +LSTM +Adversarial +FGSM +BIM +PGD +Attacks +Adversarial +Adversarial +Defense +Training4.2.1 +Fast Sign Gradient Method (FGSM) +The fast gradient sign method (FGSM) [2] creates an adversarial example by leveraging the neural +network’s gradients. The approach creates a adversarial input that maximizes the loss for an original +input by using the gradients of the loss with respect to the original input. The following equation can +be used to summarize this: +xadversarial = x + ϵ × sign((∆xJ(θ, x, y)) +(1) +where xadversarial is the adversarial input, x is the original input, y is the original target variable, ϵ is +the multiplier to ensure the perturbations are small, θ is the model parameters, and J is the loss. +4.2.2 +Basic Iterative Method (BIM) +Basic Iterative Method (BIM) [1] is an enhancement of the FGSM. It proposes repeating the FGSM +step with a small step size (α) and clipping the intermediate results after each step to verify that +they are in the same neighborhood as the original input. It initializes the initial adversarial sample +as the original input. Then it iteratively updates the adversarial sample while keeping it within the +neighborhood of the original input. The size of the neighborhood is set by the hyperparameter ϵ. +4.2.3 +Projected Gradient Descent (PGD) +Projected Gradient Descent (PGD) attack is quite similar to the BIM attack. The main difference +is that PGD starts the adversarial example at a random location in the neighborhood (set by the +multiplier ϵ) and restarts it randomly, whereas BIM starts at the original input. +4.3 +Adversarial Training +As a defense against these attacks, we performed adversarial training using similar attack methods +for all the models in all the datasets. For each model, we created a new training set by combining +the original training set and perturbed training sets using each attack. Thus for each dataset, we +performed 21 adversarial pieces of training altogether 3 (FGSM, BIM, and PGD) for 7 different +models. A new model with adversarial training is used to calculate adversarial robustness on the +original perturbed testing data. +Each adversarial training follows the principle of min-max optimization. Adversarial training +with FGSM involves a single step of inner maximization using FGSM and a single step of outer +minimization while adversarial training with BIM and PGD involves 5 steps of inner maximization +using BIM and PGD and a single of outer minimization using stochastic gradient descent. +5 +Experimental Evaluation +5.1 +Data sets +For classification, 1 data set is used as follows: +• Temperature[6]: It is derived from a Kaggle weather dataset. The data is based on hourly +temperature data for a city over ten years. We use the first 23 time steps in the window as +the predictor variables and the last time step as the target variable. +• Power Consumption[5]: It is derived from Kaggle which comes from PJM (regional trans- +mission organization in the United States). It contains hourly energy consumption data in +megawatts (MW) for the eastern part of the United State over three years. We use the first +23 time steps in the window as the predictor variables and the last time step as the target +variable. +• Humidity[6]: It is derived from a Kaggle weather dataset. The data is based on hourly +humidity data for a city over ten years. We use the first 23 time steps in the window as the +predictor variables and the last time step as the target variable. +• Wind Speed[6]: It is derived from a Kaggle weather dataset. The data is based on hourly +wind speed data for a city over ten years. We use the first 23 time steps in the window as the +predictor variables and the last time step as the target variable. +4 + +• Hurricane[4]: This corresponds to tropical cyclone intensity data obtained from the HUR- +DAT2 database [4]. For each hurricane, wind speeds (intensities) were reported at every +6-hour interval. We use the first 23 time steps in the window as the predictor variables and +the last time step as the target variable. +• Eye State[3]: This data is collected from UCI Machine Learning Repository-[3]. It is from +one continuous EEG measurement with the Emotiv EEG Neuroheadset. The duration of +the measurement was 117 seconds. The eye state ’1’ indicates the eye-closed and ’0’ the +eye-open state. +5.2 +Experimental Setup +We used different hyperparameters for the model’s architecture and adversarial attacks. For the +model’s architecture, the number of layers, hidden nodes, and epochs is hyperparameters to be tuned. +Similarly, for attacks the hyperparameters are perturbation budget ϵ, step size α, and the number of +iterations in BIM and PGD. These hyperparameters are tuned differently depending on the attack and +the model. We used ϵ between 0.1 to 0.3 and α between 0.1 and 0.2 depending on the model. +5.3 +Results +We performed adversarial attacks and adversarial training on 5 regression problems (Power Consump- +tion, Wind Speed, Temperature, Humidity, and Hurricane) and 1 classification problem (Eye State +Classification) for time series analysis. Accuracy is used as a metric to determine the performance +of the classification model. Table 2 depicts the results for our classification task. According to that, +LSTM has the best performance for initial training with an accuracy of 82.46 %. When three different +adversarial attacks (FGSM, BIM, and PGD) are applied to the model, the accuracy drops to 58.54 %, +72.83 %, and 62.43% respectively. The adversarial training increases the adversarial robustness of +the model, boosting its accuracy to 81.28%, 82.46%, and 65.13%. +For the 5 regression tasks, RMSE is used as a metric for evaluating the performance of the model. For +instance, the results of the Temperature dataset are summarized in Table 3, the GRU model during +the initial training has the best performance with an RMSE of 0.01. When three different adversarial +attacks (FGSM, BIM, and PGD) are applied to the model, the RMSE increases to 0.09, 0.09, and +0.07 respectively. The adversarial training increases the adversarial robustness of the model, reducing +RMSE to 0.04,0.03, and 0.04 respectively. Other data sets show a similar pattern. Table 4, 5, 6, 7 +illustrate the rest of the four regression problems on Power Consumption, Wind Speed Humidity, and +Hurricane dataset respectively. +Models +LSTM +Stacked +LSTM +GRU +RNN +CNN +CNN- +LSTM +ConvLSTM +Accuracy for +models +0.82 +0.82 +0.81 +0.75 +0.77 +0.76 +0.79 +Accuracy for +Adversarial +Attacks +FGSM +0.59 +0.62 +0.59 +0.65 +0.66 +0.54 +0.61 +BIM +0.72 +0.74 +0.74 +0.73 +0.74 +0.69 +0.75 +PGD +0.62 +0.67 +0.61 +0.66 +0.65 +0.59 +0.56 +Accuracy after +Adversarial +Training +FGSM +0.81 +0.82 +0.81 +0.71 +0.78 +0.73 +0.81 +BIM +0.82 +0.82 +0.81 +0.66 +0.77 +0.79 +0.81 +PGD +0.65 +0.70 +0.65 +0.63 +0.67 +0.68 +0.68 +Table 2: Adversarial attacks and defenses on different models for time series classification (Eye State) +Models +LSTM +Stacked +LSTM +GRU +RNN +CNN +CNN- +LSTM +ConvLSTM +RMSE for +models +0.03 +0.03 +0.01 +0.02 +0.02 +0.04 +0.02 +RMSE for +Adversarial +Attacks +FGSM +0.09 +0.1 +0.09 +0.09 +0.09 +0.07 +0.09 +BIM +0.09 +0.1 +0.09 +0.09 +0.09 +0.07 +0.09 +PGD +0.06 +0.08 +0.07 +0.07 +0.06 +0.06 +0.07 +RMSE after +Adversarial +Training +FGSM +0.03 +0.04 +0.04 +0.04 +0.03 +0.04 +0.03 +BIM +0.06 +0.06 +0.03 +0.05 +0.05 +0.06 +0.05 +PGD +0.04 +0.04 +0.04 +0.04 +0.05 +0.05 +0.04 +Table 3: Adversarial attacks and defenses on different models for time series regression (Temperature) +5 + +Models +LSTM +Stacked +LSTM +GRU +RNN +CNN +CNN- +LSTM +ConvLSTM +RMSE for +models +0.03 +0.03 +0.02 +0.02 +0.02 +0.03 +0.02 +RMSE for +Adversarial +Attacks +FGSM +0.26 +0.22 +0.28 +0.27 +0.24 +0.24 +0.28 +BIM +0.09 +0.08 +0.09 +0.09 +0.1 +0.1 +0.1 +PGD +0.18 +0.14 +0.25 +0.22 +0.21 +0.22 +0.18 +RMSE after +Adversarial +Training +FGSM +0.06 +0.06 +0.06 +0.07 +0.04 +0.05 +0.05 +BIM +0.04 +0.04 +0.05 +0.05 +0.03 +0.04 +0.04 +PGD +0.08 +0.07 +0.09 +0.08 +0.07 +0.07 +0.07 +Table 4: Adversarial attacks and defenses on different models for time series regression (Power +Consumption) +Models +LSTM +Stacked +LSTM +GRU +RNN +CNN +CNN- +LSTM +ConvLSTM +RMSE for +models +0.04 +0.04 +0.04 +0.03 +0.04 +0.05 +0.03 +RMSE for +Adversarial +Attacks +FGSM +0.13 +0.12 +0.16 +0.16 +0.14 +0.18 +0.14 +BIM +0.13 +0.13 +0.16 +0.16 +0.14 +0.18 +0.14 +PGD +0.12 +0.1 +0.14 +0.13 +0.12 +0.16 +0.13 +RMSE after +Adversarial +Training +FGSM +0.08 +0.08 +0.1 +0.1 +0.08 +0.14 +0.08 +BIM +0.11 +0.1 +0.12 +0.13 +0.11 +0.14 +0.13 +PGD +0.08 +0.09 +0.1 +0.1 +0.1 +0.13 +0.1 +Table 5: Adversarial attacks and defenses on different models for time series regression (Humidity) +Models +LSTM +Stacked +LSTM +GRU +RNN +CNN +CNN- +LSTM +ConvLSTM +RMSE for +models +0.04 +0.04 +0.01 +0.02 +0.02 +0.04 +0.04 +RMSE for +Adversarial +Attacks +FGSM +0.12 +0.12 +0.09 +0.09 +0.09 +0.07 +0.11 +BIM +0.12 +0.12 +0.09 +0.09 +0.09 +0.07 +0.11 +PGD +0.09 +0.09 +0.07 +0.07 +0.06 +0.06 +0.11 +RMSE after +Adversarial +Training +FGSM +0.06 +0.07 +0.04 +0.04 +0.05 +0.04 +0.08 +BIM +0.09 +0.1 +0.06 +0.05 +0.05 +0.04 +0.1 +PGD +0.06 +0.06 +0.04 +0.04 +0.05 +0.05 +0.08 +Table 6: Adversarial attacks and defenses on different models for time series regression (Wind Speed) +Models +LSTM +Stacked +LSTM +GRU +RNN +CNN +CNN- +LSTM +ConvLSTM +RMSE for +models +0.21 +0.22 +0.19 +0.19 +0.21 +0.43 +0.19 +RMSE for +Adversarial +Attacks +FGSM +0.3 +0.32 +00.39 +0.38 +0.39 +0.6 +0.36 +BIM +0.27 +0.29 +0.35 +0.34 +0.35 +0.56 +0.32 +PGD +0.28 +0.29 +0.36 +0.35 +0.36 +0.55 +0.34 +RMSE after +Adversarial +Training +FGSM +0.29 +0.28 +0.29 +0.29 +0.29 +0.55 +0.3 +BIM +0.26 +0.29 +0.23 +0.27 +0.3 +0.57 +0.27 +PGD +0.3 +0.28 +0.26 +0.28 +0.29 +0.51 +0.3 +Table 7: Adversarial attacks and defenses on different models for time series regression (Hurricane +Intensity) +6 + +Figure 2: Actual Temperature vs Predicted Temperature by LSTM model for the Temperature dataset +(a) 1 +(b) 2 +(c) 3 +(d) 4 +(e) 5 +(f) 6 +Figure 3: Time series plot for actual temperature vs predicted temperature in LSTM models after +adversarial attacks and adversarial training( [1]Adv attack with FGSM [2] Adv training with FGSM +[3] Adv attack with BIM [4] Adv training with BIM [5] Adv attack with PGD [6] Adv training with +PGD) +6 +Insights/Discussion +Based on our experiments on the data sets we used, we observed that FGSM generates a much +stronger attack than both BIM and PGD. BIM performs poorly among these three attacks. BIM’s +initial perturbation is set to the original input. It might be the reason that it can not perturb much as it +starts from the original input. Apart from attack generation, FGSM also has higher training efficiency +than both BIM and PGD. +For the adversarial defense, recovering from the FGSM perturbed model is the easiest one. But, +recovering from a PGD attack is the hardest. PGD’s random initialization and random restarts might +make it harder for recovery. +7 + +Stacked LSTM model +0.8 +ActualTemperature +Predicted Temperature +0.7 +0.6 +Temperature +0.5 +0.4 +0.3 +0.2 +0 +50 +100 +150 +200 +250 +300 +350 +400 +TimeFGsM perturbed stacked LsTM +0.8 +0.7 +0.6 +Temperature +0.5 +0.4 +0.3 +0.2 +ActualTemperature +PredictedTemperature +0.1 +0 +50 +100 +150 +200 +250 +300 +350 +400 +TimeFGsM-Stacked LsTM model (Perturbed) +0.8 +ActualTemperature +PredictedTemperature +0.7 +0.6 +Temperature +0.5 +0.4 +0.3 +0.2 +0 +50 +100 +150 +200 +250 +300 +350 +400 +TimeBIM perturbed Stacked LsTM +0.8 +0.7 +0.6 +Temperature +0.5 +0.4 +0.3 +0.2 +ActualTemperature +PredictedTemperature +0.1 +0 +50 +100 +150 +200 +250 +300 +350 +400 +TimeBIM-Stacked LsTM model (Perturbed) +0.8 +0.7 +0.6 +rature +0.5 +0.3 +ActualTemperature +0.2 +PredictedTemperature +0 +50 +100 +150 +200 +250 +300 +350 +400 +TimePGD perturbed Stacked LSTM +0.8 +1'0 +0.6 +Temperature +0.5 +0.4 +0.3 +0.2 +ActualTemperature +PredictedTemperature +0.1 +0 +50 +100 +150 +200 +250 +300 +350 +400 +TimePGD-StackedLSTMmodel(Perturbed) +0.8 +ActualTemperature +PredictedTemperature +0.7 +0.6 +Temperature +0.5 +0.4 +0.3 +0.2 +0 +50 +100 +150 +200 +250 +300 +350 +400 +TimeAmong different time series models, all-time series models are vulnerable to adversarial attacks. +GRU and RNN seem to be more vulnerable than the others. In terms of recovery, all models can +recover from the attacks, but LSTM and GRU can recover slightly better. +7 +Individual Contributions +Bidhan:Literature review of [10][11] [12]; implemented 3 models (LSTM, RNN and Stacked-LSTM); +Implemented Adversarial training part. +Asadullah Hill Galib:Literature review of [9][8] [7]; implemented 4 models (GRU, CNN, CNN- +LSTM, ConvLSTM); Implemented Adversarial attacks part. +8 +Conclusion +In this paper, we look at how vulnerable different time series models are under adversarial attacks and +how well they recover using adversarial defense. We evaluated 7-time series models on three gradient- +based attack mechanisms and adversarial training-based defense against them for both regression and +classification problems. FGSM attack has a much stronger effect on generating adversarial attacks +and efficient adversarial training. All models, especially GRU and RNN, appear to be vulnerable, +according to the findings. In addition, LSTM and GRU have higher defense recovery. In terms of +attacks, FGSM outperforms the competition. It’s more difficult to recover from a PGD attack than it +is for other types of attacks. In the future, we will extend this work by incorporating more types of +attacks and defense mechanisms. +References +[1] Kurakin, A., Goodfellow, I., Bengio, S. & Others Adversarial examples in the physical world. +(2016) +[2] Goodfellow, I., Shlens, J. & Szegedy, C. Explaining and harnessing adversarial examples. ArXiv +Preprint arXiv:1412.6572. (2014) +[3] Roesler, O. EEG Eye State. (UCI Machine Learning Repository,2013) +[4] Landsea, C. & Franklin, J. Atlantic hurricane database uncertainty and presentation of a new +database format. Monthly Weather Review. 141, 3576-3592 (2013) +[5] Mulla, +R. +Hourly +Energy +Consumption. +Kaggle. +(2018), +https://www.kaggle.com/datasets/robikscube/hourly-energy-consumption +[6] Muthukumar, J. Weather Dataset. Kaggle. (2017,12), https://www.kaggle.com/muthuj7/weather- +dataset +[7] Rathore, P., Basak, A., Nistala, S. & Runkana, V. Untargeted, Targeted and Universal Adversarial +Attacks and Defenses on Time Series. 2020 International Joint Conference On Neural Networks +(IJCNN). pp. 1-8 (2020) +[8] Wu, T., Wang, X., Qiao, S., Xian, X., Liu, Y. & Zhang, L. Small perturbations are enough: +Adversarial attacks on time series prediction. Information Sciences. 587 pp. 794-812 (2022) +[9] Siddiqui, S., Dengel, A. & Ahmed, S. Benchmarking adversarial attacks and defenses for +time-series data. International Conference On Neural Information Processing. pp. 544-554 +(2020) +[10] Fawaz, H., Forestier, G., Weber, J., Idoumghar, L. & Muller, P. Adversarial attacks on deep +neural networks for time series classification. 2019 International Joint Conference On Neural +Networks (IJCNN). pp. 1-8 (2019) +[11] Harford, S., Karim, F. & Darabi, H. Adversarial attacks on multivariate time series. ArXiv +Preprint arXiv:2004.00410. (2020) +[12] Mode, G. & Hoque, K. Adversarial examples in deep learning for multivariate time series +regression. 2020 IEEE Applied Imagery Pattern Recognition Workshop (AIPR). pp. 1-10 (2020) +[13] Sezer, O., Gudelek, M. & Ozbayoglu, A. Financial Time Series Forecasting with Deep Learning +: A Systematic Literature Review: 2005-2019. (arXiv,2019), https://arxiv.org/abs/1911.13288 +8 + diff --git a/p9E2T4oBgHgl3EQfKgZi/content/tmp_files/load_file.txt b/p9E2T4oBgHgl3EQfKgZi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..536dc4ef55d324de1123da7040888f4959542247 --- /dev/null +++ b/p9E2T4oBgHgl3EQfKgZi/content/tmp_files/load_file.txt @@ -0,0 +1,647 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf,len=646 +page_content='On the Susceptibility and Robustness of Time Series Models through Adversarial Attack and Defense Asadullah Hill Galib Computer Science and Engineering Michigan State University galibasa@msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='edu Bidhan Bashyal Computer Science and Engineering Michigan State University bashyalb@msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='edu Abstract Under adversarial attacks, time series regression and classification are vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Adversarial defense, on the other hand, can make the models more resilient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' It is important to evaluate how vulnerable different time series models are to attacks and how well they recover using defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The sensitivity to various attacks and the robustness using the defense of several time series models are investigated in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Experiments are run on seven time series models with three adversarial attacks and one adversarial defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' According to the findings, all models, particu- larly GRU and RNN, appear to be vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' LSTM and GRU also have better defense recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' FGSM exceeds the competitors in terms of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' PGD attacks are more difficult to recover from than other sorts of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' 1 Introduction Time series analysis is a vital problem in data mining with many applications including finance, weather forecasting, industrial maintenance, and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Deep Learning (DL) methods have shown great success in analyzing time series data [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' One downside of DL methods is that they can be fooled easily by generating adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Adversarial attacks in deep learning have been vastly studied for image recognition and classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Similarly, it is very important to explore adversarial attacks in the time-series analysis as time series analysis uses various DL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Designing a defense mechanism against these attacks is even essential for robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' From an adversarial standpoint, there has not been much research into DL models for time series regression and classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' DL models for time series are also susceptible under adversarial attacks [10, 11, 12, 9, 8, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Most of this research concentrate on performing adversarial attacks for either regression or classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Defense mechanism against the adversarial attacks for time series is also explored [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Mostly, this research focuses on different adversarial attacks and defenses rather than the model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' But it’s also worth considering how vulnerable different time series models are to attacks and how effectively they can recover utilizing defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' This research examines the susceptibility and robustness of various time series models to various attacks and defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' To the best of our knowledge, no previous research has explored this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Our contribution and the things we are addressing are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' In this paper, we use 7 different DL time series models, such as Long -Short Term Memory(LSTM), Stacked-LSTM (LSTM with more than one layer), Gated Recurrent Unit (GRU), Recurrent Neural Network (RNN), Convolutional Neural Network (CNN), CNN-LSTM and ConvLSTM to train for regression and classification problems for time series analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Different adversarial mechanisms such as Fast Gradient Sign Method (FGSM), Basic Iterative Method (BIM), and Projected Gradient Descent (PGD) are used to attack the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Additionally, we also implemented adversarial training as a defense mechanism to tackle the attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' We evaluate the performance of each adversarial attack Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='03703v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='LG] 9 Jan 2023 References Regression Classification Models Adversarial Attack Adversarial Defense LSTM GRU RNN CNN CNN-LSTM ConvLSTM ResNet Other FGSM BIM PGD Other Fawaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=', 2019 [10] ✓ ✓ ✓ ✓ Harford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=', 2020 [11] ✓ ✓ ✓ Mode et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=', 2020 [12] ✓ ✓ ✓ ✓ ✓ ✓ Siddiqui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=', 2020 [9] ✓ ✓ ✓ ✓ ✓ ✓ Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=', 2022[8] ✓ ✓ ✓ ✓ ✓ ✓ Rathore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=', 2020 [7] ✓ ✓ ✓ ✓ Our Study ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Table 1: Contribution of this Study over the 7 different models in 6 different datasets (5 regression and 1 classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Results suggest all models are vulnerable, especially GRU and RNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Also, LSTM and GRU show better recovery through the defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' FGSM outperforms others in terms of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Recovering from a PGD attack is harder than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Section 2 briefly discusses the related works, Section 3 introduces the problem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Section 4 covers in detail the models we used for training, adversarial attacks, and defense mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Section 5 describes data sets, experimental setup, and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Section 6 provides discussions on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Section 7 briefly provides the individual contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Section 8 concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' 2 Related Works Several studies investigate adversarial attacks and defenses for time series problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' [10] proposes to use adversarial attack techniques such as (FGSM and BIM) to reduce model performance when classifying instances at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' As a time series classifier, it employs ResNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The classifier is subject to adversarial attacks, according to its findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' It does not take into account PGD attacks and only attacks one classifier model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' It also focuses solely on time series classification rather than regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' [11] studies black-box and white-box attacks on multivariate time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Adversarial Transformation Network (ATN) and Gradient Adversarial Transformation Network (GATN) are used to create adversarial samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' It makes use of 18 datasets to attack 1-Nearest Neighbor Dynamic Time Warping (1-NN DTW) and a Fully Convolutional Network (FCN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' On all 18 datasets, both models were vulnerable to attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' It does not consider time series regression and typical attacks (FGSM, BIM, PGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Using time-series data, [9] performs extensive benchmarking of well-proven adversarial defensive approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' It uses 2 white-box attacks (FGSM and PGD) and 3 black-box attacks (Noise Attack, Boundary Attack, and Simple Black-box Attack (SIMB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' It evaluates robustness using multiple adversarial defenses (adversarial training, TRADES, Feature Denoising).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' It, on the other hand, ignores the regression problem and does not assess the robustness of different models, instead of analyzing multiple defenses with a single classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' FGSM and BIM attacks are used by another study [7] to undertake untargeted, targeted, and universal adversarial attacks on time series classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' It does not, however, make any adversarial defenses and focuses solely on the classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' [12] presents the concept of adversarial attacks (FGSM and BIM) on various deep learning models (CNN, LSTM, and GRU) for multivariate linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Although it introduces the concept of adversarial attacks on regression tasks, it does not take account into the defense against the attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The results showed that DL regression models are vulnerable to adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' On LSTM, CNN, RNN, and Multi-Head Attention Network, [8] examines two different attacks: FGSM and Adversarial attack with importance measurement (AAIM) (MHANet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' However, it exclusively concentrates on the regression problem and makes no adversarial defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' None of the studies take both regression and categorization into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Also, none of them investigate how vulnerable various models are to attacks and how well they can recover using defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' 3 Problem Statement Previous works focus only on the adversarial attacks only for both regression and the classification problems in the series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' We have implemented the adversarial attacks and defense mechanism against attacks on both the classification and regression problems in the time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Adversarial attacks refer to finding adversarial examples for well-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' For a classification task, we 2 use f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' θ) to denote a model that maps an input to a discrete label set with k classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Given a perturbation size ϵ, the adversary tries to find a perturbation δ that maximizes the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Thus, the adversarial counterpart xi of x can be expressed as: xi = x + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Some of the common attacks for generating adversarial attacks are FGSM, BIM, and PGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' We designed a defense against these attacks using adversarial training for adversarial robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' In general, Adversarial robustness is the model’s performance on test data with adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Similar attacks techniques can also be used for adversarial training by expanding it to the min-max optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Min-max optimization problem consists of two parts: inner maximization which can be single or multiple steps (depending on the attack) is used to generate maximum perturbation and outer minimization is a single step stochastic gradient descent to minimize the loss of a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' 4 Methodology Various DL algorithms are trained for regression and classification problems for time series analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' These models are attacked by generating perturbations on the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' We implemented adversarial training as a defense mechanism to tackle the attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The subsections below introduce the models used for training, methods for generating adversarial attacks, and adversarial training mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Figure 1 depicts the overview of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Figure 1: Overview of this study 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='1 Models For the time series regression and classification, we used the following 7 models: LSTM (Long short-term memory), Stacked LSTM, GRU (Gated recurrent unit), RNN (Recurrent neural network), CNN (Convolutional neural network), CNN-LSTM, and ConvLSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The models are widely used for time series problems due to their capability of capturing sequential patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' For sequential data, RNN, LSTM, and GRU are the go-to models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' CNN is often used for time-series data as its convolutional capability is useful for modeling time-series patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' CNN-LSTM and ConvLSTM combine the CNN and LSTM architectures, the latter of which is especially suitable for two-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' All the models are evaluated on time series classification and regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Adversarial attacks and defense are employed in these models for analyzing their vulnerability and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='2 Adversarial Attacks We have employed three well-known gradient-based attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Gradient-based attacks employ a perturbation for the input time series by altering the back-propagation process slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' They use the gradients with respect to the input given the output for perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' All of these attacks are performed on the models for time series classification and regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' 3 Time Series Regression Classification Stacked Models LSTM GRU RNN CNN CNN-LSTM ConvLSTM LSTM Adversarial FGSM BIM PGD Attacks Adversarial Adversarial Defense Training4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='1 Fast Sign Gradient Method (FGSM) The fast gradient sign method (FGSM) [2] creates an adversarial example by leveraging the neural network’s gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The approach creates a adversarial input that maximizes the loss for an original input by using the gradients of the loss with respect to the original input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The following equation can be used to summarize this: xadversarial = x + ϵ × sign((∆xJ(θ, x, y)) (1) where xadversarial is the adversarial input, x is the original input, y is the original target variable, ϵ is the multiplier to ensure the perturbations are small, θ is the model parameters, and J is the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='2 Basic Iterative Method (BIM) Basic Iterative Method (BIM) [1] is an enhancement of the FGSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' It proposes repeating the FGSM step with a small step size (α) and clipping the intermediate results after each step to verify that they are in the same neighborhood as the original input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' It initializes the initial adversarial sample as the original input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Then it iteratively updates the adversarial sample while keeping it within the neighborhood of the original input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The size of the neighborhood is set by the hyperparameter ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='3 Projected Gradient Descent (PGD) Projected Gradient Descent (PGD) attack is quite similar to the BIM attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The main difference is that PGD starts the adversarial example at a random location in the neighborhood (set by the multiplier ϵ) and restarts it randomly, whereas BIM starts at the original input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='3 Adversarial Training As a defense against these attacks, we performed adversarial training using similar attack methods for all the models in all the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' For each model, we created a new training set by combining the original training set and perturbed training sets using each attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Thus for each dataset, we performed 21 adversarial pieces of training altogether 3 (FGSM, BIM, and PGD) for 7 different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' A new model with adversarial training is used to calculate adversarial robustness on the original perturbed testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Each adversarial training follows the principle of min-max optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Adversarial training with FGSM involves a single step of inner maximization using FGSM and a single step of outer minimization while adversarial training with BIM and PGD involves 5 steps of inner maximization using BIM and PGD and a single of outer minimization using stochastic gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' 5 Experimental Evaluation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='1 Data sets For classification, 1 data set is used as follows: Temperature[6]: It is derived from a Kaggle weather dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The data is based on hourly temperature data for a city over ten years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' We use the first 23 time steps in the window as the predictor variables and the last time step as the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Power Consumption[5]: It is derived from Kaggle which comes from PJM (regional trans- mission organization in the United States).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' It contains hourly energy consumption data in megawatts (MW) for the eastern part of the United State over three years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' We use the first 23 time steps in the window as the predictor variables and the last time step as the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Humidity[6]: It is derived from a Kaggle weather dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The data is based on hourly humidity data for a city over ten years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' We use the first 23 time steps in the window as the predictor variables and the last time step as the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Wind Speed[6]: It is derived from a Kaggle weather dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The data is based on hourly wind speed data for a city over ten years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' We use the first 23 time steps in the window as the predictor variables and the last time step as the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' 4 Hurricane[4]: This corresponds to tropical cyclone intensity data obtained from the HUR- DAT2 database [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' For each hurricane, wind speeds (intensities) were reported at every 6-hour interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' We use the first 23 time steps in the window as the predictor variables and the last time step as the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Eye State[3]: This data is collected from UCI Machine Learning Repository-[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' It is from one continuous EEG measurement with the Emotiv EEG Neuroheadset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The duration of the measurement was 117 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The eye state ’1’ indicates the eye-closed and ’0’ the eye-open state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='2 Experimental Setup We used different hyperparameters for the model’s architecture and adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' For the model’s architecture, the number of layers, hidden nodes, and epochs is hyperparameters to be tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Similarly, for attacks the hyperparameters are perturbation budget ϵ, step size α, and the number of iterations in BIM and PGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' These hyperparameters are tuned differently depending on the attack and the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' We used ϵ between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='3 and α between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='2 depending on the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='3 Results We performed adversarial attacks and adversarial training on 5 regression problems (Power Consump- tion, Wind Speed, Temperature, Humidity, and Hurricane) and 1 classification problem (Eye State Classification) for time series analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Accuracy is used as a metric to determine the performance of the classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Table 2 depicts the results for our classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' According to that, LSTM has the best performance for initial training with an accuracy of 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='46 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' When three different adversarial attacks (FGSM, BIM, and PGD) are applied to the model, the accuracy drops to 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='54 %, 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='83 %, and 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='43% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The adversarial training increases the adversarial robustness of the model, boosting its accuracy to 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='28%, 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='46%, and 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='13%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' For the 5 regression tasks, RMSE is used as a metric for evaluating the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' For instance, the results of the Temperature dataset are summarized in Table 3, the GRU model during the initial training has the best performance with an RMSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' When three different adversarial attacks (FGSM, BIM, and PGD) are applied to the model, the RMSE increases to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='09, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='09, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='07 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' The adversarial training increases the adversarial robustness of the model, reducing RMSE to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='03, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='04 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Other data sets show a similar pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Table 4, 5, 6, 7 illustrate the rest of the four regression problems on Power Consumption, Wind Speed Humidity, and Hurricane dataset respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Models LSTM Stacked LSTM GRU RNN CNN CNN- LSTM ConvLSTM Accuracy for models 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='79 Accuracy for Adversarial Attacks FGSM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='61 BIM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='75 PGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='56 Accuracy after Adversarial Training FGSM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='81 BIM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='81 PGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='67 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='27 PGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='Table 7: Adversarial attacks and defenses on different models for time series regression (Hurricane ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='Intensity) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='Figure 2: Actual Temperature vs Predicted Temperature by LSTM model for the Temperature dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='(a) 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='(b) 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='(c) 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='(d) 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='(e) 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='(f) 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='Figure 3: Time series plot for actual temperature vs predicted temperature in LSTM models after ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='adversarial attacks and adversarial training( [1]Adv attack with FGSM [2] Adv training with FGSM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='[3] Adv attack with BIM [4] Adv training with BIM [5] Adv attack with PGD [6] Adv training with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='PGD) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='Insights/Discussion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='Based on our experiments on the data sets we used,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' we observed that FGSM generates a much stronger attack than both BIM and PGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' BIM performs poorly among these three attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' BIM’s initial perturbation is set to the original input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' It might be the reason that it can not perturb much as it starts from the original input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Apart from attack generation, FGSM also has higher training efficiency than both BIM and PGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' For the adversarial defense, recovering from the FGSM perturbed model is the easiest one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' But, recovering from a PGD attack is the hardest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' PGD’s random initialization and random restarts might make it harder for recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' 7 Stacked LSTM model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='8 ActualTemperature Predicted Temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='6 Temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='2 0 50 100 150 200 250 300 350 400 TimeFGsM perturbed stacked LsTM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='6 Temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='2 ActualTemperature PredictedTemperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='1 0 50 100 150 200 250 300 350 400 TimeFGsM-Stacked LsTM model (Perturbed) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='8 ActualTemperature PredictedTemperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='6 Temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='2 0 50 100 150 200 250 300 350 400 TimeBIM perturbed Stacked LsTM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='6 Temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='2 ActualTemperature PredictedTemperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='1 0 50 100 150 200 250 300 350 400 TimeBIM-Stacked LsTM model (Perturbed) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='6 rature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='3 ActualTemperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='2 PredictedTemperature 0 50 100 150 200 250 300 350 400 TimePGD perturbed Stacked LSTM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content="8 1'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='6 Temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='2 ActualTemperature PredictedTemperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='1 0 50 100 150 200 250 300 350 400 TimePGD-StackedLSTMmodel(Perturbed) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='8 ActualTemperature PredictedTemperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='6 Temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='2 0 50 100 150 200 250 300 350 400 TimeAmong different time series models, all-time series models are vulnerable to adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' GRU and RNN seem to be more vulnerable than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' In terms of recovery, all models can recover from the attacks, but LSTM and GRU can recover slightly better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' 7 Individual Contributions Bidhan:Literature review of [10][11] [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' implemented 3 models (LSTM, RNN and Stacked-LSTM);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Implemented Adversarial training part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Asadullah Hill Galib:Literature review of [9][8] [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' implemented 4 models (GRU, CNN, CNN- LSTM, ConvLSTM);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Implemented Adversarial attacks part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' 8 Conclusion In this paper, we look at how vulnerable different time series models are under adversarial attacks and how well they recover using adversarial defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' We evaluated 7-time series models on three gradient- based attack mechanisms and adversarial training-based defense against them for both regression and classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' FGSM attack has a much stronger effect on generating adversarial attacks and efficient adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' All models, especially GRU and RNN, appear to be vulnerable, according to the findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' In addition, LSTM and GRU have higher defense recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' In terms of attacks, FGSM outperforms the competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' It’s more difficult to recover from a PGD attack than it is for other types of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' In the future, we will extend this work by incorporating more types of attacks and defense mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' References [1] Kurakin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=', Goodfellow, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=', Bengio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' & Others Adversarial examples in the physical world.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' Financial Time Series Forecasting with Deep Learning : A Systematic Literature Review: 2005-2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content=' (arXiv,2019), https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='org/abs/1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} +page_content='13288 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQfKgZi/content/2301.03703v1.pdf'} diff --git a/rNFKT4oBgHgl3EQfIi19/content/tmp_files/2301.11734v1.pdf.txt b/rNFKT4oBgHgl3EQfIi19/content/tmp_files/2301.11734v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..03f88fc045eeba6db4f5c734b40f8314457b0694 --- /dev/null +++ b/rNFKT4oBgHgl3EQfIi19/content/tmp_files/2301.11734v1.pdf.txt @@ -0,0 +1,1829 @@ +Behaviour Discriminator: A Simple Data Filtering Method +to Improve Offline Policy Learning +Qiang Wang 1 Robert McCarthy 2 David Cordova Bulens 1 Kevin McGuinness 3 4 +Noel E. O’Connor 3 4 Francisco Roldan Sanchez 3 4 Stephen J. Redmond 1 3 +Abstract +This paper studies the problem of learning a con- +trol policy without the need for interactions with +the environment; instead, learning purely from +an existing dataset. Prior work has demonstrated +that offline learning algorithms (e.g., behavioural +cloning and offline reinforcement learning) are +more likely to discover a satisfactory policy when +trained using high-quality expert data. However, +many real-world/practical datasets can contain +significant proportions of examples generated us- +ing low-skilled agents. Therefore, we propose a +behaviour discriminator (BD) concept, a novel +and simple data filtering approach based on semi- +supervised learning, which can accurately discern +expert data from a mixed-quality dataset. Our +BD approach was used to pre-process the mixed- +skill-level datasets from the Real Robot Challenge +(RRC) III, an open competition requiring partici- +pants to solve several dexterous robotic manipu- +lation tasks using offline learning methods1; the +new BD method allowed a standard behavioural +cloning algorithm to outperform other more so- +phisticated offline learning algorithms. Moreover, +we demonstrate that the new BD pre-processing +method can be applied to a number of D4RL +benchmark problems, improving the performance +of multiple state-of-the-art offline reinforcement +learning algorithms. +1University College Dublin, Ireland 2University College Lon- +don, UK 3Insight SFI Research Centre for Data Analytics, Ireland +4Dublin City University, Ireland. Correspondence to: Stephen J. +Redmond . +This publication has emanated from research conducted with the fi- +nancial support of Science Foundation Ireland under grant numbers +17/FRL/4832 and SFI/12/RC/2289_P2 and of China Scholarship +Council (CSC No.202006540003). For the purpose of Open Ac- +cess, the author has applied a CC BY public copyright licence to +any Author Accepted Manuscript version arising from this submis- +sion. +1Featured in the NeurIPS 2022 Competition Track, more details +see https://real-robot-challenge.com/ +1. Introduction +Data-driven learning methods can discover sophisticated +control strategies with minimal human involvement, and +have demonstrated impressive performance in learning skills +across many challenging domains (Mnih et al., 2013; Sil- +ver et al., 2017; Wang et al., 2022; Nagabandi et al., 2020). +Nonetheless, data-driven methods are not often applied in +the real world due to the inefficiency of having to interact +with the environment so many times before an effective +policy can be learned. Online data-driven learning requires +time-consuming data collection (Yarats et al., 2021; Tassa +et al., 2018; McCarthy et al., 2021), which can be costly +and/or unsafe in many physical environments, including +those used for robotic manipulation. The inefficiency of +online learning can potentially be ameliorated by learning +from previously-collected data; i.e., learning a policy from a +historical dataset without the need for additional data acqui- +sition from the environment. This is termed offline policy +learning (OPL). In recent years, research methods in the +area can largely be categorised as behavioural cloning (BC) +or offline reinforcement learning (RL). As the policy learned +using OPL is learned solely from historical data, the quality +of the examples in this dataset will directly determine the +quality of the learned policy. Thus we are motivated to en- +sure the dataset used for offline learning contains examples +generated using a highly-skilled agent. +1.1. Importance of data quality in OPL +Data quality has been acknowledged as a critical factor +in traditional machine learning (Jain et al., 2020). In the +context of OPL, high-quality data is considered to be data +collected by a domain-specific expert agent (Fu et al., 2020); +in contrast, low-quality data refers to low-skill or irrelevant +behaviours with respect to the task. +Offline RL is capable of learning effective policies when +trained using mixed-quality data by taking the rewards as- +sociated with the behaviours into account. However, BC +has the risk of learning a suboptimal policy, or failing to +learn any useful policy, when the dataset is of mixed-quality +(Levine, 2022). Indeed, prior work has shown that both +BC and offline RL can learn better policies and have a +arXiv:2301.11734v1 [cs.LG] 27 Jan 2023 + +Preprint +greater chance of outperforming the demonstration agent +when trained using high-quality data, as expected (Zhou +et al., 2021; Kostrikov et al., 2021; Fujimoto & Gu, 2021). +Furthermore, it is difficult to learn using offline RL in com- +plex problem domains, with the performance of learned +policy being excessively sensitive to hyperparameter choice +(Paine et al., 2020; Fujimoto & Gu, 2021; Kostrikov et al., +2021), particularly when the dataset is of mixed- or low- +quality. Hence, high-quality data is always preferable when +using any OPL algorithm. +But data quality cannot always be guaranteed in practical +problems, particularly in noisy real-world environments; for +instance, human experts sometimes make mistakes due to +subjective emotional factors or the difficulty of the task. A +dataset might contain a mix of both high- and low-quality +data. For instance, in the RRC III competition and the D4RL +benchmark datasets (Fu et al., 2020) (shown in Figure 4), +the large mixed-quality dataset, Dm, contains a large propor- +tion of desirable high-quality expert-generated data, from +which an effective policy could be learned. However, in +both the RRC III competition and D4RL benchmarks, we +do not know which agent generated which data, nor the +skill level of the various agents, and so the high-quality data +(ideally generated by a single skilled agent) cannot easily be +separated from the larger mixed-quality dataset. Therefore, +we propose a novel filtering technique capable of recognis- +ing the high-quality subset of data in Dm, allowing us to +subsequently use this high-quality subset of data to learn an +effective control policy. +1.2. Our method +In our work, we extract a relatively small subset, Ds, from +the training dataset which we hope will contain mostly high- +quality data. This subset is selected in a naive way, by +extracting the best-ranked examples in the dataset based +solely on the reward achieved during episodes where a +task was attempted; although other features (other than +reward) could be used. (Ds could also be separately col- +lected from a known expert, but this is not the focus of our +work.) This subset of high-quality expert-generated data, +Ds, is subsequently used to seed (and ultimately anchor) +the semi-supervised iterative training of a classifier that +can discriminate between behaviours generated by expert +and non-expert agents. Once trained, this classifier, which +we term a behaviour discriminator (BD), is applied to the +entire dataset, Dm, to filter the dataset and isolate high- +quality expert-generated data (including examples from ex- +perts which might not have achieved a large reward), Df, +based on recognition of the behaviours exhibited in each +episode. The union of seed and filtered datasets (Ds ∪ Df) +are then subsequently used for OPL. Our BD methodology +addresses several practical OPL problems for both physical +and simulated domains. +2. Related work +2.1. Offline policy learning +BC and offline RL are promising approaches to OPL; we +refer readers to (Hussein et al., 2017; Levine et al., 2020) for +comprehensive surveys, and to (Fu et al., 2020; Gulcehre +et al., 2020; Fujimoto et al., 2019a) for research benchmarks. +2.1.1. BEHAVIOURAL CLONING (BC) +BC is the simplest form of imitation learning (IL) (Bain & +Sammut, 1995; Pomerleau, 1991). BC aims to find a policy +that can mimic the behaviour used to perform a given task. +The target behaviour to be cloned is usually obtained from +an expert; for instance, a human (Mandlekar et al., 2021; +Sermanet et al., 2017) or a well-performing scripted agent +(Fujimoto et al., 2019b). BC can be viewed as supervised +regression and learns a policy by mapping states to actions. +BC is very efficient when training using high-quality expert +data, and the trained agent usually performs remarkably +well (Merel et al., 2018); compared with other relatively +complex IL methods, such as GAIL (Ho & Ermon, 2016) +and inverse RL (Ng et al., 2000), BC generally results in +superior performance. However, standard BC has the fol- +lowing limitations, as it is a form of supervised learning: +1) to avoid regression ambiguity, the action conditioned on +the state must be drawn from a unimodal distribution, or +the target action mode must occupy the majority propor- +tion in the dataset (Levine, 2022); 2) The training dataset +size should be reasonably large to mitigate the covariate +shift issue (Chang et al., 2021) (i.e., the compound error +produced by the unseen data in the deployment stage); 3) +BC is typically upper-bounded by the performance of the +demonstrator (Brown et al., 2020). More recently, people +have proposed implicit BC (Florence et al., 2022) using the +energy-based model (LeCun et al., 2006) to improve the +performance of BC. +2.1.2. OFFLINE REINFORCEMENT LEARNING +Also known as batch RL, like the paradigms of standard +online RL (Kaelbling et al., 1996), the goal is to find a policy +that can maximise the expectation of the sum of discounted +rewards. Most off-policy RL algorithms are applicable of- +fline; however, due to the absence of exploration, the actual +training batch would mismatch the expected state-action +visitation of the current policy. As a result, during the pol- +icy evaluation stage, state-action pairs not contained in the +dataset would be poorly estimated; correspondingly, those +overestimated out-of-distribution (OOD) actions are learned +in the policy improvement stage (Fujimoto et al., 2019b). +Because of this OOD issue, the learned policy by classical +off-policy RL algorithms usually performs poorly in pure +offline settings. To mitigate this issue, several methods, such +as policy regularization (Kostrikov et al., 2021; Fujimoto + +Preprint +(a) +(b) +Figure 1. Behaviour discriminator (BD) block diagram. (a) Illustration of the iterative process of training BD filter. The light-grey +rounded rectangle shows the composition and pre-processing of the BD training data. Within this block, Dm refers to large mixed-quality +dataset. Ds refers to the seed dataset that represents the behaviour we wish to discriminate, which can be a subset of Dm or can be +collected separately (for example, using a skilled agent). Df refers to the subset of data remaining after filtering Dm with the BD filter. +D+ = Ds ∪ Df refers to the positive training set used to train the next iteration of the BD filter, which should increasingly contain more +expert data as the training process converges. The red rounded rectangle shows how positive and negative training samples are generating, +with the negative examples generated by intentionally mismatching actions and states from different time points and/or different data +subsets to form action-state pairs that would most likely represent poorly performing behaviours; the white rounded rectangle (solid) +refers to a random state-action pairs. See section 3.3 for a more comprehensive explanation of the sample generation process. The blue +rounded rectangle is the BD filter, which aggregates the output from K classifiers to label the data from Dm as either positive (e.g., +expert-generated) or negative (e.g., non-expert-generated). (b) The neural network structure of an individual classifier includes an encoder +to reduce the usually high-dimensional state vector, s, followed by a multilayer perceptron (MLP) classifier, taking as input both state (s) +and action (a) vectors, and whose sigmoidal output is interpreted as a probability that the episode was generated by an expert agent. +et al., 2019b; Zhou et al., 2021; Fujimoto & Gu, 2021) and +forming conservative value estimates (Kumar et al., 2020; +An et al., 2021) are proposed to reduce the deviation be- +tween the learned policy and the policy used by the agent +that generated the dataset. +2.2. Filtering data based on behaviour +Filter-like techniques have previously been proposed to sup- +port BC learning; several reward-based value/advantage +functions are learned in (Wang et al., 2020; 2018; Peng +et al., 2019; Chen et al., 2020; Siegel et al., 2020; Neumann +& Peters, 2008) to re-weight the importance of training +samples that are used to train a policy using BC. Similarly, +others have proposed Xu et al. (2022) a GAN-like archi- +tecture (Goodfellow et al., 2020), where a discriminator is +learned to regularise the BC training process. +3. Behaviour discriminator (BD) +3.1. Preliminaries +The OPL problem is formulated in the context of a Markov +decision process, M = (S, A, R, P, γ) (Puterman, 1990), +where S is the state space, A is the action space, R is the +reward function, P is environment dynamic and γ is the +discount factor. At each time step, t, the agent gets a state +st ∈ S and outputs an action at ∈ A according to a policy +π(at | st); after applying the action to the environment, the +agent will get a reward rt ∈ R and the environment state +transitions to st+1. In our various settings, we can obtain +or are given a seed dataset, Ds = {(ss +t, as +t, rs +t , ss +t+1)t=1...i}, +with i time steps and a large mixed-quality offline dataset +Dm = {(sm +t , am +t , rm +t , sm +t+1)t=1...j}, with j time steps. +3.2. Iterative process of training the BD filter +Training of the BD filter is a semi-supervised iterative pro- +cess, illustrated in Figure 1(a). The core decision component +of the BD filter is a classifier, which can distinguish whether +the input state-action pair was generated by an expert or not. +To increase the capability of the classifier, we use an ensem- +ble technique, which learns several independent classifiers +simultaneously and aggregates the individual decisions to +obtain a final decision label for the inputted state-action pair; +the structure of a classifier unit in the ensemble is shown in +Figure 1(b). The iterative process of training and using the +BD can be summarised in the following steps: +(a) Create a seed dataset, Ds (if not given), a subset of Dm, +which we expect to contain state-action pairs mostly +generated by an expert policy; Use Ds as D+ (set of + +Datasets +: Update +Sample generator +Filter +D+ +Subset 1 +D +St +at +Positive +D+ +Classifier 1 +Subset 2 +[stt=1.. +Train +D +Equation 2 +Label +{stt=1..i +[att=1]..i +D. +St' +Classifier 2 +at" +Subset K +Train +- +'Dm +Negative +St' ++p +Classifier K +Train +{st)t=1....j +{at}t=1,... +s +a +DassecintoConcatenate +S +Probability +Encoder +a +MLPPreprint +positive, i.e., expert, examples) for the first iteration; +(b) Sample K independent subsets, Dk ++, from D+ with +replacement, for k ∈ 1, ..., K; +(c) For each of the K subsets, Dk ++, generate negative state- +action samples, Dk +−, for k ∈ 1, ..., K; +(d) Form K training sets, Dk +tr = Dk ++ ∪ Dk +− for k ∈ +1, ..., K, for each of the K filter classifiers and train; +(e) Input Dm into each of the K independent classifiers, +{Fθk}k=1...K; +(f) Ensemble the outputs of the K trained classifiers to +obtain a label for each episode in Dm; +(g) Form the set Df as the subset of episodes in Dm that +were labelled positive by the ensemble of classifiers; +(h) Update the membership of D+ as the union of the seed +dataset Ds and the updated filtered subset Df; i.e., +D+ = Ds ∪ Df; +(i) Repeat step (b)-(h) until the memberships of D+ con- +verges; Use the final converged D+ for policy learning. +The complete BD training algorithm is described in Algo- +rithm 1 in Appendix A. The following sections detail each +component of BD training process. +3.3. Generating the training samples +The classifiers are trained by providing a dataset of labelled +action-state pairs, ideally generated by skilled and unskilled +policies. It is feasible to label examples in the dataset that are +mostly expert-generated, simply by ranking the action-state +pairs based on the rewards that the actions taken achieved, +and choosing some of the top-ranking pairs. However, rank- +ing by reward is not a reliable way to identify non-expert- +generated data, as too often both expert and non-expert poli- +cies produce action-state pairs that achieve a small rewards. +Thus we propose a generative method to create artificial +non-expert data (see Figure 1(a) Sample generator). +Remark 3.1. We annotate all state-action pairs in D+ (D+ = +Ds in the first iteration, but D+ = Ds∪Df in all subsequent +iterations), as positive examples, meaning expert-generated. +Remark 3.2. The negative examples D−, meaning non- +expert-generated, are created by mixing the states and ac- +tions from different sources, including from D+, from Dm, +and random samples from the state-action space. When +a state and action are drawn from the same dataset and +mixed to form a negative example of an action-state pair, +they are sampled from different time points so that it is un- +likely that the pair resemble the behaviour that an expert +would produce. Therefore, it can be written as: D− = +{(s+ +t′, a+ +t′′)n1 ∪ (s+, am)n2 ∪ (s+, ¯a)n3 ∪ (sm, a+)n4 ∪ +(sm +t′ , am +t′′)n5 ∪(sm, ¯a)n6 ∪(¯s, a+)n7 ∪(¯s, am)n8 ∪(¯s, ¯a)n9}, +where t′ ̸= t′′ ; ¯s and ¯a refers to the random states and ac- +tions. +To further reduce the chance that the generated negative +state-action pair examples are similar to the (we assume) +mostly expert data in the training dataset, D+, we consider +the similarities of the action in the state-action pairs before +and after mixing. If the action at′ that is mixed with state +st′′, where t′ ̸= t′′, is too similar to the original action at′′ +that was originally paired with that state st′′, then this mixed +action-state pair are not included in D−; here we consider +a value of the L2 norm of the difference between at′ and +at′′ less than 0.3(Ahigh − Alow), where Ahigh and Alow +are the upper and lower limits of the action space, to mean +the actions are too similar to consider the action-state pair +to represent non-expert behaviour. +3.4. Classifier structure +Figure 1(b) shows the network structure of a unit classifier +in the ensemble, which is a combination of multilayer per- +ceptrons (MLPs); it takes the state-action pair and outputs +the probability that this pair was generated by the actions +of the same expert agent that generated (we assume) most +of the data in the training dataset, D+. Experimentally, we +found that directly inputting a concatenation of raw state- +action pairs into the neural network will cause the action +to be effectively ignored, as the dimension of the observa- +tion space is typically much larger than the action space in +practical task settings. Therefore, to reduce the dimension +of the observation space, we encode the observation vec- +tor into a lower dimensional space before concatenating it +with the action vector and inputting to the MLP to obtain +a decision. The final layer of each MLP uses a sigmoid +activation function, sigmoid(x) = 1/(1 + e−x), where the +output is interpreted as a probability the state-action pair +was generated by an expert. Binary cross-entropy is chosen +as the training loss, L: +L = +E +(st,at)∼D− [− log(1 − F(st, at))] ++ +E +(st,at)∼D+ [− log F(st, at)] . +(1) +The discriminator training here is similar to that of a GAN +(Goodfellow et al., 2020) or GAIL (Ho & Ermon, 2016), +with the difference being that the negative training samples +here are heuristically-generated by mixing states and actions +from different time points, rather than being output by a +trained generator network. +3.5. Iteratively updating the BD filter training set +Since we know that any single episode of state-action pairs +are all generated by the same agent, the labels assigned to +all state-action pairs in an episode are further aggregated + +Preprint +such that the same label is applied to all data (i.e., at all +time steps) in the same episode. This is done by soft voting, +taking the mean of the individually predicted probabilities +of the state-action pairs in the episode to create a confidence +score for the episode. Subsequently, a threshold, thconf, is +applied to the confidence score to binarize it. Finally, we +ensemble (Dietterich, 2000) the binary outputs from each +of the K classifiers through equal-weight voting: +f := 1 +� K +� +k=1 +1 +�� +1 +T +T +� +t=1 +Fθk(st, at) +� +≥ thconf +� +> K +2 +� +, +(2) +where 1[·] are indicator functions; T is the number of time +steps in each episode sequence; and K is the number of +classifiers in the ensemble, set to an odd number (5 in our +work) to avoid ties. An adaptive mechanism is applied to +choose thconf; briefly, it searches for a local minimum in +the confidence score histogram that represents an approxi- +mate cut-off, above which it is assumed the histrogram is +dominated by expert-generated episodes (see Appendix C +for details). +A filtered subset Df can be easily obtained according to +the labels output by the BD filter when applied to Dm. We +update D+ by taking the union of the filtered subset and +the seed set: D+ = Df ∪ Ds. Ds is used to seed the first +iteration of training the classifier. In subsequent iterations, +it becomes an anchor for all future training sets, to prevent +the classifier from drifting away from being able to discrim- +inate the very high-ranking state-action pairs. The updated +membership of D+ is compared its membership from the +previous iteration; if the membership is not changing signif- +icantly, the BD training is considered to have converged and +the process is halted. +4. Experimental results +We aim to demonstrate the effectiveness of our proposed +BD method in this section. A wide range of continuous +benchmark tasks are included in our experiments, including +the challenging physical robotic manipulation tasks of the +RRC III competition and the D4RL benchmarks of OpenAI +gym MuJoCo locomotion tasks (Fu et al., 2020; Todorov +et al., 2012; Brockman et al., 2016). We first report the BD +filtering accuracy, and then demonstrate the performance +benefits of using the BD as a pre-processing step for OPL. +4.1. Task and dataset description +4.1.1. REAL ROBOT CHALLENGE (RRC) III +In the RRC III competition, participants are provided with +dozens of hours of robotic data collected by pre-trained poli- +cies deployed on TriFinger robots (Wüthrich et al., 2020). +Datasets are provided for two robotic tasks: push and lift. +Examples of robot environment are shown in Figure 4(a)- +4(c). In the push task, the cube must be moved to target +positions on the arena floor. In the more challenging lift task, +the cube must be lifted and maintained at a target position +and orientation. For each task, two separate datasets are +provided, one of which is of high quality (i.e., collected by +an expert policy, called the expert dataset), and the other +is collected by a mixture of different policies with varying +levels of skill (called the mixed dataset). The focus of this +paper is on learning an effective policy in an offline manner +from the relatively challenging datasets containing exam- +ples of behaviour generated using a range of policies with a +mix of skill levels. +4.1.2. MUJOCO LOCOMOTION TASKS +As shown in Figure 4(d)-4(h), the bodies being controlled +during locomotion tasks are composed of segments and +joints. Actions are applied to maintain the balance of the +body and drive it to move forward. We make use of the three +available D4RL (Fu et al., 2020) medium-expert datasets for +locomotion tasks, including halfcheetah-medium-expert-v0, +hopper-medium-expert-v0, and walker2d-medium-expert-v0 +to show the benefit of using our BD method to identify high- +skill behaviours to improve OPL. Similar to the settings of +RRC III competition, each of these datasets was collected +by two agents with different skill levels; i.e., medium and +expert skill levels. +Figure 2. The histogram of the accumulated rewards (i.e., rewards +summed over all time steps in an episode). The expert dataset +consists mostly of successful episodes achieving large reward +values and it has one distinct peak. The mixed dataset appears +to consist of two distinct peaks. However, from the long tail on +the expert distribution that overlaps significantly with the mixed +dataset, applying a reward-based threshold to the mixed dataset to +identify expert episodes is unlikely to be very successful. + +140 +Lift/expert +80 +Lift/mixed +70 +120 +hh +Lift/Expert count +.00 +50 +80 +40 +60 +30 +40 +20 +20 +10 +0 +1400 +0 +200 +400 +600 +800 +1000 +1200 +Summed reward over an episodePreprint +Table 1. The evaluated scores comparing our method to state-of-the-art offline RL algorithms applied to the RRC III competition, where +task-specific scores are given as mean ± SD. We train each policy for 1e6 time steps. In the evaluation stage, the policy is randomly +deployed on one of six available physical robots, and the goal (position and/or orientation of a cube) is randomly generated. The evaluation +for each task lasts 15 episodes. Lift/expert is a lifting task performed by an expert policy, whereas Lift/mixed is performed by a mixture of +expert and non-expert policies. +BC +CRR +TD3+BC +PLAS +IQL +BC+BD +Final submission +Lift/expert +928 ± 205 +792 ± 227 +852 ± 401 +874 ± 359 +789 ± 299 +- ± +- +1130 +± +193 +Lift/mixed +489 ± 282 +606 ± 312 +698 ± 362 +707 ± 350 +550 ± 325 +917 ± 237 +1038 +± +305 +Push/expert +626 ± 101 +611 ± 127 +623 ± 99 +618 ± 92 +607 ± 162 +- ± +- +662 +± +87 +Push/mixed +497 ± 88 +599 ± 93 +601 ± 85 +604 ± 111 +595 ± 121 +618 ± 83 +636 +± +126 +Table 2. The final official ranking of RRC III competition. The organisers pre-defined several goals in the official evaluation protocol and +evenly distributed them among the six robots; the evaluation lasted 72 episodes for each evaluation case. Our team is named excludedrice. +superiordinosaur and jealousjaguar shared joint third place. +Baseline score +660 +429 +1064 +851 +- +# +Team name +Push/expert +Push/mixed +Lift/expert +Lift/mixed +Score +1 +excludedrice +624 ± 144 +635 ± 137 +956 ± 431 +923 ± 442 +784 +2 +decimalcurlew +639 ± 112 +613 ± 134 +841 ± 415 +717 ± 383 +703 +3 +superiordinosaur +618 ± 143 +575 ± 191 +856 ± 452 +571 ± 346 +655 +jealousjaguar +639 ± 121 +561 ± 178 +855 ± 392 +506 ± 348 +640 +4.2. Applying BD +To obtain the seed dataset Ds for RRC III and three D4RL +datasets, we assume that expert is likely to have generated +the large majority of very high reward episodes. Figure 2 +shows the episodic reward distributions of both lift/expert +and lift/mixed datasets. The bimodal distribution of the +lift/mixed dataset would suggest that it might be possible to +discriminate between many of the expert- and non-expert- +generated data by thresholding on the episode reward. How- +ever, when attempting this approach, we found that naively +labelling the lift/mixed dataset based on episodic reward +in this way and then using BC to learn a policy using the +higher-reward subset (which we incorrectly presumed would +consist mostly of data generated by the expert policy) lead +to poor performance during policy evaluation. +Although, if a very high the threshold is used, it is possible +to obtain a subset of data which was predominantly gener- +ated by the expert policy. However, using this strategy, we +find these subsets are then too small to go on and learn a +good policy using BC. In our attempts, we experimented +with extracting the top 30%, 40%, 50%, and 60% reward +episodes to train BC models; however, all of them performed +poorly or failed during learning; we report these supplemen- +tary results in Appendix D. But, a small dataset of mostly +expert-generated examples is sufficiently large for use as +the seed subset, D∫, for our BD filtering method. Thus we +calculated the episodic reward for all episodes and took the +top 0.2% of high-reward episodes as our seed dataset. +(a) Push/mix +(b) Lift/mix +(c) HalfCheetah-v2 +(d) Ant-v2 +(e) Humanoid-v3 +Figure 3. Confusion matrices showing the excellent performance +of the BD filtering method in recognising expert-generated +episodes for mixed datasets of the RRC III competition (a-b) and +customised locomotion tasks from the D4RL datasets (c-e). +4.3. Filter results +After the competition, we obtained the episode labels for +lift/mixed and push/mixed datasets from RRC III organisers, +to know which episodes were generated by the which poli- +cies. We learned that each mixed dataset was collected by +two different policies with different skill levels, and each +policy collected half of the dataset. We report the filter re- + +Positive +1600 +1920 +0 +1200 +Actual +800 +Negative +0 +1920 +400 +0 +Positive +Negative +PredictedPositive +1000 +1182 +15 +750 +Actual +500 +Negative +7 +1190 +250 +Positive +Negative +PredictedPositive +1200 +494 +4 +900 +Actual +600 +Negative +7 +1495 +300 +Positive +Negative +Predicted1600 +Positive +496 +3 +1200 +Actual +800 +Negative +10 +1607 +400 +Positive +Negative +PredictedPositive +800 +211 +5 +600 +Actual +400 +Negative +11 +953 +200 +Positive +Negative +PredictedPreprint +sults for both mixed dataset in Figure 3. We also conduct a +study of the more traditional reward-based filtering method +that is widely used in offline RL as a comparator; the results +can be seen in Appendix E. +Since we cannot access the policy labels in the D4RL dataset +to know which policy was used to generate each episode +(and its skill level), in order to evaluate the BD method +explicitly, as we do for the RRC III competition, we de- +veloped a similar but much more challenging discrimina- +tion task as an alternative, purely for BD filter evaluation. +Here, we selected three locomotion domains with relatively +high-dimensional action and state spaces, including Ant-v2, +HalfCheetah-v2, and Humanoid-v3. Firstly, we train three +policies in each domain using online RL algorithms. To +make the BD’s goal of discriminating the episodes from one +policy from those generated by the other two policies, we +allow the performance of the policies (in terms of episode +reward) to be similar; that is, there is no expert policy here, +just different policies. Afterwards, these policies were used +to collect their respective datasets for OPL; we randomly +selected one policy among three learned policies to collect +a small amount of seed data and train the BD filter to recog- +nise that policy’s behaviour. To further increase the difficulty +of identifying the selected policy’s behaviour, we also in- +cluded random behaviours; i.e., state-action pairs randomly +sampled from the state-action space. Finally, to complete +the dataset, the collected data are mixed. All episodes are +annotated with ground truth labels to test the BD method’s +accuracy in detecting episodes originating from the ran- +domly selected policy that was used to seed the BD. We +emphasise that this exercise is included only to evaluate the +accuracy of BD method and does not involve any subsequent +policy learning using these novel datasets. We report the +filter results in Figure 3. Our BD approach can accurately +separate the target policy behaviour from large datasets with +an accuracy of over 99%. With the above results giving +confidence that the BD method can work for D4RL tasks, +we proceed to filter the original unlabeled D4RL datasets +with the BD method to identify high-skill behaviours and +demonstrate how this improves OPL for the D4RL datasets. +The detailed configuration of this filter evaluation task can +be seen in Table 5 in Appendix F. +4.4. Comparative evaluation +4.4.1. REAL ROBOT CHALLENGE (RRC) III +We explored several OPL algorithms, specifically be- +havioural cloning (BC) (Bain & Sammut, 1995), critic +regularized regression (CRR), TD3PlusBC (Fujimoto & +Gu, 2021), policy in latent action space (PLAS) (Zhou +et al., 2021), and implicit Q-learning (IQL) (Kostrikov et al., +2021). Here, we used the implementations of these algo- +rithms with recommended hyperparameters from d3rlpy +(Seno & Imai, 2021). We train each algorithm for 106 time +steps. We report the performance of these algorithms in +Table 1; most of these algorithms performed poorly on both +lift/mixed and lift/expert datasets, whereas the naive BC +algorithm surprisingly outperformed all the other complex +offline RL algorithms on the lift/expert dataset. Motivated by +this, we attempted to improve results on the mixed-policy +datasets by using the BD to identify a subset of mostly +expert-generated examples, which was subsequently used +to learn a policy using BC. +Our BD method improved the performance of the naive +BC algorithm, and enabled it to learn a policy with expert- +level performance on both mixed-skill datasets; see the +results of BC+BD in Table 1. Whereas complex offline RL +algorithms cannot learn an effective policy, especially for +the high-complexity lift/mixed dataset2. CRR is also a filter- +based algorithm, based on a learned advantage function, but +performs much worse than our proposed BD method. +Our final submission to the RRC III competition used a sim- +ple and effective data augmentation method complemented +by a learn & tune training method which took advantage of +spatial symmetries in the arrangement of the RRC robot’s +three fingers; this data augmentation improved the perfor- +mance our policy and has potential to generalise to other +domains where there is spatial symmetry. We detail this +approach Appendix H. We augmented all four datasets of +the RRC III. As the two expert datasets provided are of high +quality, we learned directly on these datasets without using +BD, while for the mixed datasets, we first used the BD to +identify a high-quality subset and then subsequently applied +data augmentation followed BC to learn a control policy. +The training process followed the paradigm of learn & tune; +the hyperparameters are displayed in Appendix I.1. +Table 2 reports the final official evaluated scores of the RRC +III competition; we were the only team to learn a policy +better than the baseline in the lift/mixed task. The control +policy of team decimalcurlew is TD3PlusBC (Fujimoto & +Gu, 2021), and they used spatial smoothing (Mysore et al., +2021) to process the noisy data from the physical environ- +ment. Team superiordinosaur also learned a policy using +BC, and they used feature selection to exclude redundant +features. Team jealousjaguar used IQL (Kostrikov et al., +2021) to learn their control policy, and they augmented the +data using traditional methods. The reports of all teams +can be found on the leaderboard page of RRC website3. +Their data processing methods are well-structured and en- +able their agents to acquire a better performance; however, +their method cannot avoid being hindered by examples in +2A demo video is available at https://www.youtube. +com/watch?v=f4EMizGTzsU +3https://real-robot-challenge.com/ +leaderboard + +Preprint +Table 3. Offline learning algorithm performance with BD vs without BD for three D4RL benchmark tasks. Each result includes three +random seed values and each training trail lasts 106 time steps. We evaluate each learned policy for 100 environmental episodes and +sum the scores. Subsequently, the summed score is normalised by: scorenorm = (score − scoremin)/(scoremin − scoremax), where +scoremin is the reference minimum score, i.e., the average reward of a randomly policy, and scoremax corresponds to the average +rewards of a domain-specific expert provided in the D4RL benchmark (Fu et al., 2020). +BD +BC +TD3+BC +PLAS +IQL +Halfcheetah-medium-expert-v0 + +90.8 ± 3.7 +93.1 ± 2.9 +90.48 ± 5.1 +90.1 ± 4.2 + +56.2 ± 12.4 +89.2 ± 4.1 +70.25 ± 12.0 +72.9 ± 14.9 +Hopper-medium-expert-v0 + +110.7 ± 9.8 +109.3 ± 7.3 +61.2 ± 28.6 +65.8 ± 22.2 + +47.6 ± 16.6 +77.9 ± 28.9 +33.8 ± 21.8 +31.1 ± 23.6 +Hopper-medium-expert-v0 + +108.1 ± 5.1 +111.4 ± 3.9 +109.1 ± 6.7 +109.47 ± 3.8 + +74.4 ± 22.4 +111.0 ± 6.5 +97.7 ± 21.8 +104.8 ± 15.5 +the datasets generated by the lesser-skilled policy. +4.4.2. D4RL BENCHMARKS +We apply BD on the above mentioned datasets; then, we +train several policies using different OPL algorithms and the +high-quality subset identified using the BD method. Here, +the algorithms include BC, TD3PlusBC, PLAS, and IQL; +the implementations used are the same as in the RRC III +competition. For comparison, we train the above policies +with the same hyperparameters on the unfiltered dataset. +With BD, the performances of the agents are improved on +three tasks (see Table 3). In particular, BC’s performance is +significantly improved since BD allows it to partially focus +on the action mode of expert in the raw mixed-quality data. +Examining the learning curves (shown in Figure 7), we can +see that BD improves the learning speed and stability of all +offline RL algorithms investigated. +5. Discussion +The scenarios in which we tested the new BD method are +applicable to a range of OPL problems; i.e., those where +a given large-scale mixed-quality dataset contains a large +amount of high-quality data that would be sufficient to learn +a satisfactory policy. In such a scenario, directly learning a +policy from the unfiltered dataset would not be wise. Train- +ing using only the expert-generated subset of the dataset +would increase the possibility of learning an effective policy. +However, the expert-generated episodes may not be labelled +as such, or the labels might have been lost or intentionally +poisoned (Steinhardt et al., 2017). Moreover, manual anno- +tation is sometimes infeasible due to difficult humans would +face when trying to judge the effectiveness of a policy by +observing the agent as it performs a complex task, such +as the RRC III lift/mixed task. The BD method would be +applicable in such cases; according to our results, the BD +method can accurately extract the expert/target data by ref- +erencing a small amount of seed data, with this small seed +dataset easily obtained by selecting the very highest-reward +episodes. The BD method is more effective than a com- +plex reward-based filter in our high-complexity domains. +Furthermore, we reasonably infer that our BD method can +generalise to other more practical OPL scenarios: 1) when +examples of multiple tasks are included in the dataset (Yu +et al., 2021), but a policy to perform only one task must +be learned; 2) when the dataset contains a large number of +irrelevant behaviours, which can be considered a type of +noise in the dataset that one would not want to clone when +learning a policy (Swamy et al., 2022). +Of course, some limitations of our method exist: 1) it is +not applicable when the seed dataset cannot be obtained; +2) the negative behaviour sample generator could be more +general, i.e., the negative samples could be derived from +more sources, for example, using a large dataset to train +a non-expert policy which can then be used as a negative +sample generator. These ideas will be pursued in our future +work. Furthermore, we will aim to merge the BD principle +with the RL paradigm to propose a better-performing OPL +algorithm. +6. Conclusion +This paper introduces our new proposed BD method, an +effective approach to identify behaviours generated by a +specific policy in order to improve the quality of the dataset +used for subsequent OPL. This approach allows a learning +algorithm to disregard low-skill behaviours, hence improv- +ing the performance of the learned policy. In our work, +the BD method allows a naive BC learning algorithm to +outperform other state-of-the-art offline RL algorithms in +challenging physical problem domains. Furthermore, the +BD method enabled a number of offline RL algorithms to +achieve better performance in various D4RL benchmark +tasks. + +Preprint +References +Allshire, A., MittaI, M., Lodaya, V., Makoviychuk, V., +Makoviichuk, D., Widmaier, F., Wüthrich, M., Bauer, +S., Handa, A., and Garg, A. 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BD training algorithm +Algorithm 1 BD training algorithm +Input: Mixed-quality dataset Dm, seed dataset Ds +D+ = Ds +repeat +Initialise Done = false +Randomly sample K subsets {Dk ++}k=1...K from D+ +Generate K corresponding negative subsets {Dk +−}k=1...K +Initialise K classifiers {Fθk}k=1...K with parameters of θ1, ..., θK +for k = 1 to K do +for epoch = 1 to epochs do +Sample a mini-batch of positive examples, � +D+, with n state-action pairs from Dk ++ +Sample a mini-batch of negative examples, � +D−, with n state-action pairs from Dk +− +Update θk by minimizing the loss: +θk ← θk − λ∇θk +� +E +(st,at)∼� +D− +[− log(1 − Fθk(st, at))] + +E +(st,at)∼� +D+ +[− log Fθk(st, at)] +� +end for +end for +Input data from Dm to the trained classifiers {Fθk}k=1...K +Obtain confidence score values for all episodes Dm and update confidence threshold thconf using histogram +for all Episode = {(st, at)t=1...T } in Dm do +if f := 1 +��K +k=11 +� +1 +T +�T +t=1Fθk(st, at) ≥ thconf +� +> K +2 +� +then +Annotate Episode as expert +end if +end for +Extract the filtered expert subset Df from Dm according to the labels in set f +if Df ∪ Ds ≈ D+ then +Done = true +# If the membership of the positive training set has converged, break the loop +end if +D+ = Ds ∪ Df +# Update training data for next iteration as union of seed and latest subset output by filter +until Done is true +Output: D+ +# For subsequent use with policy learning +B. Environments involved in our work +The environments of the tasks involved in our work are illustrated in Figure 4. +C. Description of how the adaptive confidence threshold is set +We use an adaptive mechanism to adjust the confidence threshold, thconf, which is used in Section 3.5 to convert the +continuous classifier output probability to a discrete binary label. Figure 5 shows an example of selecting the thconf value +over the BD training iterations for the RRC III competition lift/mix dataset. We firstly apply the ensemble classifier to Dm +to get a confidence score (probability) that each episode was generated by an expert policy. Secondly, we calculate the +histogram of these confidence scores. Finally, we use a polynomial to fit the confidence score histogram. The threshold +is set at the confidence score (i.e., probability) at which the first local minimum (trough) to the left of 0.96 on the x-axis +occurs, marked as a blue dot in the bottom row of subplots; so the threshold must be less than 0.96, so there is always some +data selected by the filter. While not done here, a lower bound (for example, 0.8) could be applied to the threshold to avoid +allowing too much data through the filter. As the iterative training process proceeds, the episodes selected by the BD filter +changes and eventually the filter output converges. Our implementation of the polynomial fit function is from NumPy, and +the local minimal search function is from SciPy. + +Preprint +(a) TriFinger robot +(b) Push +(c) Lift +(d) HalfCheetah +(e) Hopper +(f) Walker-2D +(g) Ant +(h) Humanoid +Figure 4. (a) The physical TriFinger robot from the RRC III competition, where three identical robotic fingers are equally spaced 120◦ +apart around the circular arena; the coloured cube is the object to be moved. (b) Illustration of the push task, where the translucent green +dot indicates the 2D target position on the arena floor. (c) Illustration of the lift task, where the translucent cube indicates the target 3D +position and orientation, often above the arena floor. (d)-(h) Illustrate the MuJoCo locomotion task environments. +D. Naive filtering using accumulated reward +These additional results support our reported results in Section 4.2. Here, we naively use the available rewards for every +action to filter the lift/mixed dataset and subsequently learn a policy using BC. The subset of data used to learn with BC +is simply a percentage of all available episodes which have the highest ranking reward; were the reward achieved for the +episode is simply the sum of rewards over all time steps in the episode. We report the result in Table 4. In comparison to the +results reported using our proposed BD method (see Table 1), the performance of the resulting policies is relatively poor. +Table 4. We extract the top 30%, 40%, 50%, and 60% of episodes from the dataset Dm based on the reward accumulated over each episode +for the lift/mixed dataset and then learn a policy using BC. The training and evaluation protocol is same as Table 1. Data augmentation +and Learn & tune are described in Section H. +Percentage of data high-reward episodes selected +30% +40% +50% +60% +Lift/mixed +656 ± 256 +538 ± 346 +492 ± 219 +507 ± 344 +Lift/mixed + Data augmentation + Learn & tune +701 ± 252 +517 ± 262 +515 ± 325 +485 ± 284 +E. Filter using the advantage function of the CRR algorithm +In this section, again as a comparator, we conduct a study of filtering based on the learned advantage function. We use the +CRR algorithm methods (Wang et al., 2020) to estimated the advantage, ˆA, of a state-action pair: +ˆA(st, at) = Qθ(st, at) − 1 +m +�m +j=1Qθ(st, aj), +(3) +where aj ∼ π(· | st), π(· | st) is the learned policy, and Qθ(s, a) refers to the learned critic. The binary decision is made by +a indicator function f := 1[ ˆA(st, at) > 0] + +Preprint +Figure 5. A demonstration of selecting the adaptive confidence threshold, thconf, for the lift/mixed dataset. The top row of subplots shows +the confidence score counts for all episodes in the larger dataset, Dm. To select a filtered subset of Dm, called Df, of (we assume) +mostly expert data for the next BD filter training iteration, the threshold thconf is set to the value at which the first local minimum of the +polynomial fit to the histogram occurs which is less than 0.96; i.e., start at a probability value of 0.96 and scan from right to left along the +polynomial until a local minimum is found. A 10th order polynomial was used for all experiments in this paper. As this iterative process +continues, the membership of Df (the subset of D+ which is changing, since Ds is fixed at the first iteration) changes and eventually +converges. +We first trained the binary-weighted CRR on both of the RRC III competition mixed datasets; each training trial lasts for 1 +million steps (as per the experiments described in the main text of the paper). Once the training is complete, we use the +advantage function to assign a binary weight to each sample in the dataset. Here, we also leverage the form of the dataset +described in Section 3.5. We sum the binary weights in each episode together; if the resulting sum is greater than or equalt +to half the number of the samples in the episode, i.e. �T +t=11[ ˆA(st, at) > 0] ≥ (T/2), we deem the entire episode to have +been collected by an expert policy/agent. The reward function used in this dataset was well-structured and its effectiveness +has been demonstrated in prior work (Allshire et al., 2022). +We report the filter results in Figure 6. This method of filtering is not effective, with a large proportion of non-expert data +not filtered out by the advantage function. And similarly, a large proportion of expert data is labelled as non-expert. As a +result, the performance of the policies learned from this filtered subset of data are not competitive. +F. Configurations used to train D4RL locomotion task policies purely for BD filter evaluation +In Section 4.3, we describe how a number of policies are trained such that they can be used to generated datasets purely for +the purposes of evaluating the BD method’s ability to discriminate between behaviours generated by these different policies, +using a small seed dataset from one of the learned policies as a reference. Here we describe the configurations used to train +these D4RL locomotion task policies, listed in Table 5. +G. Learning curves of offline RL algorithms on D4RL benchmark tasks +The learning curves regardsing Table 3 is shown in Figure 7. + +Iteration 1 +Iteration 0 +Iteration 2 +Iteration 3 +175 +100 +50 +160- +150- +140 +80 +40 +125 +120 +100 +100- +60 +80- +75 +A20: +40 +60 +50- +40 +20 +10 +25 - +20 +0 +0 +0 +175 +175 +50 +100 +150 +150 +80 +40 +125 +125 +100 +100- +60 +75- +75 +40 +A20 +50 +50 +20 +25 - +10 +25 +0- +0 - +0 +0 - +0.2 +0.4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.6 +0.8 +1.0 +0.6 +0.7 +0.8 +0.9 +1.0 +0.6 +0.7 +0.8 +0.9 +1.0 +Probability +Probability +Probability +Probability +th conf +Raw value +Polyfit curvePreprint +Table 5. Details of the configurations used to learn several locomotion policies to perform D4RL tasks; these policies used purely to create +datasets in order to evaluate the BD method’s performance in recognising behaviours similar to those represented in the provided seed +dataset. Three policies are trained from three different random seed numbers. Our implementations of online reinforcement learning, +including SAC (Haarnoja et al., 2018), DDPG (Lillicrap et al., 2015), TD3 (Fujimoto et al., 2018), and PPO (Schulman et al., 2017). These +implementations are taken from stable-baseline3 (Raffin et al., 2021) and we used the recommend hyperparameters. In the Humanoid - v3 +environment, we set reset_noise_scale to 10−2. +Task +Configuration +Agent 1 +Agent 2 +Agent 3 +Noise +HalfCheetah - v2 +Algorithm +DDPG +TD3 +PPO +- +Train length (Time step) +1.5 × 105 +1.5 × 105 +1.5 × 105 +- +Amount (Episode) +5 × 102 +5 × 102 +5 × 102 +5 × 102 +Mean episodic length (Time step) +103 +103 +103 +- +Mean episodic reward +3.24 × 103 +3.26 × 103 +1.93 × 103 +- +Seed amount (Episode) +5 +- +- +- +Ant - v2 +Algorithm +TD3 +TD3 +TD3 +- +Train length (Time step) +1.5 × 105 +2 × 105 +2.5 × 105 +- +Amount (Episode) +5 × 102 +5 × 102 +5 × 102 +5 × 102 +Mean episodic length (Time step) +103 +103 +103 +- +Mean episodic reward +8.72 × 102 +9.61 × 103 +1.03 × 103 +- +Seed amount (Episode) +5 +- +- +- +Humanoid - v3 +Algorithm +SAC +SAC +SAC +- +Train length(Time step) +3 × 105 +3 × 105 +3 × 105 +- +Amount (Episode) +2.95 × 102 +2.95 × 102 +2.95 × 102 +2.95 × 102 +Mean episodic length (Time step) +1.39 × 103 +1.50 × 103 +1.29 × 103 +- +Mean episodic reward +3.35 × 103 +3.36 × 103 +2.98 × 103 +- +Seed amount (Episode) +3 +- +- +- +H. Geometry-based data augmentation and the learn & tune training method +We proposed this technique in the RRC III competition to further enhance the performance of our learned policy. In prior +work, data has been augmented for OPL by editing the state vector to improve the robustness of the learned policy, using +techniques such as adding noise, scaling, dimensional dropout, state-switch, state mix-up, and adversarial transformation +(Sinha et al., 2022); however, these approaches have a limited ability diversify of the dataset, as these operations are anchored +around the same state-action pair. +In our approach, we leverage the spatial (rotational) symmetry of the robot arena (see Figure 4(a)) to transform each +component of the state and action respectively, forming new state-action pairs. However, similar to the well-known +sim-to-real gap problem (which describes how policies learned in ideal simulations often underperform in the real-world due +to lack of consideration of variances in physical properties, such as weight, shape, and friction), trying to leverage spatial +(a) Push/mixed +(b) Lift/mixed +Figure 6. The confusion matrix shows the filter result of the advantage-function-based method on mixed datasets of RRC III + +Positive +1025 +895 +1250 +Actual +1000 +Negative +750 +423 +1497 +500 +Positive +Negative +PredictedPositive +750 +554 +643 +Actual +600 +Negative +342 +855 +450 +Positive +Negative +PredictedPreprint +Figure 7. Learning curves comparing the performance of policies trained with and without the BD pre-processing step; curves are averaged +over three random seeds, with the shaded areas representing the minimum/maximum values across these three seeds. Each data point +refers to the normalised score of 10 environmental episodes. +symmetry to perform data augmentation can also fall foul of similar violated assumptions of ideal physical properties. For +example, we might assume that all three fingers of the robot are identical in every way, but we know that this is unlikely to +be true, and they might vary in ways such as having different frictional properties, the motors generating different torques in +response to a given command, or the sensitivity/calibration of the tactile sensors differing across fingers. Hence, we propose +an additional training method, called learn & tune, to mitigate this issue. The following details this approach. +H.1. Data augmentation leveraging rotational symmetry +The robotic arena and top view sketch are shown in Figure 4(a) and Figure 8. The three fingers of the robot are evenly +spaced around the center of the circular arena, with an angle difference between the nearby two fingers of 120◦. Since the +structure of each finger is theoretically identical, the correctness of the data, including the states of the object and robot, +should remain unchanged after rotating clockwise or counterclockwise around the central point of the arena by 120◦. In +effect, we will spatially rotate the entire experiment (robot, arena, and object) around the centre of the arena by integer +multiples of 120◦ in the world frame, but the indexes of each finger do not move and still references the same location +in the world frame. Since different transformations are required to perform this rotation, depending on whether we are +transforming a robot state or a spatial pose of the object, we split the state vector into robot and object state4 subvectors +4For more detail about the observation space used to obtain the object state estimates, see https://webdav.tuebingen.mpg. +de/real-robot-challenge/2022/docs/tasks.html + +With BD +Without BD +Halfcheetah / IQL +Halfcheetah / PLAS +Halfcheetah / TD3+BC +Halfcheetah/ BC +8 +8 +www +6 +6 +2 +0 +0 +0 +0 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +Hopper / IQL +Hopper / PLAS +Hopper / TD3+BC +Hopper / BC +8 +8 +0 +0 +0 +0 +20 +40 +60 +80 +100 +20 +40 +80 +100 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +60 +0 +Walker-2D / IQI +Walker-2D / PLAS +Walker-2D / TD3+BC +Walker-2D / BC +NWM +8 +8 +6 +6 +4 +2 +2 +2 +0 +0 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +0 +Time step (le6) +Time step (1e6) +Time step (1e6) +Time step (1e6)Preprint +(s = [srobotT , sobjT ]T ) and simultaneously performed the following transformations on the robot state subvector, srobot, its +associated action vector a, and on the object state subvector, sobj, to augment the data: +Figure 8. The sketch of top view of the TriFinger robot arena, and arrows illustrating how states and actions are augmented using spatial +rotation. Three straight red lines mark the default positions of three robotic fingers. The arrows of two different colours indicate the +clockwise and anticlockwise rotation directions used for augmentation. +The robot state and action data, and the object state data, are augmented by permutation and spatial rotation, respectively, as +follows: +srobot +aug (θ) = srobot(θ · k · 120◦) +(4) +aaug(θ) = a(θ · k · 120◦) +(5) +�sobj +x,aug +sobj +y,aug +� += +�cos(k · 120◦) +− sin(k · 120◦) +sin(k · 120◦) +cos(k · 120◦) +� +· +�sobj +x +sobj +y +� +, +(6) +where θ ∈ (0◦, 120◦, 240◦) and k ∈ (0, 1, 2), srobot(θ) and a(θ) represents the state and action subvectors for the finger of +the robot located at an angle of theta degrees, and sobj +x +and sobj +x +represent the x and y coordinates of the object. For example, +the entire experiment will be spatially rotated 120◦ around the z of the world frame when k = 1 (anticlockwise, looking +top-down). The z coordinate of the object remains unchanged. The original data after these rotational transformations are +concatenated with the original dataset to form a larger augmented dataset. +H.2. Learn & tune +As mentioned above, slight physical differences between the fingers will introduce inconsistencies in the dataset (robot +states being physically inconsistent with observed object states) once the above augmentation method is applied. Hence, we +first trained a BC policy on the large augmented dataset to acquire a more general policy, but then subsequently tuned this +policy using the original (unaugmented) dataset at a lower learning rate, making the final deployed policy more consistent +with the consistent physical data collected on the real robots. +H.3. Results and discussion +The effectiveness of our simple augmentation method combined with this learn & tune learning strategy is evidenced by the +scores of both lift/expert and push/expert datasets (see Table 1). As the BD method was not used for these two datasets, + +Preprint +the performance improvements when using standard BC learning can be solely attributed to the use of data augmentation +method followed by learn & tune policy refinement, which possibly mitigates the covariate shift issue (Chang et al., 2021). +In the physical world, the controlled system and the environment in which it is deployed may occasionally have spatial +symmetries; as a simple example, bimanual robots have two arms and usually a left-right mirror symmetry. Combined with +the learn & tune training method proposed here, such spatial symmetries may be useful in finding more general policies +which have been fine tuned on physically-consistent real-world data. +I. Implementation of neural networks +Our +implementations +of +BD +are +open-sourced +on +GitHub: +https://github.com/wq13552463699/ +Behaviour-Discriminator.git +I.1. Learn & tune implementation +In the learn stage of learn&tune, we used Adam (Kingma & Ba, 2014) for learning the neural network parameters with the +learning rate of 10−3, and the learn process lasted 5 × 105 steps with a batch size of 1024. The hyperparameter of the tune +stage is the same as learn, except the learning rate is 2 × 10−4. +Figure 9. The neural network structure of classifier, it illustrates the details of 1(b) +I.2. BD implementation +The BD training process for the push/mixed dataset lasted 3 iterations, 4 iterations for the lift/mixed dataset, and 2 iterations +for each MoJoCo task. For each iteration, we used Adam (Kingma & Ba, 2014) to learn the neural network parameters, with +the learning of 10−3. The training process lasted 50 epochs with a batch size of 1024. +I.3. Other implementation information +Because the RRC III dataset is large, our approach for processing the mixed datasets requires a machine with 16 GB RAM, +whereas for the expert datasets we require 32 GB. Our experiments on the mixed datasets from the RRC III competition and +MoJoCo benchmarks ran on a PC with an Intel I7-10875H CPU (2.30 GHz × 16, 16 GB RAM) and an NVIDIA 2060 GPU. +Moreover, our experiments on the expert datasets from the RRC III competition ran on high-performance cluster, where +each machine is configured with 2 Intel Xeon Gold 6152 CPUs (2.1 GHz × 22, 64 GB RAM) and 2 Nvidia Tesla V100. + +FC(Action dim*2) / BatchNorm / ReLu +Classifier neural network +FC(512) / BatchNorm / ReLu +FC(256) / BatchNorm / ReLu +FC(128) / BatchNorm / ReLu +FC(Action dim*3) / BatchNorm / ReLu +FC(128) / BatchNorm / ReLu +State +FC(2) / Softmax +Output +Action +Encoder \ No newline at end of file diff --git a/rNFKT4oBgHgl3EQfIi19/content/tmp_files/load_file.txt b/rNFKT4oBgHgl3EQfIi19/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9631b09f4aaafd0aff3663c8890f31dd921bc15e --- /dev/null +++ b/rNFKT4oBgHgl3EQfIi19/content/tmp_files/load_file.txt @@ -0,0 +1,1446 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFKT4oBgHgl3EQfIi19/content/2301.11734v1.pdf,len=1445 +page_content='Behaviour Discriminator: A Simple Data Filtering Method to Improve Offline Policy Learning Qiang Wang 1 Robert McCarthy 2 David Cordova Bulens 1 Kevin McGuinness 3 4 Noel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFKT4oBgHgl3EQfIi19/content/2301.11734v1.pdf'} +page_content=' O’Connor 3 4 Francisco Roldan Sanchez 3 4 Stephen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFKT4oBgHgl3EQfIi19/content/2301.11734v1.pdf'} +page_content=' Redmond 1 3 Abstract This paper studies the problem of learning a con- trol policy without the need for interactions with the environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFKT4oBgHgl3EQfIi19/content/2301.11734v1.pdf'} +page_content=' instead, learning purely from an existing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFKT4oBgHgl3EQfIi19/content/2301.11734v1.pdf'} +page_content=' Prior work has demonstrated that offline learning algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFKT4oBgHgl3EQfIi19/content/2301.11734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFKT4oBgHgl3EQfIi19/content/2301.11734v1.pdf'} +page_content=', behavioural cloning and offline reinforcement learning) are more likely to discover a satisfactory policy when trained using high-quality expert data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFKT4oBgHgl3EQfIi19/content/2301.11734v1.pdf'} +page_content=' However, many real-world/practical datasets can contain significant proportions of examples generated us- ing low-skilled agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFKT4oBgHgl3EQfIi19/content/2301.11734v1.pdf'} +page_content=' Therefore, we propose a behaviour discriminator (BD) concept, a novel and simple data filtering approach based on semi- supervised learning, which can accurately discern expert data from a mixed-quality dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFKT4oBgHgl3EQfIi19/content/2301.11734v1.pdf'} +page_content=' Our BD approach was used to pre-process the mixed- skill-level datasets from the Real Robot Challenge (RRC) III, an open competition requiring partici- pants to solve several dexterous robotic manipu- lation tasks using offline learning methods1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFKT4oBgHgl3EQfIi19/content/2301.11734v1.pdf'} +page_content=' the new BD method allowed a standard behavioural cloning algorithm to outperform other more so- phisticated offline learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFKT4oBgHgl3EQfIi19/content/2301.11734v1.pdf'} +page_content=' Moreover, we demonstrate that the new BD pre-processing method can be applied to a number of D4RL benchmark problems, improving the performance of multiple state-of-the-art offline reinforcement learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFKT4oBgHgl3EQfIi19/content/2301.11734v1.pdf'} +page_content=' 1University College Dublin, Ireland 2University College Lon- don, UK 3Insight SFI Research Centre for Data Analytics, Ireland 4Dublin City University, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFKT4oBgHgl3EQfIi19/content/2301.11734v1.pdf'} +page_content=' Correspondence to: Stephen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFKT4oBgHgl3EQfIi19/content/2301.11734v1.pdf'} +page_content=' Redmond zone axis, wherein +the pores were aligned along the observation direction. Conventionally, the atomic structure from the +same zone-axis was observed via HRTEM [19], but only the pore arrangements were resolved by this +method, and the atomic resolution was difficult to achieve. Herein, the electron probe current was set +to 0.5 pA to suppress the beam damage, which was approximately two orders of magnitude lower than +that of the usual STEM observation condition for analyzing typical inorganic materials. Under these +conditions, OBF STEM observations of the FAU-type zeolite sample were conducted, as shown in Fig. +2b-2e. Fig. 2b shows the experimental OBF image of the FAU-type zeolite. The OBF image of the FAU +framework structure indicated the atomic sites as bright spots, evidently for the tetrahedral (T-sites +occupied by Si or Al) and oxygen sites. An amorphous layer covering the sample surface [20] is also +recognizable in the image. Fig. 2c shows the power spectrum of the OBF image in Fig. 2b that exhibits +an information transfer up to 0.869 Å. Furthermore, Fig. 2d shows a unit cell-averaged OBF image +obtained from the original OBF image in Fig. 2b. The FAU framework structure can be observed at an +atomic resolution and conforms extremely well with the simulated image shown in the inset, indicating +that the atomic structure can be resolved without any electron irradiation damage. Fig. 2e is the cropped +image of Fig. 2d, focusing on the D6R building block of the FAU structure. It evidently shows that the +tetrahedral units are connected via corner-shared oxygen sites. In zeolites, the oxygen-bridging sites +between the tetrahedral units play an essential role in dictating the properties exhibited by the material, +such as the catalytic activity [21] and structural transformation introduced by the interactions between +the framework host and captured guests [22]. Thus, the capability of OBF STEM to visualize individual +oxygen atom sites significantly helps to understand the structure-property relationship in zeolites. It is +noteworthy that the visibility of the oxygen atom was already attained in the raw OBF image before +unit-cell averaging. The SNR of the S/TEM images of zeolites is usually enhanced by averaging the +raw data using a priori knowledge about the sample, such as the space groups of the material [23,24]. +Although this is effective for homogeneous bulk structure analyses, it cannot be applied to +heterogeneous or nonperiodic local structure analyses. Therefore, the presented direct atom imaging +capability will be helpful for zeolitic heterogeneous/nonperiodic structure analyses, such as aluminum +substitution, counter cations, and other defects. Later in this study, we have demonstrated the OBF +STEM imaging of a defect structure in the FAU-type zeolite. + + + + +6 + + +Fig. 2. Atomic-resolution OBF STEM observation of an FAU zeolite along <110> zone +axis. (a) Schematic of the FAU zeolite framework structure and projected atomic-structure +model along <110> zone-axis. Red and blue polygons represent the building units (sodalite +cages and double 6-rings (D6Rs), respectively). (b) OBF STEM image of FAU zeolite +observed at the edge of the sample. Bright spots indicate T- and oxygen-sites (scale bar: 1 nm). +(c) Fast Fourier Transform (FFT) of (b), wherein FFT spots are seen up to 0.869 Å resolution +in real space. (d) Repeat-unit-cell averaged OBF image. The inset is a simulated OBF image +calculated with the same observation condition as that in the experiment. The location of the +D6R structure, which is shown in (e), is highlighted by a dashed rectangle. (e) Magnified OBF +image of the rectangular region indicated by the red dashed line in (d). The atomic structure +models are drawn using VESTA [25]. + + +Sodalite cage +b +D6R +[110] +[001] +[110] +0.869A +Sim. +7 +We compared the OBF images with other STEM images obtained under the same dose conditions. +Fig. 3a shows the OBF, iDPC, conventional BF, and ABF images. The OBF image shows the FAU +framework structure with the highest SNR, conforming with the noise-normalized CTF calculation +shown in Fig. 1b. Although the iDPC image also reveals the basic FAU framework structure, individual +atomic sites, such as oxygen columns, are not distinguishable, as shown in the inset. For the further +analysis of these contrast characteristics, we simulated the noise components of OBF and iDPC images +(see the Materials and Methods section for details). In the OBF reconstruction, the noise level is set to +be flat as a function of spatial frequency, which is known as the noise-normalized condition. This is +equivalent to the so-called white noise. The noise fluctuation of the OBF image contrast is thus +uniformly random for the entire FOV. It is noteworthy that this noise-component image is displayed on +the same spatial scale as that of the experimental images shown in Fig. 3a for comparison. However, +for the iDPC image, the contrast fluctuation due to noise exists on a spatially larger scale than that of +the OBF image. This is confirmed by a line profile of the noise component. In the integration process +of the DPC signal to form the iDPC image, the low-spatial-frequency component of the image is much +more amplified than the higher-spatial-frequency components [26]. However, the iDPC signals +essentially do not exhibit contrast transfer around the low-frequency domains against noise, as shown +in Fig. 1b. Thus, under the low-dose condition, this amplification effect enhances the noise component +in the lower frequency regions, resulting in the appearance of long-range contrast fluctuation, as shown +in Fig. 3b. This is the reason why the iDPC image contrast appears smoother but has longer-range noise- +fluctuation than those of the other methods. We also examined the experimental image intensity +distribution of each imaging technique, as shown in Fig. S1, wherein the longer-range noise effect was +more severe owing to the wide FOV. Although the OBF image exhibits an interpretable image contrast +corresponding to the sample thickness and atomic sites, the iDPC image exhibits long-range intensity +fluctuations in the experiment as well as the simulations, as shown in Fig. 3b. This contrast is much +stronger than that of each atomic site in the zeolitic framework. This results in a poor visibility of the +atomic sites and makes it difficult to interpret the atomic structures from the obtained image. In other +STEM images, such as ABF and BF, the basic structure of the FAU framework is roughly visible, but +the detailed atomic structure analysis is challenging under the present low-dose condition. + + + + +8 + + +Fig. 3. Comparison between atomic resolution images of OBF STEM and other STEM +techniques. (a) STEM images obtained via OBF, iDPC, ABF, and conventional BF imaging +techniques (scale bar: 1 nm). All the images were recorded under the same electron dose and +optical conditions (except for the defocus) as described in the Materials and Methods section. +The insets are the cropped and enlarged versions of the original images, and the orange arrows +indicate the oxygen sites in the FAU zeolitic structure. (b) Comparison of the noise components +between the OBF and iDPC simulated images (see the Materials and Methods section for +details). The intensity profiles of noise are also shown (obtained from the orange lines). The +assumed dose is the same as that in the experiments shown in (a), and the noise components in +both the methods are obtained by the same noise-introduced segmented-detector datasets. As +indicated by the orange arrows, the iDPC noise image has longer-range fluctuations than that +of the OBF. + + + +OBF +iDPC +ABF +BF +OBF noise +iDPC noise +0.15 +0.10 +0.10 +0.05 +0.00 +-0.05 +0.05 +0.10 +0.10 +0.15 +0.15 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +60 +Distance (A) +Distance (A) +9 +Direct observation of FAU twin boundary +We applied the OBF technique to characterize the atomic structure of a twin boundary in the +FAU zeolite. In FAU-type zeolites, the framework is constructed by cubic stacking of a layered +structure unit called a ‘faujasite sheet’ [27]. When the faujasite sheets are stacked in a hexagonal +sequence, the resultant framework exhibits an EMT-type structure, known as a polymorph of an FAU- +type zeolite. There are twin boundaries between two opposite sequences of the cubic stacking in the +FAU framework that likely result in an EMT-type structure at the boundary, as schematically shown in +Fig. 4a. However, the detailed atomic structure could not be directly determined owing to the limited +spatial resolution under the low-dose condition in the previous TEM study [28]. +Fig. 4b shows the OBF image of the FAU twin boundary. This image indicates that the FAU +cubic stacking sequence is inverted at the twin boundary. The power spectrum of the image indicates +an information transfer beyond 1 Å. For further analysis, we averaged the structural units of the FAU +twin boundary, as shown in Fig. 5a. The T and oxygen atomic sites are evidently visible in the twin +boundary core, and two FAU-type domains are connected coherently at the atomic scale. Furthermore, +the atomic structure of the twin boundary is confirmed to be identical to that of the EMT-type structure. +Density functional theory (DFT) calculations were performed to evaluate the stability of the twin +boundary structure. The initial twin-atomic structure was created by stacking the faujasite sheets based +on the OBF image and then relaxed via DFT calculations. Fig. 5c shows the relaxed atomic structure +model, and Fig. 5b shows its corresponding simulated OBF image. The experimental image conforms +well with its simulated counterpart. Furthermore, the interface energy was calculated to be 7.4 mJ/m2, +which is comparable with those of the twin boundaries in face-centered cubic (FCC) metals on the +{111} plane [29], but approximately three orders of magnitude lower than those (typically) in oxide +ceramic materials, such as grain boundaries and twin boundaries [30,31]. The origin of this difference +can be explained as follows: in the {111} twin boundary of cubic zirconia, for example, the origin of +the higher interface energy is attributed to the different coordination numbers of anions around the +cation sites on the interface whereas the cation sites produce a coherent interface structure similar to +those of FCC metals [32]. In the case of zeolites, the framework is constructed by the corner-sharing of +rigid TO4 tetrahedra, which have a nearly perfect tetrahedral shape and are connected via oxygen atoms +as soft hinges, offering a rigid but stress-free atomic structure [33]. Thus, zeolites can relax their +framework structure by simply changing the bond angle between two rigid TO4 tetrahedrons (T-O-T +angle). In silicate materials, the atomic structure is energetically stable over a wide range of T-O-T +angles [34]. The observed structure of the FAU twin boundary was constructed in a similar manner, +keeping the coordination numbers of the cations and anions unchanged across the boundary. This +structural flexibility should result in extremely low excess energy at the twin boundary. Structural +information about minute strains around some defects is essential for applications such as molecular +sieves and gas separators. It may affect the diffusion process of ions and molecules adsorbed in the +zeolitic nanocavity. + + + + +10 + + +Fig. 4. Atomic-resolution OBF STEM image of FAU twin boundary. (a) The framework +model of the FAU twin boundary. The FAU cubic stacking sequence is inverted on the twin +boundary, making the EMT framework structure with hexagonal stacking. The structure +highlighted with a green-dotted box is a faujasite sheet, a layer structure unit for the FAU and +the EMT frameworks. The triangles represent the directions of the stacking sequence. (b) +Atomic-resolution OBF STEM image of the FAU twin boundary (scale bar = 1 nm). The inset +is the FFT pattern of the OBF image, which exhibits a contrast transfer beyond 1 Å. + + + +Fauiasite sheet +11 + + +Fig. 5. Comparison between experimental OBF image and simulated image based on +DFT-relaxed structure of the FAU twin boundary. (a) Unit-cell averaged experimental OBF +image obtained from the raw experimental image shown in Fig. 4(b). The averaging operation +is performed along the direction parallel to the interface, and no structural information is +assumed about the symmetry other than the translational symmetry along the boundary (scale +bar: 1 nm). (b) Simulated OBF image based on the DFT-relaxed structure shown in (c). The +image is calculated under the same condition as that of the experiment. (c) Atomic structure +model of FAU twin boundary relaxed by the DFT calculation. The blue and red balls indicate +the T- and oxygen-sites, respectively. These images/structures show good agreement in both +T- and oxygen-sites. + + + +Exp. +Sim. +DFT +12 +Discussion +We developed a highly dose-efficient STEM imaging technique, OBF STEM, for application in +low-dose atomic-resolution imaging. We demonstrated that OBF STEM can directly reveal the atomic +structures of all elements in an FAU-type zeolite, which is a beam-sensitive material, with a sub- +angstrom spatial resolution. OBF STEM can also be used to observe the lattice defects in zeolitic +framework structures. We succeeded in directly determining the atomic structure on an FAU twin +boundary, and the corresponding result was consistent with the DFT calculations. The proposed +technique can thus be used to characterize the local atomic structure in zeolites and other beam-sensitive +materials, facilitating the study of structure-property relationships in these materials. + + + + +13 +Materials and Methods +Atomic-resolution OBF STEM observation of an FAU-type zeolite +Atomic-resolution OBF STEM images were acquired using an aberration-corrected STEM +(JEOL JEM ARM-300F) equipped with a second-generation segmented annular all-field (SAAF) +detector (16-segmented type) [35]. We developed an in-house program for the real-time OBF display +function and implemented it in the SAAF system, as shown in Fig. S3. Movie S1 shows the real-time +observation of a SrTiO3 [001] sample with an accelerating voltage of 300 kV and a probe-forming +aperture of 30 mrad. All the atomic columns, including the oxygen atoms, were visualized under a low- +dose condition (probe current: 0.5 pA, i.e., two orders of magnitude less than the usual condition). This +result demonstrated the capability of OBF STEM for low-dose and live atomic-resolution imaging. We +used a real-time OBF display system to acquire all the experimental OBF images shown in the present +study. +For the TEM sample preparation of an FAU-type zeolite, a commercially available powder +sample of FAU zeolite (Tosoh Corp., Si/Al=50) was gently crushed in an agate mortar with ethanol and +dispersed onto a TEM microgrid. Before STEM observation, the sample was dehydrated overnight in +the high vacuum environment of the TEM column to suppress the irradiation damage [8]. The +accelerating voltage was set to 300 kV, which effectively reduces the irradiation damage in the zeolites +[6,36]. The probe current and probe-forming aperture were 0.5 pA and 15 mrad, respectively. Images +of the FAU bulk structure were sequentially acquired at a dwell time of 16 µs with 1024 × 1024 pixels +in the same region to suppress irradiation damage and scan distortion. Under this condition, the total +dose was 1.2×103 e-/Å2 per frame. For the FAU twin boundary observation, the dwell time was reduced +to 10 µs to further suppress the image distortion, with the total dose was 7.5×102 e-/Å2 per frame. After +the sequential image acquisition, the first five images were aligned and averaged for each data set. +Furthermore, we obtained unit-cell-averaged images for a detailed structural analysis, as shown in Figs. +2d and 5a. It can be noted that a priori knowledge about the structure group symmetry of the atomic +structure was not assumed for the image averaging except for the translational symmetry for both FAU +bulk and twin boundary analyses. +To obtain the OBF images, the camera length was set such that the edge of the STEM direct beam +disk coincided with the outermost edge of the SAAF detector. Under these conditions, the OBF image +was obtained using Equation (1): + +𝐼OBF(𝑹p) = ℱ−1 [∑ 𝐼𝑗(𝑸p)𝑊𝑗(𝑸p) +16 +𝑗=1 +] = ∑ 𝐼𝑗(𝑹p) ⊗ 𝑤𝑗(𝑹p) +16 +𝑗=1 +, +(1) + +where 𝐼OBF(𝑹p), 𝐼𝑗(𝑸p), 𝑊𝑗(𝑸p), and 𝑤𝑗(𝑹p) are the OBF image intensity, Fourier transformed +image acquired by the j-th segment 𝐼𝑗(𝑹p), frequency filter calculated for the j-th segment, and point +spread function obtained via the inverse Fourier transform of the frequency filter 𝑊𝑗(𝑸p), respectively. + + +14 +The filtering process was performed by multiplying the frequency filter in the reciprocal space 𝑸p or +convolution with the point spread function in the real space 𝑹p. The post-processed OBF image could +be obtained via either procedure, and the real-time OBF imaging synchronized with the STEM scan +was acquired using the approximated convolution process in real space [13]. For the focal condition to +obtain the STEM images, it was reported that the OBF and DPC image contrast can be theoretically +maximized upon focusing the electron probe on the mid-plane of the specimen [13,37]. In contrast, the +ABF and BF images exhibited the highest contrast upon focusing the probe on the entrance surface [16]. +Thus, we acquired the images under the optimal focal conditions for each technique. To obtain the +experimental/simulated iDPC, BF, and ABF images, the segmented/annular detector images were +synthesized using the SAAF detector datasets to reproduce the detector geometry dedicated to each +method. + +Image simulations +For the STEM image simulation, we used the MuSTEM package [38] based on the multi-slice +model [39]. 16-segmented-detector images were calculated and processed using the OBF reconstruction +algorithm to obtain the simulated OBF image. The effective source size was considered by convolution +with a 2D Gaussian with a full-width-half-maximum of 0.6 Å. The sample thickness was assumed to +be 10 nm, and the defocus Δ𝑓 was set to middle-focus condition, wherein the focal plane is located at +the mid-plane inside the sample (Δ𝑓 = −5 nm). +The STEM image simulation was also used for noise property analysis, as shown in Fig. 3b. First, +noise was added to the simulated images of each detector segment based on the Poisson statistics. The +noisy and noise-free images of each imaging method were then reconstructed. The assumed dose was +equal to that of the experimental condition, as shown in Fig. 3a. The noise component images were then +obtained by subtracting the noise-free images from their noisy counterparts, as shown in Fig. S2. The +noise-component images were normalized using the contrast range of their corresponding noise-free +image. The noise characteristics of different techniques were then compared, as shown in Fig. 3b. + +DFT calculations +To relax the FAU twin-boundary structure and calculate the interface energy, we performed DFT +calculations using the VASP code [40] with the rev-vdW-DF2 method [41], which is suitable for +calculating zeolitic atomic structures and energies [42]. For the relaxation, we first relaxed the FAU +bulk structure, whose data is available in the International Zeolite Association database [43]. The initial +FAU twin boundary structure was then created by connecting the two FAU framework models with +opposite stacking sequences. Finally, we obtained the relaxed FAU twin structure and calculated the +interface energy Δ𝐸interface as follows: + +Δ𝐸interface = 𝐸twin − 𝐸bulk +2𝐴 +, +(2) + + +15 +where 𝐴 is the cross-sectional area of the interface, and 𝐸bulk and 𝐸twin are the total energies of the +FAU bulk and twin boundary structures, respectively. + + + + +16 +References +[1] +Y. 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A part of this study was conducted at the Research Hub for Advanced +Nano Characterization, the University of Tokyo, with support from the Nanotechnology +Platform (Project No. 12024046), MEXT, Japan. +Author contributions: K.O., T.S., and N.S. designed the study and wrote the paper. K.O. +performed the STEM experiments, image simulations, and DFT calculations. K.Y. supported +the STEM observation of zeolite samples and contributed to the discussions. Y.K. supported +the development of the OBF STEM system and software. Y.I. contributed to the discussions +and suggestions. N.S. and T.S. directed the entire study. +Competing interests: The authors declare the following financial interests/personal +relationships which may be considered as potential conflict of interests: a part of the present +authors are inventors on Japanese unexamined patent application publication filed by the +University of Tokyo (No. 2021-077523). +Data and materials availability: All data necessary to evaluate the conclusions of this study +are present in the paper and/or the Supplementary Materials. The data may be provided upon +the request to the authors. + + + + +21 +Supplementary Materials + + + +Supplementary Text +Dose efficiency evaluation based on the noise-normalized CTFs +To compare the contrast transfer efficiency of different STEM techniques against the noise-level, +we calculated the noise-normalized CTFs as shown in Fig. 1b. In this note, we show how to evaluate +the theoretical dose efficiency of each technique by the noise-normalized CTFs. +Under the weak phase object approximation (WPOA), the STEM image intensity 𝐼STEM(𝑹p) is +given as follows + +𝐼STEM(𝑹p) = 𝑑0 + ℱ−1[𝜎𝑉(𝑸p)𝛽(𝑸p)], +(S1) + +where 𝑑0, ℱ−1, 𝜎, 𝑉(𝑸p), and 𝛽(𝑸p) respectively indicate background intensity, the inverse Fourier +transformation operator, the interaction parameter determined by the accelerating voltage of the electron +beam, the Fourier component of the specimen projected potential, and the CTF. Assuming that the +specimen projected potential in real space 𝑣(𝑹p) is a delta function (i.e., 𝑣(𝑹p) ≅ 𝑣0𝛿(𝑹p) where 𝑣0 +is a constant), the STEM image intensity above the point scatterer (𝑹p = 𝟎) 𝐼STEM(𝟎) is approximated +as follows + +𝐼STEM(𝟎) = 𝑑0 + 𝜎𝑣0 ∫ 𝛽(𝑸p) d𝑸p. +(S2) + +This equation indicates that we can evaluate the contrast amplitude approximately by the second term, +integration of CTF over the frequency domains, because the first term 𝑑0 is the background. Thus, we +compare the obtainable contrast against the noise-level among different STEM techniques by +integrating the noise-normalized CTFs [13,44]. In the Poisson statistics, the signal-to-noise ratio (SNR) +is proportional to the √𝜆, where 𝜆 is the electron dose. Here, the integration value of noise-normalized +CTFs can be regarded as a relative SNR between different techniques, and thus we can compare the +dose-efficiency by the squared values of the CTF integration. + Table S1 shows the dose efficiency ratio calculated by the integration of the noise-normalized +CTFs for OBF, iDPC, iCoM (integrated center-of-mass) [45,46], ABF, and conventional BF imaging +methods, as discussed above. The iCoM meghod is a kind of iDPC imaging with a pixelated detector. +Since we previously showed that OBF imaging can be extended to the pixelated detector [13], we +calculated the CTF of OBF using the pixelated detector also. As for the segmented detector, the detector + + +22 +shape is the same as literatures for OBF [35] and iDPC [17], respectively. According to the calculated +values shown in Table S1, OBF has approximately two-orders of magnitude higher dose efficiency than +ABF theoretically. Furthermore, because OBF reconstructs the phase-contrast image in a theoretically +optimized manner to obtain the highest SNR for each type of detector, the calculated dose efficiency of +OBF is higher than the iDPC or iCoM techniques that use segmented or pixelated detectors, respectively. +Since the iDPC technique is currently used for the STEM observation of beam-sensitive samples [47,48], +the OBF observation should be able to reduce the irradiation dose more or obtain a higher spatial +resolution on the same samples. Additionally, it should be noted that the OBF using a segmented +detector obtains almost the same dose efficiency as iCoM that uses a pixelated detector. Although it is +known that the pixelated detector can get significantly rich information about the sample, which could +lead to higher dose-efficiency, this type of detector still needs longer dwell time while the recent +technological progress improves the read-out speed. In the low-dose experiment, the operator must tune +experimental conditions quickly under low SNR conditions. Thus, the capability of high-dose efficiency +with a high-speed segmented detector is definitely helpful for beam-sensitive materials analysis. + + + + +23 + +Fig. S1. Comparison between OBF and iDPC images obtained from the same +experimental dataset. (a) OBF and iDPC images generated from the same dataset. The OBF +image is the same as Fig. 2b. The intensity profiles taken along (b) [001] direction and (c) [1- +10] direction respectively from the orange rectangles shown in the OBF and iDPC images. +Since the observed sample is wedge-shaped and has an amorphous layer near the edge, the +projected atomic potentials should be increased from left hand side (vacuum area) to right hand +side (thicker sample area) along the direction shown in (b). On the other hand, along the +direction shown in (c), the thickness is almost uniform and corresponding image contrast +should also be uniform. These are the case for the OBF image, but the iDPC image has strong +intensity fluctuations (indicated by orange arrows) as discussed in Fig. 3b. + + + +b +b +Image intensity (a.u.) +Image intensity (a.u.) +20 +40 +60 +80 +100 +120 +0 +20 +40 +60 +80 +100 +120 +Distance (A) +Distance (A) +'n'e) +Image intensity (a.u.) +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +Distance (A) +Distance (A) +24 +Fig. S2. Schematic of noise evaluation technique. Schematic illustration of noise evaluation +method shown in Fig. 3b. By combining the noise-free image and noisy image, noise +characteristics against the contrast range can be calculated. + + + + +25 + +Fig. S3. Display of live OBF imaging system. Captured image of Movie S1 and its description. +Movie S1 shows the real-time atomic-resolution OBF imaging of SrTiO3 [001] under a low- +dose condition. The dwell time is 10 µs, and the image is sampled with 512x512 pixels. In the +upper row, the left panel, center panel, and the right panel show ABF, OBF, and annular dark- +field (ADF) images, respectively. In the lower row, the left panel shows a center bright-field +(CBF) image, and the center and right panel shows Fourier transformed OBF and ADF images, +respectively. The images, including OBF, are synchronized with STEM probe scans and +updated in real-time. The updated area in this capture is highlighted with a dotted line in each +image. Movie S1 also shows the area scan mode, where the only selected area inside the image +is scanned and the frame rate is increased for tuning aberrations such as defocus. + + + +Arrange +ToolsSlideShowWindowHelp +PreDefense_slide_ooe_FINAL +SavedtomyMac +Playback +Share +Comments +AI +SmartArt +Picture +Styles +Shape Outline +ive OBF imaging of SrTiO3[001] (movie) +34 +ScreencaptureofOBFliveimaging(x1realtimemovie) + ABFPanel +回X +Sr +ABF +OBF +ADF +Ti+O +0 +V SrTiO3 [001] +V HT = 300 kV +V Convergence angle = 30 mrad +Beam current = 0.5 pA +(normally: ~30 pA) +CBF +LiveFFTof OBF +Live FFT of ADF +Dwell time = 10 μs (512x512) +Imageprocessing software +wasdevelopedvia C++ +OBFmethodimplementedinSTEM +Tuningfocus,aberrations, and Fov +Very useful for low-dose observation! +0:04.06 +1去一電流0.5A七低下条件觀察秸果左见世寸 +中OBF像右ADF，左ABF像，OBF像么表子思 +二=上：OBF像见5 +差调野来手法等感觉常高可能 +26 +Table S1. Comparison of dose efficiency of different STEM methods. Dose efficiency ratio +among different STEM imaging techniques based on the noise-normalized CTF calculations. +The values are normalized such that the dose-efficiency of ABF becomes one. + + + +Movie S1. +Live atomic-resolution OBF imaging of SrTiO3 [001] using the real-time OBF system. + + diff --git a/sdE3T4oBgHgl3EQfMwm8/content/tmp_files/load_file.txt b/sdE3T4oBgHgl3EQfMwm8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..06b2a5d974b2cdf1e6ae805488e174b90d12b778 --- /dev/null +++ b/sdE3T4oBgHgl3EQfMwm8/content/tmp_files/load_file.txt @@ -0,0 +1,1036 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf,len=1035 +page_content='1 Direct imaging of local atomic structures in zeolite using novel low- dose scanning transmission electron microscopy Kousuke Ooe1,3, Takehito Seki1,2*, Kaname Yoshida3, Yuji Kohno4, Yuichi Ikuhara1,3, Naoya Shibata1,3* 1Institute of Engineering Innovation, School of Engineering, the University of Tokyo, Yayoi 2-11-16, Bunkyo, Tokyo, 113-0032, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 2PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama, 332-0012, Japan 3Nanostructures Research Laboratory, Japan Fine Ceramics Center, Mutsuno 2-4-1, Atsuta, Nagoya, 456-8587, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 4JEOL Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=', 1-2-3 Musashino, Akishima, Tokyo 196-8558, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Email: seki@sigma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='u tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='jp (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' shibata@sigma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='u tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='jp (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=') Abstract Zeolites have been used in industrial applications such as catalysts, ion exchangers, and molecular sieves because of their unique porous atomic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' However, the direct observation of zeolitic local atomic structures via electron microscopy is difficult owing to their low resistance to electron irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Subsequently, the fundamental relationships between these structures and their properties remain unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' A novel low-electron-dose imaging technique, optimum bright-field scanning transmission electron microscopy (OBF STEM) has recently been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' It reconstructs images with a high signal-to-noise ratio and a dose efficiency approximately two orders of magnitude higher than that of conventional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Herein, we performed low-dose atomic-resolution OBF STEM observations of an FAU-type zeolite, effectively visualizing all the atomic sites in its framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Additionally, the complex local atomic structure of the twin boundaries in the zeolite was directly characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The results of this study facilitate the characterization of the local atomic structures in many electron-beam-sensitive materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 2 Introduction Zeolites are porous materials with regularly arranged nanosized pores, which enable a wide range of applications in catalysis, gas separation, and ion exchange [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The material properties of zeolites are closely related to the geometry of their pores and their subsequent interactions with any adsorbed guest molecules and ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' To date, diffractometric techniques have been most often used for the structural analysis of zeolites [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Although diffraction methods can accurately analyze averaged structures, obtaining local structural information related to defects, interfaces, and surfaces is extremely difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Scanning transmission electron microscopy (STEM) is a powerful technique for local structural analysis that enables the direct observation of atomic structures in electron-resistant materials at a sub-angstrom resolution [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' However, zeolites are more electron beam-sensitive than other inorganic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Thus, atomic-scale observations via electron microscopy are severely limited by electron irradiation damage [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Menter observed faujasite zeolite for the first time in 1958 via high- resolution transmission electron microscopy (HRTEM), and reported a lattice resolution of 14 Å [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Subsequently, the zeolite framework structure was observed [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In the 1990s, an aberration corrector was developed and the S/TEM resolution was significantly improved [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' These technological advances have enabled the direct observation of the framework structure and arrangement of adsorbed cations in zeolites [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' However, it remains extremely challenging to directly observe all the atomic sites in zeolites, including the Si/Al and oxygen sites, owing to the severe electron irradiation damage within zeolites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Recently, the development of new STEM electron detectors has led to more advanced imaging techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Although conventional STEM uses a single annular detector to detect transmitted/scattered electrons to form images, the recently developed segmented/pixelated detectors can simultaneously form many STEM images using electrons detected in multiple areas on the diffraction plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' By processing these multiple STEM images, information regarding the electromagnetic fields and phase information of the samples can be obtained [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' We have theoretically developed an optimum bright-field (OBF) STEM technique for low-dose imaging that enables the observation of atomic structures at the highest signal-to-noise ratio (SNR) using segmented/pixelated detectors [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 1a shows the schematic of the OBF STEM technique, which uses a segmented detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Herein, a finely focused electron probe is raster-scanned across the sample, and the transmitted/scattered electrons are detected at each raster by a multiple-segmented electron detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Subsequently, frequency filters are applied to each image obtained by the corresponding detector segment, and the filtered images are assembled to obtain the OBF image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' These filters were designed to maximize the SNR of the synthesized image based on the STEM contrast transfer function (CTF) [14] and noise-evaluation theory [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 1b presents the noise-normalized CTFs for various phase-contrast STEM techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The noise-normalized CTF represents the SNR as a function of spatial frequency and is helpful in evaluating the imaging efficiency of different techniques [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' OBF STEM, which uses a segmented detector, exhibits a much higher imaging efficiency than those of conventional techniques such as annular bright-field (ABF) and conventional bright-field (BF) imaging [16] over an entire spatial 3 frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Additionally, OBF is more efficient than integrated differential phase-contrast (iDPC) imaging [17], which is a phase-imaging technique that also uses a segmented detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' As described in Supplementary Text, OBF imaging achieves a dose efficiency approximately two orders of magnitude higher than those of the conventional STEM imaging methods and is ~24% higher than that of iDPC imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Furthermore, OBF can obtain information at much higher spatial frequencies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=', higher resolution) than those of the conventional techniques, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In this study, we obtained atomic-level structural images of zeolite via aberration-corrected STEM with an accelerating voltage of 300 kV and a probe-forming aperture of 15 mrad, wherein the information limit of the OBF contrast transfer was calculated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='66 Å in real space, indicating the present optical condition to be sufficient for obtaining images with atomic resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Furthermore, the OBF images could be reconstructed in real time [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In a typical atomic- resolution STEM operation, fine optical tuning adjustments, such as astigmatism correction, defocus correction, and field-of-view (FOV) adjustment, are performed by an operator who refers to atomic- resolution images displayed on a monitor in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In low-dose observations, fine tuning becomes much more difficult because the operator cannot observe atomic structures in the real-time images owing to poor SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' However, the photomultiplier-based segmented detector enables the dwell time of the electron probe to be as short as that of conventional detectors, and the synthesized images can also be processed as live imaging [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Thus, by implementing a real-time OBF imaging function combined with a high-speed segmented detector and rapid scanning, an operator can observe atomic structures in real time with a higher SNR and tune the optical parameters even under low-dose conditions, as shown in Movie S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' This technique facilitates the observation of beam-sensitive materials with minimal irradiation damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In this study, we used real-time OBF imaging to observe FAU-type zeolites with sub-angstrom resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' We demonstrated that OBF imaging allows the direct observation of the T (=Si, Al) and oxygen atoms in the TO4 tetrahedron building units, which constitute the FAU-type framework structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Furthermore, OBF imaging was used to directly observe the detailed atomic structure of a twin boundary, which is a common lattice defect in FAU zeolites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The results of this study highlight the capability of electron microscopy for the local structural characterization of beam-sensitive materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Reconstruction scheme of OBF STEM and dose-efficiency comparison based on noise-normalized CTFs for different STEM imaging techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' (a) Schematic illustration of OBF STEM image processing workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In OBF STEM, a segmented detector is located on the diffraction plane that collects the intensity of transmitted/diffracted electrons at each probe position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The STEM images acquired by each segment are then processed with frequency filters to extract the phase-contrast component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The frequency filters are derived via STEM CTF, which are of a complex value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Subsequently, the filters are also complex-valued and visualized as a color map representing the phase and amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' After filtering, all the images are summed, and the OBF image is synthesized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' As the filter is calculated via microscope optical information such as accelerating voltage and convergence angle of the probe as well as the CTF, OBF reconstruction does not need a priori knowledge of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' (b) Noise-normalized CTFs of OBF and various phase-contrast imaging techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' CTFs show the window of contrast transfer from samples as a function of spatial frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Noise-normalized CTF is calculated by normalizing CTFs based on the noise level at each spatial frequency within the Poisson statistics, which shows the SNR at each Fourier component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Herein, the CTFs are calculated at an accelerating voltage of 300 kV, a convergence semi-angle of 15 mrad, and a sample thickness of 10 nm: the same conditions as those of the experiments conducted in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' These CTFs are shown as radially-averaged values, and the OBF technique shows a higher noise-normalized CTF than both the conventional methods (BF and ABF) and iDPC, the recently developed phase imaging technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' r prcoe Icontrast transfer function 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='8 OBF sample 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='7 iDPC GBFimage ABF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='6 BF :1A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='66A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='2 Phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='1 -Amplitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='6 Scattering vector (A-1) 5 Results Direct imaging of atomic structures in FAU zeolite Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 2a schematically shows the FAU framework, which consists of two building blocks: sodalite cages and double 6-membered rings (D6Rs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' These two building blocks are connected with the same symmetry as that in a diamond structure and form large pores of 12 Å diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Implementing the real- time OBF imaging technique, we observed the FAU framework along the <110> zone axis, wherein the pores were aligned along the observation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Conventionally, the atomic structure from the same zone-axis was observed via HRTEM [19], but only the pore arrangements were resolved by this method, and the atomic resolution was difficult to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Herein, the electron probe current was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='5 pA to suppress the beam damage, which was approximately two orders of magnitude lower than that of the usual STEM observation condition for analyzing typical inorganic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Under these conditions, OBF STEM observations of the FAU-type zeolite sample were conducted, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 2b-2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 2b shows the experimental OBF image of the FAU-type zeolite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The OBF image of the FAU framework structure indicated the atomic sites as bright spots, evidently for the tetrahedral (T-sites occupied by Si or Al) and oxygen sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' An amorphous layer covering the sample surface [20] is also recognizable in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 2c shows the power spectrum of the OBF image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 2b that exhibits an information transfer up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='869 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 2d shows a unit cell-averaged OBF image obtained from the original OBF image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The FAU framework structure can be observed at an atomic resolution and conforms extremely well with the simulated image shown in the inset, indicating that the atomic structure can be resolved without any electron irradiation damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 2e is the cropped image of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 2d, focusing on the D6R building block of the FAU structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' It evidently shows that the tetrahedral units are connected via corner-shared oxygen sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In zeolites, the oxygen-bridging sites between the tetrahedral units play an essential role in dictating the properties exhibited by the material, such as the catalytic activity [21] and structural transformation introduced by the interactions between the framework host and captured guests [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Thus, the capability of OBF STEM to visualize individual oxygen atom sites significantly helps to understand the structure-property relationship in zeolites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' It is noteworthy that the visibility of the oxygen atom was already attained in the raw OBF image before unit-cell averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The SNR of the S/TEM images of zeolites is usually enhanced by averaging the raw data using a priori knowledge about the sample, such as the space groups of the material [23,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Although this is effective for homogeneous bulk structure analyses, it cannot be applied to heterogeneous or nonperiodic local structure analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Therefore, the presented direct atom imaging capability will be helpful for zeolitic heterogeneous/nonperiodic structure analyses, such as aluminum substitution, counter cations, and other defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Later in this study, we have demonstrated the OBF STEM imaging of a defect structure in the FAU-type zeolite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Atomic-resolution OBF STEM observation of an FAU zeolite along <110> zone axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' (a) Schematic of the FAU zeolite framework structure and projected atomic-structure model along <110> zone-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Red and blue polygons represent the building units (sodalite cages and double 6-rings (D6Rs), respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' (b) OBF STEM image of FAU zeolite observed at the edge of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Bright spots indicate T- and oxygen-sites (scale bar: 1 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' (c) Fast Fourier Transform (FFT) of (b), wherein FFT spots are seen up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='869 Å resolution in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' (d) Repeat-unit-cell averaged OBF image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The inset is a simulated OBF image calculated with the same observation condition as that in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The location of the D6R structure, which is shown in (e), is highlighted by a dashed rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' (e) Magnified OBF image of the rectangular region indicated by the red dashed line in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The atomic structure models are drawn using VESTA [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Sodalite cage b D6R [110] [001] [110] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='869A Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 7 We compared the OBF images with other STEM images obtained under the same dose conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 3a shows the OBF, iDPC, conventional BF, and ABF images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The OBF image shows the FAU framework structure with the highest SNR, conforming with the noise-normalized CTF calculation shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Although the iDPC image also reveals the basic FAU framework structure, individual atomic sites, such as oxygen columns, are not distinguishable, as shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' For the further analysis of these contrast characteristics, we simulated the noise components of OBF and iDPC images (see the Materials and Methods section for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In the OBF reconstruction, the noise level is set to be flat as a function of spatial frequency, which is known as the noise-normalized condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' This is equivalent to the so-called white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The noise fluctuation of the OBF image contrast is thus uniformly random for the entire FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' It is noteworthy that this noise-component image is displayed on the same spatial scale as that of the experimental images shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 3a for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' However, for the iDPC image, the contrast fluctuation due to noise exists on a spatially larger scale than that of the OBF image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' This is confirmed by a line profile of the noise component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In the integration process of the DPC signal to form the iDPC image, the low-spatial-frequency component of the image is much more amplified than the higher-spatial-frequency components [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' However, the iDPC signals essentially do not exhibit contrast transfer around the low-frequency domains against noise, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Thus, under the low-dose condition, this amplification effect enhances the noise component in the lower frequency regions, resulting in the appearance of long-range contrast fluctuation, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' This is the reason why the iDPC image contrast appears smoother but has longer-range noise- fluctuation than those of the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' We also examined the experimental image intensity distribution of each imaging technique, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' S1, wherein the longer-range noise effect was more severe owing to the wide FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Although the OBF image exhibits an interpretable image contrast corresponding to the sample thickness and atomic sites, the iDPC image exhibits long-range intensity fluctuations in the experiment as well as the simulations, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' This contrast is much stronger than that of each atomic site in the zeolitic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' This results in a poor visibility of the atomic sites and makes it difficult to interpret the atomic structures from the obtained image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In other STEM images, such as ABF and BF, the basic structure of the FAU framework is roughly visible, but the detailed atomic structure analysis is challenging under the present low-dose condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Comparison between atomic resolution images of OBF STEM and other STEM techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' (a) STEM images obtained via OBF, iDPC, ABF, and conventional BF imaging techniques (scale bar: 1 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' All the images were recorded under the same electron dose and optical conditions (except for the defocus) as described in the Materials and Methods section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The insets are the cropped and enlarged versions of the original images, and the orange arrows indicate the oxygen sites in the FAU zeolitic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' (b) Comparison of the noise components between the OBF and iDPC simulated images (see the Materials and Methods section for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The intensity profiles of noise are also shown (obtained from the orange lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The assumed dose is the same as that in the experiments shown in (a), and the noise components in both the methods are obtained by the same noise-introduced segmented-detector datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' As indicated by the orange arrows, the iDPC noise image has longer-range fluctuations than that of the OBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' OBF iDPC ABF BF OBF noise iDPC noise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='00 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='15 10 20 30 40 50 10 20 30 40 50 60 Distance (A) Distance (A) 9 Direct observation of FAU twin boundary We applied the OBF technique to characterize the atomic structure of a twin boundary in the FAU zeolite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In FAU-type zeolites, the framework is constructed by cubic stacking of a layered structure unit called a ‘faujasite sheet’ [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' When the faujasite sheets are stacked in a hexagonal sequence, the resultant framework exhibits an EMT-type structure, known as a polymorph of an FAU- type zeolite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' There are twin boundaries between two opposite sequences of the cubic stacking in the FAU framework that likely result in an EMT-type structure at the boundary, as schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' However, the detailed atomic structure could not be directly determined owing to the limited spatial resolution under the low-dose condition in the previous TEM study [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 4b shows the OBF image of the FAU twin boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' This image indicates that the FAU cubic stacking sequence is inverted at the twin boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The power spectrum of the image indicates an information transfer beyond 1 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' For further analysis, we averaged the structural units of the FAU twin boundary, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The T and oxygen atomic sites are evidently visible in the twin boundary core, and two FAU-type domains are connected coherently at the atomic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Furthermore, the atomic structure of the twin boundary is confirmed to be identical to that of the EMT-type structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Density functional theory (DFT) calculations were performed to evaluate the stability of the twin boundary structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The initial twin-atomic structure was created by stacking the faujasite sheets based on the OBF image and then relaxed via DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 5c shows the relaxed atomic structure model, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 5b shows its corresponding simulated OBF image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The experimental image conforms well with its simulated counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Furthermore, the interface energy was calculated to be 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='4 mJ/m2, which is comparable with those of the twin boundaries in face-centered cubic (FCC) metals on the {111} plane [29], but approximately three orders of magnitude lower than those (typically) in oxide ceramic materials, such as grain boundaries and twin boundaries [30,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The origin of this difference can be explained as follows: in the {111} twin boundary of cubic zirconia, for example, the origin of the higher interface energy is attributed to the different coordination numbers of anions around the cation sites on the interface whereas the cation sites produce a coherent interface structure similar to those of FCC metals [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In the case of zeolites, the framework is constructed by the corner-sharing of rigid TO4 tetrahedra, which have a nearly perfect tetrahedral shape and are connected via oxygen atoms as soft hinges, offering a rigid but stress-free atomic structure [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Thus, zeolites can relax their framework structure by simply changing the bond angle between two rigid TO4 tetrahedrons (T-O-T angle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In silicate materials, the atomic structure is energetically stable over a wide range of T-O-T angles [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The observed structure of the FAU twin boundary was constructed in a similar manner, keeping the coordination numbers of the cations and anions unchanged across the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' This structural flexibility should result in extremely low excess energy at the twin boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Structural information about minute strains around some defects is essential for applications such as molecular sieves and gas separators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' It may affect the diffusion process of ions and molecules adsorbed in the zeolitic nanocavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Atomic-resolution OBF STEM image of FAU twin boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' (a) The framework model of the FAU twin boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The FAU cubic stacking sequence is inverted on the twin boundary, making the EMT framework structure with hexagonal stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The structure highlighted with a green-dotted box is a faujasite sheet, a layer structure unit for the FAU and the EMT frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The triangles represent the directions of the stacking sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' (b) Atomic-resolution OBF STEM image of the FAU twin boundary (scale bar = 1 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The inset is the FFT pattern of the OBF image, which exhibits a contrast transfer beyond 1 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Fauiasite sheet 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Comparison between experimental OBF image and simulated image based on DFT-relaxed structure of the FAU twin boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' (a) Unit-cell averaged experimental OBF image obtained from the raw experimental image shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The averaging operation is performed along the direction parallel to the interface, and no structural information is assumed about the symmetry other than the translational symmetry along the boundary (scale bar: 1 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' (b) Simulated OBF image based on the DFT-relaxed structure shown in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The image is calculated under the same condition as that of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' (c) Atomic structure model of FAU twin boundary relaxed by the DFT calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The blue and red balls indicate the T- and oxygen-sites, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' These images/structures show good agreement in both T- and oxygen-sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' DFT 12 Discussion We developed a highly dose-efficient STEM imaging technique, OBF STEM, for application in low-dose atomic-resolution imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' We demonstrated that OBF STEM can directly reveal the atomic structures of all elements in an FAU-type zeolite, which is a beam-sensitive material, with a sub- angstrom spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' OBF STEM can also be used to observe the lattice defects in zeolitic framework structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' We succeeded in directly determining the atomic structure on an FAU twin boundary, and the corresponding result was consistent with the DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The proposed technique can thus be used to characterize the local atomic structure in zeolites and other beam-sensitive materials, facilitating the study of structure-property relationships in these materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 13 Materials and Methods Atomic-resolution OBF STEM observation of an FAU-type zeolite Atomic-resolution OBF STEM images were acquired using an aberration-corrected STEM (JEOL JEM ARM-300F) equipped with a second-generation segmented annular all-field (SAAF) detector (16-segmented type) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' We developed an in-house program for the real-time OBF display function and implemented it in the SAAF system, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Movie S1 shows the real-time observation of a SrTiO3 [001] sample with an accelerating voltage of 300 kV and a probe-forming aperture of 30 mrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' All the atomic columns, including the oxygen atoms, were visualized under a low- dose condition (probe current: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='5 pA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=', two orders of magnitude less than the usual condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' This result demonstrated the capability of OBF STEM for low-dose and live atomic-resolution imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' We used a real-time OBF display system to acquire all the experimental OBF images shown in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' For the TEM sample preparation of an FAU-type zeolite, a commercially available powder sample of FAU zeolite (Tosoh Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=', Si/Al=50) was gently crushed in an agate mortar with ethanol and dispersed onto a TEM microgrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Before STEM observation, the sample was dehydrated overnight in the high vacuum environment of the TEM column to suppress the irradiation damage [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The accelerating voltage was set to 300 kV, which effectively reduces the irradiation damage in the zeolites [6,36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The probe current and probe-forming aperture were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='5 pA and 15 mrad, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Images of the FAU bulk structure were sequentially acquired at a dwell time of 16 µs with 1024 × 1024 pixels in the same region to suppress irradiation damage and scan distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Under this condition, the total dose was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='2×103 e-/Å2 per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' For the FAU twin boundary observation, the dwell time was reduced to 10 µs to further suppress the image distortion, with the total dose was 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='5×102 e-/Å2 per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' After the sequential image acquisition, the first five images were aligned and averaged for each data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Furthermore, we obtained unit-cell-averaged images for a detailed structural analysis, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 2d and 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' It can be noted that a priori knowledge about the structure group symmetry of the atomic structure was not assumed for the image averaging except for the translational symmetry for both FAU bulk and twin boundary analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' To obtain the OBF images, the camera length was set such that the edge of the STEM direct beam disk coincided with the outermost edge of the SAAF detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Under these conditions, the OBF image was obtained using Equation (1): 𝐼OBF(𝑹p) = ℱ−1 [∑ 𝐼𝑗(𝑸p)𝑊𝑗(𝑸p) 16 𝑗=1 ] = ∑ 𝐼𝑗(𝑹p) ⊗ 𝑤𝑗(𝑹p) 16 𝑗=1 , (1) where 𝐼OBF(𝑹p), 𝐼𝑗(𝑸p), 𝑊𝑗(𝑸p), and 𝑤𝑗(𝑹p) are the OBF image intensity, Fourier transformed image acquired by the j-th segment 𝐼𝑗(𝑹p), frequency filter calculated for the j-th segment, and point spread function obtained via the inverse Fourier transform of the frequency filter 𝑊𝑗(𝑸p), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 14 The filtering process was performed by multiplying the frequency filter in the reciprocal space 𝑸p or convolution with the point spread function in the real space 𝑹p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The post-processed OBF image could be obtained via either procedure, and the real-time OBF imaging synchronized with the STEM scan was acquired using the approximated convolution process in real space [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' For the focal condition to obtain the STEM images, it was reported that the OBF and DPC image contrast can be theoretically maximized upon focusing the electron probe on the mid-plane of the specimen [13,37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In contrast, the ABF and BF images exhibited the highest contrast upon focusing the probe on the entrance surface [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Thus, we acquired the images under the optimal focal conditions for each technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' To obtain the experimental/simulated iDPC, BF, and ABF images, the segmented/annular detector images were synthesized using the SAAF detector datasets to reproduce the detector geometry dedicated to each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Image simulations For the STEM image simulation, we used the MuSTEM package [38] based on the multi-slice model [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 16-segmented-detector images were calculated and processed using the OBF reconstruction algorithm to obtain the simulated OBF image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The effective source size was considered by convolution with a 2D Gaussian with a full-width-half-maximum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='6 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The sample thickness was assumed to be 10 nm, and the defocus Δ𝑓 was set to middle-focus condition, wherein the focal plane is located at the mid-plane inside the sample (Δ𝑓 = −5 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The STEM image simulation was also used for noise property analysis, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' First, noise was added to the simulated images of each detector segment based on the Poisson statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The noisy and noise-free images of each imaging method were then reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The assumed dose was equal to that of the experimental condition, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The noise component images were then obtained by subtracting the noise-free images from their noisy counterparts, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The noise-component images were normalized using the contrast range of their corresponding noise-free image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The noise characteristics of different techniques were then compared, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' DFT calculations To relax the FAU twin-boundary structure and calculate the interface energy, we performed DFT calculations using the VASP code [40] with the rev-vdW-DF2 method [41], which is suitable for calculating zeolitic atomic structures and energies [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' For the relaxation, we first relaxed the FAU bulk structure, whose data is available in the International Zeolite Association database [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The initial FAU twin boundary structure was then created by connecting the two FAU framework models with opposite stacking sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Finally, we obtained the relaxed FAU twin structure and calculated the interface energy Δ𝐸interface as follows: Δ𝐸interface = 𝐸twin − 𝐸bulk 2𝐴 , (2) 15 where 𝐴 is the cross-sectional area of the interface, and 𝐸bulk and 𝐸twin are the total energies of the FAU bulk and twin boundary structures, respectively.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Deng, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Niu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Wu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Yu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Sun, Imaging biological samples by integrated differential phase contrast (iDPC) STEM technique, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 214 (2022) 107837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='jsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='107837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 20 Acknowledgements Funding: This work was supported by JSPS KAKENHI (Grant Numbers 20H05659, 20H00301, 20K15014), JST ERATO (Grant Number JPMJER2202), and a Grant-in-Aid for Specially Promoted Research “Atom-by-atom imaging of ion dynamics in nanostructures for materials innovation” (Grant Number 17H06094).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' acknowledges the support from the Grant-in-Aid for JSPS Research Fellow (Grant Numbers 19J23138, 22J01665).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' acknowledges the support from JST-PRESTO (Grant Number JPMJPR21AA) and the Kazato Research Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' A part of this study was conducted at the Research Hub for Advanced Nano Characterization, the University of Tokyo, with support from the Nanotechnology Platform (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 12024046), MEXT, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Author contributions: K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=', and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' designed the study and wrote the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' performed the STEM experiments, image simulations, and DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' supported the STEM observation of zeolite samples and contributed to the discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' supported the development of the OBF STEM system and software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' contributed to the discussions and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' directed the entire study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Competing interests: The authors declare the following financial interests/personal relationships which may be considered as potential conflict of interests: a part of the present authors are inventors on Japanese unexamined patent application publication filed by the University of Tokyo (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 2021-077523).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Data and materials availability: All data necessary to evaluate the conclusions of this study are present in the paper and/or the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The data may be provided upon the request to the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 21 Supplementary Materials Supplementary Text Dose efficiency evaluation based on the noise-normalized CTFs To compare the contrast transfer efficiency of different STEM techniques against the noise-level, we calculated the noise-normalized CTFs as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In this note, we show how to evaluate the theoretical dose efficiency of each technique by the noise-normalized CTFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Under the weak phase object approximation (WPOA), the STEM image intensity 𝐼STEM(𝑹p) is given as follows 𝐼STEM(𝑹p) = 𝑑0 + ℱ−1[𝜎𝑉(𝑸p)𝛽(𝑸p)], (S1) where 𝑑0, ℱ−1, 𝜎, 𝑉(𝑸p), and 𝛽(𝑸p) respectively indicate background intensity, the inverse Fourier transformation operator, the interaction parameter determined by the accelerating voltage of the electron beam, the Fourier component of the specimen projected potential, and the CTF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Assuming that the specimen projected potential in real space 𝑣(𝑹p) is a delta function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=', 𝑣(𝑹p) ≅ 𝑣0𝛿(𝑹p) where 𝑣0 is a constant), the STEM image intensity above the point scatterer (𝑹p = 𝟎) 𝐼STEM(𝟎) is approximated as follows 𝐼STEM(𝟎) = 𝑑0 + 𝜎𝑣0 ∫ 𝛽(𝑸p) d𝑸p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' (S2) This equation indicates that we can evaluate the contrast amplitude approximately by the second term, integration of CTF over the frequency domains, because the first term 𝑑0 is the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Thus, we compare the obtainable contrast against the noise-level among different STEM techniques by integrating the noise-normalized CTFs [13,44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In the Poisson statistics, the signal-to-noise ratio (SNR) is proportional to the √𝜆, where 𝜆 is the electron dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Here, the integration value of noise-normalized CTFs can be regarded as a relative SNR between different techniques, and thus we can compare the dose-efficiency by the squared values of the CTF integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Table S1 shows the dose efficiency ratio calculated by the integration of the noise-normalized CTFs for OBF, iDPC, iCoM (integrated center-of-mass) [45,46], ABF, and conventional BF imaging methods, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The iCoM meghod is a kind of iDPC imaging with a pixelated detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Since we previously showed that OBF imaging can be extended to the pixelated detector [13], we calculated the CTF of OBF using the pixelated detector also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' As for the segmented detector, the detector 22 shape is the same as literatures for OBF [35] and iDPC [17], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' According to the calculated values shown in Table S1, OBF has approximately two-orders of magnitude higher dose efficiency than ABF theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Furthermore, because OBF reconstructs the phase-contrast image in a theoretically optimized manner to obtain the highest SNR for each type of detector, the calculated dose efficiency of OBF is higher than the iDPC or iCoM techniques that use segmented or pixelated detectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Since the iDPC technique is currently used for the STEM observation of beam-sensitive samples [47,48], the OBF observation should be able to reduce the irradiation dose more or obtain a higher spatial resolution on the same samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Additionally, it should be noted that the OBF using a segmented detector obtains almost the same dose efficiency as iCoM that uses a pixelated detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Although it is known that the pixelated detector can get significantly rich information about the sample, which could lead to higher dose-efficiency, this type of detector still needs longer dwell time while the recent technological progress improves the read-out speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In the low-dose experiment, the operator must tune experimental conditions quickly under low SNR conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Thus, the capability of high-dose efficiency with a high-speed segmented detector is definitely helpful for beam-sensitive materials analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 23 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Comparison between OBF and iDPC images obtained from the same experimental dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' (a) OBF and iDPC images generated from the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The OBF image is the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The intensity profiles taken along (b) [001] direction and (c) [1- 10] direction respectively from the orange rectangles shown in the OBF and iDPC images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Since the observed sample is wedge-shaped and has an amorphous layer near the edge, the projected atomic potentials should be increased from left hand side (vacuum area) to right hand side (thicker sample area) along the direction shown in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' On the other hand, along the direction shown in (c), the thickness is almost uniform and corresponding image contrast should also be uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' These are the case for the OBF image, but the iDPC image has strong intensity fluctuations (indicated by orange arrows) as discussed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' b b Image intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=') Image intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=") 20 40 60 80 100 120 0 20 40 60 80 100 120 Distance (A) Distance (A) 'n'e) Image intensity (a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=') 20 40 60 80 100 20 40 60 80 100 Distance (A) Distance (A) 24 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Schematic of noise evaluation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Schematic illustration of noise evaluation method shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' By combining the noise-free image and noisy image, noise characteristics against the contrast range can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 25 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Display of live OBF imaging system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Captured image of Movie S1 and its description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Movie S1 shows the real-time atomic-resolution OBF imaging of SrTiO3 [001] under a low- dose condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The dwell time is 10 µs, and the image is sampled with 512x512 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In the upper row, the left panel, center panel, and the right panel show ABF, OBF, and annular dark- field (ADF) images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' In the lower row, the left panel shows a center bright-field (CBF) image, and the center and right panel shows Fourier transformed OBF and ADF images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The images, including OBF, are synchronized with STEM probe scans and updated in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The updated area in this capture is highlighted with a dotted line in each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Movie S1 also shows the area scan mode, where the only selected area inside the image is scanned and the frame rate is increased for tuning aberrations such as defocus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Arrange ToolsSlideShowWindowHelp PreDefense_slide_ooe_FINAL SavedtomyMac Playback Share Comments AI SmartArt Picture Styles Shape Outline ive OBF imaging of SrTiO3[001] (movie) 34 ScreencaptureofOBFliveimaging(x1realtimemovie) ABFPanel 回X Sr ABF OBF ADF Ti+O 0 V SrTiO3 [001] V HT = 300 kV V Convergence angle = 30 mrad Beam current = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='5 pA (normally: ~30 pA) CBF LiveFFTof OBF Live FFT of ADF Dwell time = 10 μs (512x512) Imageprocessing software wasdevelopedvia C++ OBFmethodimplementedinSTEM Tuningfocus,aberrations, and Fov Very useful for low-dose observation!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' 0:04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='06 1去一電流0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content='5A七低下条件觀察秸果左见世寸 中OBF像右ADF，左ABF像，OBF像么表子思 二=上：OBF像见5 差调野来手法等感觉常高可能 26 Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Comparison of dose efficiency of different STEM methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Dose efficiency ratio among different STEM imaging techniques based on the noise-normalized CTF calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' The values are normalized such that the dose-efficiency of ABF becomes one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Movie S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} +page_content=' Live atomic-resolution OBF imaging of SrTiO3 [001] using the real-time OBF system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE3T4oBgHgl3EQfMwm8/content/2301.04377v1.pdf'} diff --git a/tdA0T4oBgHgl3EQfLf9-/content/tmp_files/2301.02119v1.pdf.txt b/tdA0T4oBgHgl3EQfLf9-/content/tmp_files/2301.02119v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..124f40cd501aa5e26b6ae0071ff995fe86441077 --- /dev/null +++ b/tdA0T4oBgHgl3EQfLf9-/content/tmp_files/2301.02119v1.pdf.txt @@ -0,0 +1,3942 @@ +arXiv:2301.02119v1 [math.NA] 5 Jan 2023 +A TENSOR BIDIAGONALIZATION METHOD FOR HIGHER-ORDER +SINGULAR VALUE DECOMPOSITION WITH APPLICATIONS +A. EL HACHIMI˚, K. JBILOU:, A. RATNANI˚, AND L. REICHEL; +Abstract. The need to know a few singular triplets associated with the largest singular values of third-order +tensors arises in data compression and extraction. This paper describes a new method for their computation using +the t-product. Methods for determining a couple of singular triplets associated with the smallest singular values +also are presented. The proposed methods generalize available restarted Lanczos bidiagonalization methods for +computing a few of the largest or smallest singular triplets of a matrix. The methods of this paper use Ritz and +harmonic Ritz lateral slices to determine accurate approximations of the largest and smallest singular triplets, +respectively. Computed examples show applications to data compression and face recognition. +Key words. tensors, t-product, partial tensor bidiagonalization, restarted tensor bidiagonalization, singular +value decomposition, face recognition. +1. Introduction. The last 20 years has seen an immense growth of the amount of data +that is collected for analysis, but it is a challenging problem to extract useful information from +available data. This difficulty arises, e.g., in machine learning, data mining, and deep learning; +see, e.g., Arnold et al. [1]. The extraction of useful information from data that is represented by +a matrix often is facilitated by the singular value decomposition of the matrix. Typically, only +a few of the largest singular triplets, i.e., the largest singular values and associated right and +left singular vectors, are required to extract useful information from the matrix. A restarted +Lanczos bidiagonalization method for computing accurate approximations of these singular +triplets is described in [5], and R code written by Bryan W. Lewis is available at [6]. +In many recent applications the given data are represented by a multidimensional array. These +arrays, known as tensors, are natural generalizations of matrices. Several approaches to define +tensor-tensor products and tensor-matrix products are described in the literature, including +the n-mode product [9, 25], the t-product [22, 31], and the c-product [21, 30]. Generalizations +of the singular value decomposition (SVD) to tensors are described in [25] using the n-mode +product (the so-called HOSVD), and in [21, 22] using the tensor c-product and t-product. The +need to compute the SVD or a partial SVD of a tensor arises in a variety of applications, +including image restoration, tensor completion [10], robust tensor principal component analysis +[13], tensor compression [3], and recognition of color faces [17, 18]. These applications require +knowledge of the largest singular values and associated lateral tensor singular slices. +It is the purpose of the this paper to introduce a new restarted tensor Lanczos bidiagonaliza- +tion method for third-order tensors using the t-product for approximating a few of the largest +singular values and associated lateral tensor singular slices. This method generalizes the ap- +proach described in [5] from matrices to tensors. We remark that the Lanczos bidiagonalization +method (also known as the Golub-Kahan bidiagonalization method) for third-order tensors us- +ing the t-product has been described in [15, 16, 22, 32]; however, this bidiagonalization method +differs from the one of the present paper. +In [5] the authors also describe a restarted Lanczos bidiagonalization method for the compu- +tation of a few of the smallest singular values and associated singular vectors of a large matrix +by determining harmonic Ritz values is presented. This paper presents an analogous scheme +˚Laboratory MSDA, Mohammed VI Polytechnic University, Green City, Morocco. +:Universit´e du Littoral Cote d’Opale, LMPA, 50 rue F. Buisson, 62228 Calais-Cedex, France. +;Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA. +1 + +for third-order tensors. +The organization of this paper is as follows. +Section 2 recalls some properties of the t- +product and Section 3 reviews tensor Lanczos bidiagonalization of third-order tensors using the +t-product. Restarted tensor Lanczos bidiagonalization methods are presented for the approx- +imation of a few of the largest singular values and associated lateral tensor singular slices by +computing lateral tensor Ritz slices, as well as for approximating a few of the smallest singular +values and associated lateral tensor singular slices by evaluating harmonic lateral tensor Ritz +slices. Section 4 discusses multidimensional principal component analysis using a partial tensor +HOSVD with application to face recognition, and Section 5 presents a few computed examples. +Concluding remarks and possible extensions can be found in Section 6. +2. The tensor t-product. This section reviews results by Kilmer et al. [22, 23] and +uses notation employed there and by Kolda and Bader [25]. A third-order tensor is an array +A “ raijks P Rℓˆpˆn. Matrices and vectors are tensors of order two and one, respectively. A +slice or frame of a third-order tensor A is a section obtained by fixing any one of the three +indices. Using MATLAB notation, A pi, :, :q, A p:, j, :q, and A p:, :, kq denote the ith horizontal, +the jth lateral, and the kth frontal slices of A , respectively. The lateral slice A p:, j, :q also is +denoted by +⃗ +Aj, and the frontal slice A p:, :, kq is an ℓ ˆ p matrix that is sometimes denoted by +A pkq. A fiber of a third order tensor A is defined by fixing any two of the three indices. The +fiber A pi, j, :q is called a tube of A . We will use capital calligraphic letters A to denote third- +order tensors, capital letters A to identify matrices, bold face lower case letters a to denote +tubes, and lower case letters a stand for scalars. Further, Kℓˆp +n +“ Rℓˆpˆn denotes the space of +third-order tensors of size ℓ ˆ p ˆ n, Kℓ +n “ Rℓˆ1ˆn stands for the space of lateral slices of size +ℓ ˆ n, and Kn “ R1ˆ1ˆn denotes the space of tubes with n entries. For a third-order tensor +A P Kℓˆp +n +with frontal slices A piq, i “ 1, . . . , n, we define: +‚ The block circulant matrix associated with A : +bcircpA q “ +» +———– +A p1q +A pnq +. . . +A p2q +A p2q +A p1q +. . . +A p3q +... +... +... +... +A pnq +A pn´1q +. . . +A p1q +fi +ffiffiffifl P Kℓnˆpn. +(2.1) +‚ The operator unfold applied to A gives the matrix made up of its frontal slices, +unfoldpA q “ +» +———– +A p1q +A p2q +... +A pnq +fi +ffiffiffifl P Kℓnˆp. +We also will need the inverse operator fold such that foldpunfold pA qq “ A . +‚ The block diagonal matrix associated with A is defined as +bdiagpA q “ +» +———– +A p1q +A p2q +... +A pnq +fi +ffiffiffifl P Kℓnˆpn. +2 + +Definition 1. ([23]) Let A P Kℓˆq +n +and B P Kqˆp +n +be third-order tensors. The t-product of +A and B is defined by +A ‹ B :“ foldpbcircpA q unfoldpBqq P Kℓˆp +n +. +The block circulant matrix (2.1) can be block-diagonalized by using the discrete Fourier +transform (DFT) as follows: +bcircpA q “ +` +F H +n b Iℓ +˘ +bdiagp � +A q pFn b Ipq , +where Fn P Cnˆn is the discrete Fourier matrix, F H +n denotes its conjugate transpose, � +A stands +for the Fourier transform of A along each tube, Iℓ P Rℓˆℓ denotes the identity matrix, and b is +the Kronecker product. The matrix � +A can be computed with the fast Fourier transform (FFT) +algorithm; see [23] for details. Using MATLAB notations, we have +� +A “ fftpA , r s, 3q. +The inverse operation can be evaluated in MATLAB with the command +A “ ifftp � +A , r s, 3q. +Hence, the t-product C “ A ‹ B can be evaluated as +� +C piq “ � +A piq � +Bpiq, +i “ 1, 2, . . . , n, +(2.2) +where � +A piq, � +Bpiq, and � +C piq are the ith frontal slices of the tensors � +A , � +B, and � +C , respectively. +As already pointed out by Kilmer et al. [22], one can use symmetry properties of the DFT +when applied to real data to reduce the computational effort when evaluating the t-product +with the FFT. This is described by the following result, which can be found, e.g., in [33]. +Lemma 1. Given a real vector v P Rn, the associated DFT vector �v “ Fnv satisfies +�v1 P R, +conjp�viq “ �vn´i`2, +i “ 2, 3, . . ., +„n ` 1 +2 + +, +where conj denotes the complex conjugation operator and +„n ` 1 +2 + +denotes the integer part of +n ` 1 +2 +. +It follows that for a third-order tensor A P Kℓˆp +n +, we have +� +A p1q P Rℓˆp, +conj +´ +� +A piq¯ +“ � +A pn´i`2q, +i “ 2, 3, . . ., +„n ` 1 +2 + +. +This shows that the t-product of two third-order tensors can be determined by evaluating just +about half the number of products involved in (2.2). Algorithm 1 describes the computations. +3 + +Algorithm 1 t-product of third-order tensors. +Input: A P Kℓˆq +n +, B P Kqˆp +n +. +Output: C :“ A ‹ B P Kℓˆp +n +. +1: Compute � +A “ fftpA , r s, 3q, � +B “ fftpB, r s, 3q. +2: for i “ 1, . . . , +„n ` 1 +2 + +do +3: +� +C piq “ � +A piq � +Bpiq. +4: end for +5: for i “ +„n ` 1 +2 + +` 1, . . . , n do +6: +� +C piq “ conj +´ +� +C pn´i`2q¯ +. +7: end for +8: C “ ifft +´ +� +C , r s, 3 +¯ +. +The following definition is concerned with the t-product of a third-order tensor and a tube. +Definition 2. Let A P Kℓˆp +n +and b P Kn. Then C :“ A ‹ b P Kℓˆp +n +is obtained by applying +the inverse DFT along each tube of +� +C , where each frontal slice is determined by the standard +matrix product between each frame of � +A and �b, i.e., +� +C piq “ � +A piq�bpiq “ �bpiq � +A piq, +i “ 1, 2, . . ., n. +A third-order tensor A P Kℓˆp +n +can be written as +A “ +” +⃗ +A1, ⃗ +A2, . . . , ⃗ +Ap +ı +, +thus, for the tensors A P Kℓˆq +n +and B P Kqˆp +n +, the t-product A ‹ B can be expressed as +A ‹ B “ +” +A ‹ ⃗ +B1, A ‹ ⃗ +B2, . . . , A ‹ ⃗ +Bp +ı +, +where +A ‹ ÝÑ +Bi “ ÝÝÝÝÝÑ +pA ‹ Bqi, +i “ 1, 2, . . ., p. +The Frobenius norm of a third-order tensor A P Kℓˆp +n +is given by +}A }F :“ +g +f +f +e +ℓ,p,n +ÿ +i1,i2,i3“1 +a2 +i1,i2,i3, +and the inner product of two third-order tensors of the same size A , B P Kℓˆp +n +is defined as +xA , By :“ +ℓ,p,n +ÿ +i1,i2,i3“1 +ai1,i2,i3bi1,i2,i3. +We have the relations +}A }F “ +1 +?n +››› � +A +››› +F , +xA , By “ 1 +nx � +A , � +By. +We recall for later use the definitions of some special tensors and operations: +4 + +‚ The identity tensor Iℓ P Kℓˆℓ +n +is the tensor whose first frontal slice is the identity +matrix and all other slices have zero entries only. +‚ The transpose of a real third-order tensor, A P Kℓˆp +n +, denoted by A H P Kpˆℓ +n +, is +the tensor obtained by first transposing each one of the frontal slices of A , and then +reversing the order of the transposed frontal slices 2 through n; see [23]. Let the third- +order tensors A and B be such that the products A ‹ B and BH ‹ A H are defined. +Then, similarly to the matrix transpose, the tensor transpose satisfies pA ‹ BqH “ +BH ‹ A H. +‚ A tensor Q P Kℓˆℓ +n +is said to be orthogonal if and only if +QH ‹ Q “ Q ‹ QH “ Iℓ. +‚ A square third-order tensor A P Kℓˆℓ +n +is invertible if there is a third-order tensor +B P Kℓˆℓ +n +such that +A ‹ B “ Iℓ, +B ‹ A “ Iℓ. +In this case B is said to be the inverse of A , and is denoted by A ´1. +Definition 3. ([22]) Let +⃗ +Ai P Kℓ +n for i “ 1, 2, . . . , p be lateral slices of the tensor A P Kℓˆp +n +. +A t-linear combination of these slices is defined as +⃗ +A1 ‹ b1 ` ⃗ +A2 ‹ b2 ` . . . ` ⃗ +Ap ‹ bp, +where the bi for i “ 1, 2, . . ., p are tubes in Kn. Moreover, +span +! +⃗ +A1, ⃗ +A2, . . . , ⃗ +Ap +) +“ +# pÿ +i“1 +⃗ +Ai ‹ bi : +bi P Kn, +i “ 1, 2, . . . , p ++ +. +The tensor singular value decomposition (t-SVD) associated with the t-product, introduced +by Kilmer and Martin [23], generalizes the classical SVD of a matrix. It is described in the +next theorem. +Theorem 4. ([23]) Let A P Kℓˆp +n +be a third-order tensor. Then it can be represented as the +t-product of three third-order tensors, +A “ U ‹ S ‹ V H, +(2.3) +where U P Kℓˆℓ +n +and V P Kpˆp +n +are orthogonal tensors, and S P Kℓˆp +n +is an f-diagonal tensor, +i.e., each frontal slice of the DFT of S is a diagonal matrix. +Algorithm 2 summarizes the computation of the t-SVD of a third-order tensor with the aid +of the FFT. +5 + +Algorithm 2 The t-SVD of a third-order tensor. +Input: A P Kℓˆp +n +. +Output: U P Kℓˆℓ +n +, S P Kℓˆp +n +, V P Kpˆp +n +. +1: +� +A “ fftpA , r s, 3q. +2: for i “ 1, . . . , +„n ` 1 +2 + +do +3: +r � +U piq, � +S piq, � +V piqs “ svdp � +A piqq. +4: end for +5: for i “ 1, . . . , +„n ` 1 +2 + +` 1 do +6: +� +U piq “ conj +´ +� +U pn´i`2q¯ +, � +S piq “ conj +´ +� +S pn´i`2q¯ +, and � +V piq “ conj +´ +� +V pn´i`2q¯ +. +7: end for +8: Compute U “ ifftp � +U , r s, 3q, S “ ifftp � +S , r s, 3q, and V “ ifftp � +V , r s, 3q. +The factorization (2.3) can be expressed as +A “ U ‹ S ‹ V H “ +mintℓ,pu +ÿ +i“1 +⃗ +Ui ‹ si ‹ ⃗ +V H +i , +where the si “ S pi, i, :q are singular tubes, and +⃗ +Ui “ U p:, i, :q and ⃗ +Vi “ U p:, i, :q are right +and left lateral tensor singular slices, respectively, for i “ 1, 2, . . ., minpℓ, pq. +The triplets +tsi, ⃗ +Ui, ⃗ +Viui“1:minpℓ,pq will be referred to as singular triplets of the tensor A . +The singular +tubes are ordered so that their norms σi “ }si}F are decreasing with i, i.e., +σ1 ě σ2 ě . . . ě σminpℓ,pq ě 0. +Note that we also have the relations +A ‹ ⃗ +Vi “ ⃗ +Ui ‹ si, +A H ‹ ⃗ +Ui “ ⃗ +Vi ‹ si, +i “ 1, 2, . . . , mintℓ, pu. +We remark that the latter relations have to be modified if A has complex-valued entries. +We note for future reference that +S pi, i, 1q “ +nÿ +j“1 +1 +n +� +S pi, i, jq. +(2.4) +In the following, we will need the notion of rank of a third-order tensor. +Definition 5. Let A P Kℓˆp +n +be a third-order tensor. Then its tubal rank is defined as +rankt pA q “ cardtσi ‰ 0, +i “ 1, 2, . . ., mintℓ, puu, +where σi is the norm of the singular tube si of A and card stands for the cardinality. +The next result generalizes the Eckart-Young theorem for matrices to third-order tensors. It +is important in the context of data compression. +Theorem 6. ([3, 23]) Let the t-SVD of a third-order tensor A P Kℓˆp +n +be given by A “ +U ‹ S ‹ V H. For 1 ď k ď mintℓ, pu, define the truncated t-SVD by +Ak “ +kÿ +i“1 +⃗ +Ui ‹ si ‹ ⃗ +V H +i . +6 + +Then +Ak “ arg min +� +A PM +›››A ´ � +A +››› +F . +Where M is the set given by M “ tX ‹ Y ; with X P Klˆk +n +, Y P Kkˆp +n +u. +The matrix QR factorization also can be generalized to tensors. +Theorem 7. ([23]) Let A P Kℓˆp +n +. Then A can be factored as +A “ Q ‹ R, +(2.5) +where Q P Kℓˆℓ +n +is an orthogonal tensor and R P Kℓˆp +n +is an f-upper triangular tensor, i.e., +each frontal slice of the DFT of R is an upper triangular matrix. The factorization (2.5) is +referred to as the t-QR factorization of A . +Algorithm 3 summarizes the computation of the t-QR factorization (2.5). +The function +qr in line 3 of the algorithm computes a QR factorization of the matrix +� +A piq P Rℓˆp; thus +� +A piq “ � +Qpiq � +Rpiq, where the matrix +� +Qpiq P Rℓˆℓ is orthogonal and the matrix � +Rpiq P Rℓˆp has +an upper triangular leading principal submatrix of order ℓ. +Algorithm 3 t-QR factorization of a third-order tensor. +Input: A P Kℓˆp +n +. +Output: Q P Kℓˆℓ +n +, R P Kℓˆp +n +. +1: +� +A “ fftpA , r s, 3q. +2: for i “ 1 . . . , +„n ` 1 +2 + +do +3: +r � +Qpiq, � +Rpiqs “ qrp � +A piqq. +4: end for +5: for i “ +„n ` 1 +2 + +` 1 . . . , n do +6: +� +Qpiq “ conj +´ +� +Qpn´i`2q¯ +and � +Rpiq “ conj +´ +� +Rpn´i`2q¯ +. +7: end for +8: Compute Q “ ifftp � +Q, r s, 3q and R “ ifftp � +R, r s, 3q. +Following Kilmer et al. [22], we define orthogonality of lateral tensor slices. Let +⃗ +X and ⃗ +Y +be two lateral tensor slices in Kℓ +n and define the inner product of these slices as +� +⃗ +X , ⃗ +Y +� +:“ +⃗ +X H ‹ ⃗ +Y P Kn. +The lateral slices in the set +! +⃗ +X1, ⃗ +X2, . . . , ⃗ +Xp +) +, +(2.6) +with p ě 2, are said to be orthogonal if +� +⃗ +Xi, ⃗ +Xj +� +“ +" +αie1 +if i “ j, +0 +if i ‰ j, +7 + +where e1 is the tube in Kn, whose its first element is 1 and the remaining elements vanish, and +the αi, i “ 1, 2, . . ., p, are nonvanishing scalars. Furthermore, if αi “ 1 for all i “ 1, 2, . . ., p, +then the set (2.6) is said to be orthonormal. +Following [22], we observe that any lateral slice +⃗ +X P Kℓ +n can be normalized as +⃗ +X “ ⃗ +Y ‹ a +(2.7) +with ⃗ +Y P Kℓ +n, +››› ⃗ +Y +››› “ 1, and a P Kn. Here the tensor norm is defined as +››› ⃗ +Y +››› “ +››› +� +⃗ +Y , ⃗ +Y +�››› +F +››› ⃗ +Y +››› +F +. +Note that ⃗ +Y has unit norm if and only if +� +⃗ +Y , ⃗ +Y +� +“ e1; see [22] for more detail. Algorithm +4 summarizes the normalization process. The MATLAB function randn in the algorithm gen- +erates a vector in Rℓ with normally distributed pseudorandom entries with mean zero and +variance one. +Algorithm 4 Normalize( ⃗ +X ). +Input: +⃗ +X P Kℓ +n. +Output: +⃗ +Y P Kℓ +n of unit norm and a P Kn that satisfy (2.7). +1: ⃗� +Y “ fftp ⃗ +X , r s, 3q. +2: for i “ 1, . . . , +„n ` 1 +2 + +do +3: +�apiq “ +››››› +⃗� +Y +piq››››› +F +. +4: +if �apiq ą 0 then +5: +⃗� +Y +piq +“ +⃗� +Y +piq +�apiq +6: +else +7: +⃗� +Y +piq +“ randnpℓ, 1q; bpiq “ +››››› +⃗� +Y +piq››››› +F +, and ⃗� +Y +piq +“ +⃗� +Y +piq +bpiq . +8: +end if +9: end for +10: for i “ +„n ` 1 +2 + +` 1, . . . , n do +11: +⃗� +Y +piq +“ conj +˜ +⃗� +Y +pn´i`2q¸ +, �apiq “ conj +´ +�apn´i`2q¯ +. +12: end for +13: ⃗� +Y “ ifftp ⃗� +Y , r s, 3q, a “ ifftp�a, r s, 3q. +3. Tensor Lanczos bidiagonalization for computing the largest and smallest sin- +gular triplets. This section describes the Lanczos bidiagonalization process for tensors using +8 + +the t-product, and discusses how approximations of the largest and smallest singular triplets of +a large third-order tensor A P Kℓˆp +n +can be computed. +3.1. The tensor Lanczos bidiagonalization algorithm. The Lanczos bidiagonaliza- +tion process was introduced for matrices by Golub and Kahan [14] and therefore sometimes is +referred to as the Golub-Kahan bidiagonalization process. For a matrix A P Rℓˆp, this process +is closely related to symmetric Lanczos process applied to the real symmetric matrices AAT +and AT A, or alternatively to the symmetric matrix +„ +0 +A +AT +0 + +. +Lanczos bidiagonalization algorithms have been applied to solve numerous problems such as +large-scale least squares problem [28], the approximation of the largest or smallest singular +triplets of a large matrix [5, 19, 24], and in Tikhonov regularization of large linear discrete +ill-posed problems; see, e.g., [11, 12]. We note that the bidiagonalization method described in +[28] and applied in [11, 12] reduces a large matrix A to a small lower bidiagonal matrix, while +in [5] the matrix A is reduced to a small upper bidiagonal matrix. We will review the latter +approach. +Application of m ! mintℓ, pu steps of the Lanczos bidiagonalization process to the matrix +A P Rℓˆp with the initial unit vector p1 P Rℓ generically produces two matrices +Pm “ rp1, p2, . . . , pms P Rpˆm, +Qm “ rq1, q2, . . . , qms P Rℓˆm. +The columns of Pm and Qm form orthonormal bases for the Krylov subspaces +Km +` +AT A, p1 +˘ +“ spantp1, AT Ap1, +` +AT A +˘2 p1, . . . , +` +AT A +˘m´1 p1u, +Km +` +AAT , q1 +˘ +“ spantq1, AAT q1, +` +AAT ˘2 q1, . . . , +` +AAT ˘m´1 q1u, +respectively, where q1 “ Ap1{}Ap1}2. A matrix interpretation of the recursion relations of the +Lanczos process gives the matrix relations +APm “ QmBm, +(3.1) +AT Qm “ PmBT +m ` βmpm`1eT +m, +(3.2) +where em “ r0, . . . , 0, 1sT P Rm, βm ě 0 is a scalar, and pm`1 P Rp. The matrix Bm P Rmˆm +is upper bidiagonal and satisfies Bm “ QT +mAPm. +When considering bidiagonalization of a third-order tensor A using the t-product, the scalars +and the columns of the matrices Pm and Qm in the matrix decompositions (3.1) and (3.2) be- +come tubes and lateral slices, respectively, in the decompositions determined by the tensor +Lanczos bidiagonalization process. The application of m steps of tensor Lanczos bidiagonaliza- +tion to the third-order tensor A P Kℓˆp +n +generically computes two tensors +Pm “ +” +⃗ +P1, ⃗ +P2, . . . , ⃗ +Pm +ı +P Kpˆm +n +and Qm “ +” +⃗ +Q1, ⃗ +Q2, . . . , ⃗ +Qm +ı +P Kℓˆm +n +, +whose lateral slices form bases for the tensor Krylov subspaces Km +´ +A H ‹ A , ⃗ +P1 +¯ +and +Km +´ +A ‹ A H, ⃗ +Q1 +¯ +, respectively. They are defined by +Km +´ +A H ‹ A , ⃗ +P1 +¯ +“ spant ⃗ +P1, +` +A H ‹ A +˘ +‹ ⃗ +P1, . . . , +` +A H ‹ A +˘m´1 ‹ ⃗ +P1u, +Km +´ +A ‹ A H, ⃗ +Q1 +¯ +“ spant ⃗ +Q1, +` +A ‹ A H˘ +‹ ⃗ +Q1, . . . , +` +A ‹ A H˘m´1 ‹ ⃗ +Q1u, +9 + +where ⃗ +P1 P Kp +n is a lateral slice of unit norm, and the lateral slice ⃗ +Q1 P Kℓ +n is of unit norm and +proportional to A ‹ ⃗ +P1. Algorithm 5 describes the tensor Lanczos bidiagonalization algorithm. +Algorithm 5 Tensor Lanczos bidiagonalization using the t-product. +Input: A P Kℓˆp +n +, number of steps m ď mintℓ, pu, ⃗ +P1 P Kp +n with unit norm. +Output: +Pm +“ +r ⃗ +P1, ⃗ +P2, . . . , ⃗ +Pms +P +Kpˆm +n +and +Qm +“ +r ⃗ +Q1, ⃗ +Q2, . . . , ⃗ +Qms +P +Kℓˆm +n +with orthonormal lateral slices, Bm +P +Kmˆm +n +a bidiagonal tensor, and +⃗ +Rm +P +Kℓ +m. +1: P1 “ +” +⃗ +P1 +ı +. +2: +⃗ +Q1 “ A ‹ ⃗ +P1. +3: r ⃗ +Q1, α1s “ Normalizep ⃗ +Q1q. +4: Q1 “ +” +⃗ +Q1 +ı +, Bmp1, 1, :q “ α1. +5: for i “ 1 to m do +6: +⃗ +Ri “ A H ‹ ⃗ +Qi ´ αi ‹ ⃗ +Pi. +7: +Reorthogonalization ⃗ +Ri “ ⃗ +Ri ´ Pi ‹ pPH +i ‹ ⃗ +Riq. +8: +if i ă m then +9: +r ⃗ +Pi`1, βis “ Normalizep ⃗ +Riq. +10: +Pi`1 “ +” +Pi, ⃗ +Pi`1 +ı +, Bmpi, i ` 1, :q “ βi. +11: +⃗ +Qi`1 “ A ‹ ⃗ +Pi`1 ´ βi ‹ ⃗ +Qi. +12: +Reorthogonalization ⃗ +Qi`1 “ ⃗ +Qi`1 ´ Qi ‹ pQH +i ‹ ⃗ +Qi`1q. +13: +r ⃗ +Qi`1, αi`1s “ Normalizep ⃗ +Qi`1q. +14: +Qi`1 “ +” +Qi, ⃗ +Qi`1 +ı +, Bmpi ` 1, i ` 1, :q “ αi`1. +15: +end if +16: end for +We remark that Algorithm 5 differs from the tensor bidiagonalization algorithms described in +[22, 32] in that the former produces an upper bidiagonal tensor Bm, while the latter determine +a lower bidiagonal tensor. The use of an upper bidiagonal tensor in the present paper is inspired +by the choices in [5, 14]. Algorithm 5 is said to break down when one of the tensor slices ⃗ +Ri or +⃗ +Qi`1 vanishes. We comment below on this situation, but note that breakdown is exceedingly +rare. +Theorem 8. Generically, Algorithm 5 determines the decompositions +A ‹ Pm “ Qm ‹ Bm, +(3.3) +A H ‹ Qm “ Pm ‹ BH +m ` ⃗ +Rm ‹ ⃗E H +m , +(3.4) +with Pm P Kpˆm +n +, Qm P Kℓˆm +n +, where PH +m ‹ Pm “ Im and QH +m ‹ Qm “ Im. The tensor +⃗Em P Km +n is the canonical lateral slice whose elements are zero except for the first element of +the mth tube, which equals 1, and ⃗ +Rm P Kp +n is determined by steps 4 and 5 of Algorithm 5 such +that PH +m ‹ ⃗ +Rm “ 0. The tensor Bm P Kmˆm +n +is upper bidiagonal, each of whose frontal slices +10 + +is an upper bidiagonal matrix. Thus, +Bm “ +» +——————– +α1 +β1 +0 +. . . +0 +0 +α2 +β2 +0 +... +... +... +... +... +... +0 +. . . +. . . +αm´1 +βm´1 +0 +. . . +. . . +0 +αm +fi +ffiffiffiffiffiffifl +, +where αi and βi are tubes in Kn. +Proof. The relations (3.3) and (3.4) follow immediately from the recursion relations of Algo- +rithm 5. The orthonormality of the lateral slices of Pm and Qm can be shown by induction. +The proof is closely related to the proof of the existence of the relations (3.1) and (3.2), and +the properties of the matrices involved. The latter relations are used in [5]. +The Lanczos bidiagonalization process may suffer from loss of orthogonality of the lateral +slices of the tensors Pm and Qm. Therefore, reorthogonalization is carried out in Lines 5 and +9 in Algorithm 5. We remark that reorthogonalization makes the algorithm more costly both +in terms of storage and arithmetic floating point operations. The extra cost may be acceptable +as long as the number of steps m is fairly small; see [5, 34] for discussions in the matrix case. +Let ⃗ +Rm be the tensor whose lateral slices are defined in Line 5. Then +r ⃗ +Pm`1, βms “ Normalize +´ +⃗ +Rm +¯ +. +(3.5) +In the rare event that some βj, 1 ď j ă m, vanishes, Algorithm 5 breaks down. Then the +singular tubes of Bj are singular tubes of A , and the left and right lateral tensor singular slices +are obtained as described below. When no breakdown takes place, we can express equation +(3.4) as +A H ‹ Qm “ Pm`1 ‹ BH +m,m`1, +where Pm`1 is obtained from Pm by appending the lateral slice +⃗ +Pm`1, defined in (3.5), to +get Pm`1 “ +” +Pm, ⃗ +Pm`1 +ı +P Kpˆpm`1q +n +, and Bm,m`1 P Kmˆpm`1q +n +is obtained by appending +the lateral slice βm ‹ ⃗Em to Bm, i.e., Bm,m`1 “ +” +Bm, βm ‹ ⃗Em +ı +. +We turn to the connection between the partial Lanczos tridiagonalization of a third-order +tensor and the partial Lanczos tridiagonalization process of the tensor A H ‹A . This connection +will be used later. Multiplying (3.3) from the left by A H, we get +A H ‹ A ‹ Pm “ A H ‹ Qm ‹ Bm +“ Pm ‹ BH +m ‹ Bm ` ⃗ +Rm ‹ ⃗E H +m ‹ Bm +“ Pm ‹ BH +m ‹ Bm ` ⃗ +Rm ‹ ⃗E H +m ‹ αm. +(3.6) +Let Tm be the symmetric tridiagonal tensor defined by +Tm “ BH +m ‹ Bm P Kmˆm +n +. +Then (3.6) is a partial tensor Lanczos bidiagonalization of A H ‹ A with initial lateral slice +⃗ +P1 “ Pm ‹ ⃗E1. The lateral slices of Pm form an orthonormal basis for the tensor Krylov +subspace +Km +´ +A H ‹ A , ⃗ +P1 +¯ +“ spant ⃗ +P1, A H ‹ A ‹ ⃗ +P1, +` +A H ‹ A +˘2 ‹ ⃗ +P1, . . . , +` +A H ‹ A +˘m´1 ‹ ⃗ +P1u. +11 + +Similarly, multiplying (3.4) from the left by A , we obtain +A ‹ A H ‹ Qm “ Qm ‹ Bm ‹ BH +m ` A ‹ ⃗ +Rm ‹ ⃗E H +m . +It follows that the lateral slices of Qm form an orthonormal basis for the Krylov subspace +Km +´ +A ‹ A H, ⃗ +Q1 +¯ +“ spant ⃗ +Q1, A ‹ A H ‹ ⃗ +Q1, +` +A ‹ A H˘2 ‹ ⃗ +Q1, . . . , +` +A ‹ A H˘m´1 ‹ ⃗ +Q1u. +3.2. Approximating singular tubes and singular lateral slices. We describe an +approach to approximate the largest or smallest singular triplets (singular tubes and associated +left and right lateral singular slices) of a large tensor A P Kℓˆp +n +using restarted partial tensor +Lanczos bidiagonalization. Since the tensor A is large, computing its k largest or smallest +singular triplets by determining the t-SVD of A is very expensive. The idea is to approximate +the extreme singular triplets of the tensor A by determining the extreme singular triplets the +bidiagonal tensor Bm, where m is small. Let tsi, ⃗ +Ui, ⃗ +Viu, 1 ď i ď m, denote the singular +triplets of Bm. They satisfy +Bm ‹ ⃗ +Vi “ si ‹ ⃗ +Ui and BH +m ‹ ⃗ +Ui “ si ‹ ⃗ +Vi. +The k ď m largest singular triplets of A are approximated by the triplets tsA +i,m, ⃗ +U A +i,m, ⃗ +V A +i,mu +defined by +sA +i,m “ si, +⃗ +U A +i,m “ Qm ‹ ⃗ +Ui, +⃗ +V A +i,m “ Pm ‹ ⃗ +Vi, +i “ 1, 2, . . ., k. +(3.7) +For i “ 1, 2, . . . , k, we have +A ‹ ⃗ +V A +i,m “ A ‹ Pm ‹ ⃗ +Vi +“ Qm ‹ Bm ‹ ⃗ +Vi +“ Qm ‹ si ‹ ⃗ +Ui +“ Qm ‹ ⃗ +Ui ‹ si +“ ⃗ +U A +i,m ‹ sA +i,m. +Similarly, +A H ‹ ⃗ +U A +i,m “ A H ‹ Qm ‹ ⃗ +Ui “ +´ +Pm ‹ Bm ` ⃗ +Rm ‹ ⃗E H +m +¯ +‹ ⃗ +Ui +“ ⃗ +V A +i,m ‹ sA +i,m ` ⃗ +Rm ‹ ⃗E H +m ‹ ⃗ +Ui. +(3.8) +To accept tsA +i,m, ⃗ +U A +i,m, ⃗ +V A +i,mu as an approximate singular triplet of A , the remainder term +⃗ +Rm ‹ ⃗E H +m ‹ ⃗ +Ui should be small enough. We can bound the remainder term according to +››› ⃗ +Rm ‹ ⃗E H +m ‹ ⃗ +Ui +››› +F “ +1 +?n +››››bdiag +ˆ +� +⃗ +Rm +˙ +bdiag +ˆ � +´ +⃗E H +m +¯˙ +bdiag +ˆ +� +⃗Ui +˙›››› +F +ď +1 +?n +››››bdiag +ˆ +� +⃗ +Rm +˙›››› +F +››››bdiag +ˆ � +´ +⃗E H +m +¯˙ +bdiag +ˆ +� +⃗Ui +˙›››› +F +“ +›››bdiag +´ +⃗ +Rm +¯››› +F +››››bdiag +ˆ � +´ +⃗E H +m +¯˙ +bdiag +ˆ +� +⃗Ui +˙›››› +F +“ }βm}F +n +ÿ +s“1 +ˇˇˇˇˇ +� +´ +⃗E H +m +¯psq� +⃗Ui +psqˇˇˇˇˇ . +12 + +Analogously as in [5], we require for 1 ď s ď n that +ˇˇˇˇˇ +� +´ +⃗E H +m +¯psq� +⃗Ui +psqˇˇˇˇˇ ď δ1 ››› � +A psq››› “ δ1 ´ +s +� +A psq +1,m +¯ +“ δ +´ +s +� +A +1,m +¯psq +, +for a user-chosen parameter δ1 ą 0, where +´ +s � +A +j,m +¯psq +denotes the sth element of the jth approx- +imate singular tube of � +A . We obtain from eq. (2.4) that +››› ⃗ +Rm ‹ ⃗E H +m ‹ ⃗ +Ui +››› +F ď δ1 }βm}F +n +ÿ +s“1 +´ +s +� +A +1 +¯psq +“ nδ1 }βm}F +` +sA +1 +˘p1q “ nδ2 ` +sA +1 +˘p1q , +where δ2 “ δ1 }βm}F . +The computed approximate singular triplets tsA +i,m, ⃗ +U A +i,m, ⃗ +V A +i,mu, i “ +1, 2, . . ., k, of A are accepted as singular triplets of A if +››› ⃗ +Rm ‹ ⃗E H +m ‹ ⃗ +Ui +››› +F ď δ +` +sA +1,m +˘p1q , +i “ 1, 2, . . .k, +(3.9) +for some user-specified parameter δ ą 0. +To keep the storage requirement fairly small for large-scale problems, we would like the num- +ber of steps m of the tensor Lanczos bidiagonalization process to be small. However, when m is +small, it may not be possible to approximate the desired singular triplets sufficiently accurately +using the available Krylov subspaces Km +´ +A H ‹ A , ⃗ +Q1 +¯ +and Km +´ +A ‹ A H, ⃗ +P1 +¯ +. A remedy +for this situation is to restart the tensor Lanczos bidiagonalization process. The idea is to +repeatedly update the initial lateral slices used for the tensor Lanczos bidiagonalization pro- +cess, and in this way determine a sequence of increasingly more appropriate Krylov subspaces, +until the k desired singular triplets have been found with required accuracy. We remark that +restarting techniques have been used for computing a few desired singular triplets or eigenvalue- +eigenvector pairs of a large matrix, where properties of Ritz vectors, harmonic Ritz vectors, and +refined Ritz vectors have been exploited; see, e.g., [5, 19, 20, 35, 36] for details. +3.3. Augmentation by Ritz lateral slices. Assume that we would like to approximate +the k largest singular triplets of A P Rℓˆpˆn. +To this end, we carry out m ą k steps of +tensor Lanczos bidiagonalization as described in the previous subsection. The approximate +right singular lateral slice ⃗ +V A +i,m is a Ritz lateral slice of A H ‹ A associated with the Ritz tube +` +sA +i,m +˘2 “ sA +i,m ‹ sA +i,m for i P t1, 2, . . ., mu, and we have +A H ‹ A ‹ ⃗ +V A +i,m “ A H ‹ ⃗ +U A +i,m ‹ sA +i,m “ +´ +⃗ +V A +i,m ‹ sA +i,m ` ⃗ +Rm ‹ ⃗E H +m ‹ ⃗ +Ui +¯ +‹ sA +i,m +“ ⃗ +V A +i,m ‹ +` +sA +i,m +˘2 ` ⃗ +Rm ‹ ⃗E H +m ‹ ⃗ +Ui ‹ sA +i,m. +In what follows we will show some results that will help us to approximate the largest or +smallest singular triplets of a third-order tensor. +The idea behind these results is to find +equations that are analogous to (3.3) and (3.4), and such that the reduced tensor will contain +the k approximate singular tubes among its first k elements on the diagonal, and the right +projection tensor will contain the k right Ritz lateral slices among its first k lateral slices, and +13 + +the left projection tensor will contain the k left Ritz lateral slices among its first k lateral slices. +The following theorem will be helpful. +Theorem 9. Assume that m steps of Algorithm 5 have been applied to the third-order tensor +A P Kℓˆp +n +, and suppose that βm in (3.4) is nonvanishing. Then for k ă m, we have +A ‹ � +Pk`1 “ � +Qk`1 ‹ � +Bk`1, +(3.10) +A H ‹ � +Qk`1 “ � +Pk`1 ‹ � +BH +k`1 ` �βk`1 ‹ ⃗� +Pk`2 ‹ ⃗E H +k`1, +(3.11) +where � +Pk`1 P Kpˆpk`1q +n +and +� +Qk`1 P Kℓˆpk`1q +n +have orthonormal lateral slices, and the first k +lateral slices of � +Pm are the first k Ritz lateral slices of A , +� +Bk`1 P Kpk`1qˆpk`1q +n +is an upper +triangular tensor, ⃗� +Pk`2 P Kp +n is a lateral slice that is orthogonal to � +Pk`1, �βk`1 P Kn, and +⃗Ek`1 P Kk`1 +n +is the canonical element under the t-product. +Proof. Let the Ritz lateral slices ⃗ +V A +i,m for 1 ď i ď k be associated with the k Ritz tubes of A . +Introduce the tensor +� +Pk`1 “ +” +⃗ +V A +1,m, ⃗ +V A +2,m, . . . , ⃗ +V A +k,m, ⃗ +Pm`1 +ı +P Kpˆpk`1q +n +, +(3.12) +where ⃗ +Pm`1 is given by (3.5). Then, using the fact that A ‹ ⃗ +V A +i,m “ ⃗ +U A +i,m‹sA +i,m for i “ 1, 2, . . ., k, +we obtain +A ‹ � +Pk`1 “ +” +A ‹ ⃗ +V A +1,m, A ‹ ⃗ +V A +2,m, . . . , A ‹ ⃗ +V A +k,m, A ‹ ⃗ +Pm`1 +ı +“ +” +⃗ +U A +1,m ‹ sA +1,m, ⃗ +U A +2,m ‹ sA +2,m, . . . , ⃗ +U A +k,m ‹ sA +k,m, A ‹ ⃗ +Pm`1 +ı +. +(3.13) +Orthogonalizing the term A ‹ ⃗ +Pm`1 against t ⃗ +U A +i,mui“1:k gives +A ‹ ⃗ +Pm`1 “ +kÿ +i“1 +ρi ‹ ⃗ +U A +i,m ` ⃗� +Rk, +(3.14) +where ⃗� +Rk is orthogonal to t ⃗ +U A +i,mui“1:k, and the ρi for i P t1, 2, . . . , ku are given by +ρi “ +´ +⃗ +U A +i,m +¯H +‹ +´ +A ‹ ⃗ +Pm`1 +¯ +“ +´ +A H ‹ ⃗ +U A +i,m +¯H +‹ ⃗ +Pm`1 +“ +´ +⃗ +V A +i,m ‹ sA +i,m ` ⃗ +Rm ‹ ⃗E H +m ‹ ⃗ +Ui +¯H +‹ ⃗ +Pm`1 +“ βH +m ‹ +´ +⃗ +U H +i +‹ ⃗Em ‹ ⃗ +PH +m`1 +¯ +‹ ⃗ +Pm`1 +“ βm ‹ ⃗ +U H +i +‹ ⃗Em +“ βm ‹ +� +⃗ +Ui, ⃗Em +� +, +because βm “ βH +m. +Let ⃗� +Rk “ ⃗� +R1k ‹ �αk`1 be a normalization of ⃗� +Rk, and introduce the tensors +� +Qk`1 “ +„ +⃗ +U A +1,m, ⃗ +U A +2,m, . . . , ⃗ +U A +k,m, ⃗� +R1k + +P Kℓˆpk`1q +n +(3.15) +14 + +and +� +Bk`1 “ +» +—————– +sA +1,m +0 +. . . +0 +ρ1 +0 +sA +2,m +. . . +0 +ρ2 +... +... +... +... +... +0 +. . . +0 +sA +k,m +ρk +0 +. . . +. . . +0 +�αk`1 +fi +ffiffiffiffiffifl +P Kpk`1qˆpk`1q +n +. +(3.16) +Then, from (3.13) and (3.14), we obtain +A ‹ � +Pk`1 “ +« +⃗ +U A +1,m ‹ sA +1,m, ⃗ +U A +2,m ‹ sA +2,m, . . . , ⃗ +U A +k,m ‹ sA +k,m, +kÿ +i“1 +ρi ‹ ⃗ +U A +i,m ` ⃗� +Rk +ff +“ � +Qk`1 ‹ � +Bk`1. +(3.17) +On the other hand, as +A H ‹ � +Qk`1 “ +„ +A H ‹ ⃗ +U A +1,m, A H ‹ ⃗ +U A +2,m, . . . , A H ‹ ⃗ +U A +k,m, A H ‹ +ÝÑ +� +R1k + +, +using (3.8), we get +A H ‹ ⃗ +U A +i,m “ ⃗ +V A +i,m ‹ sA +i,m ` ⃗ +Rm ‹ ⃗E H +m ‹ ⃗ +Ui +“ ⃗ +V A +i,m ‹ sA +i,m ` ⃗ +Pm`1 ‹ βm ‹ ⃗E H +m ‹ ⃗ +Ui +“ ⃗ +V A +i,m ‹ sA +i,m ` ⃗ +Pm`1 ‹ ρH +i . +Since +� +A H ‹ ⃗� +R +1 +k, ⃗ +V A +i,m +� +“ +ˆ +⃗� +R +1 +k +˙H +‹ A ‹ ⃗ +V A +i,m “ sA +i,m ‹ +ˆ +⃗� +R +1 +k +˙H +‹ ⃗ +U A +i,m “ 0, +the tensor A H ‹ ⃗� +R1k is orthogonal to ⃗ +V A +i,m. Moreover, in view of that ⃗ +V A +i,m is orthogonal to +⃗ +Pm`1, we obtain +A H ‹ +ÝÑ +� +R1k “ γ ‹ ⃗ +Pm`1 ` ⃗ +Fk`1, +(3.18) +where ⃗ +Fk`1 is orthogonal to ⃗ +Pm`1 as well as to ⃗ +V A +i,m. Due to the orthogonality of ⃗� +Rk (or +ÝÑ +� +R1k) +to +! +⃗ +U A +i,m +) +i“1:k, the parameter γ in (3.18) is given by +γ “ +� +⃗ +Pm`1, A H ‹ +ÝÑ +� +R1k +� +“ +� +A ‹ ⃗ +Pm`1, +ÝÑ +� +R1k +� +“ +� kÿ +i“1 +ρi ‹ ⃗ +U A +i,m ` ⃗� +Rk, +ÝÑ +� +R1k +� +“ +� +⃗� +Rk, +ÝÑ +� +R1k +� +“ �αk`1. +15 + +Consequently, +A H ‹ � +Qk`1 “ +” +⃗ +V A +1,m ‹ sA +1,m ` ⃗ +Pm`1 ‹ ρH +1 , . . . , ⃗ +V A +k,m ‹ sA +k,m ` ⃗ +Pm`1 ‹ ρH +k , �αk`1 ‹ ⃗ +Pm`1 ` ⃗ +Fk`1 +ı +“ � +Pk`1 ‹ � +BH +k`1 ` ⃗ +Fk`1 ‹ ⃗E H +k`1 +“ � +Pk`1 ‹ � +BH +k`1 ` �βk`1 ‹ ⃗� +Pk`2 ‹ ⃗E H +k`1, +(3.19) +where �βk`1 and ⃗� +Pk`2 are determined by the normalization of ⃗ +Fk`1, i.e., ⃗ +Fk`1 “ �βk`1‹ ⃗� +Pk`2, +because +� +BH +k`1 “ +» +—————– +sA +1,m +0 +. . . +0 +0 +0 +sA +2,m +0 +. . . +0 +... +... +... +... +... +0 +. . . +0 +sA +k,m +0 +ρH +1 +ρH +2 +. . . +ρH +k +�αk`1 +fi +ffiffiffiffiffifl +P Kpk`1qˆpk`1q +n +. +The orthogonality of � +Pk`1 and +� +Qk`1 now follows from the orthogonality of the sequences +! +⃗ +V A +i,m +) +i“1:k and +! +⃗ +U A +i,m +) +i“1:k, respectively, given by (3.7). +In the preceding theorem we assumed βm to be nonvanishing. If, instead, βm vanishes, then +the singular tubes of Bm are singular tubes of A , and the left and right singular lateral slices of +A can be determined from those of Bm. Similarly, if �βk`1 in (3.18) vanishes, then the singular +tubes of � +Bk`1 are singular tubes of A , and the singular lateral slices of A can be determined +from � +Pk`1 and � +Qk`1. +If the �βk`1 is nonvanishing, then we append new lateral slices to � +Pk`1 and � +Qk`1 repeatedly +until iteration m ´ k. This is the subject of the following theorem. +Theorem 10. Assume that m steps of Algorithm 5 have been applied to A and that eqs. +(3.17) and (3.19) hold. If the �βk`1 are nonvanishing for 1 ď k ă m, then we have the following +relations +A ‹ � +Pm “ � +Qm ‹ � +Bm, +A H ‹ � +Qm “ � +Pm ‹ � +BH +m ` �βm ‹ ⃗� +Pm`1 ‹ ⃗E H +m , +where � +Pm P Kpˆm +n +and +� +Qm P Kℓˆm +n +have orthonormal lateral slices, � +Bm P Kmˆm +n +is an upper +triangular, �βm P Kn, ⃗� +Pm`1 P Kp +n is orthogonal to � +Pm, and ⃗Em P Km +n is the canonical element +under the t-product. The first k lateral slices of � +Pm and +� +Qm are the same as those of the +tensors � +Pk`1 and +� +Qk`1, respectively, given in Theorem 9. +Proof. Let the tensors � +Pk`1 and +� +Qk`1 defined in (3.17) and (3.19), respectively, be repre- +sented by +� +Pk`1 “ +„ +⃗� +P1, ⃗� +P2, . . . , ⃗� +Pk`1 + +P Kpˆpk`1q +n +and +� +Qk`1 “ +„ +⃗� +Q1, ⃗� +Q2, . . . , ⃗� +Qk`1 + +P Kℓˆpk`1q +n +, +16 + +and the tensor � +Pk`2 be given by +� +Pk`2 “ +„ +� +Pk`1, ⃗� +Pk`2 + +P Kpˆpk`2q +n +. +By normalizing the quantity +´ +Iℓ ´ � +Qk`1 ‹ � +QH +k`1 +¯ +‹A ‹ ⃗� +Pk`2, we obtain the lateral slice ⃗� +Qk`2 +such that �αk`2 ‹ ⃗� +Qk`2 “ +´ +Iℓ ´ � +Qk`1 ‹ � +QH +k`1 +¯ +‹ A ‹ ⃗� +Pk`2. Application of (3.11) gives +�αk`2 ‹ ⃗� +Qk`2 “ +´ +Iℓ ´ � +Qk`1 ‹ � +QH +k`1 +¯ +‹ A ‹ ⃗� +Pk`2 +“ A ‹ ⃗� +Pk`2 ´ � +Qk`1 ‹ � +QH +k`1 ‹ A ‹ ⃗� +Pk`2 +“ A ‹ ⃗� +Pk`2 ´ � +Qk`1 ‹ +ˆ +� +Bk`1 ‹ � +PH +k`1 ` �βk`1 ‹ ⃗Ek`1 ‹ ⃗� +P +H +k`2 +˙ +‹ ⃗� +Pk`2 +“ A ‹ ⃗� +Pk`2 ´ �βk`1 ‹ ⃗� +Qk`1. +(3.20) +Consider the tensors +� +Qk`2 “ +„ +� +Qk`1, ⃗� +Qk`2 + +P Kℓˆpk`2q +n +and +� +Bk`2 “ +» +——————————– +sA +1,m +0 +. . . +0 +ρ1 +0 +0 +sA +2,m +0 +. . . +ρ2 +0 +... +... +... +... +... +... +... +... +... +... +... +... +0 +. . . +0 +sA +k,m +ρk +0 +0 +. . . +. . . +0 +�αk`1 +�βk`1 +0 +. . . +. . . +. . . +0 +�αk`2 +fi +ffiffiffiffiffiffiffiffiffiffifl +P Kpk`2qˆpk`2q +n +. +Using (3.10) and (3.20), we get +A ‹ � +Pk`2 “ � +Qk`2 ‹ � +Bk`2. +To determine the lateral slice ⃗� +Pk`3, we normalize +´ +I ´ � +Pk`2 ‹ � +PH +k`2 +¯ +‹A H ‹ ⃗� +Qk`2 so that +�βk`2 ‹ ⃗� +Pk`3 “ +´ +I ´ � +Pk`2 ‹ � +PH +k`2 +¯ +‹ A H ‹ ⃗� +Qk`2 +and +�βk`2 ‹ ⃗� +Pk`3 “ A H ‹ ⃗� +Qk`2 ´ �αk`2 ‹ ⃗� +Pk`2. +(3.21) +It now follows from (3.10) and (3.21) that +A H ‹ � +Qk`2 “ � +Pk`2 ‹ � +BH +k`2 ` �βk`2 ‹ ⃗� +Pk`3 ‹ ⃗E H +k`2. +17 + +We can continue this procedure until iteration m ´ k and then obtain +A ‹ � +Pm “ � +Qm ‹ � +Bm, +A H ‹ � +Qm “ � +Pm ‹ � +BH +m ` �βm ‹ ⃗� +Pm`1 ‹ ⃗E H +m , +where � +Pm and � +Qm have orthonormal lateral slices and +� +Bm “ +» +———————————– +sA +1,m +0 +. . . +ρ1 +0 +. . . +0 +... +... +sA +k,m +ρk +�αk`1 +�βk`1 +... +... +�αm´1 +�βm´1 +�αm +fi +ffiffiffiffiffiffiffiffiffiffiffifl +P Kmˆm +n +. +This gives the desired result. +If we would like to compute the smallest singular triplets of A , then we can use the same +theorem, but instead of working with the first right singular lateral slices ⃗ +V A +i,m for 1 ď i ď k, we +use the last k right singular lateral slices in (3.12). The computations are analogous to those +described above. +3.4. Augmentation by harmonic Ritz lateral slices. When the smallest singular val- +ues of a matrix A are clustered, their computation by the restarted Lanczos bidiagonalization +method as described above may require many iterations. In this situation it may be beneficial +to instead compute approximations of the smallest singular values of A by seeking to determine +approximations of the largest singular values of the matrix +` +AT A +˘´1 without explicitly com- +puting the matrix +` +AT A +˘´1. This was done for the matrix case by using computing harmonic +Ritz vectors; see [5, 27]. Harmonic Ritz vectors furnish approximations of eigenvectors of AT A +associated with the corresponding harmonic Ritz values. +In the case of tensors, harmonic Ritz lateral slices furnish approximations of eigenvectors of +A H ‹ A associated with harmonic Ritz tubes of A H ‹ A . The harmonic Ritz tubes qθj of +A H ‹A associated with the partial tensor tridiagonalization defined in (3.6) are the eigentubes +of the generalized eigenvalue problem +´` +BH +m ‹ Bm +˘2 ` α2 +m ‹ β2 +m ‹ ⃗Em ‹ ⃗E H +m +¯ +‹ ⃗qωj “ qθj ‹ BH +m ‹ Bm ‹ ⃗qωj, +1 ď j ď m. +(3.22) +The eigenpair tqθj, ⃗qωju can be computed without forming the tensor BH +m ‹ Bm. Let +⃗ωj “ Bm ‹ ⃗qωj. +(3.23) +Using the relations +αm ‹ ⃗E H +m “ ⃗E H +m ‹ Bm and αm ‹ ⃗Em “ BH +m ‹ ⃗Em, +we can write +α2 +m ‹ β2 +m ‹ ⃗Em ‹ ⃗E H +m “ β2 +m ‹ BH +m ‹ ⃗Em ‹ ⃗E H +m ‹ Bm. +18 + +Therefore, using (3.23), the relation (3.22) can be written as +BH +m ‹ +´ +Bm ‹ BH +m ‹ Bm ` β2 +m ‹ ⃗Em ‹ ⃗E H +m ‹ Bm +¯ +‹ B´1 +m ‹ ⃗ωj “ qθj ‹ BH +m ‹ Bm ‹ B´1 +m ‹ ⃗ωj. +It follows that +´ +Bm ‹ BH +m ` β2 +m ‹ ⃗Em ‹ ⃗E H +m +¯ +‹ ⃗ωj “ qθj ‹ ⃗ωj +(3.24) +and +´ +Bm ‹ BH +m ` β2 +m ‹ ⃗Em ‹ ⃗E H +m +¯ +“ Bm,m`1 ‹ BH +m,m`1. +In this subsection, we denote the singular triplets of Bm,m`1 by ts1 +i, ⃗ +U 1 +i , ⃗ +V 1 +i u for 1 ď i ď m, +with the first k of them being the smallest singular triplets. Recall that we are interested in +determining approximations of the smallest singular triplets of A . The k smallest singular +triplets of Bm,m`1 form the tensors +U 1 +k “ +” +⃗ +U 1 +1, ⃗ +U 1 +2, . . . , ⃗ +U 1 +k +ı +P Kmˆk +n +, +V 1 +k “ +” +⃗ +V 1 +1, ⃗ +V 1 +2, . . . , ⃗ +V 1 +k +ı +P Kpm`1qˆk +n +, +S 1 +k “ +” +s1 +1 ‹ ⃗E1, s1 +2 ‹ ⃗E2, . . . , s1 +k ‹ ⃗Ek +ı +P Kkˆk +n +, +where +Bm,m`1 ‹ V 1 +k “ U 1 +k ‹ S 1 +k and BH +m,m`1 ‹ U 1 +k “ V 1 +k ‹ S 1 +k. +We obtain from the above equations that +Bm,m`1 ‹ BH +m,m`1 ‹ U 1 +k “ U 1 +k ‹ +` +S 1 +k +˘2 , +where +` +S 1 +k +˘2 “ +”` +s1 +1 +˘2 ‹ ⃗E1, . . . , +` +s1 +k +˘2 ‹ ⃗Ek +ı +. +Consequently, the eigenpair +! +ps1 +iq2 , U 1 +i +) +satisfies (3.24), and +! +ps1 +iq2 , B´1 +m ‹ U 1 +i +) +is an eigenpair +of (3.22). It follows that the harmonic Ritz lateral slice associated with qθj is given by +⃗| +Vj “ Pm ‹ ⃗qωj “ Pm ‹ B´1 +m ‹ ⃗ +U 1 +j . +(3.25) +We turn to the computation of the residual of harmonic Ritz lateral slices. Using eqs. (3.6) +and (3.24), we obtain the relations +A H ‹ A ‹ ⃗| +Vj ´ qθj ‹ ⃗| +Vj “ A H ‹ A ‹ Pm ‹ ⃗qωj ´ qθj ‹ Pm ‹ ⃗qωj +“ +´ +Pm ‹ BH +m ‹ Bm ` βm ‹ ⃗E H +m ˚ Bm +¯ +‹ ⃗qωj ´ qθj ‹ Pm ‹ ⃗qωj +“ Pm ‹ B´1 +m ‹ +` +Bm ‹ BH +m ´ θj ˚ Im +˘ +‹ ⃗ωj ` βm ‹ ⃗ +Pm`1 ‹ ⃗E H +m ‹ ⃗ωj +“ ´β2 +m ‹ Pm ‹ B´1 +m ˚ ⃗Em ‹ ⃗E H +m ‹ ⃗ωj ` βm ˚ ⃗ +Pm`1 ‹ ⃗E H +m ‹ ⃗ωj +“ ⃗E H +m ‹ ⃗ωj ‹ βm +´ +⃗ +Pm`1 ´ βm ‹ Pm ‹ B´1 +m ‹ ⃗Em +¯ +. +19 + +It follows that the residual can be expressed as +⃗| +Rm “ ⃗ +Pm`1 ´ βm ‹ Pm ‹ B´1 +m ‹ ⃗Em. +(3.26) +We now proceed analogously as in the previous subsection, i.e., we use the smallest harmonic +Ritz eigentubes of BH +m`1,m ‹ Bm`1,m and associated eigenslices to approximate the k smallest +singular triplets of A . This yields relations that are analogous to (3.3) and (3.4). The following +theorem provides the details. +Theorem 11. Apply m steps of Algorithm 5 to the third-order tensor A and assume that the +tensor Bm in (3.3) and (3.4) is invertible. Then, for k “ 1, . . . , m ´ 1, we have the relations +A ‹ | +Pk`1 “ q +Qk`1 ‹ q +Bk`1, +(3.27) +A H ‹ q +Qk`1 “ | +Pk`1 ‹ q +BH +k`1 ` qβk`1 ‹ ⃗ +} +Pk`2 ‹ ⃗E H +k`1, +(3.28) +where | +Pk`1 P Kpˆpk`1q +n +and +q +Qk`1 P Kℓˆpk`1q +n +have orthonormal lateral slices and +q +Bk`1 P +Kpk`1qˆpk`1q +n +is an upper triangular tensor, where the k first lateral slices of | +Pk`1 are a t-linear +combination of the k first harmonic Ritz lateral slices of A with +⃗ +} +Pk`2 P Kp +n is orthogonal to +| +Pk`1. Moreover, ⃗Ek`1 P Km +n is the canonical lateral slice under the t-product. +Proof. Let t ⃗| +Viui“1:k be the first k harmonic Ritz lateral slices of A . Using (3.25) and (3.26), +we get +„ +s1 +1 ‹ ⃗| +V1, s1 +2 ‹ ⃗| +V2, . . . , s1 +k ‹ ⃗| +Vk, ⃗| +Rm + +“ +” +Pm, ⃗ +Pm`1 +ı +‹ +„ +B´1 +m ‹ U 1 +k ‹ S 1 +k +´βm ‹ B´1 +m ‹ ⃗Em +0 +e + +“ Pm`1 ‹ +„ +B´1 +m ‹ U 1 +k ‹ S 1 +k +´βm ‹ B´1 +m ‹ ⃗Em +0 +e + +. +Define the tensor +Jk`1 “ +„ +B´1 +m ‹ U 1 +k ‹ S 1 +k +´βm ‹ B´1 +m ‹ ⃗Em +0 +e + +. +(3.29) +Using the reduced t-QR factorization of Jk`1, we get +Jk`1 “ Q1 +k`1 ‹ R1 +k`1, +where Q1 +k`1 P Kpm`1qˆpk`1q +n +has orthonormal lateral slices and R1 +k`1 P Kpk`1qˆpk`1q +n +is an +f-upper triangular tensor. +This factorization can be computed by a simple modification of +Algorithm 3. +Let +| +Pk`1 “ +„ +⃗ +} +P1, ⃗ +} +P2, . . . , ⃗ +} +Pk`1 + +“ Pm`1 ‹ Q1 +k`1 P Kℓˆpk`1q +n +. +(3.30) +20 + +Then +A ‹ | +Pk`1 “ A ‹ Pm`1 ‹ Q1 +k`1 +“ +” +A ‹ Pm, A ‹ ⃗ +Pm`1 +ı +‹ Q1 +k`1 +“ +” +A ‹ Pm, A ‹ ⃗ +Pm`1 +ı +‹ Jk`1 ‹ +` +R1 +k`1 +˘´1 +“ +” +A ‹ Pm ‹ B´1 +m ‹ U 1 +k ‹ S 1 +k, A ‹ ⃗ +Pm`1 ´ A ‹ Pm ‹ βm ‹ B´1 +m ‹ ⃗Em +ı +‹ +` +R1 +k`1 +˘´1 +“ +” +Qm ‹ U 1 +k ‹ S 1 +k, A ‹ ⃗ +Pm`1 ´ ⃗ +Qm ‹ βm +ı +‹ +` +R1 +k`1 +˘´1 . +Define +q +Qk “ Qm ‹ U 1 +k P Kpˆk +n +. +(3.31) +Using the orthogonality of A ‹ ⃗ +Pm`1 ´ βm ‹ ⃗ +Qm against the lateral slices of q +Qk gives +qαk`1 ‹ ⃗ +} +Qk`1 “ ´βm ‹ ⃗ +Qm ` A ‹ ⃗ +Pm`1 ´ q +Qk ‹ +» +———– +qγ1 +qγ2 +... +qγk +fi +ffiffiffifl , +(3.32) +where +›››› +⃗ +} +Qk`1 +›››› “ 1 and qαk`1 is the tube obtained from the normalization of the tensor +´βm ‹ ⃗ +Qm ` A ‹ ⃗ +Pm`1 ´ q +Qk ‹ +» +———– +qγ1 +qγ2 +... +qγk +fi +ffiffiffifl +with +q +QH +k ‹ +´ +´βm ‹ ⃗ +Qm ` A ‹ ⃗ +Pm`1 +¯ +“ +» +———– +qγ1 +qγ2 +... +qγk +fi +ffiffiffifl . +It follows from (3.31) and (3.32) that +A ‹ | +Pk`1 “ +» +—–Qm ‹ U 1 +k ‹ S 1 +k, qαk`1 ‹ ⃗ +} +Qk`1 ` q +Qk ‹ +» +—– +| +γ1 +... +qγk +fi +ffifl +fi +ffifl ‹ +` +R1 +k`1 +˘´1 +“ +„ +Qm ‹ U 1 +k, ⃗ +} +Qk`1 + +‹ +» +———– +s1 +1 +qγ1 +... +... +s1 +k +qγk +qαk`1 +fi +ffiffiffifl ‹ +` +R1 +k`1 +˘´1 . +21 + +Hence, +A ‹ | +Pk`1 “ q +Qk`1 ‹ q +Bk`1, +(3.33) +with +q +Bk`1 “ +» +———– +s1 +1 +qγ1 +... +... +s1 +k +qγk +qαk`1 +fi +ffiffiffifl ‹ +` +R1 +k`1 +˘´1 P Kpk`1qˆpk`1q +n +, +(3.34) +where q +Bk`1 is an upper triangular tensor as it is the t-product of two upper triangular tensors. +To show (3.28), we first notice that +A H ‹ q +Qk “ A H ‹ Qm ‹ U 1 +k “ Pm`1 ‹ BH +m,m`1 ‹ U 1 +k “ Pm`1 ‹ V 1 +k ‹ S 1 +k. +Using the fact that +Bm,m`1 “ +” +Bm, βm ‹ ⃗Em +ı +“ Bm ‹ +” +Im, βm ‹ B´1 +m ‹ ⃗Em +ı +, +we get +Bm,m`1 ‹ V 1 +k “ U 1 +k ‹ S 1 +k ô +” +Im, βm ‹ B´1 +m ‹ ⃗Em +ı +‹ V 1 +k “ B´1 +m ‹ U 1 +k ‹ S 1 +k. +It follows from the above result that +V 1 +k “ +„ +B´1 +m ‹ U 1 +k ‹ Sk +´βm ‹ B´1 +m ‹ ⃗Em +0 +e + +‹ +„ +Ik +⃗E H +m`1 ‹ V 1 +k + +“ Jk`1 ‹ +„ +Ik +⃗E H +m`1 ‹ V 1 +k + +. +We obtain +A H ‹ q +Qk “ A H ‹ Qm ‹ U 1 +k +“ Pm`1 ‹ Bm,m`1 ‹ U 1 +k +“ Pm`1 ‹ V 1 +k ‹ S 1 +k +“ Pm`1 ‹ Jk`1 ‹ +„ +Ik +⃗E H +m`1 ‹ V 1 +k + +‹ S 1 +k +“ Pm`1 ‹ Q1 +k`1 ‹ R1 +k`1 ‹ +„ +Ik +⃗E H +m`1 ‹ V 1 +k + +‹ S 1 +k +“ | +Pk`1 ‹ R1 +k`1 ‹ +„ +Ik +⃗E H +m`1 ‹ V 1 +k + +‹ S 1 +k. +The relation (3.33) now yields +q +QH +k ‹ A ‹ | +Pk`1 “ q +Bk,k`1 ô | +PH +k`1 ‹ A H ‹ q +Qk “ q +BH +k,k`1, +where +q +Bk,k`1 P Kpk`1qˆk +n +is the subtensor of +q +Bk`1, which is obtained by removing the last +horizontal slice of q +Bk`1. Then +| +PH +k`1 ‹ A H ‹ q +Qk “ R1 +k`1 ‹ +„ +Ik +⃗E H +m`1 ‹ V 1 +k + +‹ S 1 +k “ q +BH +k,k`1 +22 + +and +| +PH +k`1 ‹ A H ‹ ⃗ +} +Qk`1 “ q +BH +k`1 ‹ q +QH +k`1 ‹ ⃗ +} +Qk`1 “ q +BH +k`1 ‹ ⃗Ek`1 “ qαk`1 ‹ ⃗Ek`1. +Hence, +A H ‹ ⃗ +} +Qk`1 “ qαk`1 ‹ ⃗ +} +Pk`1 ` ⃗| +R1 +k`1 +(3.35) +with ⃗| +R1 +k`1 K | +Pk`1. It follows that +A H ‹ q +Qk`1 “ | +Pk`1 ‹ q +BH +k`1 ` ⃗| +R1 +k`1 ‹ ⃗E H +k`1. +Normalization of ⃗| +R1 +k`1 gives +A H ‹ q +Qk`1 “ | +Pk`1 ‹ q +BH +k`1 ` qβk`1 ‹ ⃗ +} +Pk`2 ‹ ⃗E H +k`1. +The orthonormality of the lateral slices of | +Pk`1 and +q +Qk`1 holds by the construction of these +tensors. Specifically, it follows from (3.30) that the lateral slices of | +Pk`1 are orthonormal. Due +to (3.31), the first k lateral slices of q +Qk`1 are orthonormal. +Notice that if qβk`1 given in (3.28) vanishes, then we have determined k singular triplets, i.e., +these singular triplets of A can be computed by using the singular triplets of q +Bk`1, as well as +| +Pk`1 and +q +Qk`1 defined in (3.27) and (3.28). If qβk`1 does not vanish, then we append new +lateral slices to | +Pk`1 and +q +Qk`1 in a similar way as we did in the previous subsection. The +following result is analogous to Theorem 10. +Theorem 12. Carry out m steps of Algorithm 5 and assume that eqs (3.27) and (3.28) hold +for k “ 1, 2, . . ., m´1. Further, let qβk`1 in (3.28) be nonvanishing. Then we have the following +relations +A ‹ | +Pm “ q +Qm ‹ q +Bm, +A H ‹ q +Qm “ | +Pm ‹ q +BH +m ` qβm ‹ ⃗ +} +Pm`1 ‹ ⃗E H, +where | +Pm P Kpˆm +n +and +q +Qm P Kℓˆm +n +are orthonormal tensors, +q +Bm P Kmˆm +n +is an upper +triangular tensor, qβm is a tube of n elements, ⃗ +} +Pm`1 P Kp +n is orthogonal to all the lateral slices +of | +Pm and ⃗E H P Kℓ +n is the canonical lateral slice under the t-product, where the first k lateral +slices of | +Pm and +q +Qm are the same as the lateral slices of | +Pk`1 and +q +Qk`1, respectively, given +in Theorem 11. +Proof. These results can be shown similarly as Theorem 10. +Theorem 11 requires the invertibility of Bm. Notice that this tensor is well conditioned if all +the frontal slices of � +Bm are well conditioned, i.e., if +max +1ďiďn κ +´ +� +Bpiq +m +¯ +is small, where +κp � +Bpiq +m q “ +´ +�sBm +1 +¯piq +´ +�sBm +m +¯piq . +23 + +Algorithm 6 describes computations required to compute approximations of either the k +largest singular triplets or the k smallest singular triplets of a third-order tensor A using the +methods we developed in the present or previous subsections. +Algorithm 6 Tensor Lanczos Bidiagonalization Ritz (t-LBR) algorithm for computing the +largest and the smallest singular triplets. +Input: A P Kℓˆp +n +. +m: the number of tensor Lanczos bidiagonalization steps. +⃗ +P1 P Kp +n with unit norm. +k: the number of the desired singular triplets. +δ: The tolerance to accept the singular triplets approximated. +ǫ: machine epsilon. +type: A Boolean variable for the kind of augmentation which is either ’Ritz’ for Ritz +augmentation or ’Harm’ for harmonic Ritz augmentation. +Output: The k desired singular triplets of A , tσi, ⃗ +Ui, ⃗ +Viui“1:k. +1: Compute the Partial Lanczos bidiagonalization of A by Algorithm 5. +2: Compute the t-SVD of Bm using Algorithm 2. +3: Check the convergence stated in Equation (3.9). If all the k desired singular triplets are +well approximated, then exist. +4: Compute the augmented vectors: +5: if type=’Ritz’ or kpBmq ą ǫ +1 +2 then +6: +Compute the tensors P :“ � +Pk`1, Q :“ +� +Qk`1, B :“ � +Bk`1 and the residual ⃗ +Fk from +(3.12), (3.15), (3.16) and (3.18). +7: end if +8: if type=’Harm’ and kpBmq ď ǫ +1 +2 then +9: +Compute the t-SVD of Bm,m`1. +10: +Compute the t-QR factorization of Jk`1 in (3.29). +11: +Compute the tensors P :“ | +Pk`1, Q :“ +q +Qk`1, B :“ q +Bk`1 and the residual ⃗| +Rm from +(3.30), (3.31), (3.34) and (3.35). +12: end if +13: Append m ´ k lateral slices to P and Q, and m ´ k horizontal and lateral slices to B to +obtain Pm, Qm and Bm, and determine a new residual ⃗ +Rm. +14: Go to 2. +4. Multidimensional principal component analysis for facial recognition. Prin- +cipal component analysis (PCA) is used in numerous areas of science and engineering, such as +in data denoising, image classification, and facial recognition. Some approaches to color image +classification involve conversion of color images to grayscale images to reduce the computational +burden, because color images are represented by tensors, while gray scale images can be rep- +resented by matrices; see [2, 29]. However, this conversion entails loss of information. A color +image in RGB format can be represented by a third-order tensor. This section discusses the +application of PCA to third-order tensors. +PCA when applied to gray-scale face recognition computes a set of characteristics (eigenfaces) +corresponding to the main components of the initial set of training images. Recognition is done +by projecting the training images into the eigenface subspace, in which an image of a person +24 + +is classified by comparing it with other available images in the eigenface subspace. The main +advantages of this procedure are its simplicity, speed, and insensitivity to small changes on the +faces. +When applying PCA to third-order tensors using the t-product, tubes, lateral slices, and +third-order tensors are analogues of scalars, vectors, and matrices in the eigenface technique +for classifying grayscale images. Using this identification, PCA for third-order tensors that +represent color images is structurally very similar to PCA for matrices that represent grayscale +images. The latter is described in [17]. +Let N training color images I1, I2, . . . , IN of size ℓ ˆ p ˆ n be available. They are represented +by the third-order tensors I1, I2, . . . , IN in Rℓˆpˆn. The procedure of recognizing color facial +images using third-order tensors is as follows: +1. For each image Ii for i “ 1, 2, . . ., N, we determine a lateral slice +⃗ +Xi P Rℓpˆ1ˆn by +vectorizing each frontal slice, i.e., +⃗ +X psq +i +“ vecpI psq +i +q for s “ 1, 2, . . ., n. +We then +construct a tensor, whose frontal slices are given by +⃗ +Xi, i.e., +X “ +” +⃗ +X1, ⃗ +X2, . . . , ⃗ +XN +ı +P RℓpˆNˆn. +2. Compute the mean of the frontal slices of X , i.e., +⃗ +M “ +N +ÿ +i“1 +⃗ +Xi +N , +and let +X “ r ⃗X 1, ⃗X 2, . . . , ⃗X Ns, +⃗X i “ +⃗ +Xi ´ +⃗ +M . +3. Determine the first k left singular vectors of X . +We denote them by +⃗ +U1, . . . , ⃗ +Uk. +Construct the projection subspace +Uk “ span +! +⃗ +U1, ⃗ +U2, . . . , ⃗ +Ak +) +(4.1) +and let +Uk “ +” +⃗ +U1, ⃗ +U2, . . . , ⃗ +Uk +ı +P Rℓpˆkˆn. +4. Project each face Ii onto the subspace (4.1) to obtain U H +k ‹ ⃗X i. A test image I0 also is +projected onto the same space to get U H +k ‹ +´ +⃗ +X0 ´ +⃗ +M +¯ +. Finally, determine the closest +image to the test image by computing the minimal distance between the projected test +image and all the projected training images. +The main difference between methods that use PCA for facial recognition is the way that +the first (dominant) left singular vectors of X are computed. In the present paper, we use +our proposed method to compute the dominant singular triplets that are used in PCA. The +following algorithm summarises the different steps in our approach. +25 + +Algorithm 7 Facial recognition using tensor Lanczos bidiagonalization with Ritz augmenta- +tion. +1: Input: Training set of images X (N images), mean image X , test image I0 with its +associate lateral slice +⃗ +X0 “ vecpI0q; m the number of tensor Lanczos bidiagonalization +algorithm; k the number of the desired left singular slices. +2: Output: Closest image in the database. +3: rUk, Sk, Vks “ t-LBRpX , m, kq using Algorithm 6. +4: Project X onto Uk to get P “ U H +k +‹ X . +5: Project the mean of the test image I0 onto Uk, ⃗ +P0 “ U H +k +‹ +´ +⃗ +X0 ´ +⃗ +M +¯ +“ U H +k +⃗X 0. +6: Find i “ arg min +i“1,2,...,N +››› ⃗ +P0 ´ ⃗ +Pi +››› +F . +5. Numerical experiments. This section illustrates the performance of Algorithm 6 +for detecting the largest or smallest singular triplets when applied to synthetic data, tensor +compression, and facial recognition. All computations are carried out on a laptop computer +with 2.3 GHz Intel Core i5 processors and 8 GB of memory using MATLAB 2018a. +5.1. Examples with synthetic data. We use synthetic data generated by the MATLAB +command randnpℓ, p, nq, which generates a tensor A P Rℓˆpˆn, whose entries are normally +distributed pseudorandom numbers with mean zero and variance one. +5.1.1. Largest singular values. Table 5.1 displays the error in the four largest approx- +imate singular tubes computed by augmentation by Ritz lateral slices (referred to as Ritz in +the table) and by the partial Lanczos bidiagonalization/Golub-Kahan algorithm (referred to +as GK in the table) as described in [17], but using the t-product. These errors are given by +}S pi, i, :q ´ Σpi, i, :q}F for i “ 1, 2, 3, 4 with m “ 20. Table 5.2 shows the number of iterations +required when using augmentation by Ritz lateral slices to approximate the four largest singular +triplets for tensors of different sizes and the number of Lanczos bidiagonalization steps m. +i +Methods +100 ˆ 100 ˆ 3 +500 ˆ 500 ˆ 3 +1000 ˆ 1000 ˆ 3 +100 ˆ 100 ˆ 5 +500 ˆ 500 ˆ 5 +1 +Ritz +7.13e-14 +1.60e-13 +2.27e-13 +2.85e-14 +1.63e-13 +GK +8.16e-10 +0.09 +0.01 +3.18e-08 +0.01 +2 +Ritz +9.29e-14 +1.98e-13 +1.56e-13 +5.62e-14 +1.48e-13 +GK +1.27e-05 +0.07 +0.44 +3.12e-04 +0.15 +3 +Ritz +5.01e-14 +2.70e-13 +8.93e-14 +5.41e-14 +2.66e-13 +GK +0.02 +0.95 +1.78 +6.05e-04 +0.51 +4 +Ritz +3.39e-13 +4.92e-11 +9.01e-13 +3.39e-14 +6.74e-13 +GK +0.01 +1.60 +3.37 +0.08 +2.03 +Table 5.1: The Frobenius norm }S pi, i, :q ´ Σpi, i, :q}F , where S pi, i, :q denotes the singular +tubes computed by either augmentation by Ritz lateral slices (Ritz) or by partial Lanczos bidi- +agonalization also known a partial Golub-Kahan bidiagonalization (GK), and Σpi, i, :q stands +for the singular tubes determined by the t-SVD method with m “ 20 for i “ 1, 2, 3, 4. +26 + +Ritz +100 ˆ 100 ˆ 3 +500 ˆ 500 ˆ 3 +1000 ˆ 1000 ˆ 3 +100 ˆ 100 ˆ 5 +500 ˆ 500 ˆ 5 +augmentation +iter +time +iter +time +iter +time +iter +time +iter +time +m “ 10 +15 +0.40 +29 +2.84 +41 +18.20 +13 +0.41 +29 +4.09 +m “ 20 +3 +0.15 +5 +2.14 +7 +12.88 +3 +0.18 +5 +2.91 +Table 5.2: Number of iterations (iter) needed by the Ritz augmentation method to determine +the four largest singular tubes for third-order tensors of different sizes with m “ 10, 20. The +columns with header “time” shows the CPU time in seconds. +Table 5.1 shows the Ritz augmentation method to yield much higher accuracy than the GK +method. Figures 5.1 and 5.2 display the values of some frames of the first 10 singular tubes +of third-order tensors of sizes 100 ˆ 100 ˆ 3 and 1000 ˆ 1000 ˆ 5, respectively, computed by +Ritz augmentation using Algorithm 6, the t-SVD, and partial Lanczos bidiagonalization (GK). +Each tube is denoted by S pk, k, :q P Kn, where n is equal to 3 or 5, and k “ 1, 2, . . ., 10. In +other word, for a fixed i with 1 ď i ď n, we plot S pk, k, iq P Kn for k “ 1, 2, . . . , 10. As +mentioned above, the ith computed singular triplet is accepted as an approximate singular +triplet if ⃗ +Rm ‹ ⃗E H +m ‹ ⃗ +Ui is small enough for 1 ď i ď k, where k is the number of desired +singular triplets and the ⃗ +Ui are left singular lateral slice of the current tensor Bm; see eq. (3.9). +Figure 5.3 shows the evolution of the error computed by (3.9) for the first three singular triplets +determined by Algorithm 6 when applied to a third-order tensor of size 1000 ˆ 1000 ˆ 3 for +m “ 20. +Figure 5.1: On the left, we display the values of the first frontal slices (frames) of the first 10 +singular tubes detected by t-SVD, Ritz augmentation and Partial Lanczos bidiagonalization +(GK) for a synthetic data of size 100 ˆ 100 ˆ 3 with m “ 20, and on the right we plotted the +third frontal slices of these tubes, i.e., S pk, k, iq with k “ 1, 2, . . ., 10 and i “ 1, 3. +Figures 5.1 and 5.2 illustrate that using Algorithm 6 with Ritz augmented method gives +more accurate approximations than the GK method. In particular, the frontal slices of each +tube computed with Algorithm 6 are very close to the corresponding frontal slices of the tubes +determined by the t-SVD, independently of the size of the third-order tensor. +27 + +35 +0.7 +Ritz-Xt-svdDGK +0.6 +0.5 +25 +0.4 +20 +h,3) +s +0.3 +15 +0.2 +10 +0.1 +0 +Ritz-X +t-svd..D..GK +0 +-0.1 +2 +3 +4. +5 +6 +7 +8 +6 +10 +2 +3 +4 +5 +6 +7 +8 +9 +10Figure 5.2: The left-hand side pane shows the values of the first frontal slices (frames) of +the first 10 singular tubes computed by t-SVD, Ritz augmentation, and the partial Lanczos +bidiagonalization (GK) method for a synthetic data of size 1000 ˆ 1000 ˆ 5 with m “ 20. +The right-hand side pane displays the third frontal slices of these tubes, i.e., S pk, k, iq for +k “ 1, 2, . . ., 10 and i “ 1, 3. +Figure 5.3: Evolution of the remainder term for a third-order tensor of size 1000 ˆ 1000 ˆ 3 +when computing the first three singular triplets by Algorithm 6 with Ritz augmentation. +5.1.2. Smallest singular values. This subsection illustrates the performance of Algo- +rithm 6 with Ritz augmentation (referred to as Ritz) and with harmonic Ritz augmentation +(referred to as Harm) for computing the smallest singular triplets of synthetic third-order ten- +sors of different sizes. Table 5.3 displays the error in the fourth smallest singular tubes computed +by Ritz augmentation and harmonic Ritz augmentation for m “ 20, and compares with results +determined by the t-SVD method. In Table 5.4 we show the number of iterations and the +required CPU time (in seconds) for these methods when m “ 20. +28 + +4.5 +4 +Rm*E*U +3.5 +3 +2.5 +2 +1.5 +0.5 +0 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +5.5 +6 +Iterations160 +0.6 +0.5 +Ritz +t-svd..DGK +140% +0.4 +120 +0.3 +100 +0.2 +80 +0.1 +0 +0.1 +40F +Ritz-t-svdDGK +0.2 +20 +0.3 +2 +3 +4 +5 +6 +7 +8 +9 +10 +1 +2 +3 +4 +6 +1 +8 +9 +10 +k +ki +Method +100 ˆ 100 ˆ 3 +100 ˆ 100 ˆ 5 +500 ˆ 500 ˆ 3 +500 ˆ 500 ˆ 5 +n ´ 3 +Ritz +3.82e-11 +5.22e-12 +1.34e-10 +2.50e-10 +Harm +1.03e-13 +4.64e-13 +4.66e-13 +1.07e-13 +n ´ 2 +Ritz +1.99e-14 +4.34e-13 +1.20e-14 +1.68e-11 +Harm +4.94e-15 +3.10e-13 +2.46e-14 +3.77e-14 +n ´ 1 +Ritz +8.36e-14 +4.56e-14 +1.77e-14 +6.86e-12 +Harm +1.64e-15 +6.05e-15 +2.88e-14 +1.39e-13 +n +Ritz +1.38e-15 +7.71e-16 +6.49e-15 +2.00e-12 +Harm +8.59e-16 +7.90e-16 +3.01e-15 +1.41e-14 +Table 5.3: The Frobenius norm }S pi, i, :q ´ Σpi, i, :q}F , where S pi, i, :q denotes the singular +tubes determined by Ritz augmentation or harmonic Ritz augmentation for m “ 20, and +Σpi, i, :q are tubes computed by the t-SVD method for the four smallest tubes, i.e., for i “ +n ´ 3, n ´ 2, n ´ 1, n. +Method +100 ˆ 100 ˆ 3 +500 ˆ 500 ˆ 3 +100 ˆ 100 ˆ 5 +500 ˆ 500 ˆ 5 +CPU time +iter +CPU time +iter +CPU time +iter +CPU time +iter +Ritz +0.99 +31 +231.81 +615 +1.11 +30 +425.83 +831 +Harm +0.85 +29 +227.49 +606 +1.03 +30 +355.35 +723 +Table 5.4: CPU time in seconds, and number of iterations required by Algorithm 6 with Ritz +augmentation and harmonic Ritz augmentation for m “ 20 to compute the four smallest singular +triplets of synthetic third-order tensors of different sizes. +Tables 5.3 and 5.4 show that harmonic Ritz augmentation gives higher accuracy than Ritz +augmentation when computing the smallest singular triplets. Figures 5.4 and 5.5 depict the +Frobenius norm of the remainder term ⃗ +Rm ‹ ⃗E H +m ‹ ⃗ +Ui for each iteration with Algorithm 6 with +Ritz augmentation and harmonic Ritz augmentation when approximating the last two singular +triplets for m “ 20. +29 + +Figure 5.4: The Frobenius norm of ⃗ +Rm ‹ ⃗E H +m ‹ ⃗ +Ui obtained by Algorithm 6 with Ritz aug- +mentation when approximating the two smallest singular triplets of a synthetic tensor of size +500 ˆ 500 ˆ 5 with m “ 20 at each iteration for i “ 499, 500. +Figure 5.5: The Frobenius norm of ⃗ +Rm ‹ ⃗E H +m ‹ ⃗ +Ui obtained by harmonic Ritz augmentation +when approximating the last two singular triplets of a synthetic tensor data of size 500ˆ500ˆ5 +with m “ 20, at each iteration for i “ 499, 500. +Figures 5.4 and 5.5 show the error } ⃗ +Rm ‹ ⃗E H +m ‹ ⃗ +Ui}F associated with Ritz augmentation in +Algorithm 6 to converge in a smoother way than the corresponding error for harmonic Ritz +augmentation. Both errors converge to zero as the number of iterations increases. +30 + +16 +F +14 +12 +10 +Error +8 +4 +2 +0 +0 +100 +200 +300 +400 +500 +600 +700 +800 +Iterations15 +Rm *t Em * U500 +F +Rm* Em *t U499 +10 +Error +5 +0 +0 +100 +200 +300 +400 +500 +600 +700 +800 +Iterations5.2. Application to data compression. Figure 5.6 displays examples of image compres- +sion using two color images: “house” of size 256 ˆ256 ˆ3 and “Hawaii” of size 1200 ˆ1200 ˆ3. +For each image, we compute the kth largest singular triplets using Ritz augmentation in Algo- +rithm 6, which will be referred to as “Ritz,” for different numbers k of desired singular triplets. +Figure 5.7 displays the relative error of the compressed images for k “ 5, 10, 15, 25, by using +Ritz augmentation (Ritz) and the t-SVD method. This error is measured by +}Ak ´ A }F +}A }F +, +(5.1) +where A denotes the tensor that represents the original image and Ak “ řk +i“1 ⃗ +Ui ‹ si ‹ ⃗ +V H. +Figure 5.6: Examples of image compression applied to the “house” and “Hawaii” images for +k “ 5, 10, 15, 25 slices using Algorithm 6 with Ritz augmentation. +Figure 5.7: Relative compression error (5.1) for the images “house” and “Hawaii” obtained +with Algorithm 6 with Ritz augmentation (Ritz) and the t-SVD method. +Figure 5.7 shows the relative errors obtained with Algorithm 6 with Ritz augmentation and +the t-SVD are almost the same. This means that the approximate singular tubes and the right +31 + +0.14 +0.26 +0.13 +0.25 +0.12 +0.24 +0.23 +0.11 +error +0.1 +0.22 +0.09 +0.21 +0.08 +0.2 +0.07 +0.19 +0.06 +Ritz +0.18 +t-svd +会 +Ritz +t-svd +0.05 +0.17 +5 +10 +15 +20 +25 +5 +10 +15 +20 +25 +k +kOriginal +k=5 +k = 10 +k = 15 +k = 25and left singular lateral slices determined by Algorithm 6 with Ritz augmentation are very +accurate. +5.3. Facial recognition. We illustrate the application of Algorithm 7 to facial recognition +using color images that are represented by third-order tensors. The images in our test are from +the Georgia Tech database GTDB crop [26], which contains 750 images of 50 persons, with each +person represented by 15 images that show various facial expressions and facial orientation, and +different illumination conditions. Figure 5.8 shows an example of images of one person in the +data set. +Figure 5.8: An example of a person with different facial expressions and orientations. +Each image in the data set is of size 100 ˆ 100 ˆ 3 pixels, and we use 3 randomly chosen +images of each person as test images. The remaining 600 images form our training set and define +the tensor X P R10000ˆ600ˆ3. We applied Algorithm 7 and compared the results with those +obtained by the t-SVD and also with results obtained by the‘ Golub-Kahan (GK) algorithm +using the t-product. The performance of these methods is measured by the identification rate +given by +Identification rate “ number of correctly matched images +number of test images +ˆ 100p%q. +(5.2) +Figures 5.9 and 5.10 show results obtained for k “ 1 and k “ 5 for two different persons. The +mean image is defined as in Algorithm 7. +32 + +Figure 5.9: A test for k “ 1. +Figure 5.10: A test for k “ 5. +33 + +Test image +closest image +Mean image +Eigenfacetestimage +closest image +mean image +eigenfaceFigure 5.11: Identification rates for different truncation indices k by Ritz augmentation, t-SVD +and Golub-kahan methods. +Figures 5.9 and 5.10 show that Algorithm 7 performs well for some values of the truncation +index k. In Figure 5.11, we plotted the identification rate (5.2) obtained with Algorithm 7 (Ritz +augmentation), GK for m “ k, and with the exact t-SVD method for the 150 test images. +k +2 +3 +4 +Method +Ritz +t-SVD +Ritz +t-SVD +Ritz +t-SVD +CPU time (s) +10.60 +52.82 +13.11 +63.63 +13.88 +64.77 +Table 5.5: CPU time (in seconds) for Algorithm 7 (Ritz) and for the t-SVD method for m “ 10 +and different values of the truncation index k. +Table 5.5 reports CPU times for Algorithm 7 for m “ 10 (Ritz) and for the t-SVD method +for different values of the truncation index k. The results show Algorithm 7 to be very effective +both in terms of accuracy and CPU time compared to the t-SVD and the classical Golub-Kahan +methods. +6. Conclusion and extensions. This paper presents two new methods for approximat- +ing the largest or smallest singular triplets of large third-order tensors using the t-product. We +use restarted Lanczos bidiagonalization for third-order tensors to develop the Ritz augmenta- +tion method to determine the largest or smallest singular triplets. Moreover, we propose the +harmonic Ritz augmentation method to compute the smallest singular triplets. These methods +are applied to data compression and face recognition. +REFERENCES +[1] T. Arnold, M. Kane, B. W. Lewis, A Computational Approach to Statistical Learning, CRC Press, Boca +Raton, 2018. +34 + +84 +82 +Identification rate (%) +80 +78 +76 +74 +t-svd +Ritzaugmentation +GK +72 +6 +8 +10 +12 +14 +16 +18 +20 +Truncationnumberk[2] M. N. Asif, I. S. Bajwa, S. I. Hyder, M. Naweed, Feature based image classification by using principal +component analysis, ICGST International Journal on Graphics, Vision and Image Processing. 9, 22–17 +(2009). +[3] H. Avron, L. Horesh, M. E. Kilmer, E. 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Saad, Restarting techniques for the (Jacobi-) Davidson symmetric eigenvalue methods, +Electronic Transactions on Numerical Analysis, 7, 163–181 (1998). +36 + diff --git a/tdA0T4oBgHgl3EQfLf9-/content/tmp_files/load_file.txt b/tdA0T4oBgHgl3EQfLf9-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1ad990eac5cefbb59dfd750f3a4e3770598308e9 --- /dev/null +++ b/tdA0T4oBgHgl3EQfLf9-/content/tmp_files/load_file.txt @@ -0,0 +1,1479 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf,len=1478 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='02119v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='NA] 5 Jan 2023 A TENSOR BIDIAGONALIZATION METHOD FOR HIGHER-ORDER SINGULAR VALUE DECOMPOSITION WITH APPLICATIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' EL HACHIMI˚, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' JBILOU:, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' RATNANI˚, AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' REICHEL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The need to know a few singular triplets associated with the largest singular values of third-order tensors arises in data compression and extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This paper describes a new method for their computation using the t-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Methods for determining a couple of singular triplets associated with the smallest singular values also are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The proposed methods generalize available restarted Lanczos bidiagonalization methods for computing a few of the largest or smallest singular triplets of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The methods of this paper use Ritz and harmonic Ritz lateral slices to determine accurate approximations of the largest and smallest singular triplets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Computed examples show applications to data compression and face recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' tensors, t-product, partial tensor bidiagonalization, restarted tensor bidiagonalization, singular value decomposition, face recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The last 20 years has seen an immense growth of the amount of data that is collected for analysis, but it is a challenging problem to extract useful information from available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This difficulty arises, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', in machine learning, data mining, and deep learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', Arnold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The extraction of useful information from data that is represented by a matrix often is facilitated by the singular value decomposition of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Typically, only a few of the largest singular triplets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', the largest singular values and associated right and left singular vectors, are required to extract useful information from the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A restarted Lanczos bidiagonalization method for computing accurate approximations of these singular triplets is described in [5], and R code written by Bryan W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Lewis is available at [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' In many recent applications the given data are represented by a multidimensional array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' These arrays, known as tensors, are natural generalizations of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Several approaches to define tensor-tensor products and tensor-matrix products are described in the literature, including the n-mode product [9, 25], the t-product [22, 31], and the c-product [21, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Generalizations of the singular value decomposition (SVD) to tensors are described in [25] using the n-mode product (the so-called HOSVD), and in [21, 22] using the tensor c-product and t-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The need to compute the SVD or a partial SVD of a tensor arises in a variety of applications, including image restoration, tensor completion [10], robust tensor principal component analysis [13], tensor compression [3], and recognition of color faces [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' These applications require knowledge of the largest singular values and associated lateral tensor singular slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' It is the purpose of the this paper to introduce a new restarted tensor Lanczos bidiagonaliza- tion method for third-order tensors using the t-product for approximating a few of the largest singular values and associated lateral tensor singular slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This method generalizes the ap- proach described in [5] from matrices to tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We remark that the Lanczos bidiagonalization method (also known as the Golub-Kahan bidiagonalization method) for third-order tensors us- ing the t-product has been described in [15, 16, 22, 32];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' however, this bidiagonalization method differs from the one of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' In [5] the authors also describe a restarted Lanczos bidiagonalization method for the compu- tation of a few of the smallest singular values and associated singular vectors of a large matrix by determining harmonic Ritz values is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This paper presents an analogous scheme ˚Laboratory MSDA, Mohammed VI Polytechnic University, Green City, Morocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' :Universit´e du Littoral Cote d’Opale, LMPA, 50 rue F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Buisson, 62228 Calais-Cedex, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 1 for third-order tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The organization of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Section 2 recalls some properties of the t- product and Section 3 reviews tensor Lanczos bidiagonalization of third-order tensors using the t-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Restarted tensor Lanczos bidiagonalization methods are presented for the approx- imation of a few of the largest singular values and associated lateral tensor singular slices by computing lateral tensor Ritz slices, as well as for approximating a few of the smallest singular values and associated lateral tensor singular slices by evaluating harmonic lateral tensor Ritz slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Section 4 discusses multidimensional principal component analysis using a partial tensor HOSVD with application to face recognition, and Section 5 presents a few computed examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Concluding remarks and possible extensions can be found in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The tensor t-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This section reviews results by Kilmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' [22, 23] and uses notation employed there and by Kolda and Bader [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A third-order tensor is an array A “ raijks P Rℓˆpˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Matrices and vectors are tensors of order two and one, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A slice or frame of a third-order tensor A is a section obtained by fixing any one of the three indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Using MATLAB notation, A pi, :, :q, A p:, j, :q, and A p:, :, kq denote the ith horizontal, the jth lateral, and the kth frontal slices of A , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The lateral slice A p:, j, :q also is denoted by ⃗ Aj, and the frontal slice A p:, :, kq is an ℓ ˆ p matrix that is sometimes denoted by A pkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A fiber of a third order tensor A is defined by fixing any two of the three indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The fiber A pi, j, :q is called a tube of A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We will use capital calligraphic letters A to denote third- order tensors, capital letters A to identify matrices, bold face lower case letters a to denote tubes, and lower case letters a stand for scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Further, Kℓˆp n “ Rℓˆpˆn denotes the space of third-order tensors of size ℓ ˆ p ˆ n, Kℓ n “ Rℓˆ1ˆn stands for the space of lateral slices of size ℓ ˆ n, and Kn “ R1ˆ1ˆn denotes the space of tubes with n entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' For a third-order tensor A P Kℓˆp n with frontal slices A piq, i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , n, we define: ‚ The block circulant matrix associated with A : bcircpA q “ » ———– A p1q A pnq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A p2q A p2q A p1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A p3q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A pnq A pn´1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A p1q fi ffiffiffifl P Kℓnˆpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1) ‚ The operator unfold applied to A gives the matrix made up of its frontal slices, unfoldpA q “ » ———– A p1q A p2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A pnq fi ffiffiffifl P Kℓnˆp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We also will need the inverse operator fold such that foldpunfold pA qq “ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ‚ The block diagonal matrix associated with A is defined as bdiagpA q “ » ———– A p1q A p2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A pnq fi ffiffiffifl P Kℓnˆpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 2 Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ([23]) Let A P Kℓˆq n and B P Kqˆp n be third-order tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The t-product of A and B is defined by A ‹ B :“ foldpbcircpA q unfoldpBqq P Kℓˆp n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The block circulant matrix (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1) can be block-diagonalized by using the discrete Fourier transform (DFT) as follows: bcircpA q “ ` F H n b Iℓ ˘ bdiagp � A q pFn b Ipq , where Fn P Cnˆn is the discrete Fourier matrix, F H n denotes its conjugate transpose, � A stands for the Fourier transform of A along each tube, Iℓ P Rℓˆℓ denotes the identity matrix, and b is the Kronecker product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The matrix � A can be computed with the fast Fourier transform (FFT) algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' see [23] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Using MATLAB notations, we have � A “ fftpA , r s, 3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The inverse operation can be evaluated in MATLAB with the command A “ ifftp � A , r s, 3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Hence, the t-product C “ A ‹ B can be evaluated as � C piq “ � A piq � Bpiq, i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , n, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2) where � A piq, � Bpiq, and � C piq are the ith frontal slices of the tensors � A , � B, and � C , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' As already pointed out by Kilmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' [22], one can use symmetry properties of the DFT when applied to real data to reduce the computational effort when evaluating the t-product with the FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This is described by the following result, which can be found, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Given a real vector v P Rn, the associated DFT vector �v “ Fnv satisfies �v1 P R, conjp�viq “ �vn´i`2, i “ 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', „n ` 1 2 \uf6be , where conj denotes the complex conjugation operator and „n ` 1 2 \uf6be denotes the integer part of n ` 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' It follows that for a third-order tensor A P Kℓˆp n , we have � A p1q P Rℓˆp, conj ´ � A piq¯ “ � A pn´i`2q, i “ 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', „n ` 1 2 \uf6be .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This shows that the t-product of two third-order tensors can be determined by evaluating just about half the number of products involved in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Algorithm 1 describes the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 3 Algorithm 1 t-product of third-order tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Input: A P Kℓˆq n , B P Kqˆp n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Output: C :“ A ‹ B P Kℓˆp n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 1: Compute � A “ fftpA , r s, 3q, � B “ fftpB, r s, 3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 2: for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , „n ` 1 2 \uf6be do 3: � C piq “ � A piq � Bpiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 4: end for 5: for i “ „n ` 1 2 \uf6be ` 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , n do 6: � C piq “ conj ´ � C pn´i`2q¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 7: end for 8: C “ ifft ´ � C , r s, 3 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The following definition is concerned with the t-product of a third-order tensor and a tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Let A P Kℓˆp n and b P Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Then C :“ A ‹ b P Kℓˆp n is obtained by applying the inverse DFT along each tube of � C , where each frontal slice is determined by the standard matrix product between each frame of � A and �b, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', � C piq “ � A piq�bpiq “ �bpiq � A piq, i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A third-order tensor A P Kℓˆp n can be written as A “ ” ⃗ A1, ⃗ A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ Ap ı , thus, for the tensors A P Kℓˆq n and B P Kqˆp n , the t-product A ‹ B can be expressed as A ‹ B “ ” A ‹ ⃗ B1, A ‹ ⃗ B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , A ‹ ⃗ Bp ı , where A ‹ ÝÑ Bi “ ÝÝÝÝÝÑ pA ‹ Bqi, i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The Frobenius norm of a third-order tensor A P Kℓˆp n is given by }A }F :“ g f f e ℓ,p,n ÿ i1,i2,i3“1 a2 i1,i2,i3, and the inner product of two third-order tensors of the same size A , B P Kℓˆp n is defined as xA , By :“ ℓ,p,n ÿ i1,i2,i3“1 ai1,i2,i3bi1,i2,i3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We have the relations }A }F “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='n ››› � A ››› F , xA , By “ 1 nx � A , � By.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We recall for later use the definitions of some special tensors and operations: 4 ‚ The identity tensor Iℓ P Kℓˆℓ n is the tensor whose first frontal slice is the identity matrix and all other slices have zero entries only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ‚ The transpose of a real third-order tensor, A P Kℓˆp n , denoted by A H P Kpˆℓ n , is the tensor obtained by first transposing each one of the frontal slices of A , and then reversing the order of the transposed frontal slices 2 through n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' see [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Let the third- order tensors A and B be such that the products A ‹ B and BH ‹ A H are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Then, similarly to the matrix transpose, the tensor transpose satisfies pA ‹ BqH “ BH ‹ A H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ‚ A tensor Q P Kℓˆℓ n is said to be orthogonal if and only if QH ‹ Q “ Q ‹ QH “ Iℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ‚ A square third-order tensor A P Kℓˆℓ n is invertible if there is a third-order tensor B P Kℓˆℓ n such that A ‹ B “ Iℓ, B ‹ A “ Iℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' In this case B is said to be the inverse of A , and is denoted by A ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ([22]) Let ⃗ Ai P Kℓ n for i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , p be lateral slices of the tensor A P Kℓˆp n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A t-linear combination of these slices is defined as ⃗ A1 ‹ b1 ` ⃗ A2 ‹ b2 ` .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ` ⃗ Ap ‹ bp, where the bi for i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', p are tubes in Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Moreover, span !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ⃗ A1, ⃗ A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ Ap ) “ # pÿ i“1 ⃗ Ai ‹ bi : bi P Kn, i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , p + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The tensor singular value decomposition (t-SVD) associated with the t-product, introduced by Kilmer and Martin [23], generalizes the classical SVD of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' It is described in the next theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ([23]) Let A P Kℓˆp n be a third-order tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Then it can be represented as the t-product of three third-order tensors, A “ U ‹ S ‹ V H, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3) where U P Kℓˆℓ n and V P Kpˆp n are orthogonal tensors, and S P Kℓˆp n is an f-diagonal tensor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', each frontal slice of the DFT of S is a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Algorithm 2 summarizes the computation of the t-SVD of a third-order tensor with the aid of the FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 5 Algorithm 2 The t-SVD of a third-order tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Input: A P Kℓˆp n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Output: U P Kℓˆℓ n , S P Kℓˆp n , V P Kpˆp n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 1: � A “ fftpA , r s, 3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 2: for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , „n ` 1 2 \uf6be do 3: r � U piq, � S piq, � V piqs “ svdp � A piqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 4: end for 5: for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , „n ` 1 2 \uf6be ` 1 do 6: � U piq “ conj ´ � U pn´i`2q¯ , � S piq “ conj ´ � S pn´i`2q¯ , and � V piq “ conj ´ � V pn´i`2q¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 7: end for 8: Compute U “ ifftp � U , r s, 3q, S “ ifftp � S , r s, 3q, and V “ ifftp � V , r s, 3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The factorization (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3) can be expressed as A “ U ‹ S ‹ V H “ mintℓ,pu ÿ i“1 ⃗ Ui ‹ si ‹ ⃗ V H i , where the si “ S pi, i, :q are singular tubes, and ⃗ Ui “ U p:, i, :q and ⃗ Vi “ U p:, i, :q are right and left lateral tensor singular slices, respectively, for i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', minpℓ, pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The triplets tsi, ⃗ Ui, ⃗ Viui“1:minpℓ,pq will be referred to as singular triplets of the tensor A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The singular tubes are ordered so that their norms σi “ }si}F are decreasing with i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', σ1 ě σ2 ě .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ě σminpℓ,pq ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Note that we also have the relations A ‹ ⃗ Vi “ ⃗ Ui ‹ si, A H ‹ ⃗ Ui “ ⃗ Vi ‹ si, i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , mintℓ, pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We remark that the latter relations have to be modified if A has complex-valued entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We note for future reference that S pi, i, 1q “ nÿ j“1 1 n � S pi, i, jq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4) In the following, we will need the notion of rank of a third-order tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Let A P Kℓˆp n be a third-order tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Then its tubal rank is defined as rankt pA q “ cardtσi ‰ 0, i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', mintℓ, puu, where σi is the norm of the singular tube si of A and card stands for the cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The next result generalizes the Eckart-Young theorem for matrices to third-order tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' It is important in the context of data compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ([3, 23]) Let the t-SVD of a third-order tensor A P Kℓˆp n be given by A “ U ‹ S ‹ V H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' For 1 ď k ď mintℓ, pu, define the truncated t-SVD by Ak “ kÿ i“1 ⃗ Ui ‹ si ‹ ⃗ V H i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 6 Then Ak “ arg min � A PM ›››A ´ � A ››› F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Where M is the set given by M “ tX ‹ Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' with X P Klˆk n , Y P Kkˆp n u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The matrix QR factorization also can be generalized to tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ([23]) Let A P Kℓˆp n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Then A can be factored as A “ Q ‹ R, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5) where Q P Kℓˆℓ n is an orthogonal tensor and R P Kℓˆp n is an f-upper triangular tensor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', each frontal slice of the DFT of R is an upper triangular matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The factorization (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5) is referred to as the t-QR factorization of A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Algorithm 3 summarizes the computation of the t-QR factorization (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The function qr in line 3 of the algorithm computes a QR factorization of the matrix � A piq P Rℓˆp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' thus � A piq “ � Qpiq � Rpiq, where the matrix � Qpiq P Rℓˆℓ is orthogonal and the matrix � Rpiq P Rℓˆp has an upper triangular leading principal submatrix of order ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Algorithm 3 t-QR factorization of a third-order tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Input: A P Kℓˆp n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Output: Q P Kℓˆℓ n , R P Kℓˆp n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 1: � A “ fftpA , r s, 3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 2: for i “ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , „n ` 1 2 \uf6be do 3: r � Qpiq, � Rpiqs “ qrp � A piqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 4: end for 5: for i “ „n ` 1 2 \uf6be ` 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , n do 6: � Qpiq “ conj ´ � Qpn´i`2q¯ and � Rpiq “ conj ´ � Rpn´i`2q¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 7: end for 8: Compute Q “ ifftp � Q, r s, 3q and R “ ifftp � R, r s, 3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Following Kilmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' [22], we define orthogonality of lateral tensor slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Let ⃗ X and ⃗ Y be two lateral tensor slices in Kℓ n and define the inner product of these slices as � ⃗ X , ⃗ Y � :“ ⃗ X H ‹ ⃗ Y P Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The lateral slices in the set !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ⃗ X1, ⃗ X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ Xp ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='6) with p ě 2, are said to be orthogonal if � ⃗ Xi, ⃗ Xj � “ " αie1 if i “ j, 0 if i ‰ j, 7 where e1 is the tube in Kn, whose its first element is 1 and the remaining elements vanish, and the αi, i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', p, are nonvanishing scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Furthermore, if αi “ 1 for all i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', p, then the set (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='6) is said to be orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Following [22], we observe that any lateral slice ⃗ X P Kℓ n can be normalized as ⃗ X “ ⃗ Y ‹ a (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='7) with ⃗ Y P Kℓ n, ››› ⃗ Y ››› “ 1, and a P Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Here the tensor norm is defined as ››› ⃗ Y ››› “ ››› � ⃗ Y , ⃗ Y �››› F ››› ⃗ Y ››› F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Note that ⃗ Y has unit norm if and only if � ⃗ Y , ⃗ Y � “ e1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' see [22] for more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Algorithm 4 summarizes the normalization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The MATLAB function randn in the algorithm gen- erates a vector in Rℓ with normally distributed pseudorandom entries with mean zero and variance one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Algorithm 4 Normalize( ⃗ X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Input: ⃗ X P Kℓ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Output: ⃗ Y P Kℓ n of unit norm and a P Kn that satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 1: ⃗� Y “ fftp ⃗ X , r s, 3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 2: for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , „n ` 1 2 \uf6be do 3: �apiq “ ››››› ⃗� Y piq››››› F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 4: if �apiq ą 0 then 5: ⃗� Y piq “ ⃗� Y piq �apiq 6: else 7: ⃗� Y piq “ randnpℓ, 1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' bpiq “ ››››› ⃗� Y piq››››› F , and ⃗� Y piq “ ⃗� Y piq bpiq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 8: end if 9: end for 10: for i “ „n ` 1 2 \uf6be ` 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , n do 11: ⃗� Y piq “ conj ˜ ⃗� Y pn´i`2q¸ , �apiq “ conj ´ �apn´i`2q¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 12: end for 13: ⃗� Y “ ifftp ⃗� Y , r s, 3q, a “ ifftp�a, r s, 3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Tensor Lanczos bidiagonalization for computing the largest and smallest sin- gular triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This section describes the Lanczos bidiagonalization process for tensors using 8 the t-product, and discusses how approximations of the largest and smallest singular triplets of a large third-order tensor A P Kℓˆp n can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The tensor Lanczos bidiagonalization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The Lanczos bidiagonaliza- tion process was introduced for matrices by Golub and Kahan [14] and therefore sometimes is referred to as the Golub-Kahan bidiagonalization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' For a matrix A P Rℓˆp, this process is closely related to symmetric Lanczos process applied to the real symmetric matrices AAT and AT A, or alternatively to the symmetric matrix „ 0 A AT 0 \uf6be .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Lanczos bidiagonalization algorithms have been applied to solve numerous problems such as large-scale least squares problem [28], the approximation of the largest or smallest singular triplets of a large matrix [5, 19, 24], and in Tikhonov regularization of large linear discrete ill-posed problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We note that the bidiagonalization method described in [28] and applied in [11, 12] reduces a large matrix A to a small lower bidiagonal matrix, while in [5] the matrix A is reduced to a small upper bidiagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We will review the latter approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Application of m !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' mintℓ, pu steps of the Lanczos bidiagonalization process to the matrix A P Rℓˆp with the initial unit vector p1 P Rℓ generically produces two matrices Pm “ rp1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , pms P Rpˆm, Qm “ rq1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , qms P Rℓˆm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The columns of Pm and Qm form orthonormal bases for the Krylov subspaces Km ` AT A, p1 ˘ “ spantp1, AT Ap1, ` AT A ˘2 p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ` AT A ˘m´1 p1u, Km ` AAT , q1 ˘ “ spantq1, AAT q1, ` AAT ˘2 q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ` AAT ˘m´1 q1u, respectively, where q1 “ Ap1{}Ap1}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A matrix interpretation of the recursion relations of the Lanczos process gives the matrix relations APm “ QmBm, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1) AT Qm “ PmBT m ` βmpm`1eT m, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2) where em “ r0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , 0, 1sT P Rm, βm ě 0 is a scalar, and pm`1 P Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The matrix Bm P Rmˆm is upper bidiagonal and satisfies Bm “ QT mAPm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' When considering bidiagonalization of a third-order tensor A using the t-product, the scalars and the columns of the matrices Pm and Qm in the matrix decompositions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2) be- come tubes and lateral slices, respectively, in the decompositions determined by the tensor Lanczos bidiagonalization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The application of m steps of tensor Lanczos bidiagonaliza- tion to the third-order tensor A P Kℓˆp n generically computes two tensors Pm “ ” ⃗ P1, ⃗ P2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ Pm ı P Kpˆm n and Qm “ ” ⃗ Q1, ⃗ Q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ Qm ı P Kℓˆm n , whose lateral slices form bases for the tensor Krylov subspaces Km ´ A H ‹ A , ⃗ P1 ¯ and Km ´ A ‹ A H, ⃗ Q1 ¯ , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' They are defined by Km ´ A H ‹ A , ⃗ P1 ¯ “ spant ⃗ P1, ` A H ‹ A ˘ ‹ ⃗ P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ` A H ‹ A ˘m´1 ‹ ⃗ P1u, Km ´ A ‹ A H, ⃗ Q1 ¯ “ spant ⃗ Q1, ` A ‹ A H˘ ‹ ⃗ Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ` A ‹ A H˘m´1 ‹ ⃗ Q1u, 9 where ⃗ P1 P Kp n is a lateral slice of unit norm, and the lateral slice ⃗ Q1 P Kℓ n is of unit norm and proportional to A ‹ ⃗ P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Algorithm 5 describes the tensor Lanczos bidiagonalization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Algorithm 5 Tensor Lanczos bidiagonalization using the t-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Input: A P Kℓˆp n , number of steps m ď mintℓ, pu, ⃗ P1 P Kp n with unit norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Output: Pm “ r ⃗ P1, ⃗ P2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ Pms P Kpˆm n and Qm “ r ⃗ Q1, ⃗ Q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ Qms P Kℓˆm n with orthonormal lateral slices, Bm P Kmˆm n a bidiagonal tensor, and ⃗ Rm P Kℓ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 1: P1 “ ” ⃗ P1 ı .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 2: ⃗ Q1 “ A ‹ ⃗ P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 3: r ⃗ Q1, α1s “ Normalizep ⃗ Q1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 4: Q1 “ ” ⃗ Q1 ı , Bmp1, 1, :q “ α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 5: for i “ 1 to m do 6: ⃗ Ri “ A H ‹ ⃗ Qi ´ αi ‹ ⃗ Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 7: Reorthogonalization ⃗ Ri “ ⃗ Ri ´ Pi ‹ pPH i ‹ ⃗ Riq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 8: if i ă m then 9: r ⃗ Pi`1, βis “ Normalizep ⃗ Riq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 10: Pi`1 “ ” Pi, ⃗ Pi`1 ı , Bmpi, i ` 1, :q “ βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 11: ⃗ Qi`1 “ A ‹ ⃗ Pi`1 ´ βi ‹ ⃗ Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 12: Reorthogonalization ⃗ Qi`1 “ ⃗ Qi`1 ´ Qi ‹ pQH i ‹ ⃗ Qi`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 13: r ⃗ Qi`1, αi`1s “ Normalizep ⃗ Qi`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 14: Qi`1 “ ” Qi, ⃗ Qi`1 ı , Bmpi ` 1, i ` 1, :q “ αi`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 15: end if 16: end for We remark that Algorithm 5 differs from the tensor bidiagonalization algorithms described in [22, 32] in that the former produces an upper bidiagonal tensor Bm, while the latter determine a lower bidiagonal tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The use of an upper bidiagonal tensor in the present paper is inspired by the choices in [5, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Algorithm 5 is said to break down when one of the tensor slices ⃗ Ri or ⃗ Qi`1 vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We comment below on this situation, but note that breakdown is exceedingly rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Generically, Algorithm 5 determines the decompositions A ‹ Pm “ Qm ‹ Bm, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3) A H ‹ Qm “ Pm ‹ BH m ` ⃗ Rm ‹ ⃗E H m , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4) with Pm P Kpˆm n , Qm P Kℓˆm n , where PH m ‹ Pm “ Im and QH m ‹ Qm “ Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The tensor ⃗Em P Km n is the canonical lateral slice whose elements are zero except for the first element of the mth tube, which equals 1, and ⃗ Rm P Kp n is determined by steps 4 and 5 of Algorithm 5 such that PH m ‹ ⃗ Rm “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The tensor Bm P Kmˆm n is upper bidiagonal, each of whose frontal slices 10 is an upper bidiagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Thus, Bm “ » ——————– α1 β1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 0 α2 β2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' αm´1 βm´1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 αm fi ffiffiffiffiffiffifl , where αi and βi are tubes in Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4) follow immediately from the recursion relations of Algo- rithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The orthonormality of the lateral slices of Pm and Qm can be shown by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The proof is closely related to the proof of the existence of the relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2), and the properties of the matrices involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The latter relations are used in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The Lanczos bidiagonalization process may suffer from loss of orthogonality of the lateral slices of the tensors Pm and Qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Therefore, reorthogonalization is carried out in Lines 5 and 9 in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We remark that reorthogonalization makes the algorithm more costly both in terms of storage and arithmetic floating point operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The extra cost may be acceptable as long as the number of steps m is fairly small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' see [5, 34] for discussions in the matrix case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Let ⃗ Rm be the tensor whose lateral slices are defined in Line 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Then r ⃗ Pm`1, βms “ Normalize ´ ⃗ Rm ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5) In the rare event that some βj, 1 ď j ă m, vanishes, Algorithm 5 breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Then the singular tubes of Bj are singular tubes of A , and the left and right lateral tensor singular slices are obtained as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' When no breakdown takes place, we can express equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4) as A H ‹ Qm “ Pm`1 ‹ BH m,m`1, where Pm`1 is obtained from Pm by appending the lateral slice ⃗ Pm`1, defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5), to get Pm`1 “ ” Pm, ⃗ Pm`1 ı P Kpˆpm`1q n , and Bm,m`1 P Kmˆpm`1q n is obtained by appending the lateral slice βm ‹ ⃗Em to Bm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', Bm,m`1 “ ” Bm, βm ‹ ⃗Em ı .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We turn to the connection between the partial Lanczos tridiagonalization of a third-order tensor and the partial Lanczos tridiagonalization process of the tensor A H ‹A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This connection will be used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Multiplying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3) from the left by A H, we get A H ‹ A ‹ Pm “ A H ‹ Qm ‹ Bm “ Pm ‹ BH m ‹ Bm ` ⃗ Rm ‹ ⃗E H m ‹ Bm “ Pm ‹ BH m ‹ Bm ` ⃗ Rm ‹ ⃗E H m ‹ αm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='6) Let Tm be the symmetric tridiagonal tensor defined by Tm “ BH m ‹ Bm P Kmˆm n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='6) is a partial tensor Lanczos bidiagonalization of A H ‹ A with initial lateral slice ⃗ P1 “ Pm ‹ ⃗E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The lateral slices of Pm form an orthonormal basis for the tensor Krylov subspace Km ´ A H ‹ A , ⃗ P1 ¯ “ spant ⃗ P1, A H ‹ A ‹ ⃗ P1, ` A H ‹ A ˘2 ‹ ⃗ P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ` A H ‹ A ˘m´1 ‹ ⃗ P1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 11 Similarly, multiplying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4) from the left by A , we obtain A ‹ A H ‹ Qm “ Qm ‹ Bm ‹ BH m ` A ‹ ⃗ Rm ‹ ⃗E H m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' It follows that the lateral slices of Qm form an orthonormal basis for the Krylov subspace Km ´ A ‹ A H, ⃗ Q1 ¯ “ spant ⃗ Q1, A ‹ A H ‹ ⃗ Q1, ` A ‹ A H˘2 ‹ ⃗ Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ` A ‹ A H˘m´1 ‹ ⃗ Q1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Approximating singular tubes and singular lateral slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We describe an approach to approximate the largest or smallest singular triplets (singular tubes and associated left and right lateral singular slices) of a large tensor A P Kℓˆp n using restarted partial tensor Lanczos bidiagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Since the tensor A is large, computing its k largest or smallest singular triplets by determining the t-SVD of A is very expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The idea is to approximate the extreme singular triplets of the tensor A by determining the extreme singular triplets the bidiagonal tensor Bm, where m is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Let tsi, ⃗ Ui, ⃗ Viu, 1 ď i ď m, denote the singular triplets of Bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' They satisfy Bm ‹ ⃗ Vi “ si ‹ ⃗ Ui and BH m ‹ ⃗ Ui “ si ‹ ⃗ Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The k ď m largest singular triplets of A are approximated by the triplets tsA i,m, ⃗ U A i,m, ⃗ V A i,mu defined by sA i,m “ si, ⃗ U A i,m “ Qm ‹ ⃗ Ui, ⃗ V A i,m “ Pm ‹ ⃗ Vi, i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='7) For i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , k, we have A ‹ ⃗ V A i,m “ A ‹ Pm ‹ ⃗ Vi “ Qm ‹ Bm ‹ ⃗ Vi “ Qm ‹ si ‹ ⃗ Ui “ Qm ‹ ⃗ Ui ‹ si “ ⃗ U A i,m ‹ sA i,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Similarly, A H ‹ ⃗ U A i,m “ A H ‹ Qm ‹ ⃗ Ui “ ´ Pm ‹ Bm ` ⃗ Rm ‹ ⃗E H m ¯ ‹ ⃗ Ui “ ⃗ V A i,m ‹ sA i,m ` ⃗ Rm ‹ ⃗E H m ‹ ⃗ Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='8) To accept tsA i,m, ⃗ U A i,m, ⃗ V A i,mu as an approximate singular triplet of A , the remainder term ⃗ Rm ‹ ⃗E H m ‹ ⃗ Ui should be small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We can bound the remainder term according to ››› ⃗ Rm ‹ ⃗E H m ‹ ⃗ Ui ››› F “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='n ››››bdiag ˆ � ⃗ Rm ˙ bdiag ˆ � ´ ⃗E H m ¯˙ bdiag ˆ � ⃗Ui ˙›››› F ď 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='n ››››bdiag ˆ � ⃗ Rm ˙›››› F ››››bdiag ˆ � ´ ⃗E H m ¯˙ bdiag ˆ � ⃗Ui ˙›››› F “ ›››bdiag ´ ⃗ Rm ¯››› F ››››bdiag ˆ � ´ ⃗E H m ¯˙ bdiag ˆ � ⃗Ui ˙›››› F “ }βm}F n ÿ s“1 ˇˇˇˇˇ � ´ ⃗E H m ¯psq� ⃗Ui psqˇˇˇˇˇ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 12 Analogously as in [5], we require for 1 ď s ď n that ˇˇˇˇˇ � ´ ⃗E H m ¯psq� ⃗Ui psqˇˇˇˇˇ ď δ1 ››› � A psq››› “ δ1 ´ s � A psq 1,m ¯ “ δ ´ s � A 1,m ¯psq , for a user-chosen parameter δ1 ą 0, where ´ s � A j,m ¯psq denotes the sth element of the jth approx- imate singular tube of � A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We obtain from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4) that ››› ⃗ Rm ‹ ⃗E H m ‹ ⃗ Ui ››› F ď δ1 }βm}F n ÿ s“1 ´ s � A 1 ¯psq “ nδ1 }βm}F ` sA 1 ˘p1q “ nδ2 ` sA 1 ˘p1q , where δ2 “ δ1 }βm}F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The computed approximate singular triplets tsA i,m, ⃗ U A i,m, ⃗ V A i,mu, i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', k, of A are accepted as singular triplets of A if ››› ⃗ Rm ‹ ⃗E H m ‹ ⃗ Ui ››› F ď δ ` sA 1,m ˘p1q , i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='k, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='9) for some user-specified parameter δ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' To keep the storage requirement fairly small for large-scale problems, we would like the num- ber of steps m of the tensor Lanczos bidiagonalization process to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' However, when m is small, it may not be possible to approximate the desired singular triplets sufficiently accurately using the available Krylov subspaces Km ´ A H ‹ A , ⃗ Q1 ¯ and Km ´ A ‹ A H, ⃗ P1 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A remedy for this situation is to restart the tensor Lanczos bidiagonalization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The idea is to repeatedly update the initial lateral slices used for the tensor Lanczos bidiagonalization pro- cess, and in this way determine a sequence of increasingly more appropriate Krylov subspaces, until the k desired singular triplets have been found with required accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We remark that restarting techniques have been used for computing a few desired singular triplets or eigenvalue- eigenvector pairs of a large matrix, where properties of Ritz vectors, harmonic Ritz vectors, and refined Ritz vectors have been exploited;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', [5, 19, 20, 35, 36] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Augmentation by Ritz lateral slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Assume that we would like to approximate the k largest singular triplets of A P Rℓˆpˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' To this end, we carry out m ą k steps of tensor Lanczos bidiagonalization as described in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The approximate right singular lateral slice ⃗ V A i,m is a Ritz lateral slice of A H ‹ A associated with the Ritz tube ` sA i,m ˘2 “ sA i,m ‹ sA i,m for i P t1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', mu, and we have A H ‹ A ‹ ⃗ V A i,m “ A H ‹ ⃗ U A i,m ‹ sA i,m “ ´ ⃗ V A i,m ‹ sA i,m ` ⃗ Rm ‹ ⃗E H m ‹ ⃗ Ui ¯ ‹ sA i,m “ ⃗ V A i,m ‹ ` sA i,m ˘2 ` ⃗ Rm ‹ ⃗E H m ‹ ⃗ Ui ‹ sA i,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' In what follows we will show some results that will help us to approximate the largest or smallest singular triplets of a third-order tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The idea behind these results is to find equations that are analogous to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4), and such that the reduced tensor will contain the k approximate singular tubes among its first k elements on the diagonal, and the right projection tensor will contain the k right Ritz lateral slices among its first k lateral slices, and 13 the left projection tensor will contain the k left Ritz lateral slices among its first k lateral slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The following theorem will be helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Assume that m steps of Algorithm 5 have been applied to the third-order tensor A P Kℓˆp n , and suppose that βm in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4) is nonvanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Then for k ă m, we have A ‹ � Pk`1 “ � Qk`1 ‹ � Bk`1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='10) A H ‹ � Qk`1 “ � Pk`1 ‹ � BH k`1 ` �βk`1 ‹ ⃗� Pk`2 ‹ ⃗E H k`1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='11) where � Pk`1 P Kpˆpk`1q n and � Qk`1 P Kℓˆpk`1q n have orthonormal lateral slices, and the first k lateral slices of � Pm are the first k Ritz lateral slices of A , � Bk`1 P Kpk`1qˆpk`1q n is an upper triangular tensor, ⃗� Pk`2 P Kp n is a lateral slice that is orthogonal to � Pk`1, �βk`1 P Kn, and ⃗Ek`1 P Kk`1 n is the canonical element under the t-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Let the Ritz lateral slices ⃗ V A i,m for 1 ď i ď k be associated with the k Ritz tubes of A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Introduce the tensor � Pk`1 “ ” ⃗ V A 1,m, ⃗ V A 2,m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ V A k,m, ⃗ Pm`1 ı P Kpˆpk`1q n , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='12) where ⃗ Pm`1 is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Then, using the fact that A ‹ ⃗ V A i,m “ ⃗ U A i,m‹sA i,m for i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', k, we obtain A ‹ � Pk`1 “ ” A ‹ ⃗ V A 1,m, A ‹ ⃗ V A 2,m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , A ‹ ⃗ V A k,m, A ‹ ⃗ Pm`1 ı “ ” ⃗ U A 1,m ‹ sA 1,m, ⃗ U A 2,m ‹ sA 2,m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ U A k,m ‹ sA k,m, A ‹ ⃗ Pm`1 ı .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='13) Orthogonalizing the term A ‹ ⃗ Pm`1 against t ⃗ U A i,mui“1:k gives A ‹ ⃗ Pm`1 “ kÿ i“1 ρi ‹ ⃗ U A i,m ` ⃗� Rk, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='14) where ⃗� Rk is orthogonal to t ⃗ U A i,mui“1:k, and the ρi for i P t1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ku are given by ρi “ ´ ⃗ U A i,m ¯H ‹ ´ A ‹ ⃗ Pm`1 ¯ “ ´ A H ‹ ⃗ U A i,m ¯H ‹ ⃗ Pm`1 “ ´ ⃗ V A i,m ‹ sA i,m ` ⃗ Rm ‹ ⃗E H m ‹ ⃗ Ui ¯H ‹ ⃗ Pm`1 “ βH m ‹ ´ ⃗ U H i ‹ ⃗Em ‹ ⃗ PH m`1 ¯ ‹ ⃗ Pm`1 “ βm ‹ ⃗ U H i ‹ ⃗Em “ βm ‹ � ⃗ Ui, ⃗Em � , because βm “ βH m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Let ⃗� Rk “ ⃗� R1k ‹ �αk`1 be a normalization of ⃗� Rk, and introduce the tensors � Qk`1 “ „ ⃗ U A 1,m, ⃗ U A 2,m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ U A k,m, ⃗� R1k \uf6be P Kℓˆpk`1q n (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='15) 14 and � Bk`1 “ » —————– sA 1,m 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 ρ1 0 sA 2,m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 ρ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 sA k,m ρk 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 �αk`1 fi ffiffiffiffiffifl P Kpk`1qˆpk`1q n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='16) Then, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='14), we obtain A ‹ � Pk`1 “ « ⃗ U A 1,m ‹ sA 1,m, ⃗ U A 2,m ‹ sA 2,m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ U A k,m ‹ sA k,m, kÿ i“1 ρi ‹ ⃗ U A i,m ` ⃗� Rk ff “ � Qk`1 ‹ � Bk`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='17) On the other hand, as A H ‹ � Qk`1 “ „ A H ‹ ⃗ U A 1,m, A H ‹ ⃗ U A 2,m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , A H ‹ ⃗ U A k,m, A H ‹ ÝÑ � R1k \uf6be , using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='8), we get A H ‹ ⃗ U A i,m “ ⃗ V A i,m ‹ sA i,m ` ⃗ Rm ‹ ⃗E H m ‹ ⃗ Ui “ ⃗ V A i,m ‹ sA i,m ` ⃗ Pm`1 ‹ βm ‹ ⃗E H m ‹ ⃗ Ui “ ⃗ V A i,m ‹ sA i,m ` ⃗ Pm`1 ‹ ρH i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Since � A H ‹ ⃗� R 1 k, ⃗ V A i,m � “ ˆ ⃗� R 1 k ˙H ‹ A ‹ ⃗ V A i,m “ sA i,m ‹ ˆ ⃗� R 1 k ˙H ‹ ⃗ U A i,m “ 0, the tensor A H ‹ ⃗� R1k is orthogonal to ⃗ V A i,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Moreover, in view of that ⃗ V A i,m is orthogonal to ⃗ Pm`1, we obtain A H ‹ ÝÑ � R1k “ γ ‹ ⃗ Pm`1 ` ⃗ Fk`1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='18) where ⃗ Fk`1 is orthogonal to ⃗ Pm`1 as well as to ⃗ V A i,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Due to the orthogonality of ⃗� Rk (or ÝÑ � R1k) to !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ⃗ U A i,m ) i“1:k, the parameter γ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='18) is given by γ “ � ⃗ Pm`1, A H ‹ ÝÑ � R1k � “ � A ‹ ⃗ Pm`1, ÝÑ � R1k � “ � kÿ i“1 ρi ‹ ⃗ U A i,m ` ⃗� Rk, ÝÑ � R1k � “ � ⃗� Rk, ÝÑ � R1k � “ �αk`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 15 Consequently, A H ‹ � Qk`1 “ ” ⃗ V A 1,m ‹ sA 1,m ` ⃗ Pm`1 ‹ ρH 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ V A k,m ‹ sA k,m ` ⃗ Pm`1 ‹ ρH k , �αk`1 ‹ ⃗ Pm`1 ` ⃗ Fk`1 ı “ � Pk`1 ‹ � BH k`1 ` ⃗ Fk`1 ‹ ⃗E H k`1 “ � Pk`1 ‹ � BH k`1 ` �βk`1 ‹ ⃗� Pk`2 ‹ ⃗E H k`1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='19) where �βk`1 and ⃗� Pk`2 are determined by the normalization of ⃗ Fk`1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', ⃗ Fk`1 “ �βk`1‹ ⃗� Pk`2, because � BH k`1 “ » —————– sA 1,m 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 0 0 sA 2,m 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 sA k,m 0 ρH 1 ρH 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ρH k �αk`1 fi ffiffiffiffiffifl P Kpk`1qˆpk`1q n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The orthogonality of � Pk`1 and � Qk`1 now follows from the orthogonality of the sequences !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ⃗ V A i,m ) i“1:k and !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ⃗ U A i,m ) i“1:k, respectively, given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' In the preceding theorem we assumed βm to be nonvanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' If, instead, βm vanishes, then the singular tubes of Bm are singular tubes of A , and the left and right singular lateral slices of A can be determined from those of Bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Similarly, if �βk`1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='18) vanishes, then the singular tubes of � Bk`1 are singular tubes of A , and the singular lateral slices of A can be determined from � Pk`1 and � Qk`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' If the �βk`1 is nonvanishing, then we append new lateral slices to � Pk`1 and � Qk`1 repeatedly until iteration m ´ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This is the subject of the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Assume that m steps of Algorithm 5 have been applied to A and that eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='17) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='19) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' If the �βk`1 are nonvanishing for 1 ď k ă m, then we have the following relations A ‹ � Pm “ � Qm ‹ � Bm, A H ‹ � Qm “ � Pm ‹ � BH m ` �βm ‹ ⃗� Pm`1 ‹ ⃗E H m , where � Pm P Kpˆm n and � Qm P Kℓˆm n have orthonormal lateral slices, � Bm P Kmˆm n is an upper triangular, �βm P Kn, ⃗� Pm`1 P Kp n is orthogonal to � Pm, and ⃗Em P Km n is the canonical element under the t-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The first k lateral slices of � Pm and � Qm are the same as those of the tensors � Pk`1 and � Qk`1, respectively, given in Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Let the tensors � Pk`1 and � Qk`1 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='17) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='19), respectively, be repre- sented by � Pk`1 “ „ ⃗� P1, ⃗� P2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗� Pk`1 \uf6be P Kpˆpk`1q n and � Qk`1 “ „ ⃗� Q1, ⃗� Q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗� Qk`1 \uf6be P Kℓˆpk`1q n , 16 and the tensor � Pk`2 be given by � Pk`2 “ „ � Pk`1, ⃗� Pk`2 \uf6be P Kpˆpk`2q n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' By normalizing the quantity ´ Iℓ ´ � Qk`1 ‹ � QH k`1 ¯ ‹A ‹ ⃗� Pk`2, we obtain the lateral slice ⃗� Qk`2 such that �αk`2 ‹ ⃗� Qk`2 “ ´ Iℓ ´ � Qk`1 ‹ � QH k`1 ¯ ‹ A ‹ ⃗� Pk`2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Application of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='11) gives �αk`2 ‹ ⃗� Qk`2 “ ´ Iℓ ´ � Qk`1 ‹ � QH k`1 ¯ ‹ A ‹ ⃗� Pk`2 “ A ‹ ⃗� Pk`2 ´ � Qk`1 ‹ � QH k`1 ‹ A ‹ ⃗� Pk`2 “ A ‹ ⃗� Pk`2 ´ � Qk`1 ‹ ˆ � Bk`1 ‹ � PH k`1 ` �βk`1 ‹ ⃗Ek`1 ‹ ⃗� P H k`2 ˙ ‹ ⃗� Pk`2 “ A ‹ ⃗� Pk`2 ´ �βk`1 ‹ ⃗� Qk`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='20) Consider the tensors � Qk`2 “ „ � Qk`1, ⃗� Qk`2 \uf6be P Kℓˆpk`2q n and � Bk`2 “ » ——————————– sA 1,m 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 ρ1 0 0 sA 2,m 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ρ2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 sA k,m ρk 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 �αk`1 �βk`1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 �αk`2 fi ffiffiffiffiffiffiffiffiffiffifl P Kpk`2qˆpk`2q n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='20), we get A ‹ � Pk`2 “ � Qk`2 ‹ � Bk`2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' To determine the lateral slice ⃗� Pk`3, we normalize ´ I ´ � Pk`2 ‹ � PH k`2 ¯ ‹A H ‹ ⃗� Qk`2 so that �βk`2 ‹ ⃗� Pk`3 “ ´ I ´ � Pk`2 ‹ � PH k`2 ¯ ‹ A H ‹ ⃗� Qk`2 and �βk`2 ‹ ⃗� Pk`3 “ A H ‹ ⃗� Qk`2 ´ �αk`2 ‹ ⃗� Pk`2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='21) It now follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='21) that A H ‹ � Qk`2 “ � Pk`2 ‹ � BH k`2 ` �βk`2 ‹ ⃗� Pk`3 ‹ ⃗E H k`2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 17 We can continue this procedure until iteration m ´ k and then obtain A ‹ � Pm “ � Qm ‹ � Bm, A H ‹ � Qm “ � Pm ‹ � BH m ` �βm ‹ ⃗� Pm`1 ‹ ⃗E H m , where � Pm and � Qm have orthonormal lateral slices and � Bm “ » ———————————– sA 1,m 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ρ1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' sA k,m ρk �αk`1 �βk`1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' �αm´1 �βm´1 �αm fi ffiffiffiffiffiffiffiffiffiffiffifl P Kmˆm n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This gives the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' If we would like to compute the smallest singular triplets of A , then we can use the same theorem, but instead of working with the first right singular lateral slices ⃗ V A i,m for 1 ď i ď k, we use the last k right singular lateral slices in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The computations are analogous to those described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Augmentation by harmonic Ritz lateral slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' When the smallest singular val- ues of a matrix A are clustered, their computation by the restarted Lanczos bidiagonalization method as described above may require many iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' In this situation it may be beneficial to instead compute approximations of the smallest singular values of A by seeking to determine approximations of the largest singular values of the matrix ` AT A ˘´1 without explicitly com- puting the matrix ` AT A ˘´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This was done for the matrix case by using computing harmonic Ritz vectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' see [5, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Harmonic Ritz vectors furnish approximations of eigenvectors of AT A associated with the corresponding harmonic Ritz values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' In the case of tensors, harmonic Ritz lateral slices furnish approximations of eigenvectors of A H ‹ A associated with harmonic Ritz tubes of A H ‹ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The harmonic Ritz tubes qθj of A H ‹A associated with the partial tensor tridiagonalization defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='6) are the eigentubes of the generalized eigenvalue problem ´` BH m ‹ Bm ˘2 ` α2 m ‹ β2 m ‹ ⃗Em ‹ ⃗E H m ¯ ‹ ⃗qωj “ qθj ‹ BH m ‹ Bm ‹ ⃗qωj, 1 ď j ď m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='22) The eigenpair tqθj, ⃗qωju can be computed without forming the tensor BH m ‹ Bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Let ⃗ωj “ Bm ‹ ⃗qωj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='23) Using the relations αm ‹ ⃗E H m “ ⃗E H m ‹ Bm and αm ‹ ⃗Em “ BH m ‹ ⃗Em, we can write α2 m ‹ β2 m ‹ ⃗Em ‹ ⃗E H m “ β2 m ‹ BH m ‹ ⃗Em ‹ ⃗E H m ‹ Bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 18 Therefore, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='23), the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='22) can be written as BH m ‹ ´ Bm ‹ BH m ‹ Bm ` β2 m ‹ ⃗Em ‹ ⃗E H m ‹ Bm ¯ ‹ B´1 m ‹ ⃗ωj “ qθj ‹ BH m ‹ Bm ‹ B´1 m ‹ ⃗ωj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' It follows that ´ Bm ‹ BH m ` β2 m ‹ ⃗Em ‹ ⃗E H m ¯ ‹ ⃗ωj “ qθj ‹ ⃗ωj (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='24) and ´ Bm ‹ BH m ` β2 m ‹ ⃗Em ‹ ⃗E H m ¯ “ Bm,m`1 ‹ BH m,m`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' In this subsection, we denote the singular triplets of Bm,m`1 by ts1 i, ⃗ U 1 i , ⃗ V 1 i u for 1 ď i ď m, with the first k of them being the smallest singular triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Recall that we are interested in determining approximations of the smallest singular triplets of A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The k smallest singular triplets of Bm,m`1 form the tensors U 1 k “ ” ⃗ U 1 1, ⃗ U 1 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ U 1 k ı P Kmˆk n , V 1 k “ ” ⃗ V 1 1, ⃗ V 1 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ V 1 k ı P Kpm`1qˆk n , S 1 k “ ” s1 1 ‹ ⃗E1, s1 2 ‹ ⃗E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , s1 k ‹ ⃗Ek ı P Kkˆk n , where Bm,m`1 ‹ V 1 k “ U 1 k ‹ S 1 k and BH m,m`1 ‹ U 1 k “ V 1 k ‹ S 1 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We obtain from the above equations that Bm,m`1 ‹ BH m,m`1 ‹ U 1 k “ U 1 k ‹ ` S 1 k ˘2 , where ` S 1 k ˘2 “ ”` s1 1 ˘2 ‹ ⃗E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ` s1 k ˘2 ‹ ⃗Ek ı .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Consequently, the eigenpair !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ps1 iq2 , U 1 i ) satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='24), and !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ps1 iq2 , B´1 m ‹ U 1 i ) is an eigenpair of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' It follows that the harmonic Ritz lateral slice associated with qθj is given by ⃗| Vj “ Pm ‹ ⃗qωj “ Pm ‹ B´1 m ‹ ⃗ U 1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='25) We turn to the computation of the residual of harmonic Ritz lateral slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Using eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='24), we obtain the relations A H ‹ A ‹ ⃗| Vj ´ qθj ‹ ⃗| Vj “ A H ‹ A ‹ Pm ‹ ⃗qωj ´ qθj ‹ Pm ‹ ⃗qωj “ ´ Pm ‹ BH m ‹ Bm ` βm ‹ ⃗E H m ˚ Bm ¯ ‹ ⃗qωj ´ qθj ‹ Pm ‹ ⃗qωj “ Pm ‹ B´1 m ‹ ` Bm ‹ BH m ´ θj ˚ Im ˘ ‹ ⃗ωj ` βm ‹ ⃗ Pm`1 ‹ ⃗E H m ‹ ⃗ωj “ ´β2 m ‹ Pm ‹ B´1 m ˚ ⃗Em ‹ ⃗E H m ‹ ⃗ωj ` βm ˚ ⃗ Pm`1 ‹ ⃗E H m ‹ ⃗ωj “ ⃗E H m ‹ ⃗ωj ‹ βm ´ ⃗ Pm`1 ´ βm ‹ Pm ‹ B´1 m ‹ ⃗Em ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 19 It follows that the residual can be expressed as ⃗| Rm “ ⃗ Pm`1 ´ βm ‹ Pm ‹ B´1 m ‹ ⃗Em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='26) We now proceed analogously as in the previous subsection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', we use the smallest harmonic Ritz eigentubes of BH m`1,m ‹ Bm`1,m and associated eigenslices to approximate the k smallest singular triplets of A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This yields relations that are analogous to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The following theorem provides the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Apply m steps of Algorithm 5 to the third-order tensor A and assume that the tensor Bm in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Then, for k “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , m ´ 1, we have the relations A ‹ | Pk`1 “ q Qk`1 ‹ q Bk`1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='27) A H ‹ q Qk`1 “ | Pk`1 ‹ q BH k`1 ` qβk`1 ‹ ⃗ } Pk`2 ‹ ⃗E H k`1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='28) where | Pk`1 P Kpˆpk`1q n and q Qk`1 P Kℓˆpk`1q n have orthonormal lateral slices and q Bk`1 P Kpk`1qˆpk`1q n is an upper triangular tensor, where the k first lateral slices of | Pk`1 are a t-linear combination of the k first harmonic Ritz lateral slices of A with ⃗ } Pk`2 P Kp n is orthogonal to | Pk`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Moreover, ⃗Ek`1 P Km n is the canonical lateral slice under the t-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Let t ⃗| Viui“1:k be the first k harmonic Ritz lateral slices of A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='25) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='26), we get „ s1 1 ‹ ⃗| V1, s1 2 ‹ ⃗| V2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , s1 k ‹ ⃗| Vk, ⃗| Rm \uf6be “ ” Pm, ⃗ Pm`1 ı ‹ „ B´1 m ‹ U 1 k ‹ S 1 k ´βm ‹ B´1 m ‹ ⃗Em 0 e \uf6be “ Pm`1 ‹ „ B´1 m ‹ U 1 k ‹ S 1 k ´βm ‹ B´1 m ‹ ⃗Em 0 e \uf6be .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Define the tensor Jk`1 “ „ B´1 m ‹ U 1 k ‹ S 1 k ´βm ‹ B´1 m ‹ ⃗Em 0 e \uf6be .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='29) Using the reduced t-QR factorization of Jk`1, we get Jk`1 “ Q1 k`1 ‹ R1 k`1, where Q1 k`1 P Kpm`1qˆpk`1q n has orthonormal lateral slices and R1 k`1 P Kpk`1qˆpk`1q n is an f-upper triangular tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This factorization can be computed by a simple modification of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Let | Pk`1 “ „ ⃗ } P1, ⃗ } P2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ } Pk`1 \uf6be “ Pm`1 ‹ Q1 k`1 P Kℓˆpk`1q n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='30) 20 Then A ‹ | Pk`1 “ A ‹ Pm`1 ‹ Q1 k`1 “ ” A ‹ Pm, A ‹ ⃗ Pm`1 ı ‹ Q1 k`1 “ ” A ‹ Pm, A ‹ ⃗ Pm`1 ı ‹ Jk`1 ‹ ` R1 k`1 ˘´1 “ ” A ‹ Pm ‹ B´1 m ‹ U 1 k ‹ S 1 k, A ‹ ⃗ Pm`1 ´ A ‹ Pm ‹ βm ‹ B´1 m ‹ ⃗Em ı ‹ ` R1 k`1 ˘´1 “ ” Qm ‹ U 1 k ‹ S 1 k, A ‹ ⃗ Pm`1 ´ ⃗ Qm ‹ βm ı ‹ ` R1 k`1 ˘´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Define q Qk “ Qm ‹ U 1 k P Kpˆk n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='31) Using the orthogonality of A ‹ ⃗ Pm`1 ´ βm ‹ ⃗ Qm against the lateral slices of q Qk gives qαk`1 ‹ ⃗ } Qk`1 “ ´βm ‹ ⃗ Qm ` A ‹ ⃗ Pm`1 ´ q Qk ‹ » ———– qγ1 qγ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' qγk fi ffiffiffifl , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='32) where ›››› ⃗ } Qk`1 ›››› “ 1 and qαk`1 is the tube obtained from the normalization of the tensor ´βm ‹ ⃗ Qm ` A ‹ ⃗ Pm`1 ´ q Qk ‹ » ———– qγ1 qγ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' qγk fi ffiffiffifl with q QH k ‹ ´ ´βm ‹ ⃗ Qm ` A ‹ ⃗ Pm`1 ¯ “ » ———– qγ1 qγ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' qγk fi ffiffiffifl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='31) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='32) that A ‹ | Pk`1 “ » —–Qm ‹ U 1 k ‹ S 1 k, qαk`1 ‹ ⃗ } Qk`1 ` q Qk ‹ » —– | γ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' qγk fi ffifl fi ffifl ‹ ` R1 k`1 ˘´1 “ „ Qm ‹ U 1 k, ⃗ } Qk`1 \uf6be ‹ » ———– s1 1 qγ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' s1 k qγk qαk`1 fi ffiffiffifl ‹ ` R1 k`1 ˘´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 21 Hence, A ‹ | Pk`1 “ q Qk`1 ‹ q Bk`1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='33) with q Bk`1 “ » ———– s1 1 qγ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' s1 k qγk qαk`1 fi ffiffiffifl ‹ ` R1 k`1 ˘´1 P Kpk`1qˆpk`1q n , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='34) where q Bk`1 is an upper triangular tensor as it is the t-product of two upper triangular tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' To show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='28), we first notice that A H ‹ q Qk “ A H ‹ Qm ‹ U 1 k “ Pm`1 ‹ BH m,m`1 ‹ U 1 k “ Pm`1 ‹ V 1 k ‹ S 1 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Using the fact that Bm,m`1 “ ” Bm, βm ‹ ⃗Em ı “ Bm ‹ ” Im, βm ‹ B´1 m ‹ ⃗Em ı , we get Bm,m`1 ‹ V 1 k “ U 1 k ‹ S 1 k ô ” Im, βm ‹ B´1 m ‹ ⃗Em ı ‹ V 1 k “ B´1 m ‹ U 1 k ‹ S 1 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' It follows from the above result that V 1 k “ „ B´1 m ‹ U 1 k ‹ Sk ´βm ‹ B´1 m ‹ ⃗Em 0 e \uf6be ‹ „ Ik ⃗E H m`1 ‹ V 1 k \uf6be “ Jk`1 ‹ „ Ik ⃗E H m`1 ‹ V 1 k \uf6be .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We obtain A H ‹ q Qk “ A H ‹ Qm ‹ U 1 k “ Pm`1 ‹ Bm,m`1 ‹ U 1 k “ Pm`1 ‹ V 1 k ‹ S 1 k “ Pm`1 ‹ Jk`1 ‹ „ Ik ⃗E H m`1 ‹ V 1 k \uf6be ‹ S 1 k “ Pm`1 ‹ Q1 k`1 ‹ R1 k`1 ‹ „ Ik ⃗E H m`1 ‹ V 1 k \uf6be ‹ S 1 k “ | Pk`1 ‹ R1 k`1 ‹ „ Ik ⃗E H m`1 ‹ V 1 k \uf6be ‹ S 1 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='33) now yields q QH k ‹ A ‹ | Pk`1 “ q Bk,k`1 ô | PH k`1 ‹ A H ‹ q Qk “ q BH k,k`1, where q Bk,k`1 P Kpk`1qˆk n is the subtensor of q Bk`1, which is obtained by removing the last horizontal slice of q Bk`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Then | PH k`1 ‹ A H ‹ q Qk “ R1 k`1 ‹ „ Ik ⃗E H m`1 ‹ V 1 k \uf6be ‹ S 1 k “ q BH k,k`1 22 and | PH k`1 ‹ A H ‹ ⃗ } Qk`1 “ q BH k`1 ‹ q QH k`1 ‹ ⃗ } Qk`1 “ q BH k`1 ‹ ⃗Ek`1 “ qαk`1 ‹ ⃗Ek`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Hence, A H ‹ ⃗ } Qk`1 “ qαk`1 ‹ ⃗ } Pk`1 ` ⃗| R1 k`1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='35) with ⃗| R1 k`1 K | Pk`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' It follows that A H ‹ q Qk`1 “ | Pk`1 ‹ q BH k`1 ` ⃗| R1 k`1 ‹ ⃗E H k`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Normalization of ⃗| R1 k`1 gives A H ‹ q Qk`1 “ | Pk`1 ‹ q BH k`1 ` qβk`1 ‹ ⃗ } Pk`2 ‹ ⃗E H k`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The orthonormality of the lateral slices of | Pk`1 and q Qk`1 holds by the construction of these tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Specifically, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='30) that the lateral slices of | Pk`1 are orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='31), the first k lateral slices of q Qk`1 are orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Notice that if qβk`1 given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='28) vanishes, then we have determined k singular triplets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', these singular triplets of A can be computed by using the singular triplets of q Bk`1, as well as | Pk`1 and q Qk`1 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='27) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' If qβk`1 does not vanish, then we append new lateral slices to | Pk`1 and q Qk`1 in a similar way as we did in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The following result is analogous to Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Carry out m steps of Algorithm 5 and assume that eqs (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='27) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='28) hold for k “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', m´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Further, let qβk`1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='28) be nonvanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Then we have the following relations A ‹ | Pm “ q Qm ‹ q Bm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A H ‹ q Qm “ | Pm ‹ q BH m ` qβm ‹ ⃗ } Pm`1 ‹ ⃗E H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' where | Pm P Kpˆm n and q Qm P Kℓˆm n are orthonormal tensors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' q Bm P Kmˆm n is an upper triangular tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' qβm is a tube of n elements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ⃗ } Pm`1 P Kp n is orthogonal to all the lateral slices of | Pm and ⃗E H P Kℓ n is the canonical lateral slice under the t-product,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' where the first k lateral slices of | Pm and q Qm are the same as the lateral slices of | Pk`1 and q Qk`1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' given in Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' These results can be shown similarly as Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Theorem 11 requires the invertibility of Bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Notice that this tensor is well conditioned if all the frontal slices of � Bm are well conditioned, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', if max 1ďiďn κ ´ � Bpiq m ¯ is small, where κp � Bpiq m q “ ´ �sBm 1 ¯piq ´ �sBm m ¯piq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 23 Algorithm 6 describes computations required to compute approximations of either the k largest singular triplets or the k smallest singular triplets of a third-order tensor A using the methods we developed in the present or previous subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Algorithm 6 Tensor Lanczos Bidiagonalization Ritz (t-LBR) algorithm for computing the largest and the smallest singular triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Input: A P Kℓˆp n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' m: the number of tensor Lanczos bidiagonalization steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ⃗ P1 P Kp n with unit norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' k: the number of the desired singular triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' δ: The tolerance to accept the singular triplets approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ǫ: machine epsilon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' type: A Boolean variable for the kind of augmentation which is either ’Ritz’ for Ritz augmentation or ’Harm’ for harmonic Ritz augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Output: The k desired singular triplets of A , tσi, ⃗ Ui, ⃗ Viui“1:k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 1: Compute the Partial Lanczos bidiagonalization of A by Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 2: Compute the t-SVD of Bm using Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 3: Check the convergence stated in Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' If all the k desired singular triplets are well approximated, then exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 4: Compute the augmented vectors: 5: if type=’Ritz’ or kpBmq ą ǫ 1 2 then 6: Compute the tensors P :“ � Pk`1, Q :“ � Qk`1, B :“ � Bk`1 and the residual ⃗ Fk from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='12), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='15), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='16) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 7: end if 8: if type=’Harm’ and kpBmq ď ǫ 1 2 then 9: Compute the t-SVD of Bm,m`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 10: Compute the t-QR factorization of Jk`1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 11: Compute the tensors P :“ | Pk`1, Q :“ q Qk`1, B :“ q Bk`1 and the residual ⃗| Rm from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='30), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='31), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='34) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 12: end if 13: Append m ´ k lateral slices to P and Q, and m ´ k horizontal and lateral slices to B to obtain Pm, Qm and Bm, and determine a new residual ⃗ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 14: Go to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Multidimensional principal component analysis for facial recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Prin- cipal component analysis (PCA) is used in numerous areas of science and engineering, such as in data denoising, image classification, and facial recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Some approaches to color image classification involve conversion of color images to grayscale images to reduce the computational burden, because color images are represented by tensors, while gray scale images can be rep- resented by matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' see [2, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' However, this conversion entails loss of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A color image in RGB format can be represented by a third-order tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This section discusses the application of PCA to third-order tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' PCA when applied to gray-scale face recognition computes a set of characteristics (eigenfaces) corresponding to the main components of the initial set of training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Recognition is done by projecting the training images into the eigenface subspace, in which an image of a person 24 is classified by comparing it with other available images in the eigenface subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The main advantages of this procedure are its simplicity, speed, and insensitivity to small changes on the faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' When applying PCA to third-order tensors using the t-product, tubes, lateral slices, and third-order tensors are analogues of scalars, vectors, and matrices in the eigenface technique for classifying grayscale images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Using this identification, PCA for third-order tensors that represent color images is structurally very similar to PCA for matrices that represent grayscale images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The latter is described in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Let N training color images I1, I2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , IN of size ℓ ˆ p ˆ n be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' They are represented by the third-order tensors I1, I2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , IN in Rℓˆpˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The procedure of recognizing color facial images using third-order tensors is as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' For each image Ii for i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', N, we determine a lateral slice ⃗ Xi P Rℓpˆ1ˆn by vectorizing each frontal slice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', ⃗ X psq i “ vecpI psq i q for s “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We then construct a tensor, whose frontal slices are given by ⃗ Xi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', X “ ” ⃗ X1, ⃗ X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ XN ı P RℓpˆNˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Compute the mean of the frontal slices of X , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', ⃗ M “ N ÿ i“1 ⃗ Xi N , and let X “ r ⃗X 1, ⃗X 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗X Ns, ⃗X i “ ⃗ Xi ´ ⃗ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Determine the first k left singular vectors of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We denote them by ⃗ U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Construct the projection subspace Uk “ span !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' ⃗ U1, ⃗ U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ Ak ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1) and let Uk “ ” ⃗ U1, ⃗ U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , ⃗ Uk ı P Rℓpˆkˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Project each face Ii onto the subspace (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1) to obtain U H k ‹ ⃗X i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' A test image I0 also is projected onto the same space to get U H k ‹ ´ ⃗ X0 ´ ⃗ M ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Finally, determine the closest image to the test image by computing the minimal distance between the projected test image and all the projected training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The main difference between methods that use PCA for facial recognition is the way that the first (dominant) left singular vectors of X are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' In the present paper, we use our proposed method to compute the dominant singular triplets that are used in PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The following algorithm summarises the different steps in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 25 Algorithm 7 Facial recognition using tensor Lanczos bidiagonalization with Ritz augmenta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 1: Input: Training set of images X (N images), mean image X , test image I0 with its associate lateral slice ⃗ X0 “ vecpI0q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' m the number of tensor Lanczos bidiagonalization algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' k the number of the desired left singular slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 2: Output: Closest image in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 3: rUk, Sk, Vks “ t-LBRpX , m, kq using Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 4: Project X onto Uk to get P “ U H k ‹ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 5: Project the mean of the test image I0 onto Uk, ⃗ P0 “ U H k ‹ ´ ⃗ X0 ´ ⃗ M ¯ “ U H k ⃗X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 6: Find i “ arg min i“1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=',N ››› ⃗ P0 ´ ⃗ Pi ››› F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This section illustrates the performance of Algorithm 6 for detecting the largest or smallest singular triplets when applied to synthetic data, tensor compression, and facial recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' All computations are carried out on a laptop computer with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3 GHz Intel Core i5 processors and 8 GB of memory using MATLAB 2018a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Examples with synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We use synthetic data generated by the MATLAB command randnpℓ, p, nq, which generates a tensor A P Rℓˆpˆn, whose entries are normally distributed pseudorandom numbers with mean zero and variance one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Largest singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1 displays the error in the four largest approx- imate singular tubes computed by augmentation by Ritz lateral slices (referred to as Ritz in the table) and by the partial Lanczos bidiagonalization/Golub-Kahan algorithm (referred to as GK in the table) as described in [17], but using the t-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' These errors are given by }S pi, i, :q ´ Σpi, i, :q}F for i “ 1, 2, 3, 4 with m “ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2 shows the number of iterations required when using augmentation by Ritz lateral slices to approximate the four largest singular triplets for tensors of different sizes and the number of Lanczos bidiagonalization steps m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' i Methods 100 ˆ 100 ˆ 3 500 ˆ 500 ˆ 3 1000 ˆ 1000 ˆ 3 100 ˆ 100 ˆ 5 500 ˆ 500 ˆ 5 1 Ritz 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='13e-14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='60e-13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='27e-13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='85e-14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='63e-13 GK 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='16e-10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='18e-08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='01 2 Ritz 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='29e-14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='98e-13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='56e-13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='62e-14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='48e-13 GK 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='27e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='44 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='12e-04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='15 3 Ritz 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='01e-14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='70e-13 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='93e-14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='41e-14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='66e-13 GK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='78 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='05e-04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='51 4 Ritz 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='39e-13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='92e-11 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='01e-13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='39e-14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='74e-13 GK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='03 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1: The Frobenius norm }S pi, i, :q ´ Σpi, i, :q}F , where S pi, i, :q denotes the singular tubes computed by either augmentation by Ritz lateral slices (Ritz) or by partial Lanczos bidi- agonalization also known a partial Golub-Kahan bidiagonalization (GK), and Σpi, i, :q stands for the singular tubes determined by the t-SVD method with m “ 20 for i “ 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 26 Ritz 100 ˆ 100 ˆ 3 500 ˆ 500 ˆ 3 1000 ˆ 1000 ˆ 3 100 ˆ 100 ˆ 5 500 ˆ 500 ˆ 5 augmentation iter time iter time iter time iter time iter time m “ 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='40 29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='84 41 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='20 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='41 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='09 m “ 20 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='15 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='14 7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='88 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='18 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='91 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2: Number of iterations (iter) needed by the Ritz augmentation method to determine the four largest singular tubes for third-order tensors of different sizes with m “ 10, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The columns with header “time” shows the CPU time in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1 shows the Ritz augmentation method to yield much higher accuracy than the GK method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figures 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2 display the values of some frames of the first 10 singular tubes of third-order tensors of sizes 100 ˆ 100 ˆ 3 and 1000 ˆ 1000 ˆ 5, respectively, computed by Ritz augmentation using Algorithm 6, the t-SVD, and partial Lanczos bidiagonalization (GK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Each tube is denoted by S pk, k, :q P Kn, where n is equal to 3 or 5, and k “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' In other word, for a fixed i with 1 ď i ď n, we plot S pk, k, iq P Kn for k “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' , 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' As mentioned above, the ith computed singular triplet is accepted as an approximate singular triplet if ⃗ Rm ‹ ⃗E H m ‹ ⃗ Ui is small enough for 1 ď i ď k, where k is the number of desired singular triplets and the ⃗ Ui are left singular lateral slice of the current tensor Bm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3 shows the evolution of the error computed by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='9) for the first three singular triplets determined by Algorithm 6 when applied to a third-order tensor of size 1000 ˆ 1000 ˆ 3 for m “ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1: On the left, we display the values of the first frontal slices (frames) of the first 10 singular tubes detected by t-SVD, Ritz augmentation and Partial Lanczos bidiagonalization (GK) for a synthetic data of size 100 ˆ 100 ˆ 3 with m “ 20, and on the right we plotted the third frontal slices of these tubes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', S pk, k, iq with k “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', 10 and i “ 1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figures 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2 illustrate that using Algorithm 6 with Ritz augmented method gives more accurate approximations than the GK method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' In particular, the frontal slices of each tube computed with Algorithm 6 are very close to the corresponding frontal slices of the tubes determined by the t-SVD, independently of the size of the third-order tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 27 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='7 Ritz-Xt-svdDGK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4 20 h,3) s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1 0 Ritz-X t-svd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='.D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='.GK 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1 2 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 5 6 7 8 6 10 2 3 4 5 6 7 8 9 10Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2: The left-hand side pane shows the values of the first frontal slices (frames) of the first 10 singular tubes computed by t-SVD, Ritz augmentation, and the partial Lanczos bidiagonalization (GK) method for a synthetic data of size 1000 ˆ 1000 ˆ 5 with m “ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The right-hand side pane displays the third frontal slices of these tubes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', S pk, k, iq for k “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', 10 and i “ 1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3: Evolution of the remainder term for a third-order tensor of size 1000 ˆ 1000 ˆ 3 when computing the first three singular triplets by Algorithm 6 with Ritz augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Smallest singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This subsection illustrates the performance of Algo- rithm 6 with Ritz augmentation (referred to as Ritz) and with harmonic Ritz augmentation (referred to as Harm) for computing the smallest singular triplets of synthetic third-order ten- sors of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3 displays the error in the fourth smallest singular tubes computed by Ritz augmentation and harmonic Ritz augmentation for m “ 20, and compares with results determined by the t-SVD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' In Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4 we show the number of iterations and the required CPU time (in seconds) for these methods when m “ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5 4 Rm*E*U 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5 0 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5 6 Iterations160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5 Ritz t-svd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='.DGK 140% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1 40F Ritz-t-svdDGK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3 2 3 4 5 6 7 8 9 10 1 2 3 4 6 1 8 9 10 k ki Method 100 ˆ 100 ˆ 3 100 ˆ 100 ˆ 5 500 ˆ 500 ˆ 3 500 ˆ 500 ˆ 5 n ´ 3 Ritz 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='82e-11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='22e-12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='34e-10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='50e-10 Harm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='03e-13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='64e-13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='66e-13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='07e-13 n ´ 2 Ritz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='99e-14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='34e-13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='20e-14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='68e-11 Harm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='94e-15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='10e-13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='46e-14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='77e-14 n ´ 1 Ritz 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='36e-14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='56e-14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='77e-14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='86e-12 Harm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='64e-15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='05e-15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='88e-14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='39e-13 n Ritz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='38e-15 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='71e-16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='49e-15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='00e-12 Harm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='59e-16 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='90e-16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='01e-15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='41e-14 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3: The Frobenius norm }S pi, i, :q ´ Σpi, i, :q}F , where S pi, i, :q denotes the singular tubes determined by Ritz augmentation or harmonic Ritz augmentation for m “ 20, and Σpi, i, :q are tubes computed by the t-SVD method for the four smallest tubes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=', for i “ n ´ 3, n ´ 2, n ´ 1, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Method 100 ˆ 100 ˆ 3 500 ˆ 500 ˆ 3 100 ˆ 100 ˆ 5 500 ˆ 500 ˆ 5 CPU time iter CPU time iter CPU time iter CPU time iter Ritz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='99 31 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='81 615 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='11 30 425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='83 831 Harm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='85 29 227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='49 606 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='03 30 355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='35 723 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4: CPU time in seconds, and number of iterations required by Algorithm 6 with Ritz augmentation and harmonic Ritz augmentation for m “ 20 to compute the four smallest singular triplets of synthetic third-order tensors of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Tables 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4 show that harmonic Ritz augmentation gives higher accuracy than Ritz augmentation when computing the smallest singular triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figures 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5 depict the Frobenius norm of the remainder term ⃗ Rm ‹ ⃗E H m ‹ ⃗ Ui for each iteration with Algorithm 6 with Ritz augmentation and harmonic Ritz augmentation when approximating the last two singular triplets for m “ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 29 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4: The Frobenius norm of ⃗ Rm ‹ ⃗E H m ‹ ⃗ Ui obtained by Algorithm 6 with Ritz aug- mentation when approximating the two smallest singular triplets of a synthetic tensor of size 500 ˆ 500 ˆ 5 with m “ 20 at each iteration for i “ 499, 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5: The Frobenius norm of ⃗ Rm ‹ ⃗E H m ‹ ⃗ Ui obtained by harmonic Ritz augmentation when approximating the last two singular triplets of a synthetic tensor data of size 500ˆ500ˆ5 with m “ 20, at each iteration for i “ 499, 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figures 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5 show the error } ⃗ Rm ‹ ⃗E H m ‹ ⃗ Ui}F associated with Ritz augmentation in Algorithm 6 to converge in a smoother way than the corresponding error for harmonic Ritz augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Both errors converge to zero as the number of iterations increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 30 16 F 14 12 10 Error 8 4 2 0 0 100 200 300 400 500 600 700 800 Iterations15 Rm *t Em * U500 F Rm* Em *t U499 10 Error 5 0 0 100 200 300 400 500 600 700 800 Iterations5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Application to data compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='6 displays examples of image compres- sion using two color images: “house” of size 256 ˆ256 ˆ3 and “Hawaii” of size 1200 ˆ1200 ˆ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' For each image, we compute the kth largest singular triplets using Ritz augmentation in Algo- rithm 6, which will be referred to as “Ritz,” for different numbers k of desired singular triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='7 displays the relative error of the compressed images for k “ 5, 10, 15, 25, by using Ritz augmentation (Ritz) and the t-SVD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This error is measured by }Ak ´ A }F }A }F , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1) where A denotes the tensor that represents the original image and Ak “ řk i“1 ⃗ Ui ‹ si ‹ ⃗ V H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='6: Examples of image compression applied to the “house” and “Hawaii” images for k “ 5, 10, 15, 25 slices using Algorithm 6 with Ritz augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='7: Relative compression error (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1) for the images “house” and “Hawaii” obtained with Algorithm 6 with Ritz augmentation (Ritz) and the t-SVD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='7 shows the relative errors obtained with Algorithm 6 with Ritz augmentation and the t-SVD are almost the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This means that the approximate singular tubes and the right 31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='11 error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='06 Ritz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='18 t-svd 会 Ritz t-svd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='17 5 10 15 20 25 5 10 15 20 25 k kOriginal k=5 k = 10 k = 15 k = 25and left singular lateral slices determined by Algorithm 6 with Ritz augmentation are very accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Facial recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We illustrate the application of Algorithm 7 to facial recognition using color images that are represented by third-order tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The images in our test are from the Georgia Tech database GTDB crop [26], which contains 750 images of 50 persons, with each person represented by 15 images that show various facial expressions and facial orientation, and different illumination conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='8 shows an example of images of one person in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='8: An example of a person with different facial expressions and orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Each image in the data set is of size 100 ˆ 100 ˆ 3 pixels, and we use 3 randomly chosen images of each person as test images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The remaining 600 images form our training set and define the tensor X P R10000ˆ600ˆ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We applied Algorithm 7 and compared the results with those obtained by the t-SVD and also with results obtained by the‘ Golub-Kahan (GK) algorithm using the t-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The performance of these methods is measured by the identification rate given by Identification rate “ number of correctly matched images number of test images ˆ 100p%q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2) Figures 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='9 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='10 show results obtained for k “ 1 and k “ 5 for two different persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The mean image is defined as in Algorithm 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 32 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='9: A test for k “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='10: A test for k “ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 33 Test image closest image Mean image Eigenfacetestimage closest image mean image eigenfaceFigure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='11: Identification rates for different truncation indices k by Ritz augmentation, t-SVD and Golub-kahan methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Figures 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='9 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='10 show that Algorithm 7 performs well for some values of the truncation index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' In Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='11, we plotted the identification rate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='2) obtained with Algorithm 7 (Ritz augmentation), GK for m “ k, and with the exact t-SVD method for the 150 test images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' k 2 3 4 Method Ritz t-SVD Ritz t-SVD Ritz t-SVD CPU time (s) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='60 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='82 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='11 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='63 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='88 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='77 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5: CPU time (in seconds) for Algorithm 7 (Ritz) and for the t-SVD method for m “ 10 and different values of the truncation index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content='5 reports CPU times for Algorithm 7 for m “ 10 (Ritz) and for the t-SVD method for different values of the truncation index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' The results show Algorithm 7 to be very effective both in terms of accuracy and CPU time compared to the t-SVD and the classical Golub-Kahan methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Conclusion and extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' This paper presents two new methods for approximat- ing the largest or smallest singular triplets of large third-order tensors using the t-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' We use restarted Lanczos bidiagonalization for third-order tensors to develop the Ritz augmenta- tion method to determine the largest or smallest singular triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Moreover, we propose the harmonic Ritz augmentation method to compute the smallest singular triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' These methods are applied to data compression and face recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' REFERENCES [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Arnold, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' Kane, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdA0T4oBgHgl3EQfLf9-/content/2301.02119v1.pdf'} 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mode 100644 index 0000000000000000000000000000000000000000..17aadedd411c19cd4c01a5f9de80dd76f3156b24 --- /dev/null +++ b/v9AyT4oBgHgl3EQfnPij/content/tmp_files/2301.00486v1.pdf.txt @@ -0,0 +1,3130 @@ +arXiv:2301.00486v1 [cs.IT] 1 Jan 2023 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 +1 +Time-Entanglement QKD: Secret Key Rates +and Information Reconciliation Coding +Joseph J. Boutros, Senior Member, IEEE, and Emina Soljanin, Fellow, IEEE +Abstract +In time entanglement-based quantum key distribution (QKD), Alice and Bob extract the raw key bits from the +(identical) arrival times of entangled photon pairs by time-binning. Each of them individually discretizes time into +bins and groups them into frames. They retain only the frames with a single occupied bin. Thus, Alice and Bob can +use the position of the occupied bin within a frame to generate random key bits, as in PPM modulation. Because +of entanglement, their occupied bins and their keys should be identical. However, practical photon detectors suffer +from time jitter errors. These errors cause discrepancies between Alice’s and Bob’s keys. Alice sends information +to Bob through the public channel to reconcile the keys. The amount of information determines the secret key rate. +This paper computes the secret key rates possible with detector jitter errors and constructs codes for information +reconciliation to approach these rates. +Index Terms +Quantum key distribution, secret key rates, mutual information, time entanglement, time binning, jitter errors, +soft-decision decoding. +I. INTRODUCTION +Secret key distribution protocols establish a shared sequence of bits between two (or more) distant +parties, Alice and Bob, in the presence of an eavesdropper, Eve. The key consists of uniformly random +independent bits known only to Alice and Bob. Quantum Key Distribution (QKD) starts by communicating +quantum states over a quantum channel. The role of the quantum step is to 1) ensure that no eavesdropping +goes undetected and 2) provide a source of perfect randomness in the entanglement-based systems. +Joseph J. Boutros is with the Department of Electrical and Computer Engineering, Texas A& M University, 23874 Doha, Qatar, e-mail: +boutros@ieee.org (see https://www.josephboutros.org). +Emina Soljanin is with the Department of Electrical and Computer Engineering, Rutgers, the State University of New Jersey, Piscataway, +NJ 08854, USA, e-mail: (see https://www.ece.rutgers.edu/emina-soljanin). +This research is based upon work supported by the National Science Foundation under Grant # FET-2007203 + +2 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 +There has been a significant effort to provide high key rates over long distances (see recent surveys [1], +[2]). QKD schemes based on time-entangled photons have emerged as a promising technique primarily +because each entangled photon pair can carry multiple key bits and thus potentially provide a higher +secure key rate over long distances [3], [4]. +Time-entanglement-based QKD (TE-QKD) schemes use Spontaneous Parametric Down-Conversion +(SPDC) to generate entangled photon pairs according to a Poisson Process. One of the photons goes +to Alice, and the other to Bob. Therefore, Alice and Bob ideally detect their photons simultaneously +with exponentially distributed photon inter-arrival times. The most common single-photon detectors are +Superconducting Nanowire Single-Photon Detectors (SNSPDs), which exhibit properties closest to ideal +sensors. They have low dark count rates, meaning they rarely report photon detection without a photon +arrival. Furthermore, they have low detector downtime d and slight detector timing jitter that manifests as +Gaussian noise with zero mean and variance σ2 +d. Unfortunately, these imperfections are non-negligible: 1) +detector jitters and dark counts cause disagreements between Alice’s and Bob’s keys, and 2) the downtime +introduces memory within the raw key bits. The secret key rate loss due to the non-ideal properties of +these detectors has been studied most recently in [5]. +At a high level, there are two main QKD steps. In the first step, Alice and Bob generate raw key bits +using a quantum channel. Their respective raw keys may disagree at some positions, be partly known to +Eve, and may not be uniformly random because of the aforementioned non-ideal detector properties. In +the second step, Alice and Bob process the raw key to establish a shared secret key. They communicate +through the public classical channel to reconcile differences between their raw keys, amplify the privacy +of the key concerning Eve’s knowledge, and compress their sequences to achieve uniform randomness. +At the end of the protocol, Alice and Bob 1) have identical uniformly random (binary) sequences and 2) +are confident the shared sequence is known only to them. Therefore the secret key is private and hard to +guess. This paper focuses on the information reconciliation step. +Alice and Bob obtain correlated streams of bits (raw keys) by detecting the arrival times of their +entangled photons. However, they must communicate over a public channel to agree on a key, i.e., reconcile +their differences. Here, we consider one-way information reconciliation schemes in which Alice sends +information about her sequence to Bob, who uses it to remove the differences between his and Alice’s +raw keys. After the information reconciliation, Alice and Bob share Alice’s initial raw key. However, the +shared key is not secret because of the public channel communication. Alice and Bob perform privacy + +3 +amplification to correct that, establishing secrecy but shortening the key. Since Alice and Bob base their +secret key generation on correlated photon arrival times, they follow what is known as the source model +in Information Theory [6, Ch. 22.3]. The secrecy capacity for this model when the eavesdropper has +access to public communication but does not have correlated prior information is equal to the mutual +information between Alice’s and Bob’s observations (see, e.g., [6, p. 567]). The secrecy capacity is an +achievable upper bound on the post-privacy amplification rate. +Alice and Bob generate their secret keys from the correlated random photon arrivals. There are many +ways to extract keys from this correlated information. One popular method is similar to Pulse Position +Modulation (PPM); see, e.g., [7] and references therein. (Some recently proposed adaptive schemes avoid +discarding frames with multiple occupied bins [8], [9].) In PPM, Alice and Bob synchronize their clocks +and discretize their timelines into time frames N time bins. In PPM, Alice and Bob agree to retain +only time frames in which they both detect a single photon arrival and discard all other frames. This +single photon is said to occupy a time bin depending on where within the frame it arrives. Since photon +inter-arrival times follow an exponential distribution, each bin is occupied independently of other bins. +Therefore, the number of raw key bits that PPM decoding can extract from each frame equals log N. +This paper focuses on practical photon detectors that suffer from time jitter errors. Since these errors +cause discrepancies between Alice’s and Bob’s keys, Alice must send information to Bob through the +public channel to reconcile the keys. The amount of information determines the secret key rate. This paper +computes the secret key rates possible with detector jitter errors and constructs codes for information +reconciliation to approach these rates. +This paper is organized as follows: Sec. II introduces notation and lists the paper’s main contributions. +Sec. III presents the TE-QKD channel model. Sec. IV computes the rates of raw key disagreement caused +by detection jitter, and Sec. V derives the correlations between Alice’s and Bob’s raw keys. Sec. VI +computes achievable information rates and the secrecy capacity of the TE-QKD channel. Sec. VII proposes +and tests several coding schemes for information reconciliation. +II. NOTATION AND MAIN CONTRIBUTIONS +The number N of bins per time frame could be any positive integer greater than or equal to 2, our +propositions, lemmas, and theorems have no other constraint on N. However, our numerical examples are +given for N = 2m, m integer, m ≥ 1. The set ZN denotes the set of N integers {0, 1, . . . , N − 1}. The +notation ⌊x⌋, known as the floor of x for x ∈ R, is the largest integer smaller than or equal to x. + +4 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 +Letters such as X, Y , ˜X, and ˜Y denote continuous random variables, while ˆX and ˆY are discrete random +variables. Then, p(ˆy|ˆx) denotes the conditional probability P( ˆY = ˆy| ˆX = ˆx). Also, p(y|ˆx) denotes the +conditional density pY | ˆ +X(y|ˆx). +We use Bourbaki’s notation for intervals on the real line, where a and b are two real numbers: the closed +interval [a, b] = {x ∈ R : a ≤ x ≤ b}, the half-open intervals [a, b[= [a, b] \ {b} and ]a, b] = [a, b] \ {a}, +and the open interval ]a, b[= [a, b] \ {a, b}. +We use the standard Bachmann-Landau big O notation: +The formal definition of f(σ) = O(g(σ)) is: ∃α > 0, ∃σ0 > 0, ∀σ < σ0, |f(σ)| ≤ α|g(σ)|. +In this paper, an expression such as 1−O(g(σ)) or 1+O(g(σ)) implicitly assumes that g(σ) > 0 in some +open interval ]0, σ0[. Furthermore, we will frequently use γ = 1/σ2, a signal-to-noise ratio defined as the +inverse of the jitter variance, then we could write f(γ) = O(g(γ)) in a similar situation when γ → ∞. +Two functions f : R → R and g : R → R are asymptotically equivalent if limγ→∞ +f(γ) +g(γ) = 1. In that case, +we write f(γ) ∼ g(γ). +The function Q(x) = 1 +2 erfc( x +√ +2) = O(exp(−x2/2)) is the Gaussian tail function. Recall the definition +Q(x) = +� ∞ +x φ(t)dt, where φ(t) = +1 +√ +2π exp(−t2/2) is the standard normal density. Furthermore, we recall +the binary entropy function, H2(x) = −x log(x) − (1 − x) log(1 − x), and the symmetric ternary entropy +function, H3(x) = −(1 − 2x) log(1 − 2x) − 2x log(x). +The main contributions of this paper constitute a full characterization of the time-entanglement QKD +channel, from information theory and coding theory point of view: +• We derive the error rates of the TE-QKD channel, and prove that the TE-QKD channel behaves like +an 1/2-diversity Nakagami fading channel, see Proposition 1. +• We find the exact a priori probability of bins given that both Alice’s and Bob’s frames are valid, see +Lemma 2. +• We establish the exact conditional density of Bob’s photon position given Alice’s photon bin, for +a soft-output TE-QKD channel, see Theorem 1. The output density expression is also determined, +see (32). +• We determine the expression of the transition probabilities of the discrete (hard-output) TE-QKD +channel, see Corollary 1. +• We give the exact expression of the a posteriori probability for the soft-output TE-QKD channel, +see Theorem 2. + +5 +• We derive the exact formula for the mutual information I( ˆX; ˆY ) (hard-output) and find simplified +expressions in the small-noise regime, see (33), (34), (40), and Proposition 2-c. +• The exact formula for the mutual information I( ˆX; Y ) (soft-output) is given, see (41). We also +determine all densities needed to compute the maximal rate I(X; Y ) and we give a nice log-formula +expression in the small-noise regime, see Theorem 3 and Corollary 2. +• The last section, Section VII, shows new results with huge coding gains obtained by short and +moderate-length error-correcting codes such as RS, BCH, and LDPC codes under algebraic hard- +decision decoding and probabilistic soft-decision decoding. +III. PPM CHANNEL MODEL +Let ˜X and ˜Y represent the time-position of the received photons at Alice’s and Bob’s sides, respectively. +An illustration of this QKD scheme is given in Figure 1. +Alice +Bob +0 +1 +2 +Y +frame +optical channel +optical channel +Entangled photons +0 +1 +2 +X +frame +N−1 +N−1 +photon position +photon position +Fig. 1. +QKD based on time entanglement with N bins per frame, log2(N) binary digits per bin. +We adopt the following mathematical model for the positions of two time-entangled photons: +˜X = U + Z1, +˜Y = U + Z2, +(1) +where Z1 and Z2 are independent identically distributed N (0, σ2) additive Gaussian noises modeling the +detection jitter. U is a real uniform random variable in the interval [0, N[, where the integer N = 2m is +the number of bins per frame, and m is the number of bits per photon. Alice and Bob communicate via +a public channel and agree on a valid frame when ˜X and ˜Y fall in the interval [0, N[. They reject empty +frames and frames with more than one received photon. Under the model defined by (1), the probability +of a frame to be valid for both Alice and Bob is P( ˜X, ˜Y ∈ [0, N[). Let X and Y denote the instances of + +6 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 +˜X and ˜Y within the interval [0, N[, and let ˆX and ˆY be the bin number inside a frame, i.e., +˜X = X, +for ˜X ∈ [0, N[, +˜Y = Y, +for ˜Y ∈ [0, N[, +(2) +ˆX = ⌊X⌋ ∈ ZN, +ˆY = ⌊Y ⌋ ∈ ZN. +(3) +From an information theoretical perspective, we distinguish two communication channels between Alice +and Bob: (a) an algebraic (hard) output channel, (b) a real (soft) output channel, both having a discrete +N-ary input ˆX as shown in Figure 2. +(a) hard output +(b) soft output +ˆX +ˆX +ˆY +p(ˆy|ˆx) +Y +p(y|ˆx) +Fig. 2. +Channel models for hard-decision decoding (a) and soft-decision decoding (b). +Without error-correcting codes, the information rate on these channels is log2(N) = m bits per channel +use (bpcu). The main channel parameter γ is a signal-to-noise ratio parameter (SNR) defined as +γ = Es +σ2 = 1 +σ2, +(4) +where the average energy per symbol Es = 1 is a normalized energy cost per transmitted photon. Another +QKD channel parameter is γ, referred to as the normalized signal-to-noise ratio, where the standard +deviation of the additive Gaussian noise is normalized by the frame length N, hence its definition is +γ = +1 +(σ/N)2 = N2 +σ2 , +γ(dB) = γ(dB) + 20 log10(N). +(5) +We express the probability of error and the information rate as functions of N and the SNR γ or the +normalized SNR γ. The bin width within a frame is set to 1 to simplify the analysis, i.e., the frame width +is N in all sections except for Section VI-B. The conversion of this mathematical model into a physical +model representing a laboratory experiment is straightforward after introducing a time scale to convert γ +and N into physical parameters. In Section VI-B, the number of bins is infinite (it’s a continuum of bins), +the frame has a unit length and γ = γ in that special QKD channel with both soft input and soft output. +IV. RATE OF RAW KEY DISAGREEMENT UNDER DETECTION JITTER +We consider the probability of error Pe(γ) = P( ˆX ̸= ˆY ). The probability Pe characterizes the quality of +channel (a) in Figure 2 defined by its transition probabilities p(ˆy|ˆx). The latter will be entirely determined +in Section VI. In the current section, we are interested in determining the expression of Pe(γ) as a function + +7 +of the signal-to-noise ratio γ, for a given number of bins N per frame. +Let πi = P( ˆX = i), i ∈ ZN, be the a priori probability of the unique frame photon to fall in bin +number i. Then, the exact expression of the probability of error is +Pe(γ) = +N−1 +� +i=0 +πi +N−1 +� +j=0 +j̸=i +p(ˆy = j|ˆx = i) = 1 +N +N−1 +� +i=0 +P( ˆY ̸= ˆX| ˆU = i), +(6) +where ˆU = ⌊U⌋. Since U is uniform in [0, N), we get P( ˆU = i) = P(U ∈ [i, i + 1)) = 1 +N which explains +the factor in the last equality above. As a first step, in the current section, we solve Pe(γ) from the most +right equality in (6) via the conditioning over ˆU. To avoid cumbersome expressions, exact expressions as +established in Sections V&VI, we assume that γ is large enough (σ2 is small enough) so we can neglect +the border effects in the frame. Hence, we make no difference here between ˜X and X (resp. ˜Y and Y ), +and we use the approximation that both X and Y are i.i.d. Gaussian when conditioning on U. +Proposition 1. The probability of symbol error Pe(γ) = P( ˆX ̸= ˆY ) as a function of the SNR γ and the +number N of bins per frame is given by the expression +Pe(γ) = +2 +√π × +� +1 − 1 +N +� +× γ− 1 +2 + O(exp(−γ +4)). +(7) +Proof: Set V = U − ˆU, so V is Uniform[0, 1]. Let p(i → j|v) be the probability of falling in bin +j given that ˆU = i and V = v, where i, j ∈ ZN. A symbol error occurs if X = U + Z1 remains in bin +i but Y = U + Z2 leaves to bin j, j ̸= i. The probability of such an event is p(i → i|v) × p(i → j|v), +given that both additive Gaussian noises Z1 and Z2 are independent. Also, an error occurs if both X and +Y leaves to two different bins ℓ and j, with probability p(i → ℓ|v) × p(i → j|v). Then, the conditional +symbol error probability becomes +Pe(i, v) = 2 + + +N−1 +� +j=0 +j̸=i +p(i → i|v)p(i → j|v) + +N−1 +� +ℓ=0 +ℓ̸=i +N−1 +� +j=0 +j̸=i,j̸=ℓ +p(i → ℓ|v)p(i → j|v) + + . +The factor of 2 is due to the symmetry if the two letters X and Y are switched. As illustrated in Figure 3, +we will neglect bins beyond the left and the right bin. The neglected bins are at least at distance 1.0 from +the bin ˆU = i. They correspond to a probability of error Q(1/σ) = O(exp(−1/(2σ2))) = O(exp(−γ +2)). + +8 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 +To further simplify the notations, define p1, p2, and p3, where +p1 = p(i → i|v) = 1 − Q +�v +σ +� +− Q +�1 − v +σ +� +, +(8) +p2 = p(i → i − 1|v) = Q +�v +σ +� +, +(9) +and +p3 = p(i → i + 1|v) = Q +�1 − v +σ +� +, +(10) +we obtain +Pe(i, v) = O(exp(−γ +2)) + + + + + + + + + + +2[p1p2 + p1p3 + p2p3], +for i = 1 . . . N − 1, +2p1p3, +for i = 0, +2p1p2, +for i = N − 1. +neglected +neglected +left bin +right bin +1 − v +v +bin ˆU = i +V = v ∼ Uniform[0, 1] +p1 +p2 +p3 +i − 1 +i +i + 1 +i + 2 +Fig. 3. +Illustration of the probability of error in bin position. +Now, integrate over v, +P( ˆY ̸= ˆX| ˆU = i) = +� 1 +0 +Pe(i, v) dv. +Then, apply (6) and use +� 1 +0 p1p2dv = +� 1 +0 p1p3dv to finally reach +Pe(γ) = 1 +N +N−1 +� +i=0 +� 1 +0 +Pe(i, v) dv +(11) += 4 +� +1 − 1 +N +� � 1 +0 +p1p2 dv + 2 +� +1 − 1 +N +� � 1 +0 +p2p3 dv + O(exp(−γ +2)). +(12) +The two integrals in (12) include three types of integrals. Let us process them step by step. +I1 = +� 1 +0 +Q +�v +σ +� +dv = σ(1 − e−1/2σ2) +√ +2π ++ Q +� 1 +σ +� += +σ +√ +2π + O(exp(−γ +2)). + +9 +I2 = +� 1 +0 +� +Q +�v +σ +��2 +dv = 2 +√ +2σ − 2σ(1 − 2Q( +√ +2/σ)) + 4√πQ2(1/σ) − 4 +√ +2σe−1/2σ2Q(1/σ) +4√π += ( +√ +2 − 1)σ +2√π ++ O(exp(−γ)). +I3 = +� 1 +0 +Q +�v +σ +� +Q +�1 − v +σ +� +dv ≤ +� 1 +0 +exp +�−v2 − (1 − v)2 +2σ2 +� +dv = O(exp(−γ +4)), +since v2 + (1 − v)2 ≥ 1 +2 for v ∈ [0, 1]. I1 and I2 were solved via integration by parts using the fact that +dQ(x) +dx += −φ(x). I3 has no simpler form. Our upper bound of I3 brings a sufficient answer to the current +proposition. After substituting I1, I2, and I3 into (12), we get (7) as stated by the proposition, where +σ = γ− 1 +2. +The expression +2 +√π × +� +1 − 1 +N +� +×γ− 1 +2 perfectly fits the Monte Carlo simulation of P( ˆY ̸= ˆX) even for a +signal-to-noise ratio as low as 20dB (error rate close to 10−1). Figure 4 shows the plots of the probability +of error Pe(γ) for different number of bins per frames, from 1 bit per photon up to 4 bits per photon. +The plots of the probability of error versus the normalized SNR, Pe(γ), are obtained from Figure 4 after +shifting right each curve by 20 log10(N) decibels. +10-5 +10-4 +10-3 +10-2 +10-1 + 0 + 20 + 40 + 60 + 80 + 100 +Probability of Error per Symbol +SNR (dB) +N=2 +N=4 +N=8 +N=16 +Q(0.5/sigma)=O(exp(-gamma/4)) +Fig. 4. +Probability of symbol error versus SNR, log2(N) bits per photon, no coding. +V. CORRELATION BETWEEN RAW KEYS +The conditional densities of ˜X is directly derived from (1), +p ˜ +X|U(˜x|u) = +1 +√ +2πσ2 exp +� +−(˜x − u)2 +2σ2 +� +, +˜x ∈ R. +(13) + +10 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 +Conditioned on U = u, ˜X and ˜Y are independent. Then, after integrating (13), +P( ˜X, ˜Y ∈ [0, N[|u) = P( ˜X ∈ [0, N[|u)2 = +� +Q +� +−u +σ +� +− Q +�N − u +σ +��2 +. +So the probability of both Alice’s and Bob’s frames are valid is +P( ˜X, ˜Y ∈ [0, N[) = +� N +0 +� +Q +� +−u +σ +� +− Q +�N − u +σ +��2 +pU(u) du += 1 +N +� N +0 +� +Q +� +−u +σ +� +− Q +�N − u +σ +��2 +du. +(14) +The density of ˜X is also derived by integrating over u, which is equivalent to convolving the densities +of U and Z1, we get +p ˜ +X(˜x) = +� N +0 +p ˜ +X|U(˜x|u) · 1 +N du = 1 +N +� +Q +�−˜x +σ +� +− Q +�N − ˜x +σ +�� +, +˜x ∈ R. +(15) +Since X is a version of ˜X truncated to the interval [0, N[, conditioning on U + Z1 ∈ [0, N[, the density +of X is determined by scaling the density of ˜X, namely +pX|U(x|u) = +p ˜ +X|U(x|u) +� N +0 p ˜ +X|U(t|u) dt += +1 +√ +2πσ2 exp +� +−(x−u)2 +2σ2 +� +� +Q +� +− u +σ +� +− Q +� N−u +σ +��, +x, u ∈ [0, N[, +(16) +and +pX(x) = +p ˜ +X(x) +� N +0 p ˜ +X(t) dt += +Q +� +− x +σ +� +− Q +�N−x +σ +� +� N +0 +� +Q +� +− t +σ +� +− Q +�N−t +σ +�� +dt +, +x ∈ [0, N[. +(17) +By symmetry from (1), p ˜Y |U(˜y|u), p ˜Y (˜y), pY |U(y|u), and pY (y) have expressions identical to (13), (15), +(16), and (17) respectively, for ˜y ∈ R and y ∈ [0, N[. The bins a priori probabilities πi = P( ˆX = i) = +P(X ∈ [i, i + 1]) become, +πi = P( ˆX = i) = +� i+1 +i +pX(x) dx = +� i+1 +i +� +Q +�−x +σ +� +− Q +� N−x +σ +�� +dx +� N +0 +� +Q +� +− t +σ +� +− Q +�N−t +σ +�� +dt +, +i ∈ ZN. +(18) +At high SNR, for σ2 ≪ 1, we have πi ≈ 1/N, ∀i, because the truncation to the interval [0, N[ has less effect +in the small-noise regime. Numerical examples are given in Table I, for N = 8 bins per frame. The entropy +of ˆX is very stable, as listed in the last column of the table, H( ˆX) = − �N−1 +i=0 πi log2(πi) ≈ log2(N) at +low and high signal-to-noise ratios. +TABLE I +A PRIORI PROBABILITIES OF PHOTON BINS FOR N = 8 BINS PER FRAME. +SNR +π0, . . . , π7 +H( ˆX) (bits) +10 dB +0.112796, 0.129062, 0.129071, 0.129071, 0.129071, 0.129071, 0.129062, 0.112796 +2.997655 +25 dB +0.122885, 0.125705, 0.125705, 0.125705, 0.125705, 0.125705, 0.125705, 0.122885 +2.999931 +40 dB +0.124626, 0.125125, 0.125125, 0.125125, 0.125125, 0.125125, 0.125125, 0.124626 +2.999998 + +11 +The following lemma helps understand the analytic behavior of (14)-(18) at high SNR, when σ2 ≪ 1. +Lemma 1. Let fσ(x) = Q +� +− x +σ +� +−Q +� 1−x +σ +� +. For σ > 0 and γ = 1/σ2, given the properties of the Gaussian +tail function Q(x), the difference function fσ(x) satisfies +a) ∀x ∈ R, fσ(x) = fσ(1 − x) ∈]0, 1[. Also, fσ(0) = fσ(1) = 1 +2 − O(exp(−γ +2)). +b) For x ∈]0, 1[, fσ(x) = 1 − O(exp(− min2(x, 1 − x) · γ +2)). +c) For x < 0, we have fσ(x) = O(exp(−x2 · γ +2)), and fσ(x) = O(exp(−(x − 1)2 · γ +2)) for x > 1. +d) Integrating fσ and f 2 +σ, we get +� 1 +0 fσ(x) dx += +1 − +� +2 +π · +1 +√γ + O(exp(−γ +2)) += +1 − O( 1 +√γ) and +� 1 +0 f 2 +σ(x) dx = 1 − 1+ +√ +2 +√π · +1 +√γ + O(exp(−γ +4)) = 1 − O( 1 +√γ). +e) +� (i+1)/N +i/N +fσ(x) dx = 1 +N − O( 1 +√γ) for i = 0 and i = N − 1 (the two extreme bins in a frame of N bins). +� (i+1)/N +i/N +fσ(x) dx = 1 +N + O(exp(−βγ)) for i = 1 . . . N − 2 (the inner bins), where the exponent constant +is β = 1 +2 min2( i +N , 1 − i+1 +N ). +Proof: For a), let G be a standard normal random variable. The finite interval [−x, 1 − x] is never +reduced to a single point. We get fσ(x) = P(G ∈ [−x, 1 − x]) ∈]0, 1[. Then, fσ(1 − x) = Q +� +−(1−x) +σ +� +− +Q +� x +σ +� += 1 − Q +� 1−x +σ +� +− 1 + Q +� +− x +σ +� += fσ(x), using the property Q(−x) = 1 − Q(x). Finally fσ(0) = +Q(0) − Q +� 1 +σ +� += 1 +2 − O(exp(−γ +2)). +For b), we write fσ(x) = 1−Q +� x +σ +� +−Q +� 1−x +σ +� +. Then Q +� x +σ +� ++Q +�1−x +σ +� +≤ 1 +2 exp(−x2/(2σ2))+ 1 +2 exp(−(1− +x)2/(2σ2)) ≤ exp(− min2(x, 1−x)γ/2) which yields the announced result. This inequality is only useful +to us for x ∈]0, 1[ to keep the exponential decay. +For c), x < 0, so 1 − x > −x > 0. Then fσ(x) ≤ Q +�−x +σ +� +≤ 1 +2 exp(−x2/(2σ2)) = O(exp(−x2γ/2)). The +proof is similar for x > 1. +As mentioned for I1 in the proof of Proposition 1, the anti-derivative of Q(ax), a, x ∈ R, is determined +after integration by parts. We get +� +Q(ax)dx = xQ(ax) − +1 +√ +2πa2 exp(−a2x2/2) + c, +where c is the integration constant. +(19) +For d), +� 1 +0 fσ(x) dx = +� 1 +0 +� +1 − Q +� x +σ +� +− Q +�1−x +σ +�� +dx = 1 − 2I1 = 1 − +� +2 +π · +1 +√γ + O(exp(−γ +2)), where I1 +is solved thanks to (19). +As mentioned for I2 in the proof of Proposition 1, the anti-derivative of [Q(ax)]2, a, x ∈ R, is also +determined by integration by parts and the application of (19). We get +� +Q2(ax)dx = xQ2(ax) − +� +2 +πa2Q(ax) exp(−a2x2/2) + +1 +√ +πa2Q(ax +√ +2) + c. +(20) + +12 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 +Then, +� 1 +0 f 2 +σ(x)dx = 1 − 4I1 + 2I2 + 2I3 = 1 − 4 × +σ +√ +2π + 2 × +√ +2−1 +2√π σ + O(exp(−γ +4)), where I2 is solved +thanks to (20) and I3 = O(exp(−γ +4)) as shown before. This completes the proof of d). +The proof of e) is mainly based on (19), after taking care of the bin position within the frame. We have +I4 = +� (i+1)/N +i/N +fσ(x) dx = +� (i+1)/N +i/N +� +1 − Q +�x +σ +� +− Q +�1 − x +σ +�� +dx += 1 +N − +� (i+1)/N +i/N +Q +�x +σ +� +dx − +� 1−i/N +1−(i+1)/N +Q +�x +σ +� +dx += 1 +N − +�(i + 1) +N +Q +�i + 1 +Nσ +� +− i +N Q +� i +Nσ +� +− +σ +√ +2πe¯ γ +2 +(i+1)2 +N2 ++ +σ +√ +2πe¯ γ +2 +i2 +N2 +� +− +� +(1 − i +N )Q +�1 − i +N +σ +� +− (1 − i + 1 +N )Q +�1 − i+1 +N +σ +� +− +σ +√ +2πe¯ γ +2 (1− i +N )2 + +σ +√ +2πe¯ γ +2 (1− i+1 +N )2� +. +If i = 0 or i = N − 1, I4 = +1 +N − +σ +√ +2π = 1 − O( 1 +√γ), all terms with exponential decay are absorbed +by the O( 1 +√γ). For middle bins, i = 1 . . . N − 2, I4 = +1 +N + O(e− γ +2 +i2 +N2 ) + O(e− γ +2 (1− i+1 +N )2), all terms of +higher decay are absorbed by these two big O. Hence, I4 = +1 +N + O(e−βγ), where the exponent constant +is β = 1 +2 min2(i/N, 1 − (i + 1)/N). +The convergence of fσ(x) is not uniform in the interval [0, 1]. The point-wise convergence of fσ(x) +to 0 (outside [0, 1]) or to 1 (inside [0, 1]) is very slow in the neighborhood of the points x = 0 and x = 1. +At high SNR, the difference of the two Q() functions behaves as a square function and its integral slowly +approaches 1 at a rate of 1/√γ. +Applying Lemma 1 to (14)-(18), after substituting u/N to u and σ/N to σ, proves the following +equalities where ˜x, x, u ∈]0, N[ and i ∈ ZN: +P( ˜X, ˜Y ∈ [0, N[) = 1 − O( +1 +� +N2 · γ +) = 1 − O( 1 +√γ ), +p ˜ +X(˜x) = 1 +N − O(exp(−min2( ˜x +N , 1 − ˜x +N ) · γ +2)), +pX|U(x|u) = +1 +√ +2πσ2 exp +� +−(x − u)2 +2σ2 +� +· (1 + O(exp(−min2( u +N , 1 − u +N ) · γ +2)), +pX(x) = 1 +N · (1 − O(exp(−min2( x +N , 1 − x +N ) · γ +2)) · (1 + O( 1 +√γ )), +πi = ( 1 +N ± O(g(γ))) · (1 + O( 1 +√γ )), +where the vanishing rate of g(γ) depends on i as stated by the Lemma. The high SNR behavior of many +expressions below could be determined via the application of the results listed in Lemma 1. +To complete our analysis of the QKD channel between Alice and Bob, it is necessary to find the +likelihoods pY | ˆ +X(y|ˆx) and the transition probabilities p ˆY | ˆ +X(ˆy|ˆx) for the soft-output and the hard-output + +13 +mathematical models illustrated in Figure 2. We proceed in a similar manner as from (13) to (17), by +first integrating over U, then truncating over the interval [0, N[. +Lemma 2. Given Alice’s frame is valid, i.e. ˜X ∈ [0, N[, the density of U becomes +pU| ˜ +X∈[0,N[(u) = +Q +�−u +σ +� +− Q +�N−u +σ +� +� N +0 +� +Q +�−t +σ +� +− Q +�N−t +σ +�� +dt += pU| ˜Y ∈[0,N[(u), +u ∈ [0, N[, +(21) +where pU| ˜Y ∈[0,N[(u) is the density of U given that Bob’s frame is valid. Furthermore, the a priori +probabilities {ˆπi}N−1 +i=0 when both frames are valid are given by +ˆπi = P( ˆX = i| ˜Y ∈ [0, N)) = +� N +0 +� +Q +�i−u +σ +� +− Q +� i+1−u +σ +�� +· +� +Q +� −u +σ +� +− Q +�N−u +σ +�� +du +� N +0 +� +Q +�−u +σ +� +− Q +� N−u +σ +��2 du. +(22) +Proof: Let us apply Bayes’ rule, while dropping the subscripts to simplify the notation: +p(u| ˜X ∈ [0, N[) = P( ˜X ∈ [0, N)|u) × pU(u) +P( ˜X ∈ [0, N[) +. +From (13), we get P( ˜X ∈ [0, N[|u) = Q +� −u +σ +� +− Q +�N−u +σ +� +. +From (15), we get P( ˜X ∈ [0, N[= 1 +N +� N +0 +� +Q +�−t +σ +� +− Q +�N−t +σ +�� +dt. +Finally, plugging pU(u) = 1/N leads to the result announced by the lemma in (21). The equality +pU| ˜ +X∈[0,N[(u) = pU| ˜Y ∈[0,N[(u) is the result of the symmetry between Alice and Bob in our model. +The a priori probability ˆπi is derived after establishing the density of ˜X conditioned on a valid frame +for Bob, ˜Y ∈ [0, N). +p ˜ +X| ˜Y ∈[0,N)(˜x) = +� N +0 +p ˜ +X|U, ˜Y ∈[0,N[(˜x|u) · pU| ˜Y ∈[0,N[(u) du += +� N +0 +p ˜ +X|U(˜x|u) · pU| ˜Y ∈[0,N[(u) du, +(23) +where the two factors are given by (13) and (21) respectively. The a priori probability ˆπi = P(X ∈ +[i, i + 1)| ˜Y ∈ [0, N)) becomes, for i = 0, . . . , N − 1, +ˆπi = +� i+1 +i +pX| ˜Y ∈[0,N[(x) dx = +� i+1 +i +p ˜ +X| ˜Y ∈[0,N[(x) +� N +0 p ˜ +X| ˜Y ∈[0,N)(˜x) d˜x +dx += +� i+1 +x=i +� N +u=0 p ˜ +X|U(x|u) · pU| ˜Y ∈[0,N[(u) du dx +� N +˜x=0 +� N +u=0 p ˜ +X|U(˜x|u) · pU| ˜Y ∈[0,N[(u) du d˜x += +� N +u=0 +� +Q +�i−u +σ +� +− Q +�i+1−u +σ +�� +· pU| ˜Y ∈[0,N)(u) du dx +� N +u=0 +� +Q +� −u +σ +� +− Q +� N−u +σ +�� +· pU| ˜Y ∈[0,N)(u) du d˜x +. +We obtain (22) after replacing pU| ˜Y ∈[0,N[(u) by its expression from (21). + +14 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 +Lemma 2 tells that invalidating the cases where the photon falls outside the frame converts the uniform +density pU(u) = +1 +N into a non-uniform density in (21). Furthermore, the a priori probability πi of (18) +becomes ˆπi of (22) when adding the condition that Bob’s frame is valid. πi and ˆπi already take into account +that Alice has a valid frame. The next lemma leads to establishing the channel likelihood expression. +Lemma 3. The conditional density of U given ˆX = i is +p(u| ˆX = i) = +Q +� i−u +σ +� +− Q +�i+1−u +σ +� +� N +0 +� +Q +�i−t +σ +� +− Q +� i+1−t +σ +�� +dt +, +u ∈ [0, N), +(24) +for i = 0, . . . , N − 1. Furthermore, when Bob gets a valid frame, the density of U conditioned on Alice’s +bin number i is +p(u| ˆX = i, ˜Y ∈ [0, N)) = +� +Q +�i−u +σ +� +− Q +�i+1−u +σ +�� +· +� +Q +�−u +σ +� +− Q +� N−u +σ +�� +� N +0 +� +Q +� i−t +σ +� +− Q +� i+1−t +σ +�� +· +� +Q +� −t +σ +� +− Q +� N−t +σ +�� +dt +. +(25) +Proof: The existence of X and ˆX, e.g. when writing ˆX = i, requires that ˜X ∈ [0, N). This hidden +assumption should not be forgotten. By applying Bayes’ rule, +p(u| ˆX = i) = p(u| ˆX = i, ˜X ∈ [0, N)) = P( ˆX = i|u, ˜X ∈ [0, N)) × p(u| ˜X ∈ [0, N)) +πi +. +The first term in the numerator can be developed as follows +P( ˆX = i|u, ˜X ∈ [0, N) = P(X ∈ [i, i + 1)|u, ˜X ∈ [0, N)) = P( ˜X ∈ [i, i + 1)|u) +P( ˜X ∈ [0, N)|u) += Q +� i−u +σ +� +− Q +�i+1−u +σ +� +Q +�−u +σ +� +− Q +� N−u +σ +� . +The second term in the numerator is given in (21) in Lemma 2. After substituting the expression of πi +from (18), we get +p(u| ˆX = i) = +Q +�i−u +σ +� +− Q +� i+1−u +σ +� +� i+1 +i +� +Q +�−t +σ +� +− Q +� N−t +σ +�� +dt +, +The reader is invited to prove via a change of variable that +� i+1 +i +� +Q +�−t +σ +� +− Q +�N − t +σ +�� +dt = +� N +0 +� +Q +�i − t +σ +� +− Q +�i + 1 − t +σ +�� +dt. +(26) +which leads to the result announced by the lemma in (24). +The proof of (25) follows similar steps as for the proof of (24). Firstly, using Bayes’ rule and (14) we +get a conditional density of U, +p(u| ˜X, ˜Y ∈ [0, N)) = +� +Q +� −u +σ +� +− Q +�N−u +σ +��2 +� N +0 +� +Q +� −t +σ +� +− Q +�N−t +σ +��2 dt +. +(27) + +15 +Secondly, we solve the conditional probability of Alice’s photon bins, +P( ˆX = i|u, ˜X, ˜Y ∈ [0, N)) = P( ˜X ∈ [i, i + 1), ˜Y ∈ [0, N)|u) +P( ˜X, ˜Y ∈ [0, N)|u) += +� +Q +� i−u +σ +� +− Q +�i+1−u +σ +�� +· +� +Q +�−u +σ +� +− Q +� N−u +σ +�� +� +Q +�−u +σ +� +− Q +� N−u +σ +��2 +. +Finally, we use the above expressions of P( ˆX = i|u, ˜X, ˜Y ∈ [0, N)) and p(u| ˆX, ˜Y ∈ [0, N)), and (22) +from Lemma 2 in +p(u| ˆX = i, ˜Y ∈ [0, N)) = P( ˆX = i|u, ˜X, ˜Y ∈ [0, N)) × p(u| ˜X, ˜Y ∈ [0, n)) +ˆπi +to reach (25) in this lemma. +The existence of Y assumes that ˜Y ∈ [0, N), as we mentioned for X in the proof of Lemma 3. We +deliberately remind the reader of the condition ˜Y ∈ [0, N) in the subscript of the likelihood function in +the next statement. +Theorem 1. Under the assumption that both Alice and Bob got valid frames, the soft-output QKD channel +model likelihoods, p(y|ˆx) = pY | ˆ +X, ˜Y ∈[0,N)(y|ˆx), have the following expression +pY | ˆ +X, ˜Y ∈[0,N)(y|ˆx = i) = +� N +0 +1 +√ +2πσ2 exp +� +−(y−u)2 +2σ2 +� +· +� +Q +�i−u +σ +� +− Q +� i+1−u +σ +�� +du +� N +0 +� +Q +� −t +σ +� +− Q +�N−t +σ +�� +· +� +Q +� i−t +σ +� +− Q +� i+1−t +σ +�� +dt +, +(28) +for i = 0, . . . , N − 1, y ∈ [0, N). For simplicity, the likelihood in (28) will be denoted by p(y| ˆX = i) in +next sections. +Proof: We drop the subscripts in the density functions, when possible, to simplify the notations. We +start by a marginalization before truncating p(˜y|u). +p(y| ˆX = i, ˜Y ∈ [0, N)) = +� N +0 +p(y, u| ˆX = i, ˜Y ∈ [0, N)) du += +� N +0 +p(y|u, ˆX = i, ˜Y ∈ [0, N)) · p(u| ˆX = i, ˜Y ∈ [0, N)) du += +� N +0 +p(y|u) · p(u| ˆX = i, ˜Y ∈ [0, N)) du, +(29) +The left factor p(y|u) inside the integral in (29) is given by the truncation of the density in (13) (replace +x by y) and the right factor was solved by Lemma (3). +p(y| ˆX = i, ˜Y ∈ [0, N)) = +� N +0 +p(˜y = y|u) +� N +0 p(˜y|u)d˜y +· p(u| ˆX = i, ˜Y ∈ [0, N)) du, + +16 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 += +� N +0 +1 +√ +2πσ2 exp +� +−(y−u)2 +2σ2 +� +Q +� −u +σ +� +− Q +�N−u +σ +� · +� +Q +� i−u +σ +� +− Q +�i+1−u +σ +�� +· +� +Q +�−u +σ +� +− Q +�N−u +σ +�� +� N +0 +� +Q +�i−t +σ +� +− Q +�i+1−t +σ +�� +· +� +Q +� −t +σ +� +− Q +�N−t +σ +�� +dt +du. +After simplifying the term Q +�−u +σ +� +− Q +� N−u +σ +� +we reach the announced result. +The transition probabilities pi,j = P( ˆY = j| ˆX = i) of the hard-output QKD channel model are directly +derived by integrating the conditional density function of the soft output Y established by the previous +theorem. +Corollary 1. The probability that Bob’s photon falls in bin j given that Alice’s photon fell in bin i is +given by +pij = P( ˆY = j| ˆX = i) = +� N +0 +� +Q +�j−u +σ +� +− Q +�j+1−u +σ +�� +· +� +Q +� i−u +σ +� +− Q +�i+1−u +σ +�� +du +� N +0 +� +Q +�−t +σ +� +− Q +�N−t +σ +�� +· +� +Q +� i−t +σ +� +− Q +�i+1−t +σ +�� +dt +, +i, j ∈ ZN. +(30) +Proof: Integrate (28) over Bob’s photon position y from j to j + 1, then switch the two integrals to +get the result announced by this corollary. +We complete this section by establishing the expression of the a posteriori probability useful for soft- +decision decoding, e.g., for belief-propagation decoding of low-density parity-check codes, for ordered- +statistics decoding of linear block codes, or Viterbi decoding of convolutional codes. Let APP(i) = +APP( ˆX = i) = P( ˆX = i|Y = y) be the a posteriori probability of Alice’s photon bin number i, +for i = 0 . . . N − 1. The next theorem gives the expression APP(i), which is used in our proposed +coding/decoding schemes in Section VII. +Theorem 2. Given the photon position Y = y on Bob’s side, the probability for Alice’s photon to belong +to bin number i is +APP(i) = +� N +0 +1 +√ +2πσ2e− (y−u)2 +2σ2 +· +� +Q +�i−u +σ +� +− Q +� i+1−u +σ +�� +du +� N +0 +1 +√ +2πσ2 e− (y−t)2 +2σ2 +· +� +Q +� −t +σ +� +− Q +� N−t +σ +�� +dt +, +i ∈ ZN. +(31) +Proof: Keeping in mind that ˜X, ˜Y ∈ [0, N), apply Bayes’ rule to get +P( ˆX = i|Y = y) = p(y| ˆX = i) · P( ˆX = i) +p(y) +. +The result announced by the theorem is then found in three steps. +(i) Use (28) from Theorem 1 for p(y| ˆX = i). +(ii) Use (22) for the a priori P( ˆX = i). +(iii) Finally, p(y) = pY | ˜ +X∈[0,N)(y) = p ˜Y | ˜ +X∈[0,N)(˜y = y)/ +� N +0 p ˜Y | ˜ +X∈[0,N)(t) dt after truncating the density of +˜Y . The density p ˜Y | ˜ +X∈[0,N)(˜y) is found in (23) while switching the letters x (resp. X) and y (resp. Y ), in + +17 +conjunction with (13) and (21), +p(y) = +� N +0 +1 +√ +2πσ2e− (y−u)2 +2σ2 +· +� +Q +�−u +σ +� +− Q +� N−u +σ +�� +du +� N +0 +� +Q +�−t +σ +� +− Q +� N−t +σ +��2 dt +. +(32) +For consistency, the reader could check that p(y) given at the end of the proof of Theorem 2 is also +equal to �N−1 +i=0 ˆπi · p(y| ˆX = i) from (22) and (28). Figures 5 and 6 plot the likelihoods p(y| ˆX = i) at +low SNR γ = 10 dB (low photon detector precision) and a relatively higher SNR γ = 25 dB (higher +photon detector precision), respectively. At low SNR, p(y| ˆX = i) has a Gaussian shape. The shape tends +to become square at high SNR. The a posteriori probabilities APP(i), i ∈ ZN, have a plot similar to the +channel likelihoods. + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 +Probability Density Function +Photon Position +p(y|0) +p(y|1) +p(y|2) +p(y|3) +p(y|4) +p(y|5) +p(y|6) +p(y|7) +p(y) +Fig. 5. +Soft-Output channel likelihoods, N=8 bins per frame, SNR=10 dB. + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 +Probability Density Function +Photon Position +p(y|0) +p(y|1) +p(y|2) +p(y|3) +p(y|4) +p(y|5) +p(y|6) +p(y|7) +p(y) +Fig. 6. +Soft-Output channel likelihoods, N=8 bins per frame, SNR=25 dB. + +18 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 +VI. SECRET KEY INFORMATION RATES +A. Mutual Information between Raw Keys +Firstly, we consider the mutual information of the algebraic hard-output channel defined by the transition +probability p(ˆy|ˆx) = P( ˆY = ˆy| ˆX = ˆx). In Corollary 1, the expression of pij = p( ˆY = j| ˆX = i) was +established. Hence, we can directly compute the mutual information as follows: +I( ˆX; ˆY ) = H( ˆY ) − H( ˆY | ˆX) += − +N−1 +� +i=0 +P( ˆY = i) log(P( ˆY = i)) ++ +N−1 +� +i=0 +P( ˆX = i) +N−1 +� +j=0 +P( ˆY = j| ˆX = i) log(P( ˆY = j| ˆX = i)) += − +N−1 +� +i=0 +ˆπi log(ˆπi) + +N−1 +� +i=0 +ˆπi +N−1 +� +j=0 +pij log(pij), +(33) +where a priori ˆπi of ˆX and ˆY is found in (22). The plot of I( ˆX; ˆY ) expressed in bits versus the signal-to- +noise ratio is depicted in Figure 7, for 2, 3, and 4 coded bits per transmitted photon. As expected, the curves +go towards the asymptote H( ˆX) at high signal-to-noise ratio. In fact, the entropy − �N−1 +i=0 ˆπi log(ˆπi) is +very stable even at low SNR and could be well approximated by log(N). The summation in (33) could +be truncated to neighboring bins or to bins within an integer distance less than D, +I( ˆX; ˆY ) ≈ log(N) + 1 +N +� +|i−j|≤D +pij log(pij). +(34) +The simplification (34) is an excellent approximation down to γ ≥ 10 dB for D = 1 only and it extends +to γ ≥ 5 dB for D = 2. The next proposition gives more insight into the behavior of the a priori and the +transition probabilities, and the discrete channel mutual information in the low-noise regime. +Proposition 2. At high signal-to-noise ratio, when σ2 ≪ 1, we have the following results: +a) The transition probability of the hard-output QKD channel established in Corollary 1 satisfies: +At the two extremal bins, i = 0 and i = N − 1, we have +p0,1 = +σ +√π + O(e− γ +4 ) +1 − 1+ +√ +2 +2√π · σ + O(e− γ +4 ) +, +(35) +where p0,1 = pN−1,N−2 = 1 − p0,0 = 1 − pN−1,N−1, and σ = 1/√γ. + +19 + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +-5 + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 +N=4 +N=8 +N=16 +Bits per Photon +SNR (dB) +Fig. 7. +Mutual information I( ˆX; ˆY ) of the algebraic hard-output TE-QKD channel, for N = 4, 8, 16 bins per frame. +At the middle bins, i = 2 . . . N − 2, we have +p1,2 = +σ +√π + O(e− γ +4 ) +1 − O(e− γ +4 ) , +(36) +where p1,2 = pi,i+1 = pi,i−1 = (1−pi,i)/2. All other transition probabilities pi,j for |i−j| ≥ 2 are O(e− γ +4 ) +and can be forced to 0 in any numerical calculation at high SNR. +b) The a priori probabilities established in Lemma 2 satisfy +At the two extremal bins, i = 0 and i = N − 1, we have +ˆπ0 = ˆπN−1 = +1 − 1+ +√ +2 +2√π · σ + O(e− γ +4 ) +N · +� +1 − 1+ +√ +2 +√π · σ + O(e− γ +4 ) +�, +(37) +where the numerator includes σ but the denominator involves σ = σ/N. For the middle bins, with +i = 2 . . . N − 2, we have +ˆπi = +1 − O(e− γ +4 ) +N · +� +1 − 1+ +√ +2 +√π · σ + O(e− γ +4 ) +�. +(38) +c) Following a) and b), the mutual information of the discrete-input discrete-output QKD channel given +by (33) becomes +I( ˆX; ˆY ) = +N − 2βσ +N(1 − 2βσ) log [N(1 − 2βσ)] − 2(1 − βσ) +N(1 − 2βσ) log(1 − βσ) +− 2(1 − βσ) +N(1 − 2βσ)H2 +� σ/√π +1 − βσ +� +− +(N − 2) +N(1 − 2βσ)H3 +� σ +√π +� ++ O(e− γ +4 ), +(39) + +20 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 +where β = 1+ +√ +2 +2√π , σ = 1/√γ, and σ = σ/N. +Proof: For the sake of space, we only show the detailed proof for the denominator of p0,1 (also equal +to the numerator of ˆπ0). All other results are found using similar calculus techniques. The denominator +of p0,1 from Corollary 1 (i = 0, j = 1) is equal to the integral +I5 = +� N +0 +� +Q +� −t +σ +� +− Q +�N−t +σ +�� +· +� +Q +� −t +σ +� +− Q +� 1−t +σ +�� +dt += +� N +0 +� +1 − Q +� t +σ +� +− Q +�N−t +σ +�� +· +� +Q +� t−1 +σ +� +− Q +� t +σ +�� +dt += +� N +0 +Q +�t−1 +σ +� +dt − +� N +0 +Q +� t +σ +� +dt + +� N +0 +Q2 � t +σ +� +dt − +� N +0 +Q +� t +σ +� +Q +� t−1 +σ +� +dt +− +� N +0 +Q +� N−t +σ +� +Q +�t−1 +σ +� +dt + +� N +0 +Q +� N−t +σ +� +Q +� t +σ +� +dt += +� 0 +−1 +Q +� t +σ +� +dt +� +�� +� +(i) +− +� N +N−1 +Q +� t +σ +� +dt +� +�� +� +(ii) ++ +� N +0 +Q2 � t +σ +� +dt +� +�� +� +(iii) +− +� 1 +0 +Q +� t +σ +� +(1 − Q +�1−t +σ +� +) dt +� +�� +� +(iv) +− +� N +1 +Q +� t +σ +� +Q +�t−1 +σ +� +dt +� +�� +� +(v) +− +� 1 +0 +Q +� N−t +σ +� +(1 − Q +� 1−t +σ +� +) dt +� +�� +� +(vi) +− +� N +1 +Q +� N−t +σ +� +Q +�t−1 +σ +� +dt +� +�� +� +(vii) ++ +� N +0 +Q +� N−t +σ +� +Q +� t +σ +� +dt +� +�� +� +(viii) +. +Now we solve the elementary integrals (i)-(viii) one by one. +Using (19), (i) = 0 − +σ +√ +2π + Q( −1 +σ ) + +σ +√ +2πe− γ +2 = 1 − +σ +√ +2π + O(e− γ +2 ). Using the fact that Q(x) is a +monotone decreasing function, then (ii) = O(e−(N−1)2 γ +2 ). The third integral is directly solved via (20): +(iii) = NQ2( N +σ ) − σ +� +2 +πQ( N +σ )e−N2 γ +2 + +σ +√πQ( N +√ +2 +σ ) − 0 + σ +� +2 +π · 1 +2 − +σ +√π · 1 +2. Then we find (iii) = +�√ +2−1 +2√π +� +σ + O(e−N2 γ +2 ). (iv) = I1 − I3 = +σ +√ +2π + O(e− γ +4 ). For (v), t2 + (t − 1)2 ≥ 1 in the interval +[1, N], then we have (v) = O(e− γ +2 ). The first part of (vi) is O(e−(N−1)2 γ +2 ) and the second part is also +O(e−(N−1)2 γ +2 ) because (N − t)2 + (1 − t)2 ≥ (N − 1)2 in the interval [0, 1]. So (vi) = O(e−(N−1)2 γ +2 ). +Applying similar arguments, we get (vii) = O(e−(N−1)2 γ +4 ) and (viii) = O(e−N2 γ +4 ). Combining (i)-(viii) +yields I5 = 1 − 1+ +√ +2 +2√π σ + O(e− γ +4 ) as announced. +At high signal-to-noise ratio, Proposition 2-a) shows how fast pi,j converges to +σ +√π. The latter is a +one-sided probability of error and it is half the double-sided probability of error stated in Proposition 1. +As expected, ˆπi converges to 1/N much faster for inner bins as found in Proposition 2-b). The high- +SNR expression of I( ˆX; ˆY ) established in Proposition 2-c) perfectly fits the exact mutual information + +21 +of the discrete channel down to γ = 10 dB and then diverges at low SNR below 10 dB. The binary +entropy function represents the extremal bins error. The ternary entropy function carries the inner bins +error. Expression (39) is a quick method to evaluate I( ˆX; ˆY ) at moderate and high signal-to-noise ratios +without performing any integration. +One could ask how good is the approximated mutual information if the TE-QKD discrete channel is +assumed to have a circular transition probability matrix. Under the assumptions of Proposition 1, we take: +1- X and Y are Gaussian, 2- all bins are equiprobable, and 3- the error probability of the discrete-input +discrete-output channel is dominated by events where X and Y are separated by one or two bins only. +According to Theorem 7.2.1 in [10, Ch. 7.2], the expression for a circular discrete channel is +I( ˆX; ˆY ) ≈ log(N) + +2 +� +j=−2 +p0j log2(p0j), +(40) +where p01 = p0,−1 ≈ +� 1 +0 2p1p2 dv = 2(I1 − I2 − I3), p02 = p0,−2 ≈ +� 1 +0 2p2p3 dv = 2I3, and p00 = +1 − 2p01 − 2p02. All three high-SNR approximations (39), (34) with D = 2, and (40) are respectively +shown in dotted lines from top to bottom on Figure 7 for N = 8 bins per frame. (39) and (34) follows the +exact mutual information I( ˆX; ˆY ) at high SNR. (40) is not tight enough at N = 8 but becomes tighter +for N ≥ 16 bins per frame. +The second step in this section is to compute the mutual information for the soft-output TE-QKD +channel. We chose to write I( ˆX; Y ) = H( ˆX) − H( ˆX|Y ). The second expression after flipping X and Y +based on differential entropy is also equivalent from numerical stability point of view and has all its terms +established in the previous section. We prefer the mutual information where the high-SNR asymptote is +visible, hence +I( ˆX; Y ) = H( ˆX) − H( ˆX|Y ) = H( ˆX) + +N−1 +� +i=0 +ˆπi +� N +0 +p(y|ˆx = i) log2(APP(i)) dy, +(41) +where the a priori ˆπi is from (22), the likelihood p(y|ˆx = i) is from (28), and the a posteriori APP(i) is +from (31). Figure 8 shows the mutual information I( ˆX; Y ) versus normalized SNR γ for different number +of bins per frame. The red upper envelope is established by Theorem 3 in the next section. It corresponds +to the maximal mutual information achievable on the TE-QKD channel. + +22 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 10 + 20 + 30 + 40 + 50 + 60 +N=4 +N=8 +N=16 +N=32 +N=64 +Bits per Photon +Normalized SNR (dB) +Capacity +N bins per frame +Fig. 8. +Mutual information I( ˆX; Y ) of the soft-output QKD channel, for N = 4, 8, 16 bins per frame. +B. Maximal Secrecy Rate +The random variables X, ˆX, and Y form a Markov chain X → ˆX → Y . Therefore, the data processing +inequality [10, Ch. 2] yields +I( ˆX; Y ) ≤ I(X; Y ), +∀ N ≥ 2. +(42) +Consequently, for any value of the number N of bins per frame, the rate of our channel pY | ˆ +X(y|ˆx) is +always bounded from above by the rate of the continuous-input continuous-output channel pY |X(y|x) +corresponding to a continuum of zero-measure bins in Alice’s frame. Thus, by determining the mutual +information I(X; Y ) we get the maximal secrecy rate of the photon channel between Alice and Bob. +Without loss of generality, assume that the frame size is 1, instead of N. Now, the problem is to find +I(X; Y ) where X and Y are truncated versions of the original photon positions, X += +˜X ∈ [0, 1), +Y += +˜Y ∈ [0, 1). The model in (1) becomes ˜X = U + N (0, σ2), ˜Y = U + N (0, σ2), U is uniform +in [0, 1), and the two additive Gaussian noises are independent. The normalized signal-to-noise ratio is +naturally defined by γ = γ = 1/σ2 under this context of infinite number of bins and a frame of unit length. + +23 +Theorem 3. The maximal secrecy rate I(X; Y ) is given by +I(X; Y ) = h(Y ) − h(Y |X) +(43) += − +� 1 +0 +p(y) log(p(y)) dy + +� 1 +0 +p(x) +� 1 +0 +p(y|x) log(p(y|x)) dxdy +(44) +where p(x) and p(y) are from (32) after replacing the frame size N by 1, +p(x) = pX| ˜ +X, ˜Y ∈[0,1)(x) = +� 1 +0 +1 +√ +2πσ2 e− (x−u)2 +2σ2 +� +Q +�−u +σ +� +− Q +� 1−u +σ +�� +du +� 1 +0 +� +Q +� −t +σ +� +− Q +� 1−t +σ +��2 dt +, +(45) +p(y) = pY | ˜ +X, ˜Y ∈[0,1)(y) = +� 1 +0 +1 +√ +2πσ2 e− (y−u)2 +2σ2 +� +Q +�−u +σ +� +− Q +� 1−u +σ +�� +du +� 1 +0 +� +Q +�−t +σ +� +− Q +�1−t +σ +��2 dt +, +(46) +p(y|x) = +1 +√ +4πσ2e− (y−x)2 +4σ2 +� +Q +� +0−(x+y)/2 +σ/ +√ +2 +� +− Q +� +1−(x+y)/2 +σ/ +√ +2 +�� +� 1 +0 +1 +√ +2πσ2 e− (x−u)2 +2σ2 +� +Q +� −u +σ +� +− Q +�1−u +σ +�� +du +. +(47) +Proof: We complete the proof by finding the expression of the conditional density p(y|x). Indeed, +we can write after marginalizing and applying Bayes’ rule +p(y|x) = pY |X, ˜ +X, ˜Y ∈[0,1)(y|x) = +� 1 +0 +p(y, u|x) du = +� 1 +0 +p(u)p(x|u)p(y|u) +p(x) +du. +(48) +In the above integral expression we used the fact that p(x, y|u) = p(x|u)p(y|u) as a result of the model +defined by (1). In (48), both p(x|u) and p(y|u) are from (16), p(x) is from (32), and finally p(u) = +pU| ˜ +X, ˜Y ∈[0,1)(u) is found in (27), all after substituting 1 to N. After simplifying the integrand of (48), we +get p(y|x) as stated by (47). +Corollary 2. At high signal-to-noise ratio, i.e. σ2 ≪ 1 or equivalently γ = +1 +σ2 ≫ 1, the maximal secrecy +rate satisfies +I(X; Y ) = (1 + O( 1 +√γ)) · 1 +2 log +� γ +4πe +� ++ O +� +( 1 +√γ)α� +∼ 1 +2 log +� γ +4πe +� +, +∀α ∈]0, 1[. +(49) +Proof: The proof is based on a Babylonian approach with heavy calculus. Let us first give a sketch +on how the limit is guessed. By applying Lemma 1 and some extra algebra, when γ ≫ 1, we get that +p(x) → 1, p(y) → 1, and p(y|x) → +1 +√ +4πσ2 e− (y−x)2 +4σ2 , in (45), (46), and (47) respectively, for x, y ∈]0, 1[. +Then, the differential entropy h(Y ) → 0, h(Y |X) → h(N (0, 2σ2)) = +1 +2 log(4πeσ2), so the maximal +secrecy rate satisfies I(X; Y ) = h(Y ) − h(Y |X) → 1 +2 log +� γ +4πe +� +. This ends a simple but a clear sketch on +how all involved densities and the maximal mutual information are converging at high SNR. + +24 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 +The denominator of p(x) and p(y) is given in Lemma 1 at high SNR as 1 − O( 1 +√γ). The numerator +of p(y|x) is fσ/ +√ +2((x + y)/2) = 1 − O(exp(− min2 ·γ)) for (x + y)/2 ∈]0, 1[, where min is min((x + +y)/2, 1 − (x + y)/2) from Lemma 1. Also, fσ/ +√ +2(0) = fσ/ +√ +2(1) = 1 +2 − O(exp(−γ)). The last item to +solve to get the result of this corollary is the integral in the numerator of p(x), the numerator of p(y), +and the denominator of p(y|x). Define the following integral +I6 = I6(x) = I6(1 − x) = +� 1 +0 +1 +√ +2πσ2 e− (x−u)2 +2σ2 +� +Q +� −u +σ +� +− Q +� 1−u +σ +�� +du, +x ∈ [0, 1]. +(50) +The integral I6 has no closed-form expression. Firstly, we study I6(x) at x = 0 (identical at x = 1). +I6(0) = +� 1 +0 +1 +√ +2πσ2 e− u2 +2σ2 � +1 − Q +� u +σ +� +− Q +�1−u +σ +�� +du += 1 +2 − Q +� 1 +σ +� +− +� 1 +0 +1 +√ +2πσ2 e− u2 +2σ2 Q +� u +σ +� +du − +� 1 +0 +1 +√ +2πσ2 e− u2 +2σ2 Q +�1−u +σ +� +du += 1 +2 − O(e− γ +2 ) − 1 +8 − 1 +2Q2 � 1 +σ +� +− O(e− γ +4 ) = +3 +8 − O(e− γ +4 ). +(51) +Secondly, we study I6(x) for x ∈]0, 1[. We use calculus tools similar to those used in the proofs of +Lemma 1 and Proposition 2 to obtain +I6(x) = Q +� −x +σ +� +− Q +�1−x +σ +� +− +� 1 +0 +1 +√ +2πσ2 e− (x−u)2 +2σ2 Q +� u +σ +� +du − +� 1 +0 +1 +√ +2πσ2 e− (x−u)2 +2σ2 Q +� 1−u +σ +� +du += fσ(x) − O(e− x2 +4σ2 ) − O(e− (1−x)2 +4σ2 ) = 1 − O(e− min2(x,1−x)· γ +4 ). +(52) +Now we are ready to transform the expression of I(X; Y ) in the small-noise regime given that the behavior +of all densities is solved: +p(x) = +I6(x) +1 − O( 1 +√γ), +p(y) = +I6(y) +1 − O( 1 +√γ), +and p(y|x) = e− (y−x)2 +4σ2 +√ +4πσ2 · +fσ/ +√ +2((x + y)/2) +I6(x) +. +For x, y ∈ [0, 1], we distinguish between the behavior of I6(x), I6(y), and fσ/ +√ +2((x + y)/2) near the +extremal points 0 and 1 and inside the interval. Hence, we decompose the interval as [0, 1] = [0, δ] ∪ +[δ, 1 − δ] ∪ [1 − δ, 1] for integration. The parameter δ should vanish at high SNR and should guarantee +that I6 approaches 1, then we find from (52) that δ2γ should go to 0, which leads to δ = ( 1 +√γ)α, where +0 < α < 1. Finally, (44) is decomposed via this δ into +I(X; Y ) = − +� 1−δ +δ +p(y) log(p(y)) dy − 2 +� δ +0 +p(y) log(p(y)) dy ++ +�� +x,y∈[δ,1−δ] +p(x)p(y|x) log(p(y|x)) + +�� +x,y /∈[δ,1−δ] +p(x)p(y|x) log(p(y|x)) +(53) += (1 + O( 1 +√γ)) · 1 +2 log +� γ +4πe +� ++ O(δ). +(54) + +25 +The cumbersome calculus details proving the last equality are not included for the sake of space. +The Gaussian differential entropy (49) is very close to I(X; Y ) above 2 bits per photon and becomes +very accurate beyond 3 bits per photon where it coincides with the red upper envelope in Figure 8 at +a high signal-to-noise ratio. The double variance 2σ2 in (49), originally found in (47), comes from the +superposition of the variances of Z1 and Z2 in the system model defined by (1). After canceling U, the +model becomes ˜Y = ˜X + Z1 − Z2. ˜X and Z1 are correlated, making the density expression relatively +complicated when conditioning on X. In the small-noise regime, this correlation fades away, and the +variance 2σ2 of the total additive Gaussian noise Z1 − Z2 dominates the mutual information as in (49). +At low and very low signal-to-noise ratios, one should use exact density expressions from Theorem 3 and +proceed via numerical integration to get exact values of I(X; Y ) and the corresponding SNR limits if the +user accepts to apply a relatively low coding rate which is not the trend in TE-QKD where coding rates +above 1/2 are preferred which places us in the moderate and the high SNR region. +VII. KEY-RECONCILIATION CODES +Following the complete characterization in Sections V-VI-B of the time-entanglement QKD channel +model described in Section III, we now introduce error-correcting codes to bring the error-rate performance +as close as possible to the information theoretic limits corresponding to maximal achievable rates. +A. Reed-Solomon Codes +We consider the famous family of Reed-Solomon codes with an application to a frame of N = 2m bins, +i.e. m coded bits per photon. In order to chose a high enough error-correction capacity, an RS code over +Fq is considered, where q is large enough. Each finite field element corresponds to log2(q)/m photons. +For simplicity, assume that q = 2ℓm, for some positive integer ℓ. The RS code has length n = q − 1 +(primitive) and dimension k = n − 2t, so the targeted rate is m × k +n information bits per photon. One +codeword of this C[n, k, t]q RS code requires the transmission of a total of n × log2(q)/m photons to +Alice and n × log2(q)/m photons to Bob, all with valid frames. After receiving the n × log2(q)/m valid +frames, Alice converts the n×log2(q) bits received on the quantum channel into a length-n word denoted +by c + e, where c ∈ C and e ∈ Fn +q . Similarly, Bob converts his n × log2(q) received bits into c + e′, where +e′ ∈ Fn +q . In the hard-output channel model of Section III, c + e is written at the input ˆX and c + e′ is read +from the output ˆY . On the public channel, Alice sends to Bob the syndrome s = (c + e)Ht, s ∈ Fn−k +q +, + +26 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 +where H is the parity-check matrix of C. Given s and given c + e′, the reconciliation performed by Bob +is equivalent to finding Alice’s word c + e. Bob proceeds as follows: +• Compute a syndrome s′ = (c + e′)Ht. +• Feed s′ − s to an algebraic (Berlekamp-Massey [11]) decoder to find e′ − e. +• Subtract the error e′ − e from Bob’s word to get c + e′ − (e′ − e) = c + e the Fn +q word possessed by +Alice. Replace all subtractions by additions in usual finite fields of characteristic 2. +The performance of RS C[n = 63, k = 43, t = 10]q=64 code is shown in Figure 9. One codeword requires +the transmission of a total of 126 photons, where one field element carries two photons. The results show +a large gain, e.g., about 58 dB of gain for a bit error-rate Peb = 10−5 after reconciliation. +10-5 +10-4 +10-3 +10-2 +10-1 + 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90 + 100 +Probability of Error +SNR (dB) +Uncoded Pes +Uncoded Peb +RS-coded Pes +RS-coded Peb +1/SNR1/2 +Q(0.5/sigma) +Fig. 9. +Performance of the RS code [63, 43, t = 10]64 on the hard-output time-entanglement QKD channel, for N = 8 bins per frame, +transmitting 2.05 information bits per photon. +The analysis of the algebraic decoder is easy thanks to its bounded-distance decoding in the Hamming +space. A decoding error occurs each time the channel adds more than t errors in Fq. A simple union +bound is obtained by summing from t + 1 to n errors. We proceed in the following steps to establish this +bound for the RS code: +a) The uncoded symbol error probability over the TE-QKD channel is Pe(γ) = +2 +√π(1 − 1 +N ) 1 +√γ. +b) For the RS code, the input probability of error per finite-field element is Pin(γ) = 1 − (1 − Pe(γ))ℓ. + +27 +c) The bound on the probability of error in Fq after decoding becomes +PeRS(γ) = +n +� +i=t+1 +i +n +�n +i +� +P i +in(γ)(1 − Pin(γ))n−i. +(55) +d) The symbol (per photon) error probability after decoding is then PeOut(γ) = 1 − (1 − PeRS(γ))1/ℓ. +e) The probability of error per bit after Reed-Solomon decoding, given a Gray labeling of the bins, is +well estimated by PebRS(γ) = +1 +log2(N)PeOut(γ). +The probability or error PebRS obtained from (55) perfectly fits the Monte Carlo method in the area where +this method is tractable on a computer, i.e. for error rates in the interval [10−7, 10−1]. At PebRS = 10−10, +the coding gain over the uncoded probability of error per bit is 158 dB! Such a huge gain is explained +by the diversity order of the TE-QKD channel. The diversity order is defined as limγ→∞ +− log(Pe) +log(γ) +[12, +Chapters 13-14]. From Proposition 1 we know that the TE-QKD has a diversity order of 1 +2, it behaves like +a half-diversity Nakagami fading channel. The error-correcting code increases the diversity order which +is equivalent to increasing the slope of Pe(γ). An additive Gaussian noise channel without fading has +infinite diversity, with or without coding, making all curves look parallel. In presence of fading, a high +diversity converts the channel into a Gaussian channel [13]. In practice, a diversity order beyond 8 could +be barely distinguished from the local slope of e−γ on a Gaussian channel. In our case, from (55), we +deduce that the diversity order after algebraic RS decoding is (t + 1)/2. There is no asymptotic coding +on the TE-QKD channel. The coding gain increases if measured at a lower probability of error. +B. Binary BCH Codes +The TE-QKD channel does not generate error bursts. Errors are independent and the most common +event is one erroneous bit per photon before decoding. In other words, the binary-burst error-correcting +capability of Reed-Solomon codes is not exploited. Hence, we suggest to utilize a binary BCH code of +the same binary length as the RS[63, 43]64, which is 63 ×6 = 378 binary digits. We start from a primitive +length of 511 and shorten down to 378. At t = 13 the binary BCH code has a dimension k = 261. This +BCH[378, 261, t = 13]2 code yields a diversity order (t + 1)/2 = 7 better than the 5.5 order of the RS +code shown in the previous section. The number of information bits per photon is 261/378 × 3 = 2.07 +bits for N = 8 bins per frame. +Without adding any extra figure to this sub-section, the Monte Carlo simulation and the analytical bound +show that the binary BCH[378, 261, t = 13] code beats the RS[63, 43]64 code by 3 dB in signal-to-noise + +28 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 +ratio at PebRS = PebBCH = 10−5. To get the coding gain at a lower probability of error, we propose the +following very tight union bound: +a) The uncoded symbol error probability over the TE-QKD channel is Pe(γ) = +2 +√π(1 − 1 +N ) 1 +√γ. Below +10−1 a maximum of one bit error occurs in a block of m = log2(N) coded bits thanks to Gray labeling. +There are n/m such blocks per BCH codeword involving individual binary errors. +b) The bound on the probability of error in F2 after BCH decoding becomes +PebBCH(γ) = +n/m +� +i=t+1 +i +n +�n/m +i +� +P i +e(γ)(1 − Pe(γ))n/m−i. +(56) +At PebRS = PebBCH = 10−10, the binary BCH[378, 261, t = 13] code beats the RS[63, 43]64 code by +5 dB. This value corresponds to a 163 dB of BCH coding gain with respect to the uncoded photons at +N = 8 bins per frame. Notice that the reconciliation at Bob’s side for BCH codes (binary or non-binary) +is identical to the reconciliation described in the previous section for Reed-Solomon codes where the +syndrome s′ − s is fed to a Berlekamp-Massey decoder. +C. Graph-Based LDPC Codes +The big impact of LDPC codes on the performance of polarization-based QKD systems was already +demonstrated in [14] for the reconciliation of discrete random variables, with a BSC channel model. +Low-density parity-check codes [15][16] are very flexible in terms of length, coding rate, and decoding +methods. As usual, the LDPC code parity-check matrix is the adjacency matrix of a bipartite Tanner +graph with n variable nodes and n − k check nodes, assuming that the graph is (dv, dc)-regular. For finite +fields Fq with q > 2, non-zero elements of the adjacency matrix are replaced by elements from Fq \ {0}. +The standard method for decoding LDPC codes is belief propagation (BP), i.e. iterative probabilistic +decoding. Codes over a large field Fq or a large ring Z/qZ could be considered [17] in order to minimize +the loss during the symbol-to-bit soft values conversion. It is also possible to use joint local-global LDPC +codes with optimized bin mapping to achieve good performance [18] or apply multilevel-coding as in [19], +although these papers consider a different QKD channel model. In this paper, we show the impact of LDPC +codes on TE-QKD with a (3, 9)-regular binary LDPC code only. The coding rate is 2/3 guaranteeing 2 +exchanged bits per photon when the frame has 8 bins, however we consider a short length n = 384 +(64×6) comparable to the RS and BCH codes given in the previous sub-sections, and a longer code with +n = 9999 to illustrate a performance close enough to Shannon limit. + +29 +The symbol/bin APP is found via (31), where APP(i) = APP( ˆX = i) is the a posteriori probability +of bin number i, i ∈ ZN. Then the APP of binary digit bℓ, where ℓ ∈ Zm, m = log2(N), is derived by +the following marginalization +APP(bℓ) = +� +i∈ZN : bℓ +APP( ˆX = i). +(57) +The above marginalization depends on the type of binary labeling. Our paper is restricted to N bins +per frame with a Gray labeling of log2(N) bits per bin. Figure 10 shows the bit error-rate versus γ for +the binary LDPC code on the TE-QKD soft-output channel at n = 384 bits and n = 9999 bits. They +respectively gain 12 and 16 dB with respect to the BCH[378, 261] code, at a bit error probability of 10−5. If +compared to the uncoded TE-QKD, the coding gain is 73 dB and 77 dB respectively. At length n = 9999, +the LDPC code is on top of the Shannon limit for a TE-QKD hard-output channel (γlimit = 12.61 dB) +and is 2 dB only from the Shannon limit of the soft-output TE-QKD channel (γlimit = 10.45 dB). We see +no reason for using longer LDPC codes to catch an extra 1-2 dB given that the total coding gain with +respect to the no-coding case already equals 77 dB! +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 + 10 + 15 + 20 + 25 + 30 + 35 +Bit Error Probability +SNR (dB) +RS[63,43]64 +BCH[378,261] +LDPC[384,256] +LDPC[9999,6666] +Shannon Limit (Hard) +Shannon Limit (Soft) +Fig. 10. +Performance of the (3, 9)-regular binary LDPC code at length n = 384 bits and n = 9999 bits on the soft-output time-entanglement +QKD channel, for N = 8 bins per frame, transmitting 2.0 information bits per photon. +In practice, if a lab system implementation requires a less complex expression for APP( ˆX = i) without +the erfc()/Q() function and without integration, (31) can be simplified by assuming that the Gaussian +density has the effect of a Dirac impulse at small σ and using the ∝ symbol (proportional to) since the + +30 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 +denominator does not depend on the index i, we get: +APP(i) ∝ +� N +0 +1 +√ +2πσ2e− (y−u)2 +2σ2 +· +� +Q +�i − u +σ +� +− Q +�i + 1 − u +σ +�� +du +∝ +� +Q +�i − y +σ +� +− Q +�i + 1 − y +σ +�� +. +Then, depending on the sign of the arguments i − y and i + 1 − y, we approximate Q(x) by 1 +2e− x2 +2 (if +x ≥ 0) and by 1− 1 +2e− x2 +2 (if x < 0). Let j = ⌊Y ⌋ be the bin position of Y on Bob’s side, i.e. Y ∈ [j, j+1[. +The simplified APP expressions become: +If i = j, +APP(i) ∝ +� +1 − 1 +2e− (y−i)2 +2σ2 +− 1 +2e− (y−i−1)2 +2σ2 +� +, +(58) +If i ̸= j, +APP(i) ∝ sign(j − i) · +� +1 +2e− (y−i−1)2 +2σ2 +− 1 +2e− (y−i)2 +2σ2 +� +. +(59) +When (58)-(59) are utilized in the BP decoder of the binary LDPC code over the TE-QKD soft-output +channel, the loss is limited to 0.25-0.30 dB with respect to the exact expression (31). This is a minuscule +loss when dealing with coding gains above 50 dB. +Notice that we are not showing a performance of the LDPC code over a hard-output channel. Indeed, +optimal BP decoding is identical whether the channel output is soft or not, i.e., the BP decoder is the +same decoder on both a Gaussian-like channel and a BSC-like channel. The gap between hard and soft +is about 8.5 dB for the LDPC[384, 256] and about 4 dB for the LDPC[9999, 6666] at a bit error rate of +10−5. Suppose the system implementation possesses an optimal BP decoder, but the exact photon position +is unavailable; only the bin number is available. In such a case, the lab implementation is forced to use +LDPC codes on a hard-output channel, and the binary digits APP expression (57) becomes +APP(bℓ) ∝ +� +i∈ZN : bℓ +ˆπi × pi,j, +(60) +where j = ⌊Y ⌋, ˆπi is given by (22) or (37)-(38) at small σ, and pi,j is given by (30) or (35)-(36) in +the small σ regime. Coding theorists and practitioners could also use convolutional codes, turbo codes, +polar codes, and other binary or non-binary algebraic codes with short or moderate length to achieve large +coding gains on the TE-QKD channel. +D. A summary of capacity limits at different frame sizes +We complete the current section by a table summarizing important information theoretical limits on the +time-entanglement QKD channel, with both hard and soft output. Shannon limit in terms of SNR is the + +31 +value of the non-normalized signal-to-noise ratio γ such that mutual information is equal to the targeted +information exchange rate, I( ˆX; ˆY ) = k +n log2(N) for a hard output and I( ˆX; Y ) = k +n log2(N) for a soft +output. Table II has seven columns with parameters covering 8 bins per frame up to 64 bins per frame. +The last two rows correspond to SNR and standard deviation values achieved by the BCH and the LDPC +codes as found in sub-sections VII-B and VII-C. +TABLE II +INFORMATION THEORETICAL (SHANNON) LIMITS FOR TE-QKD. +N +Bits +R = k/n +SNR limit +σ/N +SNR limit +σ/N +bins per frame +per photon +code rate +hard +hard +soft +soft +8 +2.0 +2/3 +12.61 dB +0.029269 +10.45 dB +0.037533 +16 +3.0 +3/4 +13.29 dB +0.013532 +10.85 dB +0.017922 +32 +3.0 +3/5 +3.88 dB +0.019992 +3.46 dB +0.020982 +32 +4.0 +4/5 +13.61 dB +0.0065215 +11.04 dB +0.0087670 +64 +4.0 +2/3 +4.01 dB +0.0098474 +3.58 dB +0.010347 +64 +5.0 +5/6 +13.77 dB +0.0032012 +11.13 dB +0.0043383 +8 +2.0 +2/3 +28.49 dB +0.0047034 +BCH, n=378 +Peb = 10−5 +achieved +8 +2.0 +2/3 +12.47 dB +0.029745 +LDPC, n=9999 +Peb = 10−5 +achieved +The signal-noise ratio soft-decoding limits listed in Table II appear to be close to two values, one SNR +around 10-11 dB and a lower SNR around 3.5 dB. The hard-decoding limits are higher than soft-decoding +limits, because I( ˆX; ˆY ) ≤ I( ˆX; Y ), the gap depends on the frame size and the coding rate. Of course, the +hard-soft gap vanishes at small coding rates (below 1/2) and increases at high coding rates when mutual +information approaches the asymptote log2(N). +The two typical values of soft-decoding SNR limits are explained or interpreted for small σ via (49): +1 +2 log +� γ +4πe +� += log2(N) − b, +(61) +where γ = N2γ and b is a backoff value. Here, b = 1 bit or b = 2 bits in Table II. Then, solving (61) +yields γ = (4πe)/22b. We get γ = 9.31 dB for b = 1 and γ = 3.29 for b = 2. The difference with the +values in the 6th column of Table II is due to I( ˆX; Y ) going away from the envelope I(X; Y ) to follow +its own asymptote. We hope that SNR limits given in Table II will be useful to physicists and coding +theorists working in this QKD field. + +32 +FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 +VIII. CONCLUSIONS +We focused on the time entanglement-based QKD when the photon arrival detectors suffer from time +jitter. We presented a rigorous analysis of secret key information rates and proposed and tested several +codes for information reconciliation to approach the maximum secret key rates. These achievable secret key +rates are much higher than the maximum achievable by polarization entanglement-based QKD. However, +practical photon detectors suffer from other impairments, e.g., dark currents and downtime, which may +cause further rate loss. These impairments should be a subject of future work. +REFERENCES +[1] E. Diamanti, H.-K. Lo, B. Qi, and Z. Yuan, “Practical challenges in quantum key distribution,” npj Quantum Information, vol. 2, Nov. +2016. +[2] R. Nandal, A. Nandal, K. Joshi, and A. Rathee, “A survey and comparison of some of the most prominent QKD protocols,” SSRN +Electronic Journal, 01 2021. +[3] C. Lee, D. Bunandar, Z. Zhang, G. R. Steinbrecher, P. B. Dixon, F. N. C. Wong, J. H. Shapiro, S. A. Hamilton, and D. Englund, +“High-rate field demonstration of large-alphabet quantum key distribution,” 2016. +[4] M. C. Sarihan, K.-C. Chang, X. Cheng, Y. S. Lee, T. Zhong, H. Zhou, Z. Zhang, F. N. Wong, J. H. Shapiro, and C. W. Wong, “High +dimensional quantum key distribution with biphoton frequency combs through energy-time entanglement,” in Conference on Lasers +and Electro-Optics, p. FTh1A.3, Optical Society of America, 2019. +[5] D. B. IV, C. Cheng, and E. Soljanin, “Information rates with non ideal photon detectors in time-entanglement based QKD,” IEEE +Trans. Commun., submitted, arXiv preprint arXiv:2207.04146, July 2022. +[6] A. E. Gamal and Y. Kim, Network Information Theory. Cambridge University Press, 2011. +[7] T. Zhong, H. Zhou, R. D. Horansky, C. Lee, V. B. Verma, A. E. Lita, A. Restelli, J. C. Bienfang, R. P. Mirin, T. Gerrits, S. W. Nam, +F. Marsili, M. D. Shaw, Z. Zhang, L. Wang, D. Englund, G. W. Wornell, J. H. Shapiro, and F. N. C. Wong, “Photon-efficient quantum +key distribution using time-energy entanglement with high-dimensional encoding,” NEW JOURNAL OF PHYSICS, 2015. +[8] H. Zhou and G. W. Wornell, “Adaptive pulse-position modulation for high-dimensional quantum key distribution,” in Proceedings of +the 2013 IEEE International Symposium on Information Theory, Istanbul, Turkey, July 7-12, pp. 359–363, 2013. +[9] E. Karimi, E. Soljanin, and P. Whiting, “Increasing the raw key rate in energy-time entanglement based quantum key distribution,” in +54th Asilomar Conf. on Signals, Systems, and Computers, ACSCC 2020, Pacific Grove, CA, USA, November 1-4, pp. 433–438, 2020. +[10] T. M. Cover and J. A. Thomas, Elements of Information Theory. John Wiley & Sons, 2006. +[11] R. E. Blahut, Algebraic codes for data transmission. Cambridge University Press, 2003. +[12] J. G. Proakis and M. Salehi, Digital communications. USA: McGraw-Hill, 5th ed., 2008. +[13] J. J. Boutros and E. Viterbo, “Signal space diversity: a power- and bandwidth-efficient diversity technique for the rayleigh fading +channel,” IEEE Trans. on Information Theory, vol. 42, pp. 502–518, July 1998. +[14] D. Elkouss, A. Leverrier, R. Alléaume, and J. J. Boutros, “Efficient reconciliation protocol for discrete-variable quantum key distribution,” +IEEE Information Theory Symposium, ISIT 2009, Seoul, Korea, June 2009. +[15] R. G. Gallager, Low-Density Parity-Check Codes. Cambridge, MA: MIT Press, 1963. +[16] T. Richardson and R. Urbanke, Modern Coding Theory. Cambridge University Press, 2008. + +33 +[17] J. J. Boutros, U. Erez, J. V. Wonterghem, G. I. Shamir, and G. Zémor, “Geometric shaping: low-density coding of gaussian-like +constellations,” IEEE Information Theory Workshop, ITW 2018, Guangzhou, China, Nov. 2018. +[18] S. Yang, M. C. Sarihan, K. Chang, C. W. Wong, and L. Dolecek, “Efficient information reconciliation for energy-time entanglement +quantum key distribution,” in 53rd Asilomar Conference on Signals, Systems, and Computers, ACSCC 2019, Pacific Grove, CA, USA, +pp. 1364–1368, Nov. 2019. +[19] H. Zhou, L. Wang, and G. W. Wornell, “Layered schemes for large-alphabet secret key distribution,” in 2013 Information Theory and +Applications Workshop, ITA 2013, San Diego, CA, USA, pp. 1–10, Feb. 2013. + diff --git a/v9AyT4oBgHgl3EQfnPij/content/tmp_files/load_file.txt b/v9AyT4oBgHgl3EQfnPij/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aac1ce830af08b990962a118fe7c25e88ce1df19 --- /dev/null +++ b/v9AyT4oBgHgl3EQfnPij/content/tmp_files/load_file.txt @@ -0,0 +1,1125 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf,len=1124 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='00486v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='IT] 1 Jan 2023 FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 1 Time-Entanglement QKD: Secret Key Rates and Information Reconciliation Coding Joseph J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Boutros, Senior Member, IEEE, and Emina Soljanin, Fellow, IEEE Abstract In time entanglement-based quantum key distribution (QKD), Alice and Bob extract the raw key bits from the (identical) arrival times of entangled photon pairs by time-binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Each of them individually discretizes time into bins and groups them into frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' They retain only the frames with a single occupied bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Thus, Alice and Bob can use the position of the occupied bin within a frame to generate random key bits, as in PPM modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Because of entanglement, their occupied bins and their keys should be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' However, practical photon detectors suffer from time jitter errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' These errors cause discrepancies between Alice’s and Bob’s keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Alice sends information to Bob through the public channel to reconcile the keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The amount of information determines the secret key rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' This paper computes the secret key rates possible with detector jitter errors and constructs codes for information reconciliation to approach these rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Index Terms Quantum key distribution, secret key rates, mutual information, time entanglement, time binning, jitter errors, soft-decision decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' INTRODUCTION Secret key distribution protocols establish a shared sequence of bits between two (or more) distant parties, Alice and Bob, in the presence of an eavesdropper, Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The key consists of uniformly random independent bits known only to Alice and Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Quantum Key Distribution (QKD) starts by communicating quantum states over a quantum channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The role of the quantum step is to 1) ensure that no eavesdropping goes undetected and 2) provide a source of perfect randomness in the entanglement-based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Joseph J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Boutros is with the Department of Electrical and Computer Engineering, Texas A& M University, 23874 Doha, Qatar, e-mail: boutros@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='org (see https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='josephboutros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Emina Soljanin is with the Department of Electrical and Computer Engineering, Rutgers, the State University of New Jersey, Piscataway, NJ 08854, USA, e-mail: (see https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='ece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='rutgers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='edu/emina-soljanin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' This research is based upon work supported by the National Science Foundation under Grant # FET-2007203 2 FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 There has been a significant effort to provide high key rates over long distances (see recent surveys [1], [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' QKD schemes based on time-entangled photons have emerged as a promising technique primarily because each entangled photon pair can carry multiple key bits and thus potentially provide a higher secure key rate over long distances [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Time-entanglement-based QKD (TE-QKD) schemes use Spontaneous Parametric Down-Conversion (SPDC) to generate entangled photon pairs according to a Poisson Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' One of the photons goes to Alice, and the other to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Therefore, Alice and Bob ideally detect their photons simultaneously with exponentially distributed photon inter-arrival times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The most common single-photon detectors are Superconducting Nanowire Single-Photon Detectors (SNSPDs), which exhibit properties closest to ideal sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' They have low dark count rates, meaning they rarely report photon detection without a photon arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Furthermore, they have low detector downtime d and slight detector timing jitter that manifests as Gaussian noise with zero mean and variance σ2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Unfortunately, these imperfections are non-negligible: 1) detector jitters and dark counts cause disagreements between Alice’s and Bob’s keys, and 2) the downtime introduces memory within the raw key bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The secret key rate loss due to the non-ideal properties of these detectors has been studied most recently in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' At a high level, there are two main QKD steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In the first step, Alice and Bob generate raw key bits using a quantum channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Their respective raw keys may disagree at some positions, be partly known to Eve, and may not be uniformly random because of the aforementioned non-ideal detector properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In the second step, Alice and Bob process the raw key to establish a shared secret key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' They communicate through the public classical channel to reconcile differences between their raw keys, amplify the privacy of the key concerning Eve’s knowledge, and compress their sequences to achieve uniform randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' At the end of the protocol, Alice and Bob 1) have identical uniformly random (binary) sequences and 2) are confident the shared sequence is known only to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Therefore the secret key is private and hard to guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' This paper focuses on the information reconciliation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Alice and Bob obtain correlated streams of bits (raw keys) by detecting the arrival times of their entangled photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' However, they must communicate over a public channel to agree on a key, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=', reconcile their differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Here, we consider one-way information reconciliation schemes in which Alice sends information about her sequence to Bob, who uses it to remove the differences between his and Alice’s raw keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' After the information reconciliation, Alice and Bob share Alice’s initial raw key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' However, the shared key is not secret because of the public channel communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Alice and Bob perform privacy 3 amplification to correct that, establishing secrecy but shortening the key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Since Alice and Bob base their secret key generation on correlated photon arrival times, they follow what is known as the source model in Information Theory [6, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The secrecy capacity for this model when the eavesdropper has access to public communication but does not have correlated prior information is equal to the mutual information between Alice’s and Bob’s observations (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=', [6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 567]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The secrecy capacity is an achievable upper bound on the post-privacy amplification rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Alice and Bob generate their secret keys from the correlated random photon arrivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' There are many ways to extract keys from this correlated information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' One popular method is similar to Pulse Position Modulation (PPM);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=', [7] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (Some recently proposed adaptive schemes avoid discarding frames with multiple occupied bins [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=') In PPM, Alice and Bob synchronize their clocks and discretize their timelines into time frames N time bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In PPM, Alice and Bob agree to retain only time frames in which they both detect a single photon arrival and discard all other frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' This single photon is said to occupy a time bin depending on where within the frame it arrives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Since photon inter-arrival times follow an exponential distribution, each bin is occupied independently of other bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Therefore, the number of raw key bits that PPM decoding can extract from each frame equals log N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' This paper focuses on practical photon detectors that suffer from time jitter errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Since these errors cause discrepancies between Alice’s and Bob’s keys, Alice must send information to Bob through the public channel to reconcile the keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The amount of information determines the secret key rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' This paper computes the secret key rates possible with detector jitter errors and constructs codes for information reconciliation to approach these rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' This paper is organized as follows: Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' II introduces notation and lists the paper’s main contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' III presents the TE-QKD channel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' IV computes the rates of raw key disagreement caused by detection jitter, and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' V derives the correlations between Alice’s and Bob’s raw keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' VI computes achievable information rates and the secrecy capacity of the TE-QKD channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' VII proposes and tests several coding schemes for information reconciliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' NOTATION AND MAIN CONTRIBUTIONS The number N of bins per time frame could be any positive integer greater than or equal to 2, our propositions, lemmas, and theorems have no other constraint on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' However, our numerical examples are given for N = 2m, m integer, m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The set ZN denotes the set of N integers {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' , N − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The notation ⌊x⌋, known as the floor of x for x ∈ R, is the largest integer smaller than or equal to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 4 FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 Letters such as X, Y , ˜X, and ˜Y denote continuous random variables, while ˆX and ˆY are discrete random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Then, p(ˆy|ˆx) denotes the conditional probability P( ˆY = ˆy| ˆX = ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Also, p(y|ˆx) denotes the conditional density pY | ˆ X(y|ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We use Bourbaki’s notation for intervals on the real line, where a and b are two real numbers: the closed interval [a, b] = {x ∈ R : a ≤ x ≤ b}, the half-open intervals [a, b[= [a, b] \\ {b} and ]a, b] = [a, b] \\ {a}, and the open interval ]a, b[= [a, b] \\ {a, b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We use the standard Bachmann-Landau big O notation: The formal definition of f(σ) = O(g(σ)) is: ∃α > 0, ∃σ0 > 0, ∀σ < σ0, |f(σ)| ≤ α|g(σ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In this paper, an expression such as 1−O(g(σ)) or 1+O(g(σ)) implicitly assumes that g(σ) > 0 in some open interval ]0, σ0[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Furthermore, we will frequently use γ = 1/σ2, a signal-to-noise ratio defined as the inverse of the jitter variance, then we could write f(γ) = O(g(γ)) in a similar situation when γ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Two functions f : R → R and g : R → R are asymptotically equivalent if limγ→∞ f(γ) g(γ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In that case, we write f(γ) ∼ g(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The function Q(x) = 1 2 erfc( x √ 2) = O(exp(−x2/2)) is the Gaussian tail function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Recall the definition Q(x) = � ∞ x φ(t)dt, where φ(t) = 1 √ 2π exp(−t2/2) is the standard normal density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Furthermore, we recall the binary entropy function, H2(x) = −x log(x) − (1 − x) log(1 − x), and the symmetric ternary entropy function, H3(x) = −(1 − 2x) log(1 − 2x) − 2x log(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The main contributions of this paper constitute a full characterization of the time-entanglement QKD channel, from information theory and coding theory point of view: We derive the error rates of the TE-QKD channel, and prove that the TE-QKD channel behaves like an 1/2-diversity Nakagami fading channel, see Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We find the exact a priori probability of bins given that both Alice’s and Bob’s frames are valid, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We establish the exact conditional density of Bob’s photon position given Alice’s photon bin, for a soft-output TE-QKD channel, see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The output density expression is also determined, see (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We determine the expression of the transition probabilities of the discrete (hard-output) TE-QKD channel, see Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We give the exact expression of the a posteriori probability for the soft-output TE-QKD channel, see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 5 We derive the exact formula for the mutual information I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˆY ) (hard-output) and find simplified expressions in the small-noise regime, see (33), (34), (40), and Proposition 2-c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The exact formula for the mutual information I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) (soft-output) is given, see (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We also determine all densities needed to compute the maximal rate I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) and we give a nice log-formula expression in the small-noise regime, see Theorem 3 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The last section, Section VII, shows new results with huge coding gains obtained by short and moderate-length error-correcting codes such as RS, BCH, and LDPC codes under algebraic hard- decision decoding and probabilistic soft-decision decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' PPM CHANNEL MODEL Let ˜X and ˜Y represent the time-position of the received photons at Alice’s and Bob’s sides, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' An illustration of this QKD scheme is given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Alice Bob 0 1 2 Y frame optical channel optical channel Entangled photons 0 1 2 X frame N−1 N−1 photon position photon position Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' QKD based on time entanglement with N bins per frame, log2(N) binary digits per bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We adopt the following mathematical model for the positions of two time-entangled photons: ˜X = U + Z1, ˜Y = U + Z2, (1) where Z1 and Z2 are independent identically distributed N (0, σ2) additive Gaussian noises modeling the detection jitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' U is a real uniform random variable in the interval [0, N[, where the integer N = 2m is the number of bins per frame, and m is the number of bits per photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Alice and Bob communicate via a public channel and agree on a valid frame when ˜X and ˜Y fall in the interval [0, N[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' They reject empty frames and frames with more than one received photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Under the model defined by (1), the probability of a frame to be valid for both Alice and Bob is P( ˜X, ˜Y ∈ [0, N[).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Let X and Y denote the instances of 6 FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 ˜X and ˜Y within the interval [0, N[, and let ˆX and ˆY be the bin number inside a frame, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=', ˜X = X, for ˜X ∈ [0, N[, ˜Y = Y, for ˜Y ∈ [0, N[, (2) ˆX = ⌊X⌋ ∈ ZN, ˆY = ⌊Y ⌋ ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (3) From an information theoretical perspective, we distinguish two communication channels between Alice and Bob: (a) an algebraic (hard) output channel, (b) a real (soft) output channel, both having a discrete N-ary input ˆX as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (a) hard output (b) soft output ˆX ˆX ˆY p(ˆy|ˆx) Y p(y|ˆx) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Channel models for hard-decision decoding (a) and soft-decision decoding (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Without error-correcting codes, the information rate on these channels is log2(N) = m bits per channel use (bpcu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The main channel parameter γ is a signal-to-noise ratio parameter (SNR) defined as γ = Es σ2 = 1 σ2, (4) where the average energy per symbol Es = 1 is a normalized energy cost per transmitted photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Another QKD channel parameter is γ, referred to as the normalized signal-to-noise ratio, where the standard deviation of the additive Gaussian noise is normalized by the frame length N, hence its definition is γ = 1 (σ/N)2 = N2 σ2 , γ(dB) = γ(dB) + 20 log10(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (5) We express the probability of error and the information rate as functions of N and the SNR γ or the normalized SNR γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The bin width within a frame is set to 1 to simplify the analysis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=', the frame width is N in all sections except for Section VI-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The conversion of this mathematical model into a physical model representing a laboratory experiment is straightforward after introducing a time scale to convert γ and N into physical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In Section VI-B, the number of bins is infinite (it’s a continuum of bins), the frame has a unit length and γ = γ in that special QKD channel with both soft input and soft output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' RATE OF RAW KEY DISAGREEMENT UNDER DETECTION JITTER We consider the probability of error Pe(γ) = P( ˆX ̸= ˆY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The probability Pe characterizes the quality of channel (a) in Figure 2 defined by its transition probabilities p(ˆy|ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The latter will be entirely determined in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In the current section, we are interested in determining the expression of Pe(γ) as a function 7 of the signal-to-noise ratio γ, for a given number of bins N per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Let πi = P( ˆX = i), i ∈ ZN, be the a priori probability of the unique frame photon to fall in bin number i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Then, the exact expression of the probability of error is Pe(γ) = N−1 � i=0 πi N−1 � j=0 j̸=i p(ˆy = j|ˆx = i) = 1 N N−1 � i=0 P( ˆY ̸= ˆX| ˆU = i), (6) where ˆU = ⌊U⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Since U is uniform in [0, N), we get P( ˆU = i) = P(U ∈ [i, i + 1)) = 1 N which explains the factor in the last equality above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' As a first step, in the current section, we solve Pe(γ) from the most right equality in (6) via the conditioning over ˆU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' To avoid cumbersome expressions, exact expressions as established in Sections V&VI, we assume that γ is large enough (σ2 is small enough) so we can neglect the border effects in the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Hence, we make no difference here between ˜X and X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˜Y and Y ), and we use the approximation that both X and Y are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Gaussian when conditioning on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The probability of symbol error Pe(γ) = P( ˆX ̸= ˆY ) as a function of the SNR γ and the number N of bins per frame is given by the expression Pe(γ) = 2 √π × � 1 − 1 N � × γ− 1 2 + O(exp(−γ 4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (7) Proof: Set V = U − ˆU, so V is Uniform[0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Let p(i → j|v) be the probability of falling in bin j given that ˆU = i and V = v, where i, j ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' A symbol error occurs if X = U + Z1 remains in bin i but Y = U + Z2 leaves to bin j, j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The probability of such an event is p(i → i|v) × p(i → j|v), given that both additive Gaussian noises Z1 and Z2 are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Also, an error occurs if both X and Y leaves to two different bins ℓ and j, with probability p(i → ℓ|v) × p(i → j|v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Then, the conditional symbol error probability becomes Pe(i, v) = 2 \uf8ee \uf8ef\uf8ef\uf8f0 N−1 � j=0 j̸=i p(i → i|v)p(i → j|v) + N−1 � ℓ=0 ℓ̸=i N−1 � j=0 j̸=i,j̸=ℓ p(i → ℓ|v)p(i → j|v) \uf8f9 \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The factor of 2 is due to the symmetry if the two letters X and Y are switched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' As illustrated in Figure 3, we will neglect bins beyond the left and the right bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The neglected bins are at least at distance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0 from the bin ˆU = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' They correspond to a probability of error Q(1/σ) = O(exp(−1/(2σ2))) = O(exp(−γ 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 8 FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 To further simplify the notations, define p1, p2, and p3, where p1 = p(i → i|v) = 1 − Q �v σ � − Q �1 − v σ � , (8) p2 = p(i → i − 1|v) = Q �v σ � , (9) and p3 = p(i → i + 1|v) = Q �1 − v σ � , (10) we obtain Pe(i, v) = O(exp(−γ 2)) + \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 2[p1p2 + p1p3 + p2p3], for i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' N − 1, 2p1p3, for i = 0, 2p1p2, for i = N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' neglected neglected left bin right bin 1 − v v bin ˆU = i V = v ∼ Uniform[0, 1] p1 p2 p3 i − 1 i i + 1 i + 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Illustration of the probability of error in bin position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Now, integrate over v, P( ˆY ̸= ˆX| ˆU = i) = � 1 0 Pe(i, v) dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Then, apply (6) and use � 1 0 p1p2dv = � 1 0 p1p3dv to finally reach Pe(γ) = 1 N N−1 � i=0 � 1 0 Pe(i, v) dv (11) = 4 � 1 − 1 N � � 1 0 p1p2 dv + 2 � 1 − 1 N � � 1 0 p2p3 dv + O(exp(−γ 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (12) The two integrals in (12) include three types of integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Let us process them step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' I1 = � 1 0 Q �v σ � dv = σ(1 − e−1/2σ2) √ 2π + Q � 1 σ � = σ √ 2π + O(exp(−γ 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 9 I2 = � 1 0 � Q �v σ ��2 dv = 2 √ 2σ − 2σ(1 − 2Q( √ 2/σ)) + 4√πQ2(1/σ) − 4 √ 2σe−1/2σ2Q(1/σ) 4√π = ( √ 2 − 1)σ 2√π + O(exp(−γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' I3 = � 1 0 Q �v σ � Q �1 − v σ � dv ≤ � 1 0 exp �−v2 − (1 − v)2 2σ2 � dv = O(exp(−γ 4)), since v2 + (1 − v)2 ≥ 1 2 for v ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' I1 and I2 were solved via integration by parts using the fact that dQ(x) dx = −φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' I3 has no simpler form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Our upper bound of I3 brings a sufficient answer to the current proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' After substituting I1, I2, and I3 into (12), we get (7) as stated by the proposition, where σ = γ− 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The expression 2 √π × � 1 − 1 N � ×γ− 1 2 perfectly fits the Monte Carlo simulation of P( ˆY ̸= ˆX) even for a signal-to-noise ratio as low as 20dB (error rate close to 10−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Figure 4 shows the plots of the probability of error Pe(γ) for different number of bins per frames, from 1 bit per photon up to 4 bits per photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The plots of the probability of error versus the normalized SNR, Pe(γ), are obtained from Figure 4 after shifting right each curve by 20 log10(N) decibels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 10-5 10-4 10-3 10-2 10-1 0 20 40 60 80 100 Probability of Error per Symbol SNR (dB) N=2 N=4 N=8 N=16 Q(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='5/sigma)=O(exp(-gamma/4)) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Probability of symbol error versus SNR, log2(N) bits per photon, no coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' CORRELATION BETWEEN RAW KEYS The conditional densities of ˜X is directly derived from (1), p ˜ X|U(˜x|u) = 1 √ 2πσ2 exp � −(˜x − u)2 2σ2 � , ˜x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (13) 10 FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 Conditioned on U = u, ˜X and ˜Y are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Then, after integrating (13), P( ˜X, ˜Y ∈ [0, N[|u) = P( ˜X ∈ [0, N[|u)2 = � Q � −u σ � − Q �N − u σ ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' So the probability of both Alice’s and Bob’s frames are valid is P( ˜X, ˜Y ∈ [0, N[) = � N 0 � Q � −u σ � − Q �N − u σ ��2 pU(u) du = 1 N � N 0 � Q � −u σ � − Q �N − u σ ��2 du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (14) The density of ˜X is also derived by integrating over u, which is equivalent to convolving the densities of U and Z1, we get p ˜ X(˜x) = � N 0 p ˜ X|U(˜x|u) · 1 N du = 1 N � Q �−˜x σ � − Q �N − ˜x σ �� , ˜x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (15) Since X is a version of ˜X truncated to the interval [0, N[, conditioning on U + Z1 ∈ [0, N[, the density of X is determined by scaling the density of ˜X, namely pX|U(x|u) = p ˜ X|U(x|u) � N 0 p ˜ X|U(t|u) dt = 1 √ 2πσ2 exp � −(x−u)2 2σ2 � � Q � − u σ � − Q � N−u σ ��, x, u ∈ [0, N[, (16) and pX(x) = p ˜ X(x) � N 0 p ˜ X(t) dt = Q � − x σ � − Q �N−x σ � � N 0 � Q � − t σ � − Q �N−t σ �� dt , x ∈ [0, N[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (17) By symmetry from (1), p ˜Y |U(˜y|u), p ˜Y (˜y), pY |U(y|u), and pY (y) have expressions identical to (13), (15), (16), and (17) respectively, for ˜y ∈ R and y ∈ [0, N[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The bins a priori probabilities πi = P( ˆX = i) = P(X ∈ [i, i + 1]) become, πi = P( ˆX = i) = � i+1 i pX(x) dx = � i+1 i � Q �−x σ � − Q � N−x σ �� dx � N 0 � Q � − t σ � − Q �N−t σ �� dt , i ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (18) At high SNR, for σ2 ≪ 1, we have πi ≈ 1/N, ∀i, because the truncation to the interval [0, N[ has less effect in the small-noise regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Numerical examples are given in Table I, for N = 8 bins per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The entropy of ˆX is very stable, as listed in the last column of the table, H( ˆX) = − �N−1 i=0 πi log2(πi) ≈ log2(N) at low and high signal-to-noise ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' TABLE I A PRIORI PROBABILITIES OF PHOTON BINS FOR N = 8 BINS PER FRAME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' SNR π0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' , π7 H( ˆX) (bits) 10 dB 0.' metadata={'source': 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+page_content='125705, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='125705, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='125705, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='122885 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='999931 40 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='124626, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='125125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='125125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='125125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='125125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='125125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='125125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='124626 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='999998 11 The following lemma helps understand the analytic behavior of (14)-(18) at high SNR, when σ2 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Let fσ(x) = Q � − x σ � −Q � 1−x σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' For σ > 0 and γ = 1/σ2, given the properties of the Gaussian tail function Q(x), the difference function fσ(x) satisfies a) ∀x ∈ R, fσ(x) = fσ(1 − x) ∈]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Also, fσ(0) = fσ(1) = 1 2 − O(exp(−γ 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' b) For x ∈]0, 1[, fσ(x) = 1 − O(exp(− min2(x, 1 − x) · γ 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' c) For x < 0, we have fσ(x) = O(exp(−x2 · γ 2)), and fσ(x) = O(exp(−(x − 1)2 · γ 2)) for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' d) Integrating fσ and f 2 σ, we get � 1 0 fσ(x) dx = 1 − � 2 π · 1 √γ + O(exp(−γ 2)) = 1 − O( 1 √γ) and � 1 0 f 2 σ(x) dx = 1 − 1+ √ 2 √π · 1 √γ + O(exp(−γ 4)) = 1 − O( 1 √γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' e) � (i+1)/N i/N fσ(x) dx = 1 N − O( 1 √γ) for i = 0 and i = N − 1 (the two extreme bins in a frame of N bins).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' � (i+1)/N i/N fσ(x) dx = 1 N + O(exp(−βγ)) for i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' N − 2 (the inner bins), where the exponent constant is β = 1 2 min2( i N , 1 − i+1 N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Proof: For a), let G be a standard normal random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The finite interval [−x, 1 − x] is never reduced to a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We get fσ(x) = P(G ∈ [−x, 1 − x]) ∈]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Then, fσ(1 − x) = Q � −(1−x) σ � − Q � x σ � = 1 − Q � 1−x σ � − 1 + Q � − x σ � = fσ(x), using the property Q(−x) = 1 − Q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Finally fσ(0) = Q(0) − Q � 1 σ � = 1 2 − O(exp(−γ 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' For b), we write fσ(x) = 1−Q � x σ � −Q � 1−x σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Then Q � x σ � +Q �1−x σ � ≤ 1 2 exp(−x2/(2σ2))+ 1 2 exp(−(1− x)2/(2σ2)) ≤ exp(− min2(x, 1−x)γ/2) which yields the announced result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' This inequality is only useful to us for x ∈]0, 1[ to keep the exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' For c), x < 0, so 1 − x > −x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Then fσ(x) ≤ Q �−x σ � ≤ 1 2 exp(−x2/(2σ2)) = O(exp(−x2γ/2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The proof is similar for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' As mentioned for I1 in the proof of Proposition 1, the anti-derivative of Q(ax), a, x ∈ R, is determined after integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We get � Q(ax)dx = xQ(ax) − 1 √ 2πa2 exp(−a2x2/2) + c, where c is the integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (19) For d), � 1 0 fσ(x) dx = � 1 0 � 1 − Q � x σ � − Q �1−x σ �� dx = 1 − 2I1 = 1 − � 2 π · 1 √γ + O(exp(−γ 2)), where I1 is solved thanks to (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' As mentioned for I2 in the proof of Proposition 1, the anti-derivative of [Q(ax)]2, a, x ∈ R, is also determined by integration by parts and the application of (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We get � Q2(ax)dx = xQ2(ax) − � 2 πa2Q(ax) exp(−a2x2/2) + 1 √ πa2Q(ax √ 2) + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (20) 12 FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 Then, � 1 0 f 2 σ(x)dx = 1 − 4I1 + 2I2 + 2I3 = 1 − 4 × σ √ 2π + 2 × √ 2−1 2√π σ + O(exp(−γ 4)), where I2 is solved thanks to (20) and I3 = O(exp(−γ 4)) as shown before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' This completes the proof of d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The proof of e) is mainly based on (19), after taking care of the bin position within the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We have I4 = � (i+1)/N i/N fσ(x) dx = � (i+1)/N i/N � 1 − Q �x σ � − Q �1 − x σ �� dx = 1 N − � (i+1)/N i/N Q �x σ � dx − � 1−i/N 1−(i+1)/N Q �x σ � dx = 1 N − �(i + 1) N Q �i + 1 Nσ � − i N Q � i Nσ � − σ √ 2πe¯ γ 2 (i+1)2 N2 + σ √ 2πe¯ γ 2 i2 N2 � − � (1 − i N )Q �1 − i N σ � − (1 − i + 1 N )Q �1 − i+1 N σ � − σ √ 2πe¯ γ 2 (1− i N )2 + σ √ 2πe¯ γ 2 (1− i+1 N )2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' If i = 0 or i = N − 1, I4 = 1 N − σ √ 2π = 1 − O( 1 √γ), all terms with exponential decay are absorbed by the O( 1 √γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' For middle bins, i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' N − 2, I4 = 1 N + O(e− γ 2 i2 N2 ) + O(e− γ 2 (1− i+1 N )2), all terms of higher decay are absorbed by these two big O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Hence, I4 = 1 N + O(e−βγ), where the exponent constant is β = 1 2 min2(i/N, 1 − (i + 1)/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The convergence of fσ(x) is not uniform in the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The point-wise convergence of fσ(x) to 0 (outside [0, 1]) or to 1 (inside [0, 1]) is very slow in the neighborhood of the points x = 0 and x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' At high SNR, the difference of the two Q() functions behaves as a square function and its integral slowly approaches 1 at a rate of 1/√γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Applying Lemma 1 to (14)-(18),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' after substituting u/N to u and σ/N to σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' proves the following equalities where ˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' u ∈]0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' N[ and i ∈ ZN: P( ˜X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˜Y ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' N[) = 1 − O( 1 � N2 · γ ) = 1 − O( 1 √γ ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' p ˜ X(˜x) = 1 N − O(exp(−min2( ˜x N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 1 − ˜x N ) · γ 2)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' pX|U(x|u) = 1 √ 2πσ2 exp � −(x − u)2 2σ2 � (1 + O(exp(−min2( u N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 1 − u N ) · γ 2)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' pX(x) = 1 N · (1 − O(exp(−min2( x N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 1 − x N ) · γ 2)) · (1 + O( 1 √γ )),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' πi = ( 1 N ± O(g(γ))) · (1 + O( 1 √γ )),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' where the vanishing rate of g(γ) depends on i as stated by the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The high SNR behavior of many expressions below could be determined via the application of the results listed in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' To complete our analysis of the QKD channel between Alice and Bob, it is necessary to find the likelihoods pY | ˆ X(y|ˆx) and the transition probabilities p ˆY | ˆ X(ˆy|ˆx) for the soft-output and the hard-output 13 mathematical models illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We proceed in a similar manner as from (13) to (17), by first integrating over U, then truncating over the interval [0, N[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Given Alice’s frame is valid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˜X ∈ [0, N[, the density of U becomes pU| ˜ X∈[0,N[(u) = Q �−u σ � − Q �N−u σ � � N 0 � Q �−t σ � − Q �N−t σ �� dt = pU| ˜Y ∈[0,N[(u), u ∈ [0, N[, (21) where pU| ˜Y ∈[0,N[(u) is the density of U given that Bob’s frame is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Furthermore, the a priori probabilities {ˆπi}N−1 i=0 when both frames are valid are given by ˆπi = P( ˆX = i| ˜Y ∈ [0, N)) = � N 0 � Q �i−u σ � − Q � i+1−u σ �� � Q � −u σ � − Q �N−u σ �� du � N 0 � Q �−u σ � − Q � N−u σ ��2 du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (22) Proof: Let us apply Bayes’ rule, while dropping the subscripts to simplify the notation: p(u| ˜X ∈ [0, N[) = P( ˜X ∈ [0, N)|u) × pU(u) P( ˜X ∈ [0, N[) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' From (13), we get P( ˜X ∈ [0, N[|u) = Q � −u σ � − Q �N−u σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' From (15), we get P( ˜X ∈ [0, N[= 1 N � N 0 � Q �−t σ � − Q �N−t σ �� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Finally, plugging pU(u) = 1/N leads to the result announced by the lemma in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The equality pU| ˜ X∈[0,N[(u) = pU| ˜Y ∈[0,N[(u) is the result of the symmetry between Alice and Bob in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The a priori probability ˆπi is derived after establishing the density of ˜X conditioned on a valid frame for Bob, ˜Y ∈ [0, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' p ˜ X| ˜Y ∈[0,N)(˜x) = � N 0 p ˜ X|U, ˜Y ∈[0,N[(˜x|u) · pU| ˜Y ∈[0,N[(u) du = � N 0 p ˜ X|U(˜x|u) · pU| ˜Y ∈[0,N[(u) du, (23) where the two factors are given by (13) and (21) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The a priori probability ˆπi = P(X ∈ [i, i + 1)| ˜Y ∈ [0, N)) becomes, for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' , N − 1, ˆπi = � i+1 i pX| ˜Y ∈[0,N[(x) dx = � i+1 i p ˜ X| ˜Y ∈[0,N[(x) � N 0 p ˜ X| ˜Y ∈[0,N)(˜x) d˜x dx = � i+1 x=i � N u=0 p ˜ X|U(x|u) · pU| ˜Y ∈[0,N[(u) du dx � N ˜x=0 � N u=0 p ˜ X|U(˜x|u) · pU| ˜Y ∈[0,N[(u) du d˜x = � N u=0 � Q �i−u σ � − Q �i+1−u σ �� pU| ˜Y ∈[0,N)(u) du dx � N u=0 � Q � −u σ � − Q � N−u σ �� pU| ˜Y ∈[0,N)(u) du d˜x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We obtain (22) after replacing pU| ˜Y ∈[0,N[(u) by its expression from (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 14 FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 Lemma 2 tells that invalidating the cases where the photon falls outside the frame converts the uniform density pU(u) = 1 N into a non-uniform density in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Furthermore, the a priori probability πi of (18) becomes ˆπi of (22) when adding the condition that Bob’s frame is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' πi and ˆπi already take into account that Alice has a valid frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The next lemma leads to establishing the channel likelihood expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The conditional density of U given ˆX = i is p(u| ˆX = i) = Q � i−u σ � − Q �i+1−u σ � � N 0 � Q �i−t σ � − Q � i+1−t σ �� dt , u ∈ [0, N), (24) for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Furthermore, when Bob gets a valid frame, the density of U conditioned on Alice’s bin number i is p(u| ˆX = i, ˜Y ∈ [0, N)) = � Q �i−u σ � − Q �i+1−u σ �� � Q �−u σ � − Q � N−u σ �� � N 0 � Q � i−t σ � − Q � i+1−t σ �� � Q � −t σ � − Q � N−t σ �� dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (25) Proof: The existence of X and ˆX, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' when writing ˆX = i, requires that ˜X ∈ [0, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' This hidden assumption should not be forgotten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' By applying Bayes’ rule, p(u| ˆX = i) = p(u| ˆX = i, ˜X ∈ [0, N)) = P( ˆX = i|u, ˜X ∈ [0, N)) × p(u| ˜X ∈ [0, N)) πi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The first term in the numerator can be developed as follows P( ˆX = i|u, ˜X ∈ [0, N) = P(X ∈ [i, i + 1)|u, ˜X ∈ [0, N)) = P( ˜X ∈ [i, i + 1)|u) P( ˜X ∈ [0, N)|u) = Q � i−u σ � − Q �i+1−u σ � Q �−u σ � − Q � N−u σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The second term in the numerator is given in (21) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' After substituting the expression of πi from (18), we get p(u| ˆX = i) = Q �i−u σ � − Q � i+1−u σ � � i+1 i � Q �−t σ � − Q � N−t σ �� dt , The reader is invited to prove via a change of variable that � i+1 i � Q �−t σ � − Q �N − t σ �� dt = � N 0 � Q �i − t σ � − Q �i + 1 − t σ �� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (26) which leads to the result announced by the lemma in (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The proof of (25) follows similar steps as for the proof of (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Firstly, using Bayes’ rule and (14) we get a conditional density of U, p(u| ˜X, ˜Y ∈ [0, N)) = � Q � −u σ � − Q �N−u σ ��2 � N 0 � Q � −t σ � − Q �N−t σ ��2 dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (27) 15 Secondly, we solve the conditional probability of Alice’s photon bins, P( ˆX = i|u, ˜X, ˜Y ∈ [0, N)) = P( ˜X ∈ [i, i + 1), ˜Y ∈ [0, N)|u) P( ˜X, ˜Y ∈ [0, N)|u) = � Q � i−u σ � − Q �i+1−u σ �� � Q �−u σ � − Q � N−u σ �� � Q �−u σ � − Q � N−u σ ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Finally, we use the above expressions of P( ˆX = i|u, ˜X, ˜Y ∈ [0, N)) and p(u| ˆX, ˜Y ∈ [0, N)), and (22) from Lemma 2 in p(u| ˆX = i, ˜Y ∈ [0, N)) = P( ˆX = i|u, ˜X, ˜Y ∈ [0, N)) × p(u| ˜X, ˜Y ∈ [0, n)) ˆπi to reach (25) in this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The existence of Y assumes that ˜Y ∈ [0, N), as we mentioned for X in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We deliberately remind the reader of the condition ˜Y ∈ [0, N) in the subscript of the likelihood function in the next statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Under the assumption that both Alice and Bob got valid frames, the soft-output QKD channel model likelihoods, p(y|ˆx) = pY | ˆ X, ˜Y ∈[0,N)(y|ˆx), have the following expression pY | ˆ X, ˜Y ∈[0,N)(y|ˆx = i) = � N 0 1 √ 2πσ2 exp � −(y−u)2 2σ2 � � Q �i−u σ � − Q � i+1−u σ �� du � N 0 � Q � −t σ � − Q �N−t σ �� � Q � i−t σ � − Q � i+1−t σ �� dt , (28) for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' , N − 1, y ∈ [0, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' For simplicity, the likelihood in (28) will be denoted by p(y| ˆX = i) in next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Proof: We drop the subscripts in the density functions, when possible, to simplify the notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We start by a marginalization before truncating p(˜y|u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' p(y| ˆX = i, ˜Y ∈ [0, N)) = � N 0 p(y, u| ˆX = i, ˜Y ∈ [0, N)) du = � N 0 p(y|u, ˆX = i, ˜Y ∈ [0, N)) · p(u| ˆX = i, ˜Y ∈ [0, N)) du = � N 0 p(y|u) · p(u| ˆX = i, ˜Y ∈ [0, N)) du, (29) The left factor p(y|u) inside the integral in (29) is given by the truncation of the density in (13) (replace x by y) and the right factor was solved by Lemma (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' p(y| ˆX = i, ˜Y ∈ [0, N)) = � N 0 p(˜y = y|u) � N 0 p(˜y|u)d˜y p(u| ˆX = i, ˜Y ∈ [0, N)) du, 16 FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 = � N 0 1 √ 2πσ2 exp � −(y−u)2 2σ2 � Q � −u σ � − Q �N−u σ � · � Q � i−u σ � − Q �i+1−u σ �� � Q �−u σ � − Q �N−u σ �� � N 0 � Q �i−t σ � − Q �i+1−t σ �� � Q � −t σ � − Q �N−t σ �� dt du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' After simplifying the term Q �−u σ � − Q � N−u σ � we reach the announced result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The transition probabilities pi,j = P( ˆY = j| ˆX = i) of the hard-output QKD channel model are directly derived by integrating the conditional density function of the soft output Y established by the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The probability that Bob’s photon falls in bin j given that Alice’s photon fell in bin i is given by pij = P( ˆY = j| ˆX = i) = � N 0 � Q �j−u σ � − Q �j+1−u σ �� � Q � i−u σ � − Q �i+1−u σ �� du � N 0 � Q �−t σ � − Q �N−t σ �� � Q � i−t σ � − Q �i+1−t σ �� dt , i, j ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (30) Proof: Integrate (28) over Bob’s photon position y from j to j + 1, then switch the two integrals to get the result announced by this corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We complete this section by establishing the expression of the a posteriori probability useful for soft- decision decoding, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=', for belief-propagation decoding of low-density parity-check codes, for ordered- statistics decoding of linear block codes, or Viterbi decoding of convolutional codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Let APP(i) = APP( ˆX = i) = P( ˆX = i|Y = y) be the a posteriori probability of Alice’s photon bin number i, for i = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The next theorem gives the expression APP(i), which is used in our proposed coding/decoding schemes in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Given the photon position Y = y on Bob’s side, the probability for Alice’s photon to belong to bin number i is APP(i) = � N 0 1 √ 2πσ2e− (y−u)2 2σ2 � Q �i−u σ � − Q � i+1−u σ �� du � N 0 1 √ 2πσ2 e− (y−t)2 2σ2 � Q � −t σ � − Q � N−t σ �� dt , i ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (31) Proof: Keeping in mind that ˜X, ˜Y ∈ [0, N), apply Bayes’ rule to get P( ˆX = i|Y = y) = p(y| ˆX = i) · P( ˆX = i) p(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The result announced by the theorem is then found in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (i) Use (28) from Theorem 1 for p(y| ˆX = i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (ii) Use (22) for the a priori P( ˆX = i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (iii) Finally, p(y) = pY | ˜ X∈[0,N)(y) = p ˜Y | ˜ X∈[0,N)(˜y = y)/ � N 0 p ˜Y | ˜ X∈[0,N)(t) dt after truncating the density of ˜Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The density p ˜Y | ˜ X∈[0,N)(˜y) is found in (23) while switching the letters x (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' X) and y (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ), in 17 conjunction with (13) and (21), p(y) = � N 0 1 √ 2πσ2e− (y−u)2 2σ2 � Q �−u σ � − Q � N−u σ �� du � N 0 � Q �−t σ � − Q � N−t σ ��2 dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (32) For consistency, the reader could check that p(y) given at the end of the proof of Theorem 2 is also equal to �N−1 i=0 ˆπi · p(y| ˆX = i) from (22) and (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Figures 5 and 6 plot the likelihoods p(y| ˆX = i) at low SNR γ = 10 dB (low photon detector precision) and a relatively higher SNR γ = 25 dB (higher photon detector precision), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' At low SNR, p(y| ˆX = i) has a Gaussian shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The shape tends to become square at high SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The a posteriori probabilities APP(i), i ∈ ZN, have a plot similar to the channel likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='8 1 0 1 2 3 4 5 6 7 8 Probability Density Function Photon Position p(y|0) p(y|1) p(y|2) p(y|3) p(y|4) p(y|5) p(y|6) p(y|7) p(y) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Soft-Output channel likelihoods, N=8 bins per frame, SNR=10 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='8 1 0 1 2 3 4 5 6 7 8 Probability Density Function Photon Position p(y|0) p(y|1) p(y|2) p(y|3) p(y|4) p(y|5) p(y|6) p(y|7) p(y) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Soft-Output channel likelihoods, N=8 bins per frame, SNR=25 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 18 FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' SECRET KEY INFORMATION RATES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Mutual Information between Raw Keys Firstly, we consider the mutual information of the algebraic hard-output channel defined by the transition probability p(ˆy|ˆx) = P( ˆY = ˆy| ˆX = ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In Corollary 1, the expression of pij = p( ˆY = j| ˆX = i) was established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Hence, we can directly compute the mutual information as follows: I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˆY ) = H( ˆY ) − H( ˆY | ˆX) = − N−1 � i=0 P( ˆY = i) log(P( ˆY = i)) + N−1 � i=0 P( ˆX = i) N−1 � j=0 P( ˆY = j| ˆX = i) log(P( ˆY = j| ˆX = i)) = − N−1 � i=0 ˆπi log(ˆπi) + N−1 � i=0 ˆπi N−1 � j=0 pij log(pij), (33) where a priori ˆπi of ˆX and ˆY is found in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The plot of I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˆY ) expressed in bits versus the signal-to- noise ratio is depicted in Figure 7, for 2, 3, and 4 coded bits per transmitted photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' As expected, the curves go towards the asymptote H( ˆX) at high signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In fact, the entropy − �N−1 i=0 ˆπi log(ˆπi) is very stable even at low SNR and could be well approximated by log(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The summation in (33) could be truncated to neighboring bins or to bins within an integer distance less than D, I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˆY ) ≈ log(N) + 1 N � |i−j|≤D pij log(pij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (34) The simplification (34) is an excellent approximation down to γ ≥ 10 dB for D = 1 only and it extends to γ ≥ 5 dB for D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The next proposition gives more insight into the behavior of the a priori and the transition probabilities, and the discrete channel mutual information in the low-noise regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' At high signal-to-noise ratio, when σ2 ≪ 1, we have the following results: a) The transition probability of the hard-output QKD channel established in Corollary 1 satisfies: At the two extremal bins, i = 0 and i = N − 1, we have p0,1 = σ √π + O(e− γ 4 ) 1 − 1+ √ 2 2√π · σ + O(e− γ 4 ) , (35) where p0,1 = pN−1,N−2 = 1 − p0,0 = 1 − pN−1,N−1, and σ = 1/√γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 19 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='5 4 5 0 5 10 15 20 25 30 35 40 N=4 N=8 N=16 Bits per Photon SNR (dB) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Mutual information I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˆY ) of the algebraic hard-output TE-QKD channel, for N = 4, 8, 16 bins per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' At the middle bins, i = 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' N − 2, we have p1,2 = σ √π + O(e− γ 4 ) 1 − O(e− γ 4 ) , (36) where p1,2 = pi,i+1 = pi,i−1 = (1−pi,i)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' All other transition probabilities pi,j for |i−j| ≥ 2 are O(e− γ 4 ) and can be forced to 0 in any numerical calculation at high SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' b) The a priori probabilities established in Lemma 2 satisfy At the two extremal bins, i = 0 and i = N − 1, we have ˆπ0 = ˆπN−1 = 1 − 1+ √ 2 2√π · σ + O(e− γ 4 ) N · � 1 − 1+ √ 2 √π · σ + O(e− γ 4 ) �, (37) where the numerator includes σ but the denominator involves σ = σ/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' For the middle bins, with i = 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' N − 2, we have ˆπi = 1 − O(e− γ 4 ) N · � 1 − 1+ √ 2 √π · σ + O(e− γ 4 ) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (38) c) Following a) and b), the mutual information of the discrete-input discrete-output QKD channel given by (33) becomes I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˆY ) = N − 2βσ N(1 − 2βσ) log [N(1 − 2βσ)] − 2(1 − βσ) N(1 − 2βσ) log(1 − βσ) − 2(1 − βσ) N(1 − 2βσ)H2 � σ/√π 1 − βσ � − (N − 2) N(1 − 2βσ)H3 � σ √π � + O(e− γ 4 ), (39) 20 FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 where β = 1+ √ 2 2√π , σ = 1/√γ, and σ = σ/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Proof: For the sake of space, we only show the detailed proof for the denominator of p0,1 (also equal to the numerator of ˆπ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' All other results are found using similar calculus techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The denominator of p0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='1 from Corollary 1 (i = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' j = 1) is equal to the integral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='I5 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='�t−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� N−t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='(1 − Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� 1−t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=') dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='(vi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� N−t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='�t−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='(vii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� N−t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='(viii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Now we solve the elementary integrals (i)-(viii) one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Using (19), (i) = 0 − σ √ 2π + Q( −1 σ ) + σ √ 2πe− γ 2 = 1 − σ √ 2π + O(e− γ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Using the fact that Q(x) is a monotone decreasing function, then (ii) = O(e−(N−1)2 γ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The third integral is directly solved via (20): (iii) = NQ2( N σ ) − σ � 2 πQ( N σ )e−N2 γ 2 + σ √πQ( N √ 2 σ ) − 0 + σ � 2 π · 1 2 − σ √π · 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Then we find (iii) = �√ 2−1 2√π � σ + O(e−N2 γ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (iv) = I1 − I3 = σ √ 2π + O(e− γ 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' For (v), t2 + (t − 1)2 ≥ 1 in the interval [1, N], then we have (v) = O(e− γ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The first part of (vi) is O(e−(N−1)2 γ 2 ) and the second part is also O(e−(N−1)2 γ 2 ) because (N − t)2 + (1 − t)2 ≥ (N − 1)2 in the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' So (vi) = O(e−(N−1)2 γ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Applying similar arguments, we get (vii) = O(e−(N−1)2 γ 4 ) and (viii) = O(e−N2 γ 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Combining (i)-(viii) yields I5 = 1 − 1+ √ 2 2√π σ + O(e− γ 4 ) as announced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' At high signal-to-noise ratio, Proposition 2-a) shows how fast pi,j converges to σ √π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The latter is a one-sided probability of error and it is half the double-sided probability of error stated in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' As expected, ˆπi converges to 1/N much faster for inner bins as found in Proposition 2-b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The high- SNR expression of I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˆY ) established in Proposition 2-c) perfectly fits the exact mutual information 21 of the discrete channel down to γ = 10 dB and then diverges at low SNR below 10 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The binary entropy function represents the extremal bins error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The ternary entropy function carries the inner bins error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Expression (39) is a quick method to evaluate I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˆY ) at moderate and high signal-to-noise ratios without performing any integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' One could ask how good is the approximated mutual information if the TE-QKD discrete channel is assumed to have a circular transition probability matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Under the assumptions of Proposition 1, we take: 1- X and Y are Gaussian, 2- all bins are equiprobable, and 3- the error probability of the discrete-input discrete-output channel is dominated by events where X and Y are separated by one or two bins only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' According to Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='1 in [10, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='2], the expression for a circular discrete channel is I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˆY ) ≈ log(N) + 2 � j=−2 p0j log2(p0j), (40) where p01 = p0,−1 ≈ � 1 0 2p1p2 dv = 2(I1 − I2 − I3), p02 = p0,−2 ≈ � 1 0 2p2p3 dv = 2I3, and p00 = 1 − 2p01 − 2p02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' All three high-SNR approximations (39), (34) with D = 2, and (40) are respectively shown in dotted lines from top to bottom on Figure 7 for N = 8 bins per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (39) and (34) follows the exact mutual information I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˆY ) at high SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (40) is not tight enough at N = 8 but becomes tighter for N ≥ 16 bins per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The second step in this section is to compute the mutual information for the soft-output TE-QKD channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We chose to write I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) = H( ˆX) − H( ˆX|Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The second expression after flipping X and Y based on differential entropy is also equivalent from numerical stability point of view and has all its terms established in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We prefer the mutual information where the high-SNR asymptote is visible, hence I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) = H( ˆX) − H( ˆX|Y ) = H( ˆX) + N−1 � i=0 ˆπi � N 0 p(y|ˆx = i) log2(APP(i)) dy, (41) where the a priori ˆπi is from (22), the likelihood p(y|ˆx = i) is from (28), and the a posteriori APP(i) is from (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Figure 8 shows the mutual information I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) versus normalized SNR γ for different number of bins per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The red upper envelope is established by Theorem 3 in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' It corresponds to the maximal mutual information achievable on the TE-QKD channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 22 FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 0 1 2 3 4 5 6 7 10 20 30 40 50 60 N=4 N=8 N=16 N=32 N=64 Bits per Photon Normalized SNR (dB) Capacity N bins per frame Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Mutual information I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) of the soft-output QKD channel, for N = 4, 8, 16 bins per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Maximal Secrecy Rate The random variables X, ˆX, and Y form a Markov chain X → ˆX → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Therefore, the data processing inequality [10, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 2] yields I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) ≤ I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ), ∀ N ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (42) Consequently, for any value of the number N of bins per frame, the rate of our channel pY | ˆ X(y|ˆx) is always bounded from above by the rate of the continuous-input continuous-output channel pY |X(y|x) corresponding to a continuum of zero-measure bins in Alice’s frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Thus, by determining the mutual information I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) we get the maximal secrecy rate of the photon channel between Alice and Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Without loss of generality, assume that the frame size is 1, instead of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Now, the problem is to find I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) where X and Y are truncated versions of the original photon positions, X = ˜X ∈ [0, 1), Y = ˜Y ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The model in (1) becomes ˜X = U + N (0, σ2), ˜Y = U + N (0, σ2), U is uniform in [0, 1), and the two additive Gaussian noises are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The normalized signal-to-noise ratio is naturally defined by γ = γ = 1/σ2 under this context of infinite number of bins and a frame of unit length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 23 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The maximal secrecy rate I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) is given by I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) = h(Y ) − h(Y |X) (43) = − � 1 0 p(y) log(p(y)) dy + � 1 0 p(x) � 1 0 p(y|x) log(p(y|x)) dxdy (44) where p(x) and p(y) are from (32) after replacing the frame size N by 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' p(x) = pX| ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˜Y ∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='1)(x) = � 1 0 1 √ 2πσ2 e− (x−u)2 2σ2 � Q �−u σ � − Q � 1−u σ �� du � 1 0 � Q � −t σ � − Q � 1−t σ ��2 dt ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (45) p(y) = pY | ˜ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˜Y ∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='1)(y) = � 1 0 1 √ 2πσ2 e− (y−u)2 2σ2 � Q �−u σ � − Q � 1−u σ �� du � 1 0 � Q �−t σ � − Q �1−t σ ��2 dt ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (46) p(y|x) = 1 √ 4πσ2e− (y−x)2 4σ2 � Q � 0−(x+y)/2 σ/ √ 2 � − Q � 1−(x+y)/2 σ/ √ 2 �� � 1 0 1 √ 2πσ2 e− (x−u)2 2σ2 � Q � −u σ � − Q �1−u σ �� du .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (47) Proof: We complete the proof by finding the expression of the conditional density p(y|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Indeed, we can write after marginalizing and applying Bayes’ rule p(y|x) = pY |X, ˜ X, ˜Y ∈[0,1)(y|x) = � 1 0 p(y, u|x) du = � 1 0 p(u)p(x|u)p(y|u) p(x) du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (48) In the above integral expression we used the fact that p(x, y|u) = p(x|u)p(y|u) as a result of the model defined by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In (48), both p(x|u) and p(y|u) are from (16), p(x) is from (32), and finally p(u) = pU| ˜ X, ˜Y ∈[0,1)(u) is found in (27), all after substituting 1 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' After simplifying the integrand of (48), we get p(y|x) as stated by (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' At high signal-to-noise ratio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' σ2 ≪ 1 or equivalently γ = 1 σ2 ≫ 1, the maximal secrecy rate satisfies I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) = (1 + O( 1 √γ)) · 1 2 log � γ 4πe � + O � ( 1 √γ)α� ∼ 1 2 log � γ 4πe � , ∀α ∈]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (49) Proof: The proof is based on a Babylonian approach with heavy calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Let us first give a sketch on how the limit is guessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' By applying Lemma 1 and some extra algebra, when γ ≫ 1, we get that p(x) → 1, p(y) → 1, and p(y|x) → 1 √ 4πσ2 e− (y−x)2 4σ2 , in (45), (46), and (47) respectively, for x, y ∈]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Then, the differential entropy h(Y ) → 0, h(Y |X) → h(N (0, 2σ2)) = 1 2 log(4πeσ2), so the maximal secrecy rate satisfies I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) = h(Y ) − h(Y |X) → 1 2 log � γ 4πe � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' This ends a simple but a clear sketch on how all involved densities and the maximal mutual information are converging at high SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 24 FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 The denominator of p(x) and p(y) is given in Lemma 1 at high SNR as 1 − O( 1 √γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The numerator of p(y|x) is fσ/ √ 2((x + y)/2) = 1 − O(exp(− min2 ·γ)) for (x + y)/2 ∈]0, 1[, where min is min((x + y)/2, 1 − (x + y)/2) from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Also, fσ/ √ 2(0) = fσ/ √ 2(1) = 1 2 − O(exp(−γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The last item to solve to get the result of this corollary is the integral in the numerator of p(x), the numerator of p(y), and the denominator of p(y|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Define the following integral I6 = I6(x) = I6(1 − x) = � 1 0 1 √ 2πσ2 e− (x−u)2 2σ2 � Q � −u σ � − Q � 1−u σ �� du, x ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (50) The integral I6 has no closed-form expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Firstly, we study I6(x) at x = 0 (identical at x = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' I6(0) = � 1 0 1 √ 2πσ2 e− u2 2σ2 � 1 − Q � u σ � − Q �1−u σ �� du = 1 2 − Q � 1 σ � − � 1 0 1 √ 2πσ2 e− u2 2σ2 Q � u σ � du − � 1 0 1 √ 2πσ2 e− u2 2σ2 Q �1−u σ � du = 1 2 − O(e− γ 2 ) − 1 8 − 1 2Q2 � 1 σ � − O(e− γ 4 ) = 3 8 − O(e− γ 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (51) Secondly, we study I6(x) for x ∈]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We use calculus tools similar to those used in the proofs of Lemma 1 and Proposition 2 to obtain I6(x) = Q � −x σ � − Q �1−x σ � − � 1 0 1 √ 2πσ2 e− (x−u)2 2σ2 Q � u σ � du − � 1 0 1 √ 2πσ2 e− (x−u)2 2σ2 Q � 1−u σ � du = fσ(x) − O(e− x2 4σ2 ) − O(e− (1−x)2 4σ2 ) = 1 − O(e− min2(x,1−x)· γ 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (52) Now we are ready to transform the expression of I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) in the small-noise regime given that the behavior of all densities is solved: p(x) = I6(x) 1 − O( 1 √γ), p(y) = I6(y) 1 − O( 1 √γ), and p(y|x) = e− (y−x)2 4σ2 √ 4πσ2 · fσ/ √ 2((x + y)/2) I6(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' For x, y ∈ [0, 1], we distinguish between the behavior of I6(x), I6(y), and fσ/ √ 2((x + y)/2) near the extremal points 0 and 1 and inside the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Hence, we decompose the interval as [0, 1] = [0, δ] ∪ [δ, 1 − δ] ∪ [1 − δ, 1] for integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The parameter δ should vanish at high SNR and should guarantee that I6 approaches 1, then we find from (52) that δ2γ should go to 0, which leads to δ = ( 1 √γ)α, where 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Finally, (44) is decomposed via this δ into I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) = − � 1−δ δ p(y) log(p(y)) dy − 2 � δ 0 p(y) log(p(y)) dy + �� x,y∈[δ,1−δ] p(x)p(y|x) log(p(y|x)) + �� x,y /∈[δ,1−δ] p(x)p(y|x) log(p(y|x)) (53) = (1 + O( 1 √γ)) · 1 2 log � γ 4πe � + O(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (54) 25 The cumbersome calculus details proving the last equality are not included for the sake of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The Gaussian differential entropy (49) is very close to I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) above 2 bits per photon and becomes very accurate beyond 3 bits per photon where it coincides with the red upper envelope in Figure 8 at a high signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The double variance 2σ2 in (49), originally found in (47), comes from the superposition of the variances of Z1 and Z2 in the system model defined by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' After canceling U, the model becomes ˜Y = ˜X + Z1 − Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˜X and Z1 are correlated, making the density expression relatively complicated when conditioning on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In the small-noise regime, this correlation fades away, and the variance 2σ2 of the total additive Gaussian noise Z1 − Z2 dominates the mutual information as in (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' At low and very low signal-to-noise ratios, one should use exact density expressions from Theorem 3 and proceed via numerical integration to get exact values of I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) and the corresponding SNR limits if the user accepts to apply a relatively low coding rate which is not the trend in TE-QKD where coding rates above 1/2 are preferred which places us in the moderate and the high SNR region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' KEY-RECONCILIATION CODES Following the complete characterization in Sections V-VI-B of the time-entanglement QKD channel model described in Section III, we now introduce error-correcting codes to bring the error-rate performance as close as possible to the information theoretic limits corresponding to maximal achievable rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Reed-Solomon Codes We consider the famous family of Reed-Solomon codes with an application to a frame of N = 2m bins, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' m coded bits per photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In order to chose a high enough error-correction capacity, an RS code over Fq is considered, where q is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Each finite field element corresponds to log2(q)/m photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' For simplicity, assume that q = 2ℓm, for some positive integer ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The RS code has length n = q − 1 (primitive) and dimension k = n − 2t, so the targeted rate is m × k n information bits per photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' One codeword of this C[n, k, t]q RS code requires the transmission of a total of n × log2(q)/m photons to Alice and n × log2(q)/m photons to Bob, all with valid frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' After receiving the n × log2(q)/m valid frames, Alice converts the n×log2(q) bits received on the quantum channel into a length-n word denoted by c + e, where c ∈ C and e ∈ Fn q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Similarly, Bob converts his n × log2(q) received bits into c + e′, where e′ ∈ Fn q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In the hard-output channel model of Section III, c + e is written at the input ˆX and c + e′ is read from the output ˆY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' On the public channel, Alice sends to Bob the syndrome s = (c + e)Ht, s ∈ Fn−k q , 26 FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 where H is the parity-check matrix of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Given s and given c + e′, the reconciliation performed by Bob is equivalent to finding Alice’s word c + e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Bob proceeds as follows: Compute a syndrome s′ = (c + e′)Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Feed s′ − s to an algebraic (Berlekamp-Massey [11]) decoder to find e′ − e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Subtract the error e′ − e from Bob’s word to get c + e′ − (e′ − e) = c + e the Fn q word possessed by Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Replace all subtractions by additions in usual finite fields of characteristic 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The performance of RS C[n = 63, k = 43, t = 10]q=64 code is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' One codeword requires the transmission of a total of 126 photons, where one field element carries two photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The results show a large gain, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=', about 58 dB of gain for a bit error-rate Peb = 10−5 after reconciliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 10-5 10-4 10-3 10-2 10-1 10 20 30 40 50 60 70 80 90 100 Probability of Error SNR (dB) Uncoded Pes Uncoded Peb RS-coded Pes RS-coded Peb 1/SNR1/2 Q(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='5/sigma) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Performance of the RS code [63, 43, t = 10]64 on the hard-output time-entanglement QKD channel, for N = 8 bins per frame, transmitting 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='05 information bits per photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The analysis of the algebraic decoder is easy thanks to its bounded-distance decoding in the Hamming space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' A decoding error occurs each time the channel adds more than t errors in Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' A simple union bound is obtained by summing from t + 1 to n errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We proceed in the following steps to establish this bound for the RS code: a) The uncoded symbol error probability over the TE-QKD channel is Pe(γ) = 2 √π(1 − 1 N ) 1 √γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' b) For the RS code, the input probability of error per finite-field element is Pin(γ) = 1 − (1 − Pe(γ))ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 27 c) The bound on the probability of error in Fq after decoding becomes PeRS(γ) = n � i=t+1 i n �n i � P i in(γ)(1 − Pin(γ))n−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (55) d) The symbol (per photon) error probability after decoding is then PeOut(γ) = 1 − (1 − PeRS(γ))1/ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' e) The probability of error per bit after Reed-Solomon decoding, given a Gray labeling of the bins, is well estimated by PebRS(γ) = 1 log2(N)PeOut(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The probability or error PebRS obtained from (55) perfectly fits the Monte Carlo method in the area where this method is tractable on a computer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' for error rates in the interval [10−7, 10−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' At PebRS = 10−10, the coding gain over the uncoded probability of error per bit is 158 dB!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Such a huge gain is explained by the diversity order of the TE-QKD channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The diversity order is defined as limγ→∞ − log(Pe) log(γ) [12, Chapters 13-14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' From Proposition 1 we know that the TE-QKD has a diversity order of 1 2, it behaves like a half-diversity Nakagami fading channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The error-correcting code increases the diversity order which is equivalent to increasing the slope of Pe(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' An additive Gaussian noise channel without fading has infinite diversity, with or without coding, making all curves look parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In presence of fading, a high diversity converts the channel into a Gaussian channel [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In practice, a diversity order beyond 8 could be barely distinguished from the local slope of e−γ on a Gaussian channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In our case, from (55), we deduce that the diversity order after algebraic RS decoding is (t + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' There is no asymptotic coding on the TE-QKD channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The coding gain increases if measured at a lower probability of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Binary BCH Codes The TE-QKD channel does not generate error bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Errors are independent and the most common event is one erroneous bit per photon before decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In other words, the binary-burst error-correcting capability of Reed-Solomon codes is not exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Hence, we suggest to utilize a binary BCH code of the same binary length as the RS[63, 43]64, which is 63 ×6 = 378 binary digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We start from a primitive length of 511 and shorten down to 378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' At t = 13 the binary BCH code has a dimension k = 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' This BCH[378, 261, t = 13]2 code yields a diversity order (t + 1)/2 = 7 better than the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='5 order of the RS code shown in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The number of information bits per photon is 261/378 × 3 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='07 bits for N = 8 bins per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Without adding any extra figure to this sub-section, the Monte Carlo simulation and the analytical bound show that the binary BCH[378, 261, t = 13] code beats the RS[63, 43]64 code by 3 dB in signal-to-noise 28 FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 ratio at PebRS = PebBCH = 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' To get the coding gain at a lower probability of error, we propose the following very tight union bound: a) The uncoded symbol error probability over the TE-QKD channel is Pe(γ) = 2 √π(1 − 1 N ) 1 √γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Below 10−1 a maximum of one bit error occurs in a block of m = log2(N) coded bits thanks to Gray labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' There are n/m such blocks per BCH codeword involving individual binary errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' b) The bound on the probability of error in F2 after BCH decoding becomes PebBCH(γ) = n/m � i=t+1 i n �n/m i � P i e(γ)(1 − Pe(γ))n/m−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (56) At PebRS = PebBCH = 10−10, the binary BCH[378, 261, t = 13] code beats the RS[63, 43]64 code by 5 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' This value corresponds to a 163 dB of BCH coding gain with respect to the uncoded photons at N = 8 bins per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Notice that the reconciliation at Bob’s side for BCH codes (binary or non-binary) is identical to the reconciliation described in the previous section for Reed-Solomon codes where the syndrome s′ − s is fed to a Berlekamp-Massey decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Graph-Based LDPC Codes The big impact of LDPC codes on the performance of polarization-based QKD systems was already demonstrated in [14] for the reconciliation of discrete random variables, with a BSC channel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Low-density parity-check codes [15][16] are very flexible in terms of length, coding rate, and decoding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' As usual, the LDPC code parity-check matrix is the adjacency matrix of a bipartite Tanner graph with n variable nodes and n − k check nodes, assuming that the graph is (dv, dc)-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' For finite fields Fq with q > 2, non-zero elements of the adjacency matrix are replaced by elements from Fq \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The standard method for decoding LDPC codes is belief propagation (BP), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' iterative probabilistic decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Codes over a large field Fq or a large ring Z/qZ could be considered [17] in order to minimize the loss during the symbol-to-bit soft values conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' It is also possible to use joint local-global LDPC codes with optimized bin mapping to achieve good performance [18] or apply multilevel-coding as in [19], although these papers consider a different QKD channel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In this paper, we show the impact of LDPC codes on TE-QKD with a (3, 9)-regular binary LDPC code only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The coding rate is 2/3 guaranteeing 2 exchanged bits per photon when the frame has 8 bins, however we consider a short length n = 384 (64×6) comparable to the RS and BCH codes given in the previous sub-sections, and a longer code with n = 9999 to illustrate a performance close enough to Shannon limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 29 The symbol/bin APP is found via (31), where APP(i) = APP( ˆX = i) is the a posteriori probability of bin number i, i ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Then the APP of binary digit bℓ, where ℓ ∈ Zm, m = log2(N), is derived by the following marginalization APP(bℓ) = � i∈ZN : bℓ APP( ˆX = i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (57) The above marginalization depends on the type of binary labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Our paper is restricted to N bins per frame with a Gray labeling of log2(N) bits per bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Figure 10 shows the bit error-rate versus γ for the binary LDPC code on the TE-QKD soft-output channel at n = 384 bits and n = 9999 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' They respectively gain 12 and 16 dB with respect to the BCH[378, 261] code, at a bit error probability of 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' If compared to the uncoded TE-QKD, the coding gain is 73 dB and 77 dB respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' At length n = 9999, the LDPC code is on top of the Shannon limit for a TE-QKD hard-output channel (γlimit = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='61 dB) and is 2 dB only from the Shannon limit of the soft-output TE-QKD channel (γlimit = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='45 dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We see no reason for using longer LDPC codes to catch an extra 1-2 dB given that the total coding gain with respect to the no-coding case already equals 77 dB!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 10−7 10−6 10−5 10−4 10−3 10−2 10−1 10 15 20 25 30 35 Bit Error Probability SNR (dB) RS[63,43]64 BCH[378,261] LDPC[384,256] LDPC[9999,6666] Shannon Limit (Hard) Shannon Limit (Soft) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Performance of the (3, 9)-regular binary LDPC code at length n = 384 bits and n = 9999 bits on the soft-output time-entanglement QKD channel, for N = 8 bins per frame, transmitting 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0 information bits per photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In practice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' if a lab system implementation requires a less complex expression for APP( ˆX = i) without the erfc()/Q() function and without integration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (31) can be simplified by assuming that the Gaussian density has the effect of a Dirac impulse at small σ and using the ∝ symbol (proportional to) since the 30 FOR PUBLICATION IN THE IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 30 DECEMBER 2022 denominator does not depend on the index i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' we get: APP(i) ∝ � N 0 1 √ 2πσ2e− (y−u)2 2σ2 � Q �i − u σ � − Q �i + 1 − u σ �� du ∝ � Q �i − y σ � − Q �i + 1 − y σ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Then, depending on the sign of the arguments i − y and i + 1 − y, we approximate Q(x) by 1 2e− x2 2 (if x ≥ 0) and by 1− 1 2e− x2 2 (if x < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Let j = ⌊Y ⌋ be the bin position of Y on Bob’s side, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ∈ [j, j+1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The simplified APP expressions become: If i = j, APP(i) ∝ � 1 − 1 2e− (y−i)2 2σ2 − 1 2e− (y−i−1)2 2σ2 � , (58) If i ̸= j, APP(i) ∝ sign(j − i) · � 1 2e− (y−i−1)2 2σ2 − 1 2e− (y−i)2 2σ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' (59) When (58)-(59) are utilized in the BP decoder of the binary LDPC code over the TE-QKD soft-output channel, the loss is limited to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='25-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='30 dB with respect to the exact expression (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' This is a minuscule loss when dealing with coding gains above 50 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Notice that we are not showing a performance of the LDPC code over a hard-output channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Indeed, optimal BP decoding is identical whether the channel output is soft or not, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=', the BP decoder is the same decoder on both a Gaussian-like channel and a BSC-like channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The gap between hard and soft is about 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='5 dB for the LDPC[384, 256] and about 4 dB for the LDPC[9999, 6666] at a bit error rate of 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Suppose the system implementation possesses an optimal BP decoder, but the exact photon position is unavailable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' only the bin number is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' In such a case, the lab implementation is forced to use LDPC codes on a hard-output channel, and the binary digits APP expression (57) becomes APP(bℓ) ∝ � i∈ZN : bℓ ˆπi × pi,j, (60) where j = ⌊Y ⌋, ˆπi is given by (22) or (37)-(38) at small σ, and pi,j is given by (30) or (35)-(36) in the small σ regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Coding theorists and practitioners could also use convolutional codes, turbo codes, polar codes, and other binary or non-binary algebraic codes with short or moderate length to achieve large coding gains on the TE-QKD channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' A summary of capacity limits at different frame sizes We complete the current section by a table summarizing important information theoretical limits on the time-entanglement QKD channel, with both hard and soft output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Shannon limit in terms of SNR is the 31 value of the non-normalized signal-to-noise ratio γ such that mutual information is equal to the targeted information exchange rate, I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˆY ) = k n log2(N) for a hard output and I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) = k n log2(N) for a soft output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Table II has seven columns with parameters covering 8 bins per frame up to 64 bins per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The last two rows correspond to SNR and standard deviation values achieved by the BCH and the LDPC codes as found in sub-sections VII-B and VII-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' TABLE II INFORMATION THEORETICAL (SHANNON) LIMITS FOR TE-QKD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' N Bits R = k/n SNR limit σ/N SNR limit σ/N bins per frame per photon code rate hard hard soft soft 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0 2/3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='61 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='029269 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='45 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='037533 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0 3/4 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='29 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='013532 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='85 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='017922 32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0 3/5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='88 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='019992 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='46 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='020982 32 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0 4/5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='61 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0065215 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='04 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0087670 64 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0 2/3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='01 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0098474 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='58 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='010347 64 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0 5/6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='77 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0032012 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='13 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0043383 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0 2/3 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='49 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0047034 BCH, n=378 Peb = 10−5 achieved 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='0 2/3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='47 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='029745 LDPC, n=9999 Peb = 10−5 achieved The signal-noise ratio soft-decoding limits listed in Table II appear to be close to two values, one SNR around 10-11 dB and a lower SNR around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='5 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The hard-decoding limits are higher than soft-decoding limits, because I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' ˆY ) ≤ I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ), the gap depends on the frame size and the coding rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Of course, the hard-soft gap vanishes at small coding rates (below 1/2) and increases at high coding rates when mutual information approaches the asymptote log2(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The two typical values of soft-decoding SNR limits are explained or interpreted for small σ via (49): 1 2 log � γ 4πe � = log2(N) − b, (61) where γ = N2γ and b is a backoff value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Here, b = 1 bit or b = 2 bits in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Then, solving (61) yields γ = (4πe)/22b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We get γ = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='31 dB for b = 1 and γ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='29 for b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' The difference with the values in the 6th column of Table II is due to I( ˆX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) going away from the envelope I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' Y ) to follow its own asymptote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We hope that SNR limits given in Table II will be useful to physicists and coding theorists working in this QKD field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' 32 FOR PUBLICATION IN THE IEEE, 30 DECEMBER 2022 VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' CONCLUSIONS We focused on the time entanglement-based QKD when the photon arrival detectors suffer from time jitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' We presented a rigorous analysis of secret key information rates and proposed and tested several codes for information reconciliation to approach the maximum secret key rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' These achievable secret key rates are much higher than the maximum achievable by polarization entanglement-based QKD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content=' However, practical photon detectors suffer from other impairments, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfnPij/content/2301.00486v1.pdf'} +page_content='g.' metadata={'source': 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International Institute for Carbon-Neutral Energy Research (WPI-I2CNER), Kyushu +University, Fukuoka 819-0395, Japan +2 Institute for Materials Science, University of Stuttgart, Pfaffenwaldring 55, 70569 Stuttgart, +Germany +3 Graduate School of Engineering, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan +4 Hefei General Machinery Research Institute, Hefei 230031, China + + +Despite high interest in compact and safe storage of hydrogen in the solid-state hydride form, the +design of alloys that can reversibly and quickly store hydrogen at room temperature under pressures +close to atmospheric pressure is a long-lasting challenge. In this study, first-principles calculations +are combined with experiments to develop high-entropy alloys (HEAs) for room-temperature +hydrogen storage. TixZr2-xCrMnFeNi (x = 0.4-1.6) alloys with the Laves phase structure and low +hydrogen binding energies of -0.1 to -0.15 eV are designed and synthesized. The HEAs reversibly +store hydrogen in the form of Laves phase hydrides at room temperature, while (de)hydrogenation +pressure systematically reduces with increasing the zirconium fraction in good agreement with the +binding energy calculations. The kinetics of hydrogenation are fast, the hydrogenation occurs without +any activation or catalytic treatment, the hydrogen storage performance remains stable for at least +1000 cycles, and the storage capacity is higher than that for commercial LaNi5. The current findings +demonstrate that a combination of theoretical calculations and experiments is a promising pathway +to design new high-entropy hydrides with high performance for hydrogen storage. +Keywords: Solid-state hydrogen storage; Density functional theory (DFT); High-entropy alloy +(HEA); Metal hydrides; Laves phase. + +* Corresponding author: K. Edalati (E-mail: kaveh.edalati@kyudai.jp; Tel: +81-92-802-6744) + + + + + +2 + +1.Introduction +Excessive consumption of fossil fuels and CO2 emission caused by their utilization have led +to the crisis of global warming. Nowadays, finding clean fuels which do not emit CO2 is a serious +challenge for scientists and industry leaders. Hydrogen is the cleanest fuel and has attracted attention +as a substitute for fossil fuels; however, besides the necessity for clean production of hydrogen, its +safe and compact storage is a significant challenge [1]. Storage of hydrogen in the form of gas and +liquid is conventionally used for various applications. However, the amount of stored hydrogen in +the form of gas in typical commercial tanks with 225 liters and 20 MPa pressure is just 4 Kg [2] . +Therefore, this typical method has limitations in terms of volumetric and gravimetric storage +densities, although there are recent trends to increase the storage pressure to 70 MPa by using special +tanks [1]. In the liquid storage method, the volumetric and gravimetric storage densities are higher, +and the safety is better than for the gas storage method, but liquifying hydrogen at low temperatures +makes the method expensive and evaporation losses can also occur [2]. Solid-sate hydrogen storage +particularly in the form of metal hydrides provides the most compact and safest technology to store +hydrogen [2]. +To realize the application of metal hydrides for hydrogen storage, they should have several +features such as the capability for reversible absorption and desorption of hydrogen at ambient +temperature, high cycling stability, fast kinetics, appropriate storage pressure near atmospheric +pressure and high gravimetric capacity (high gravimetric capacity for stationary applications is not +as critical as it is for mobile applications) [3]. Magnesium hydride and complex hydrides are well- +known materials with high storage capacity, but they suffer from high thermodynamic stability, and +thus, they function only at high temperatures [1]. So far, a limited number of materials such as TiFe +(TiFeH2) and LaNi5 (LaNi5H6) have been introduced for room-temperature hydrogen storage [3-6], +but they exhibit other shortcomings such as the activation problem in TiFe, and a high price and low +storage capacity in LaNi5 [3-6]. Therefore, there are still significant demands to introduce new metal +hydride systems that can satisfy the requirements for hydrogen storage at ambient temperature. +High-entropy materials, which are solid solutions of at least five principal elements with a +configurational entropy higher than 1.5R (R: gas constant), have attracted attention in recent years +for various applications including hydrogen storage [7]. The presence of several elements in a single +phase allows to manipulate the electronic structure, hydrogen binding energy and accordingly +hydrogen storage temperature and pressure by careful selection of principal elements and their +concentrations [7]. TiVZrNbHf [8-10], TiVCrNbMo [8], TiVCrNbTa [8], Ti0.2Zr0.2Hf0.2Mo0.1Nb0.3 +[11], +Ti0.2Zr0.2Hf0.2Mo0.2Nb0.2 +[11], +Ti0.2Zr0.2Hf0.2Mo0.3Nb0.1 +[11], +TiZrVNbCr +[12], +V30Ti30Cr25Fe10Nb5 [13], V35Ti30Cr25Fe5Mn5 [13], Mg0.10Ti0.30V0.25Zr0.10Nb0.25 [14], TiZrNbFeNi +[15], TiZrNbCrFe [16], MgAlTiFeNi [17], Al0.10Ti0.30V0.25Zr0.10Nb0.25 [18], Mg12Al11Ti33Mn11Nb33 +[19], +MgVAlCrNi +[20], +MgVTiCrFe +[21], +AlCrFeMnNiW +[22], +TiZrHfScMo +[23], +MgZrTiFe0.5Co0.5Ni0.5 [24] and LaNiFeVMn [25] are some of the HEAs which have been +investigated for hydrogen storage. However, as discussed in a recent review paper [7], these HEAs +have drawbacks such as either high-temperature requirement for hydrogen storage, poor hydrogen +storage reversibility, poor activation, or high storage pressure [8-25], which limit their applications. +Although the research on high-entropy hydrogen storage materials is still in its early stages, designing +these alloys by theoretical and computational methods is expected to provide a pathway to discover +new materials that can quickly and reversibly store hydrogen under ambient conditions. +In this work, first-principles calculations are combined with experiments to design HEAs for +room-temperature hydrogen storage. The designated alloys, TixZr2-xCrMnFeNi (x = 0.4-1.6) with the + +3 + +Laves phase structure and low hydrogen binding energies of -0.1 to -0.15 eV, show fast and reversible +hydrogen storage at ambient temperature under pressures adjustable to the atmospheric pressure by +changing the amounts of titanium and zirconium. This simultaneous application of theoretical and +experimental studies to high-entropy hydrogen storage materials confirms the significance of this +strategy in developing new HEAs that can satisfy the requirements for stationary hydrogen storage +applications. + +2. Materials and methods +2.1. Empirical material design +The key issue in designing room-temperature hydrogen storage materials is to adjust the +hydrogen binding energy to a negative value close to zero [26]. An earlier study on first-principles +calculations of Mg-based alloys suggested that binding energies of about -0.1 eV per hydrogen atom +can be an appropriate target to achieve room temperature hydrogen storage [26]. Moreover, the +hydrogen binding energy should be slightly more negative than -0.1 eV to reduce the equilibrium +hydrogen storage pressure close to ambient pressure. Although such a concept has not been used to +design HEAs so far, three empirical criteria were suggested by the current authors to achieve +hydrogen storage at low temperatures in HEAs [27]: (i) AB2-type atomic configuration (A: elements +which react with hydrogen; B: elements with low affinity with hydrogen), (ii) C14 Laves phase +structure formation in the alloy and hydride; and (iii) valence electron concentration (VEC) of 6.4. +• +Hydrogen storage materials are usually a mixture of A-type elements (such as lanthanum, +magnesium, titanium, etc.) and B-type elements (such as nickel, iron, manganese, etc.) [1]. The +A-type elements have negative hydrogen binding energies and produce stable hydrides, while +B-type elements have positive binding energies and usually do not absorb hydrogen, as +schematically shown in Fig. 1a using the data reported in the literature [3,26,28]. Therefore, a +combination of A-type and B-type elements can lead to the formation of alloys with an +appropriately low hydrogen binding energy for room-temperature hydrogen storage such as TiFe +and LaNi5 [1]. In this regard, it was found that AB2-type HEAs have a high potential for low- +temperature hydrogen storage [27], while AB-type and A3B2-type systems are other candidates, +yet with less potential [15,16]. +• +The Laves phase alloys are considered as potential materials for hydrogen storage with high +cycling stability for reversible hydrogenation and dehydrogenation and fast kinetics [29]. +Another benefit of Laves phase alloys is that they can have lower cost compared to rare-earth- +based alloys such as LaNi5 [29]. It was shown by both experiments and the CALPHAD +(calculation of phase diagrams) method that the Laves phases can be formed in some high- +entropy hydrogen storage systems such as Ti-Zr-Cr-Mn-Fe-Ni [27], Ti-Zr-Nb-Fe-Ni [15] and +Ti-Zr-Nb-Cr-Fe [16]. +• +VEC is another key parameter that can be considered in designing alloys for reversible hydrogen +storage at room temperature. Metals with low VEC such as lithium, magnesium and titanium +usually produce stable hydrides, which can release hydrogen only at high temperatures, while +metals with high VEC such as cobalt, nickel and copper usually exhibit low affinity with +hydrogen. It was suggested that HEAs with a VEC value of 6.4 can desorb hydrogen at +temperatures close to room temperature [8]. Although adjusting VEC is hard in simple binary +or ternary alloys, it can be adjusted much easier in HEAs by changing the type and fraction of +the principal elements. +• +The hydrogen binding energy is the most important parameter that needs to be adjusted for +reversible hydrogen storage at room temperature and under atmospheric pressures. The + +4 + +dehydrogenation temperature increases and the equilibrium hydrogen pressure decreases with +increasing the absolute value of negative binding energy. Only a limited number of materials +such as LaNi5 and TiFe show appropriate hydrogen binding energies for room-temperature +hydrogen storage. As schematically shown in Fig. 1a, the target is to set the hydrogen binding +energy to a value slightly more negative than -0.1 eV by adjusting the fraction of elements [26]. + + + + +Fig. 1. (a) Schematic illustration of hydrogen binding energy on hydrogenation and dehydrogenation +of different materials. (b) One of SQS models of HEA Ti0.5Zr1.5CrMnFeNi and corresponding +hydride Ti0.5Zr1.5CrMnFeNiH6 (visualized using VESTA [33]), considered in present ab initio +simulations. + + +Therefore, this study focuses on HEAs with the AB2-type Laves phase structure and a VEC +value of 6.4; and among the available options, the Ti-Zr-Cr-Mn-Fe-Ni system can fit all these +requirements [27]. The hydrogen binding energy is adjusted by changing the fraction of zirconium + +a +TARGET +TDeh (K): +>773 630 +460 +3o0,NoHydrogenation +Ti Mg +Mg2NiTiFeiMnCr +LaNis Ni +Fe +ttif ++1 +HydrogenBindingEnergy(eV) +Tio.5Zr1.5CrMnFeNi +Hydrogenation +Tio.5Zr1.5CrMnFeNiH6 +OMn +OFe +ONi.H5 + +in balance with titanium, because zirconium has a larger atomic radius than the other elements in this +alloying system, and thus, adjusting its fraction is supposed to have the most significant effect on +lattice volume and binding energy. Three compositions of TixZr2-xCrMnFeNi (x = 0.5, 1.0, 1.5) were +theoretically studied by first-principles binding energy calculations and four compositions of TixZr2- +xCrMnFeNi (x = 0.4, 0.8, 1.2, 1.6) were experimentally examined by hydrogen storage +characterizations. + +2.2. First-principles calculation methods +2.2.1. Crystal structure modeling of alloys +The AB2-type hexagonal C14 Laves phases have the space group of P63/mmc (No. 194), in +which the A-type atoms occupy the 4f Wyckoff sites, and the B-type atoms occupy the 2a and the 6h +Wyckoff sites [29,30]. These phases have in total 12 atoms in their unit cell. If A and B atoms in the +AB2-type Laves phases are approximated by close-packed rigid spheres with the radii rA and rB, +respectively, the ratio rA/rB is given by (3/2)1/2 ≈ 1.225 [30]. These atomic positions in the close- +packing case, which are considered the ideal positions, lead to a c/a ratio of (8/3)1/2 ≈ 1.633 (a and c +are lattice parameters). +The C14 TixZr2−xCrMnFeNi Laves phase alloys (x = 0.5, 1.0, 1.5) were modeled using 48- +atom supercells with a 2 × 2 × 1 expansion of the primitive cell of the C14 phase. These three +compositions, which slightly differ from the experimental compositions, were selected to investigate +the dependence of the hydrogen binding energy on the titanium fraction with 48-atom systems and +reasonable computation time. The A sites were occupied by titanium and zirconium, and the B sites +were occupied by chromium, manganese, iron, and nickel. The sublattice chemical disorder was +modeled using special quasi-random structure (SQS) configurations [31]. Correlation functions of +the first several nearest-neighbor doublet, triplet, and quartet clusters were optimized to be close to +the ideal values of fully random configurations using the simulated annealing approach implemented +in the ICET code [32]. To achieve better statistics, six different configurations were considered for +each composition by permuting the elements. Fig. 1b (upper image) shows one example of modeled +structure of a HEA which was modeled using SQS [31] and visualized using VESTA [33]. + +2.2.2. Crystal structure modeling of hydrides +To model the structure of the high-entropy hydrides, crystallographic information and reports +from the literature were considered. In the Laves phase alloys, there are 17 tetrahedral interstitial +sites per formula unit AB2, which could be occupied by hydrogen atoms [34]. These interstitial sites +are surrounded by either four B atoms (B4), one A and three B atoms (AB3), or two A and two B +atoms (A2B2). For many conventional Laves phases composed of the elements as utilized in the +present study, previous ab initio simulations found that the A2B2 sites are the most energetically +preferable sites for hydrogen atoms [35-39], and indeed many experiments found hydrogen atoms at +the A2B2 sites [40-48]. This may be intuitively understood because the A2B2 sites have larger volumes +than the B4 and AB3 sites when the atoms A and B are on the ideal Laves lattice sites, as summarized +in Table 1. The hydrogen atoms in Laves phases should also be repulsive to each other and should +not occupy very close interstitial sites, as empirically suggested [49] and confirmed by first-principles +calculations for some Laves phases [36,39]. +Since previous experimental studies reported that the AB2-type high-entropy Laves phase +produces a Laves phase hydride with the composition AB2H3 [27], the same hydrogen fraction was +considered for modeling in the present study. It should be noted that for the present 48-metal-atom +supercell models, there are in total 272 interstitial sites, all of which have different local chemical + +6 + +environments. Therefore, even for the fixed composition AB2H3, there are in total over 1053 possible +ways of hydrogen occupations, which are obviously impossible to test in a brute-force manner. +Therefore, based on the previous experiments and simulations mentioned above [34-48], we a priori +assumed that all hydrogen atoms occupy the A2B2 sites. Moreover, while there are four symmetrically +inequivalent A2B2 sites in the C14 hexagonal Laves phase [34], it was assumed that all hydrogen +atoms occupy the sites with the Wyckoff symbol 12k, because these sites do not share the faces of +the interstitial tetrahedra and thus the hydrogen atoms occupying them are not too close to each other +[34]. These assumptions uniquely determine the hydrogen-occupied sites for each supercell model. +Fig. 1b shows one of the thus obtained models considered in the present ab initio simulations. + + +Table 1. Numbers of tetrahedral interstitial sites with different local environments per formula unit +and ratios of their individual volumes to alloy volume per formula unit for AB2 Laves phases with +ideal atomic positions. +Tetrahedra Site +Number +Volume Ratio +B4 +1 +1/24 (~0.0417) +AB3 +4 +5/96 (~0.0521) +A2B2 +12 +1/16 (= 0.0625) + + +2.2.3. Hydrogen binding energy calculations +Ab initio density functional theory (DFT) calculations were performed using the VASP code +[50-52] with the plane-wave basis projector augmented wave (PAW) method [53]. The exchange- +correlation energy was obtained within the generalized gradient approximation (GGA) of the +Perdew-Burke-Ernzerhof (PBE) form [54]. The plane-wave cutoff energy was set to 400 eV. +Reciprocal spaces were sampled by a Γ-centered 4 × 4 × 6 k-point mesh for the 48-metal-atom +supercell models and the Methfessel-Paxton method [55] with the smearing width of 0.1 eV. The 3d +4s orbitals of titanium, chromium, manganese, iron, and nickel and the 4s 4p 4d 5s orbitals of +zirconium were treated as the valence states. Total energies were minimized until they converged +within 1 × 10-3 eV per simulation cell for each ionic step. All calculations were performed by +considering spin polarization, a fact that was also experimentally examined by magnetic +measurements, as discussed in Appendix A and Fig. A. +To obtain the energy-volume curves, seven volumes in the ranges of 144 to 180 Å3/u.c. and +186 to 222 Å3/u.c. (u.c.: unit cell) were considered for the systems with and without hydrogen atoms, +respectively. For each composition, the obtained energies of six SQS-based models for the given +volumes were then fitted to the Vinet equation of state [56,57] to obtain the volume and the energy +in the equilibrium state. Metal and hydrogen atoms were initially put on the ideal Laves-phase lattice +sites and the geometric centers (centroids) of the A2B2 12k interstitial sites, respectively. The atomic +positions were then relaxed with fixing the cell shape and volume until all the forces on the atoms +converged within 5 × 10-2 eV/Å. The hydrogen binding energy per atom was then computed as + +𝛥𝐸H = 1 +3 [𝐸(AB2H3) − 𝐸(AB2) − 3 +2 𝐸(H2)] +(1) + +where E(H2), E(AB2), and E(AB2H3) are the energies of H2, AB2, and AB2H3 per formula unit, +respectively. The energy of the H2 molecule was computed in a 20 × 20 × 20 Å3 simulation cell and + +7 + +for the Γ point in the reciprocal space. The obtained hydrogen-hydrogen distance was 0.751 Å, in +reasonable agreement with the experimental value of 0.74144 Å [58]. + +2.3. Experimental Procedures +The ingots of HEAs with compositions TixZr2-xCrMnFeNi (x = 0.4, 0.6, 1.2 and 1.6) with ~10 +g mass were prepared by arc melting using pieces of pure titanium (99.99%), zirconium (99.5%), +chromium (99.99%), manganese (99.95%), iron (99.97%) and nickel (99.9%). The pieces were +melted and mixed in a water-cooled copper crucible under an argon atmosphere. To increase the +homogeneity of the alloys, the mixture was remelted six times. Ingots produced by arc melting were +characterized by various methods, as described below. +First, the crystal structure was examined by X-ray diffraction (XRD) using Cu Kα irradiation +with a filament current of 40 mA and an acceleration voltage of 45 kV. The Rietveld method using +the PDXL software was used to identify the phases and determine their lattice parameters. +Second, the microstructure of the samples was investigated by a scanning electron microscope +(SEM) equipped with energy-dispersive X-ray spectroscopy (EDS) and electron backscatter +diffraction (EBSD) at 15 kV. The samples for SEM were prepared by cutting a piece of the ingot +using electric discharge machining, followed by mechanical grinding using sandpapers, polishing +using buff and 9 µm and 3 µm diamond suspensions, and final polishing by buff and colloidal silica +with 60 nm particle size. +Third, the nanostructure of the samples was examined using transmission electron +microscopy (TEM) and scanning-transmission electron microscopy (STEM) at 200 kV using high- +resolution imaging, fast Fourier transform (FFT) analysis and EDS analysis. For TEM and STEM +analyses, a small piece of the samples was crushed in ethanol and dispersed on a carbon grid. +Fourth, hydrogen storage was examined using a Sieverts-type gas absorption apparatus at +room temperature (298 K). For this experiment, ~1 g of the alloys was crushed in an air atmosphere +to achieve particle sizes below 75 μm. The crushed samples were examined in terms of pressure- +composition-temperature (PCT) isotherms for three hydriding-dehydriding cycles followed by a +hydrogenation kinetic measurement for one cycle. The samples before PCT measurements were +subjected to evacuation at room temperature for 3 h to remove air and moisture, but no thermal +activation treatment was performed. The crystal structure of the hydrides after hydrogen storage was +also examined by conducting the XRD analysis immediately after hydriding. Moreover, cyclic +hydrogenation-dehydrogenation measurements were conducted for 1000 cycles (10 min +hydrogenation under an initial hydrogen pressure of 3.7 MPa followed by 20 min evacuation at 298 +K) to examine the cycling stability of the alloys. + + + +8 + + + +Fig. 2. Ab initio calculated energy-volume curves fitted to the Vinet equation of state without and +with relaxation of atomic positions for (a, c, e) hydrogen-free HEAs TixZr2-xCrMnFeNi and (b, d, f) +corresponding high-entropy hydrides. Arrows show equilibrium volumes for fixed and relaxed +atomic positions. + + +3. Results +3.1. Calculated hydrogen binding energy +Figs. 2a, c and e show the ab initio computed energy-volume curves before and after +relaxation of atomic positions for the TixZr2-xCrMnFeNi alloys with x = 0.5, 1.0 and 1.5, respectively. + +a +3.0 +b +3.0 +Tio.5Zr1.5CrMnFeNi +Tio.5Zr1.5CrMnFeNiH6 +(eV) +2.5 +2.5 +Fixed +-.. Fixed +Energy +2.0 +Relaxed +2.0 +Relaxed +1.5 +1.5 +Cell +1.0 +1.0 +Unit +0.5 +0.5 +0 +0 +140 +150 +160 +170 +180 +185 +195 +205 +215 +225 +Unit Cell Volume (A3) +c +3.0 +d +3.0 +Ti1.oZr1.oCrMnFeNi +Ti1.oZr1.oCrMnFeNiH6 +(eV) +2.5 +Unit Cell Energy (eV) +2.5 +.. Fixed +Fixed +Energy +Relaxed +2.0 +Relaxed +2.0 +1.5 +1.5 +Unit Cell +1.0 +1.0 +0.5 +0.5 +0 +0 +140 +150 +160 +170 +180 +185 +195 +205 +215 +225 +Unit Cell Volume (A" +e +3.0 +3.0 +Ti1.Zro.5CrMnFeNi +Ti1.5Zro.5CrMnFeNiH6 +(eV) +2.5 +2.5 +Fixed +.Fixed +Cell Energy +2.0 +Relaxed +IEnergy +2.0 +Relaxed +1.5 +1.5 +Unit Cell +1.0 +1.0 +Unit +0.5 +0.5 +0 +140 +150 +160 +170 +180 +185 +195 +205 +215 +2259 + +The equilibrium unit cell volume decreases with increasing the fraction of titanium. Moreover, for +all the compositions, the equilibrium unit cell volume decreases after the relaxation, as summarized +in Table 2. Fig. 2b, d and f show the ab initio computed energy-volume curves before and after +relaxation of atomic positions for the TixZr2-xCrMnFeNiH6 hydrides with x = 0.5, 1.0 and 1.5, +respectively. In these hydrides, hydrogen atoms occupy all the 12k A2B2 interstitial sites. Similar to +the HEAs, the unit cell volume of the hydride decreases with increasing the titanium fraction, but as +summarized in Table 2, the main difference is that the unit cell volumes for the hydrides are 21-23% +larger compared to the corresponding alloys (i.e., larger by 3 Å3 per hydrogen atom). Such a volume +expansion is in a similar order to those experimentally reported for various binary and ternary +hydrides [59] and also similar to the one previously calculated by ab initio for the Zr(Cr0.5Ni0.5)2 alloy +with the C14 Laves phase (3.2 Å3 per hydrogen atom) [38]. Another difference between the alloys +and hydrides is that the impact of atomic-position relaxation on the equilibrium volume and energy +changes is much larger for the hydrides compared to the alloys. This is likely because the positions +of metal atoms more strongly deviate from the ideal lattice sites due to the presence of hydrogen +atoms, as detailed in Appendix B and Fig. B. +The hydrogen binding energy per hydrogen atom for relaxed atomic positions is given in +Table 2 for the three high-entropy hydrides TixZr2-xCrMnFeNiH6. The hydrogen binding energies are +-0.126, -0.105 and -0.074 eV per hydrogen atom for x = 0.5, 1.0 and 1.5, respectively. These values +are substantially lower than those at the A2B2 sites in the Laves phases consisting of similar elements +but chemically less disordered such as -0.27 eV per hydrogen atom for TiCr2 [36], and -0.20 to -0.28 +eV per hydrogen atom for Zr(Cr0.5Ni0.5)2 [38]. This implies that the present Laves HEAs can more +easily desorb the hydrogen atoms, and thus, can be more practical for applications where low- +temperature hydrogen storage is needed. Fig. 3 visualizes the dependence of the hydrogen binding +energy as a function of the titanium content for relaxed atomic positions. With increasing the titanium +content, the hydrogen binding energy becomes less negative, and thus the hydrides become less +stable. Therefore, it is expected that the alloys with lower titanium content can desorb hydrogen at +lower temperatures and lower pressures. The hydrogen binding energies for Ti0.5Zr1.5CrMnFeNiH6 +and Ti1.0Zr1.0CrMnFeNiH6 are slightly more negative than -0.1 eV, suggesting the potential of these +two alloys for room-temperature hydrogen storage [29]. The strong dependence of the hydrogen +binding energy on the composition also indicates the possibility to tailor the properties of hydrogen +storage alloys by modifying the compositions of HEAs [60]. + + +Table 2. Ab initio calculated equilibrium volumes for fixed and relaxed atomic positions, unit cell +energy difference between fixed and relaxed conditions, and hydrogen binding energy for HEAs +TixZr2-xCrMnFeNi and their corresponding TixZr2-xCrMnFeNiH6 hydrides where hydrogen atoms +occupy all the 12k A2B2 interstitial sites. +Alloy/Hydride +Volume (Å3/u.c.) +ERelaxed-EFixed +ΔEH +Fixed +Relaxed +(Å3/u.c.) +(eV / H atom) +Ti0.5Zr1.5CrMnFeNi +169.0 +168.3 +-0.18 + +Ti1.0Zr1.0CrMnFeNi +164.0 +163.0 +-0.28 + +Ti1.5Zr0.5CrMnFeNi +158.3 +157.3 +-0.26 + +Ti0.5Zr1.5CrMnFeNiH6 +207.8 +204.2 +-0.87 +- 0.126 +Ti1.0Zr1.0CrMnFeNiH6 +202.4 +198.6 +-0.99 +- 0.105 +Ti1.5Zr0.5CrMnFeNiH6 +196.5 +193.5 +-0.87 +- 0.074 + + + +10 + + + +Fig. 3. Ab initio calculated hydrogen binding energy as a function of titanium content for high- +entropy hydrides TixZr2-xCrMnFeNiH6. Atomic positions were relaxed for these calculations. + + +3.2. Structural features of alloys and hydrides +Fig. 4 summarizes the results of the XRD analysis. Fig. 4a shows the XRD profiles of the +HEAs, TixZr2-xCrMnFeNi (x = 0.4-1.6). All samples have mainly a hexagonal C14 Laves phase +structure with the P63/mmc space group, while the presence of weak peaks at 42-43˚ for the Ti-rich +alloys suggests that minor amounts of a cubic phase can also be present [27]. Fig. 4b shows a +magnified view of the XRD profiles for the (110) peak of the Laves phase located between 35.5˚ and +37.5 ˚. This figure demonstrates a systematic shift of this peak to higher angles with increasing the +titanium fraction, indicating that a lattice contraction occurs by increasing the titanium fraction. The +occurrence of lattice contraction by increasing the titanium fraction is shown more clearly in Table +3, where the calculated lattice parameters using the Rietveld method are given. It should be noted +that the plots of calculated-observed intensities in the Rietveld analysis indicate a maximum intensity +difference of 10%, which is a reasonable level. Such a difference should be partly due to the inherent +distortion in high-entropy materials [7] and partly due to the presence of a small amount of a cubic +phase [27]. Fig. 4c compares the XRD pattern of Ti0.4Zr1.6CrMnFeNi before and after hydriding. The +hydride phase also has a hexagonal C14 crystal structure, while the shift of the XRD profile to lower +angles after hydriding suggests that the lattice is expanded in the presence of hydrogen atoms in the +lattice. Fig. 4d summarizes the lattice parameters calculated from the XRD analysis for both alloys +and hydrides. Note that the data for Ti1.0Zr1.0CrMnFeNi was calculated from the XRD profiles +reported in an earlier publication [27]. It should be also noted that the crystal structure of the +Ti1.2Zr0.8CrMnFeNiH6 and Ti1.6Zr0.4CrMnFeNiH6 hydrides could not be identified because these two +hydrides released hydrogen very fast under ambient conditions and before the XRD analysis. Both a +and c lattice parameters increase with increasing the number of zirconium atoms, which have a larger +atomic radius than titanium atoms [61]. Moreover, the lattice parameters increase with the addition +of hydrogen atoms to the materials. + + +(eV/atom) +0 +Ti,Zr2-xCrMnFeNiH6 +-0.02 +Energy ( +-0.04 +-0.06 +-0.08 +-0.10 +-0.12 +-0.14 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +x in Ti,Zr2.vCrMnFeNiH11 + + + +Fig. 4. (a, b) XRD profiles of hydrogen-free HEAs TixZr2-xCrMnFeNi where (b) shows magnified +view of (110) peak of C14 Laves phase, (c) XRD profiles of hydrogen-free HEA Ti0.4Zr1.6CrMnFeNi +and corresponding high-entropy hydride, and (d) lattice parameters of hydrogen-free HEAs TixZr2- +xCrMnFeNi and corresponding high-entropy hydrides determined from XRD profiles using Rietveld +method. Data for Ti1.0Zr1.0CrMnFeNi in (d) were taken from literature [27]. + + +Table 3. Lattice parameters of C14 Laves phase obtained experimentally using XRD analysis for +HEAs TixZr2-xCrMnFeNi. +Alloy +a (Å) +c (Å) +Ti0.4Zr1.6CrMnFeNi +4.982 ± 0.008 +8.148 ± 0.020 +Ti0.8Zr1.2CrMnFeNi +4.945 ± 0.014 +8.093 ± 0.030 +Ti1.2Zr0.8CrMnFeNi +4.908 ± 0.010 +8.002 ± 0.020 +Ti1.6Zr0.4CrMnFeNi +4.871 ± 0.011 +7.942 ± 0.020 + + +a +TixZr2-xCrMnFeNi +b +TixZr2-xCrMnFeNi +?C14 +Normalized Intensity +C14 (110) +X=1.6·: += 1.6 +Normalized +X=1.2 ·^α 。 +X= 1.2 +X=0.8- +X=0.8 +x=0.4 ^ +X = 0.4 +30 +40 +50 +60 +70 +80 +35.5 +36.0 +36.5 +37.0 +37.5 +Diffraction Angle, 20 (deg.) +Diffraction Angle, 20 (deg.) +c +d +8.7 +Ti0.4Zr16CrMnFeNi +-O-- Hydride +·C14 +Alloy +A +8.4 +Hydride +8.1 +7.8 +5.4 +--Hydride +卜 Alloy +5.2 +A +Alloy +a +5.0 +4.8 +30 +40 +50 +60 +70 +80 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +Diffraction Angle, 20 (deg.) +x in Ti,Zr2-x +CrMnFeNi(H6)12 + + + +Fig. 5. Unit cell volume of C14 Laves phase obtained experimentally using XRD analysis and +calculated using ab initio without (fixed) and with relaxation of atomic positions for (a) HEAs TixZr2- +xCrMnFeNi and (b) corresponding high-entropy hydrides. + + +It is now possible to compare the unit cell volumes obtained experimentally and theoretically +to validate the models used for the first-principles calculations. Fig. 5 compares the equilibrium +volumes obtained by DFT with those obtained experimentally for the (a) alloys and (b) hydrides with +different fractions of titanium. The experimental and theoretical data show similar trends and the unit +cell volume decreases with increasing the titanium content for both alloys and hydrides. The linear +change in the unit cell volume is qualitatively consistent with Vegard’s law by considering the +difference in the atomic radius of titanium and zirconium [61]. The ab initio calculated volumes are +slightly smaller than those obtained in experiments: 3.5% for the alloys and 2.5% for the hydrides. +Such an underestimation of volume using GGA (like PW91 [62,63] or PBE [54]) is found often in +systems containing magnetic 3d elements such as pure bcc iron [64-66] and the HEA CrMnFeCoNi +[67,68]. The small differences between the experimental and theoretical cell volumes confirm the +validity of the models used for the simulation of the HEAs and hydrides. + + + +a +180 +TixZr2-xCrMnFeNi +Unit Cell Volume (A) +175 +170 +165 +Experiment +160 +DFT(Fixed) +DFT (Relaxed) +155 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +x in TixZr2-xCrMnFeNi +b +215 +TixZr2-xCrMnFeNiH6 +Unit Cell Volume (A") +210 +205 +200 +Experiment +195 +DFT (Fixed) +DFT (Relaxed) +190 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +x in Ti,Zr2-xCrMnFeNiH613 + + + +Fig. 6. Distribution of elements and their atomic fractions in HEA Ti0.4Zr1.6CrMnFeNi examined by +EDS elemental mappings using (a) SEM and (b) STEM. + + +3.3. Microstructure of alloys +The microstructure of Ti0.4Zr1.6CrMnFeNi, taken as a representative alloy with the lowest +hydrogen binding energy, is shown in Figs. 6 and 7. Fig. 6 shows (a) SEM-EDS and (b) STEM-EDS +analyses. The distribution of elements is reasonably uniform at the micrometer and nanometer levels. +Moreover, the fraction of elements is reasonably consistent with the nominal composition within the +detection limits of EDS analysis. The EDS analysis confirms that arc melting can be successfully +employed to synthesize the HEAs. An SEM micrograph and corresponding EBSD crystal orientation +and phase mappings are shown in Figs. 7a-c, respectively. Fig. 7b illustrates that the HEA includes +mainly large and elongated grains with sizes of several hundred micrometers and Fig. 7c confirms +that the alloy contains mainly a C14 Laves phase in good agreement with the XRD analysis. Figs. +7d-f show the high-resolution TEM images of the HEA. The C14 Laves phase was the only phase +that could be detected by high-resolution TEM images in good agreement with the XRD and EBSD +analyses. Most of the examined regions such as the one in Fig. 7d are free from defects, while high- +angle grain boundaries and dislocations are visible in some regions, as shown in Figs. 7e and 7f, +respectively. These microstructural features are rather similar to the microstructures of other Laves +phases synthesized by arc melting [15,16,27]. + + +a +Element +at% +b +Element +at% +Ti +6.8 ± 0.6 +Ti +6.4 ± 1.4 +Zr +27.4 ± 0.4 +Zr +25.6±3.2 +Cr +17.4 ± 0.0 +Cr +15.9 ± 3.2 +Mn +15.9 ± 0.0 +Mn +19.5±0.8 +10μm +Fe +16.5 ± 0.1 +Fe +17.3 ± 0.6 +SEM +500 nm +STEM +Ni +16.0 ± 0.0 +Ni +15.3 ±0.4 +Ti +Zr +Ti +Zr +Mn +Cr +Mn +Fe +Fe +Ni14 + + + + +Fig. 7. (a) SEM micrograph and corresponding (b) crystal orientation and (c) phase mappings +achieved by EBSD with beam step size of 1 μm; and (d-f) high-resolution TEM lattice images for (d) +C14 Laves phase lattice, (e) high-angle grain boundary and (f) dislocation for HEA +Ti0.4Zr1.6CrMnFeNi. + + +3.4. Room temperature hydrogen storage +Figs. 8a-d show the PCT absorption/desorption isotherms for three cycles at 298 K for the +TixZr2-xCrMnFeNi alloys with x = 0.4, 0.8, 1.2 and 1.6, respectively. Fig. 8a indicates that the +Ti0.4Zr1.6CrMnFeNi alloy reversibly absorbs and desorbs 1.7 wt% of hydrogen with good cycling +performance. The maximum hydrogen-to-metal (H/M) ratio for the alloy reaches 1.06, confirming +that the composition of the hydride can be reasonably considered as Ti0.4Zr1.6CrMnFeNiH6. PCT +isotherms for the Ti0.8Zr1.2CrMnFeNi alloy in Fig. 8b show a similar trend as for Ti0.4Zr1.6CrMnFeNi. +Further, the alloy can store 1.6 wt% of hydrogen with a hydrogen-to-metal ratio of 1, corresponding +to the Ti0.8Zr1.2CrMnFeNiH6 hydride. The main difference between Figs. 8a and 8b is that the alloy +with less titanium exhibits a lower plateau pressure which should be due to its stronger hydrogen +binding energy, as demonstrated in Fig. 3 using ab initio simulations. Fig. 8c shows the PCT +isotherms for the Ti1.2Zr0.8CrMnFeNi sample. This HEA absorbs 1.4 wt% hydrogen with a hydrogen- +to-metal ratio of 0.8. Since the plateau pressure for this alloy is high, its complete hydrogenation does +not occur by increasing the pressure to 9 MPa (i.e., the upper limit of pressure in the authors’ gas +absorption facility). As shown in Fig. 8d, the Ti1.6Zr0.4CrMnFeNi alloy absorbs a minor amount of + +a +C +C14 1010 +100μm +100.um +100.μm +SEM +C14 +00012110 +[112]] +[121 +C14 [311] +[122] [014] +[112] +C14 +2 nm +.nm +C14 [641] +311 +12115 + +0.1 wt% hydrogen, suggesting that the plateau pressure of this alloy is higher than 9 MPa because of +its weak hydrogen binding energy, as discussed in Fig. 3. + + + + +Fig. 8. PCT absorption/desorption isotherms at room temperature for HEAs (a) Ti0.4Zr1.6CrMnFeNi, +(b) Ti0.8Zr1.2CrMnFeNi, (c) Ti1.2Zr0.8CrMnFeNi and (d) Ti1.6Zr0.4CrMnFeNi; and (e) comparison of +third cycle of PCT absorption isotherms for HEAs. Data for Ti1.0Zr1.0CrMnFeNi in (e) were taken +from literature [27]. + + +Fig. 8e provides a clear comparison between the PCT isotherms in the third absorption cycle +for the TixZr2-xCrMnFeNi alloys with x = 0.4, 0.8, 1.0, 1.2 and 1.6, where the data for the + +a +Hydrogen to Metal Ratio, H/M +b +HydrogentoMetalRatio,H/M +0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +Tio.4Zr1.CrMnFeNi +Tio.8Zr12CrMnFeNi +10 +10 + (MPa) +Pressure (MPa) +1 +Pressure +0.1 +0.1 +H2 +0.01 +1st Cycle +0.01 +O- 1st Cycle +— 2nd Cycle +- 2nd Cycle +3rd Cycle + 3rd Cycle +0.001 +0.001 +0 +0.3 +0.6 +0.9 +1.2 +1.5 +1.8 +0 +0.3 +0.6 +0.9 +1.2 +1.5 +1.8 +Hydrogen Content (wt%) +Hydrogen Content (wt%) +c +Hydrogen to Metal Ratio, H/M +d +Hydrogen to Metal Ratio, H/M +0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +Ti1.2Zro.:CrMnFeNi +Ti1.6Zro.4CrMnFeNi +Pa) +10 + (MPa) +10 +Pressure +0.1 +0.1 +0.01 +O- 1st Cycle +0.01 +-O- 1st Cycle +H2 +2nd Cycle +—2nd Cycle +3rdCycle +- 3rd Cycle +0.001 +0.001 +0 +0.3 +0.6 +0.9 +1.2 +1.5 +1.8 +0 +0.3 +0.6 +0.9 +1.2 +1.5 +1.8 +Hydrogen Content (wt%) +Hydrogen Content (wt%) +e +3rd Absorption Cycle at 298 K +10 + (MPa) +Pressure +0.1 +Ti1.6Zro.4CrMnFeNi + Ti1.2Zro.8CrMnFeNi +0.01 +Ti.oZr1.oCrMnFeNi +-Tio.Zr1.2CrMnFeNi +O- Tio.4Zr1.6CrMnFeNi +0.001 +0 +0.3 +0.6 +0.9 +1.2 +1.5 +1.8 +Hydrogen Content (wt%)16 + +Ti1.0Zr1.0CrMnFeNi alloy were taken from the literature [27]. The plateau pressure systematically +increases by increasing the fraction of titanium, while the Ti0.4Zr1.6CrMnFeNi and +Ti0.8Zr1.2CrMnFeNi alloys have plateau pressures close to ambient pressure which renders them +appropriate for practical applications. Such an increase in the plateau pressure by increasing the +titanium content can be well explained by the influence of titanium on the hydrogen binding energy, +as shown in Fig. 3. +Fig. 9 demonstrates the kinetic measurements of hydrogen storage under an initial hydrogen +pressure of 3.7 MPa at room temperature for the HEAs. The alloys absorb hydrogen very fast within +almost 30 seconds, although the total amount of hydrogen decreases with increasing the fraction of +titanium, i.e., with increasing the plateau pressure. The amount of stored hydrogen in +Ti0.4Zr1.6CrMnFeNi in both kinetic and PCT measurements is higher than the capacity of commercial +room-temperature hydrogen storage materials [1-6]. + + + + +Fig. 9. Hydrogenation kinetic curves at room temperature for HEAs TixZr2-xCrMnFeNi. + + +Fig. 10 summarizes the cyclic hydrogen storage measurements for the Ti0.4Zr1.6CrMnFeNi +for up to 1000 cycles, where (a) shows the amount of hydrogen absorbed in each cycle within 10 min +under an initial hydrogen pressure of 3.7 MPa at 298 K and (b) shows the corresponding PCT +absorption/desorption isotherms for cycles 4, 30, 100 and 1000. The amount of absorbed hydrogen +remains almost constant within 1000 cycles, and the PCT isotherm also does not show any significant +change after 1000 hydrogenation-dehydrogenation cycles. The current results confirm the excellent +cycling performance and stability of this HEA, which is a critical issue in the commercialization of +hydrogen storage materials [1]. + +4. Discussion +HEAs are the most recent alloys that have been explored for hydrogen storage [8-25]. The +presence of multi-principal elements in the lattice of these alloys makes it possible to tune their +electronic structures, crystal structures and physical properties in a much more straightforward way +as compared to conventional alloys and intermetallics [7]. Since the capability to store hydrogen +strongly depends on the electronic structure [1-3], the flexibility in controlling the electronic structure +of HEAs introduces them as potential materials for hydrogen storage applications. However, the +investigations on high-entropy hydrogen storage materials are still in their infancy, and only a few +attempts have been pursued to combine theoretical calculations and experiments to develop new + +Hydrogen Content (wt%) +2.0 +TixZr2-xCrMnFeNi +x = 0.4 +1.5 +x = 0.8 +PH2 = 3.7 MPa +1.0 +TH2 = 298 K +x = 1.2 +0.5 +X = 1.6 +0 +0 +0.5 +1.0 +1.5 +2.0 +Time (min)17 + +HEAs with appropriate electronic structures for room-temperature hydrogen storage [7]. The current +study addresses this challenge and combines the existing empirical knowledge on the development +of HEAs for low-temperature hydrogen storage with first-principles calculations to design alloys that +can reversibly store hydrogen at ambient temperature under pressures close to atmospheric pressure. +The empirical knowledge suggests that HEAs with AB2-type Laves phase structure and a VEC close +to 6.4 have a high potential for low-temperature hydrogen storage [8,15,16,27]. Previous first- +principles calculations on conventional Mg-based alloys also suggest that a hydrogen binding energy +of -0.1 eV or slightly more negative can lead to room-temperature hydrogen storage [26]. The TixZr1- +xCrMnFeNi alloys were therefore selected for the present study because they satisfy all empirical +requirements and because their theoretical hydrogen binding energy can be tuned to negative values +smaller than -0.1 eV by changing the fraction of titanium and zirconium atoms. + + + + +Fig. 10. (a) Hydrogenation cycling tests for 1000 cycles at room temperature and (b) corresponding +PCT absorption/desorption isotherms for cycles 4, 30, 100 and 1000 for HEA Ti0.4Zr1.6CrMnFeNi. + + +The designed alloys reversibly adsorb and desorb hydrogen at room temperature, while the +equilibrium pressure can be easily tuned to appropriately low values by reducing the hydrogen +binding energy to more negative values via increasing the fraction of zirconium. The kinetics of +hydrogen storage is quite fast in these alloys due to their Laves phase structure [29]. Moreover, such +fast storage kinetics occur without any thermal activation or catalyst addition, in contrast to many +other storage materials including TiFe which suffer from the activation problem at room temperature + +a +Hydrogen Content (wt%) +2.0 +Tio.4Zr1.6CrMnFeNi +1.5 +1.0 +PH2 = 3.7 MPa +0.5 +TH2 = 298 K +0 +0 +200 +400 +600 +800 +1000 +Cycle Number +b +Hydrogen to Metal Ratio, H/M +0 +0.2 +0.4 +0.6 +0.8 +1.0 +Tio.4Zr1.6CrMnFeNi +10 +Pressure (MPa) +0.1 +4th Cycle +30th Cycle +H2 +0.01 +100thCycle +1000thCycle +0.001 +0 +0.3 +0.6 +0.9 +1.2 +1.5 +1.8 +Hydrogen Content (wt%)18 + +[3,5]. Another advantage is that the good hydrogen storage performance of the present HEAs was +achieved even though they were handled and crushed under an air atmosphere. Such a good air +resistance cannot be achieved easily for other room-temperature hydrogen storage materials without +conducting chemical modification [3] mechanochemical process [69] or mechanical treatment +through severe plastic deformation [70]. Among the selected alloys in this study, the amount of stored +hydrogen in Ti0.4Zr1.6CrMnFeNi and Ti0.8Zr1.2CrMnFeNi alloys is higher than the capacity of +commercial rare-earth-based LaNi5 for room-temperature hydrogen storage [4,71,72]. The working +pressure of these two alloys is also close to atmospheric pressure, in good agreement with their +hydrogen binding energies, suggesting their high potential for stationary applications [1,2]. +Therefore, a combination of first-principles calculations and experimental studies is an effective +approach to design and synthesize new high-entropy hydrides for room-temperature hydrogen +storage. Such an approach can also contribute to the advancement of nickel-metal-hydride (Ni-MH) +batteries because HEAs were recently reported to have high potential as anode materials of Ni-MH +batteries [73]. + +5. Conclusion +This study demonstrates the successful design and synthesis of high-entropy alloys having +the capability to store hydrogen at room temperature and under pressures close to atmospheric +pressures. The AB2-type Laves phase TixZr2-xCrMnFeNi (x = 0.4-1.6) alloys with a valence electron +concentration of 6.4 and low hydrogen binding energies (negative values close to -0.1 eV) were +designed theoretically by using first-principles calculations and fabricated experimentally by the +conventional arc melting method. These alloys reversibly absorb and desorb up to 1.7 wt% of +hydrogen at room temperature (298 K) with fast kinetics, while their (de)hydrogenation pressure is +systematically reduced by strengthening the hydrogen binding energy through increasing the +zirconium fraction. To the authors’ knowledge, this study is the first demonstration of adjusting the +hydrogen storage temperature and pressure of high-entropy alloys to ambient conditions by +employing the concept of binding-energy engineering. The concept introduced in the current study +can be universally employed to discover many hydrogen storage materials for practical applications. + +Acknowledgments +The authors thank Dr. Fritz Körmann of Max-Planck-Institut für Eisenforschung GmbH, +Germany, and Prof. Ricardo Floriano of the University of Campinas, Brazil, for fruitful discussion. +This work is supported in part by Grants-in-Aid for Scientific Research on Innovative Areas from +the MEXT, Japan (JP19H05176 & JP21H00150), in part by the European Research Council (ERC) +under the European Union’s Horizon 2020 Research and Innovation Programme (grant agreement +No 865855), in part by the State of Baden-Württemberg through bwHPC, and in part by the German +Research Foundation (DFG) through grant number INST 40/467-1 FUGG (JUSTUS cluster). + + +Appendix A +Fig. A shows the variations of magnetization versus magnetic field for (a) Ti0.4Zr1.6CrMnFeNi +and (b) Ti1.6Zr0.4CrMnFeNi at 5 K and 300 K obtained using a superconducting quantum interference +device (SQUID) magnetometer. Both alloys are paramagnetic at cryogenic and ambient temperatures +and do not show a clear ferromagnetic behavior. The ab initio and experimental results may be +comprehensively interpreted as follows; in the real alloys, each metal atom has a small magnetic +moment on average, but they are randomly orientated even at cryogenic temperatures. + +19 + + + + + +Fig. A. Magnetization versus magnetic field examined using SQUID at 5 K and 300 K for HEAs (a) +Ti0.4Zr1.6CrMnFeNi and (b) Ti1.6Zr0.4CrMnFeNi. + + +Appendix B +Atomic displacements are often considered to reflect the local lattice distortion in a HEA +[74,75]. The variation of atomic displacements and lattice distortion in a HEA is particularly +correlated with the magnitude of solid solution strengthening [76,77], although such distortions can +also influence the functional properties of HEAs such as their activity for hydrogen storage [7-24]. +Here, we quantify the atomic displacements from the ideal lattice sites for the TixZr2-xCrMnFeNi +alloys and TixZr2-xCrMnFeNiH6 hydrides (x = 0.5, 1.0 and 1.5) by ab initio DFT calculations. Fig. B +shows the ab initio computed mean atomic displacements from the ideal lattice sites for each metal +element as a function of unit cell volume. Overall, the atomic displacements become substantially +larger after hydriding which is likely due to the presence of hydrogen atoms at interstitial sites. For +the B-type elements, chromium shows the largest atomic displacements, followed by manganese, +iron, and nickel. The magnitude of atomic displacements for the B-type elements is likely related to +the binding energies to hydrogen atoms. This justification is consistent with the hydrogen binding +energies in cubic C15 ZrX2 alloys (X = V, Cr, Mn, Fe, Co, Ni) reported in a previous ab initio study +[35]. For the A-type elements, the atomic displacements of titanium increase with increasing the unit + +a +3 +Tio.4Zr1.6CrMnFeNi +(emu/g) +2 +0 +300 K +1 +-2 +5K +-3 +-60 +-40 +-20 +0 +20 +40 +60 +Magnetic Field (Oe) +X 103 +b +3 +Ti1.6Zro.4CrMnFeNi +emul +2 +0 +300 K +1 +5K +-2 +-3 +-60 +-40 +-20 +0 +20 +40 +60 +MagneticField(Oe) +X 10320 + +cell volume more strongly than for zirconium. This is likely because titanium has a smaller atomic +radius than zirconium [61]. That is, at a large volume, titanium atoms have a larger space than +zirconium atoms and thus can deviate from the ideal lattice sites more easily than zirconium atoms. + + + + +Fig. B. Ab initio calculated mean atomic displacements from ideal lattice sites for each metal element +as a function of unit cell volumes after relaxation of atomic positions for (a, c, e) HEAs TixZr2- +xCrMnFeNi and (b, d, f) corresponding high-entropy hydrides. + + + +a +0.20 +Tio.5Zr1.5CrMnFeNi +Tio.5Zr1.5CrMnFeNiH6 +A +Displacement +0.15 +0.15 +0.10 +0.10 +Mean +0.05 +Mean +0.05 +OTiZr△CrMnVFeNi +OTiZr △CrMnVFeNi +0 +0 +140 +150 +160 +170 +180 +185 +195 +205 +215 +225 +Unit Cell Volume (A") +Unit Cell Volume (A3 +c +0.20 +0.20 +Ti1.Zr1.oCrMnFeNi +Ti1.oZr1.oCrMnFeNiHe +A +blacement +0.15 +0.15 +0.10 +0.10 +Displ +Mean +0.05 +Mean +0.05 +OTi Zr△CrMnFe Ni +OTiZr △CrMnFeNi +0 +0 +U +140 +150 +160 +170 +180 +185 +195 +205 +215 +225 +Unit Cell Volume (A3 +Unit Cell Volume (A") +e +0.20 +20 +5CrMnFeNi +id +Ti1.5Zro.5CrMnFeNiH6 +A +lacement +0.15 +0.15 +0.10 +0.10 +Displ +0.05 +Mean +0.05 +OTi Zr△CrMnFeNi +OTi Zr△CrMnVFe Ni +0 +0 +140 +150 +160 +170 +180 +185 +195 +205 +215 +225 +Unit Cell Volume (A21 + +References +[1] J.B. von Colbe, J.R. 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Mater. 31 (2019) 1807142. + diff --git a/wtE0T4oBgHgl3EQf-QJD/content/tmp_files/load_file.txt b/wtE0T4oBgHgl3EQf-QJD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa918d418164b19cbb33ac72b83e99f281285b43 --- /dev/null +++ b/wtE0T4oBgHgl3EQf-QJD/content/tmp_files/load_file.txt @@ -0,0 +1,1748 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf,len=1747 +page_content='1 Acta Materialia 236 (2022) 118117 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='actamat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='118117 High-Entropy Hydrides for Fast and Reversible Hydrogen Storage at Room Temperature: Binding-Energy Engineering via First-Principles Calculations and Experiments Abbas Mohammadi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Yuji Ikeda2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Parisa Edalati1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Masaki Mito3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Blazej Grabowski2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Hai-Wen Li4 and Kaveh Edalati1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='* 1 WPI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' International Institute for Carbon-Neutral Energy Research (WPI-I2CNER),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Kyushu University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fukuoka 819-0395,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Japan 2 Institute for Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' University of Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Pfaffenwaldring 55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 70569 Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Germany 3 Graduate School of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Kyushu Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Kitakyushu 804-8550,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Japan 4 Hefei General Machinery Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Hefei 230031,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' China Despite high interest in compact and safe storage of hydrogen in the solid-state hydride form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' the design of alloys that can reversibly and quickly store hydrogen at room temperature under pressures close to atmospheric pressure is a long-lasting challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' In this study, first-principles calculations are combined with experiments to develop high-entropy alloys (HEAs) for room-temperature hydrogen storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' TixZr2-xCrMnFeNi (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6) alloys with the Laves phase structure and low hydrogen binding energies of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='15 eV are designed and synthesized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The HEAs reversibly store hydrogen in the form of Laves phase hydrides at room temperature, while (de)hydrogenation pressure systematically reduces with increasing the zirconium fraction in good agreement with the binding energy calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The kinetics of hydrogenation are fast, the hydrogenation occurs without any activation or catalytic treatment, the hydrogen storage performance remains stable for at least 1000 cycles, and the storage capacity is higher than that for commercial LaNi5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The current findings demonstrate that a combination of theoretical calculations and experiments is a promising pathway to design new high-entropy hydrides with high performance for hydrogen storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Keywords: Solid-state hydrogen storage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Density functional theory (DFT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' High-entropy alloy (HEA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Metal hydrides;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Laves phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Corresponding author: K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Edalati (E-mail: kaveh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='edalati@kyudai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='jp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Tel: +81-92-802-6744) 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='Introduction Excessive consumption of fossil fuels and CO2 emission caused by their utilization have led to the crisis of global warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Nowadays, finding clean fuels which do not emit CO2 is a serious challenge for scientists and industry leaders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Hydrogen is the cleanest fuel and has attracted attention as a substitute for fossil fuels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' however, besides the necessity for clean production of hydrogen, its safe and compact storage is a significant challenge [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Storage of hydrogen in the form of gas and liquid is conventionally used for various applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' However, the amount of stored hydrogen in the form of gas in typical commercial tanks with 225 liters and 20 MPa pressure is just 4 Kg [2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Therefore, this typical method has limitations in terms of volumetric and gravimetric storage densities, although there are recent trends to increase the storage pressure to 70 MPa by using special tanks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' In the liquid storage method, the volumetric and gravimetric storage densities are higher, and the safety is better than for the gas storage method, but liquifying hydrogen at low temperatures makes the method expensive and evaporation losses can also occur [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Solid-sate hydrogen storage particularly in the form of metal hydrides provides the most compact and safest technology to store hydrogen [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' To realize the application of metal hydrides for hydrogen storage, they should have several features such as the capability for reversible absorption and desorption of hydrogen at ambient temperature, high cycling stability, fast kinetics, appropriate storage pressure near atmospheric pressure and high gravimetric capacity (high gravimetric capacity for stationary applications is not as critical as it is for mobile applications) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Magnesium hydride and complex hydrides are well- known materials with high storage capacity, but they suffer from high thermodynamic stability, and thus, they function only at high temperatures [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' So far, a limited number of materials such as TiFe (TiFeH2) and LaNi5 (LaNi5H6) have been introduced for room-temperature hydrogen storage [3-6], but they exhibit other shortcomings such as the activation problem in TiFe, and a high price and low storage capacity in LaNi5 [3-6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Therefore, there are still significant demands to introduce new metal hydride systems that can satisfy the requirements for hydrogen storage at ambient temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' High-entropy materials, which are solid solutions of at least five principal elements with a configurational entropy higher than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5R (R: gas constant), have attracted attention in recent years for various applications including hydrogen storage [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The presence of several elements in a single phase allows to manipulate the electronic structure, hydrogen binding energy and accordingly hydrogen storage temperature and pressure by careful selection of principal elements and their concentrations [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' TiVZrNbHf [8-10], TiVCrNbMo [8], TiVCrNbTa [8], Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2Hf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='3 [11], Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2Hf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 [11], Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2Hf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='3Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 [11], TiZrVNbCr [12], V30Ti30Cr25Fe10Nb5 [13], V35Ti30Cr25Fe5Mn5 [13], Mg0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='10Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='30V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='25Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='10Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='25 [14], TiZrNbFeNi [15], TiZrNbCrFe [16], MgAlTiFeNi [17], Al0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='10Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='30V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='25Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='10Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='25 [18], Mg12Al11Ti33Mn11Nb33 [19], MgVAlCrNi [20], MgVTiCrFe [21], AlCrFeMnNiW [22], TiZrHfScMo [23], MgZrTiFe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 [24] and LaNiFeVMn [25] are some of the HEAs which have been investigated for hydrogen storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' However, as discussed in a recent review paper [7], these HEAs have drawbacks such as either high-temperature requirement for hydrogen storage, poor hydrogen storage reversibility, poor activation, or high storage pressure [8-25], which limit their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Although the research on high-entropy hydrogen storage materials is still in its early stages, designing these alloys by theoretical and computational methods is expected to provide a pathway to discover new materials that can quickly and reversibly store hydrogen under ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' In this work, first-principles calculations are combined with experiments to design HEAs for room-temperature hydrogen storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The designated alloys, TixZr2-xCrMnFeNi (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6) with the 3 Laves phase structure and low hydrogen binding energies of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='15 eV, show fast and reversible hydrogen storage at ambient temperature under pressures adjustable to the atmospheric pressure by changing the amounts of titanium and zirconium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' This simultaneous application of theoretical and experimental studies to high-entropy hydrogen storage materials confirms the significance of this strategy in developing new HEAs that can satisfy the requirements for stationary hydrogen storage applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Materials and methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Empirical material design The key issue in designing room-temperature hydrogen storage materials is to adjust the hydrogen binding energy to a negative value close to zero [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' An earlier study on first-principles calculations of Mg-based alloys suggested that binding energies of about -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 eV per hydrogen atom can be an appropriate target to achieve room temperature hydrogen storage [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Moreover, the hydrogen binding energy should be slightly more negative than -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 eV to reduce the equilibrium hydrogen storage pressure close to ambient pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Although such a concept has not been used to design HEAs so far, three empirical criteria were suggested by the current authors to achieve hydrogen storage at low temperatures in HEAs [27]: (i) AB2-type atomic configuration (A: elements which react with hydrogen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' B: elements with low affinity with hydrogen), (ii) C14 Laves phase structure formation in the alloy and hydride;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' and (iii) valence electron concentration (VEC) of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Hydrogen storage materials are usually a mixture of A-type elements (such as lanthanum, magnesium, titanium, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=') and B-type elements (such as nickel, iron, manganese, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=') [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The A-type elements have negative hydrogen binding energies and produce stable hydrides, while B-type elements have positive binding energies and usually do not absorb hydrogen, as schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 1a using the data reported in the literature [3,26,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Therefore, a combination of A-type and B-type elements can lead to the formation of alloys with an appropriately low hydrogen binding energy for room-temperature hydrogen storage such as TiFe and LaNi5 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' In this regard, it was found that AB2-type HEAs have a high potential for low- temperature hydrogen storage [27], while AB-type and A3B2-type systems are other candidates, yet with less potential [15,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The Laves phase alloys are considered as potential materials for hydrogen storage with high cycling stability for reversible hydrogenation and dehydrogenation and fast kinetics [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Another benefit of Laves phase alloys is that they can have lower cost compared to rare-earth- based alloys such as LaNi5 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' It was shown by both experiments and the CALPHAD (calculation of phase diagrams) method that the Laves phases can be formed in some high- entropy hydrogen storage systems such as Ti-Zr-Cr-Mn-Fe-Ni [27], Ti-Zr-Nb-Fe-Ni [15] and Ti-Zr-Nb-Cr-Fe [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' VEC is another key parameter that can be considered in designing alloys for reversible hydrogen storage at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Metals with low VEC such as lithium, magnesium and titanium usually produce stable hydrides, which can release hydrogen only at high temperatures, while metals with high VEC such as cobalt, nickel and copper usually exhibit low affinity with hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' It was suggested that HEAs with a VEC value of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 can desorb hydrogen at temperatures close to room temperature [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Although adjusting VEC is hard in simple binary or ternary alloys, it can be adjusted much easier in HEAs by changing the type and fraction of the principal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The hydrogen binding energy is the most important parameter that needs to be adjusted for reversible hydrogen storage at room temperature and under atmospheric pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The 4 dehydrogenation temperature increases and the equilibrium hydrogen pressure decreases with increasing the absolute value of negative binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Only a limited number of materials such as LaNi5 and TiFe show appropriate hydrogen binding energies for room-temperature hydrogen storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' As schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 1a, the target is to set the hydrogen binding energy to a value slightly more negative than -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 eV by adjusting the fraction of elements [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' (a) Schematic illustration of hydrogen binding energy on hydrogenation and dehydrogenation of different materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' (b) One of SQS models of HEA Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5CrMnFeNi and corresponding hydride Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5CrMnFeNiH6 (visualized using VESTA [33]), considered in present ab initio simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Therefore, this study focuses on HEAs with the AB2-type Laves phase structure and a VEC value of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' and among the available options, the Ti-Zr-Cr-Mn-Fe-Ni system can fit all these requirements [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The hydrogen binding energy is adjusted by changing the fraction of zirconium a TARGET TDeh (K): >773 630 460 3o0,NoHydrogenation Ti Mg Mg2NiTiFeiMnCr LaNis Ni Fe ttif +1 HydrogenBindingEnergy(eV) Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5CrMnFeNi Hydrogenation Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5CrMnFeNiH6 OMn OFe ONi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='H5 in balance with titanium, because zirconium has a larger atomic radius than the other elements in this alloying system, and thus, adjusting its fraction is supposed to have the most significant effect on lattice volume and binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Three compositions of TixZr2-xCrMnFeNi (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5) were theoretically studied by first-principles binding energy calculations and four compositions of TixZr2- xCrMnFeNi (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6) were experimentally examined by hydrogen storage characterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' First-principles calculation methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Crystal structure modeling of alloys The AB2-type hexagonal C14 Laves phases have the space group of P63/mmc (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 194), in which the A-type atoms occupy the 4f Wyckoff sites, and the B-type atoms occupy the 2a and the 6h Wyckoff sites [29,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' These phases have in total 12 atoms in their unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' If A and B atoms in the AB2-type Laves phases are approximated by close-packed rigid spheres with the radii rA and rB, respectively, the ratio rA/rB is given by (3/2)1/2 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='225 [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' These atomic positions in the close- packing case, which are considered the ideal positions, lead to a c/a ratio of (8/3)1/2 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='633 (a and c are lattice parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The C14 TixZr2−xCrMnFeNi Laves phase alloys (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5) were modeled using 48- atom supercells with a 2 × 2 × 1 expansion of the primitive cell of the C14 phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' These three compositions, which slightly differ from the experimental compositions, were selected to investigate the dependence of the hydrogen binding energy on the titanium fraction with 48-atom systems and reasonable computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The A sites were occupied by titanium and zirconium, and the B sites were occupied by chromium, manganese, iron, and nickel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The sublattice chemical disorder was modeled using special quasi-random structure (SQS) configurations [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Correlation functions of the first several nearest-neighbor doublet, triplet, and quartet clusters were optimized to be close to the ideal values of fully random configurations using the simulated annealing approach implemented in the ICET code [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' To achieve better statistics, six different configurations were considered for each composition by permuting the elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 1b (upper image) shows one example of modeled structure of a HEA which was modeled using SQS [31] and visualized using VESTA [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Crystal structure modeling of hydrides To model the structure of the high-entropy hydrides, crystallographic information and reports from the literature were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' In the Laves phase alloys, there are 17 tetrahedral interstitial sites per formula unit AB2, which could be occupied by hydrogen atoms [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' These interstitial sites are surrounded by either four B atoms (B4), one A and three B atoms (AB3), or two A and two B atoms (A2B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' For many conventional Laves phases composed of the elements as utilized in the present study, previous ab initio simulations found that the A2B2 sites are the most energetically preferable sites for hydrogen atoms [35-39], and indeed many experiments found hydrogen atoms at the A2B2 sites [40-48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' This may be intuitively understood because the A2B2 sites have larger volumes than the B4 and AB3 sites when the atoms A and B are on the ideal Laves lattice sites, as summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The hydrogen atoms in Laves phases should also be repulsive to each other and should not occupy very close interstitial sites, as empirically suggested [49] and confirmed by first-principles calculations for some Laves phases [36,39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Since previous experimental studies reported that the AB2-type high-entropy Laves phase produces a Laves phase hydride with the composition AB2H3 [27], the same hydrogen fraction was considered for modeling in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' It should be noted that for the present 48-metal-atom supercell models, there are in total 272 interstitial sites, all of which have different local chemical 6 environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Therefore, even for the fixed composition AB2H3, there are in total over 1053 possible ways of hydrogen occupations, which are obviously impossible to test in a brute-force manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Therefore, based on the previous experiments and simulations mentioned above [34-48], we a priori assumed that all hydrogen atoms occupy the A2B2 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Moreover, while there are four symmetrically inequivalent A2B2 sites in the C14 hexagonal Laves phase [34], it was assumed that all hydrogen atoms occupy the sites with the Wyckoff symbol 12k, because these sites do not share the faces of the interstitial tetrahedra and thus the hydrogen atoms occupying them are not too close to each other [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' These assumptions uniquely determine the hydrogen-occupied sites for each supercell model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 1b shows one of the thus obtained models considered in the present ab initio simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Numbers of tetrahedral interstitial sites with different local environments per formula unit and ratios of their individual volumes to alloy volume per formula unit for AB2 Laves phases with ideal atomic positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Tetrahedra Site Number Volume Ratio B4 1 1/24 (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0417) AB3 4 5/96 (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0521) A2B2 12 1/16 (= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0625) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Hydrogen binding energy calculations Ab initio density functional theory (DFT) calculations were performed using the VASP code [50-52] with the plane-wave basis projector augmented wave (PAW) method [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The exchange- correlation energy was obtained within the generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof (PBE) form [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The plane-wave cutoff energy was set to 400 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Reciprocal spaces were sampled by a Γ-centered 4 × 4 × 6 k-point mesh for the 48-metal-atom supercell models and the Methfessel-Paxton method [55] with the smearing width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The 3d 4s orbitals of titanium, chromium, manganese, iron, and nickel and the 4s 4p 4d 5s orbitals of zirconium were treated as the valence states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Total energies were minimized until they converged within 1 × 10-3 eV per simulation cell for each ionic step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' All calculations were performed by considering spin polarization, a fact that was also experimentally examined by magnetic measurements, as discussed in Appendix A and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' To obtain the energy-volume curves, seven volumes in the ranges of 144 to 180 Å3/u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' and 186 to 222 Å3/u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' (u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' : unit cell) were considered for the systems with and without hydrogen atoms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' For each composition, the obtained energies of six SQS-based models for the given volumes were then fitted to the Vinet equation of state [56,57] to obtain the volume and the energy in the equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Metal and hydrogen atoms were initially put on the ideal Laves-phase lattice sites and the geometric centers (centroids) of the A2B2 12k interstitial sites, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The atomic positions were then relaxed with fixing the cell shape and volume until all the forces on the atoms converged within 5 × 10-2 eV/Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The hydrogen binding energy per atom was then computed as 𝛥𝐸H = 1 3 [𝐸(AB2H3) − 𝐸(AB2) − 3 2 𝐸(H2)] (1) where E(H2), E(AB2), and E(AB2H3) are the energies of H2, AB2, and AB2H3 per formula unit, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The energy of the H2 molecule was computed in a 20 × 20 × 20 Å3 simulation cell and 7 for the Γ point in the reciprocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The obtained hydrogen-hydrogen distance was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='751 Å, in reasonable agreement with the experimental value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='74144 Å [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Experimental Procedures The ingots of HEAs with compositions TixZr2-xCrMnFeNi (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6) with ~10 g mass were prepared by arc melting using pieces of pure titanium (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='99%), zirconium (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5%), chromium (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='99%), manganese (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='95%), iron (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='97%) and nickel (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='9%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The pieces were melted and mixed in a water-cooled copper crucible under an argon atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' To increase the homogeneity of the alloys, the mixture was remelted six times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Ingots produced by arc melting were characterized by various methods, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' First, the crystal structure was examined by X-ray diffraction (XRD) using Cu Kα irradiation with a filament current of 40 mA and an acceleration voltage of 45 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The Rietveld method using the PDXL software was used to identify the phases and determine their lattice parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Second, the microstructure of the samples was investigated by a scanning electron microscope (SEM) equipped with energy-dispersive X-ray spectroscopy (EDS) and electron backscatter diffraction (EBSD) at 15 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The samples for SEM were prepared by cutting a piece of the ingot using electric discharge machining, followed by mechanical grinding using sandpapers, polishing using buff and 9 µm and 3 µm diamond suspensions, and final polishing by buff and colloidal silica with 60 nm particle size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Third, the nanostructure of the samples was examined using transmission electron microscopy (TEM) and scanning-transmission electron microscopy (STEM) at 200 kV using high- resolution imaging, fast Fourier transform (FFT) analysis and EDS analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' For TEM and STEM analyses, a small piece of the samples was crushed in ethanol and dispersed on a carbon grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fourth, hydrogen storage was examined using a Sieverts-type gas absorption apparatus at room temperature (298 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' For this experiment, ~1 g of the alloys was crushed in an air atmosphere to achieve particle sizes below 75 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The crushed samples were examined in terms of pressure- composition-temperature (PCT) isotherms for three hydriding-dehydriding cycles followed by a hydrogenation kinetic measurement for one cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The samples before PCT measurements were subjected to evacuation at room temperature for 3 h to remove air and moisture, but no thermal activation treatment was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The crystal structure of the hydrides after hydrogen storage was also examined by conducting the XRD analysis immediately after hydriding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Moreover, cyclic hydrogenation-dehydrogenation measurements were conducted for 1000 cycles (10 min hydrogenation under an initial hydrogen pressure of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='7 MPa followed by 20 min evacuation at 298 K) to examine the cycling stability of the alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Ab initio calculated energy-volume curves fitted to the Vinet equation of state without and with relaxation of atomic positions for (a, c, e) hydrogen-free HEAs TixZr2-xCrMnFeNi and (b, d, f) corresponding high-entropy hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Arrows show equilibrium volumes for fixed and relaxed atomic positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Calculated hydrogen binding energy Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 2a, c and e show the ab initio computed energy-volume curves before and after relaxation of atomic positions for the TixZr2-xCrMnFeNi alloys with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 b 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5CrMnFeNi Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5CrMnFeNiH6 (eV) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 Fixed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='. Fixed Energy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Relaxed 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Relaxed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 Cell 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Unit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 0 0 140 150 160 170 180 185 195 205 215 225 Unit Cell Volume (A3) c 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 d 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='oZr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='oCrMnFeNi Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='oZr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='oCrMnFeNiH6 (eV) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 Unit Cell Energy (eV) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='. Fixed Fixed Energy Relaxed 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Relaxed 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 Unit Cell 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 0 0 140 150 160 170 180 185 195 205 215 225 Unit Cell Volume (A" e 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='Zro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5CrMnFeNi Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Zro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5CrMnFeNiH6 (eV) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 Fixed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='Fixed Cell Energy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Relaxed IEnergy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Relaxed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 Unit Cell 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Unit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 0 140 150 160 170 180 185 195 205 215 2259 The equilibrium unit cell volume decreases with increasing the fraction of titanium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Moreover, for all the compositions, the equilibrium unit cell volume decreases after the relaxation, as summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 2b, d and f show the ab initio computed energy-volume curves before and after relaxation of atomic positions for the TixZr2-xCrMnFeNiH6 hydrides with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' In these hydrides, hydrogen atoms occupy all the 12k A2B2 interstitial sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Similar to the HEAs, the unit cell volume of the hydride decreases with increasing the titanium fraction, but as summarized in Table 2, the main difference is that the unit cell volumes for the hydrides are 21-23% larger compared to the corresponding alloys (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=', larger by 3 Å3 per hydrogen atom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Such a volume expansion is in a similar order to those experimentally reported for various binary and ternary hydrides [59] and also similar to the one previously calculated by ab initio for the Zr(Cr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5)2 alloy with the C14 Laves phase (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 Å3 per hydrogen atom) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Another difference between the alloys and hydrides is that the impact of atomic-position relaxation on the equilibrium volume and energy changes is much larger for the hydrides compared to the alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' This is likely because the positions of metal atoms more strongly deviate from the ideal lattice sites due to the presence of hydrogen atoms, as detailed in Appendix B and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The hydrogen binding energy per hydrogen atom for relaxed atomic positions is given in Table 2 for the three high-entropy hydrides TixZr2-xCrMnFeNiH6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The hydrogen binding energies are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='126, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='105 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='074 eV per hydrogen atom for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' These values are substantially lower than those at the A2B2 sites in the Laves phases consisting of similar elements but chemically less disordered such as -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='27 eV per hydrogen atom for TiCr2 [36], and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='20 to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='28 eV per hydrogen atom for Zr(Cr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5)2 [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' This implies that the present Laves HEAs can more easily desorb the hydrogen atoms, and thus, can be more practical for applications where low- temperature hydrogen storage is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 3 visualizes the dependence of the hydrogen binding energy as a function of the titanium content for relaxed atomic positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' With increasing the titanium content, the hydrogen binding energy becomes less negative, and thus the hydrides become less stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Therefore, it is expected that the alloys with lower titanium content can desorb hydrogen at lower temperatures and lower pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The hydrogen binding energies for Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5CrMnFeNiH6 and Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0CrMnFeNiH6 are slightly more negative than -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 eV, suggesting the potential of these two alloys for room-temperature hydrogen storage [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The strong dependence of the hydrogen binding energy on the composition also indicates the possibility to tailor the properties of hydrogen storage alloys by modifying the compositions of HEAs [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Ab initio calculated equilibrium volumes for fixed and relaxed atomic positions, unit cell energy difference between fixed and relaxed conditions, and hydrogen binding energy for HEAs TixZr2-xCrMnFeNi and their corresponding TixZr2-xCrMnFeNiH6 hydrides where hydrogen atoms occupy all the 12k A2B2 interstitial sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Alloy/Hydride Volume (Å3/u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=') ERelaxed-EFixed ΔEH Fixed Relaxed (Å3/u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=') (eV / H atom) Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5CrMnFeNi 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='18 Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0CrMnFeNi 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='28 Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5CrMnFeNi 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='3 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='26 Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5CrMnFeNiH6 207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='126 Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0CrMnFeNiH6 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='105 Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5CrMnFeNiH6 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='074 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Ab initio calculated hydrogen binding energy as a function of titanium content for high- entropy hydrides TixZr2-xCrMnFeNiH6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Atomic positions were relaxed for these calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Structural features of alloys and hydrides Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 4 summarizes the results of the XRD analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 4a shows the XRD profiles of the HEAs, TixZr2-xCrMnFeNi (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' All samples have mainly a hexagonal C14 Laves phase structure with the P63/mmc space group, while the presence of weak peaks at 42-43˚ for the Ti-rich alloys suggests that minor amounts of a cubic phase can also be present [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 4b shows a magnified view of the XRD profiles for the (110) peak of the Laves phase located between 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5˚ and 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 ˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' This figure demonstrates a systematic shift of this peak to higher angles with increasing the titanium fraction, indicating that a lattice contraction occurs by increasing the titanium fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The occurrence of lattice contraction by increasing the titanium fraction is shown more clearly in Table 3, where the calculated lattice parameters using the Rietveld method are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' It should be noted that the plots of calculated-observed intensities in the Rietveld analysis indicate a maximum intensity difference of 10%, which is a reasonable level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Such a difference should be partly due to the inherent distortion in high-entropy materials [7] and partly due to the presence of a small amount of a cubic phase [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 4c compares the XRD pattern of Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi before and after hydriding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The hydride phase also has a hexagonal C14 crystal structure, while the shift of the XRD profile to lower angles after hydriding suggests that the lattice is expanded in the presence of hydrogen atoms in the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 4d summarizes the lattice parameters calculated from the XRD analysis for both alloys and hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Note that the data for Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0CrMnFeNi was calculated from the XRD profiles reported in an earlier publication [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' It should be also noted that the crystal structure of the Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8CrMnFeNiH6 and Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4CrMnFeNiH6 hydrides could not be identified because these two hydrides released hydrogen very fast under ambient conditions and before the XRD analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Both a and c lattice parameters increase with increasing the number of zirconium atoms, which have a larger atomic radius than titanium atoms [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Moreover, the lattice parameters increase with the addition of hydrogen atoms to the materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' (eV/atom) 0 Ti,Zr2-xCrMnFeNiH6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='02 Energy ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 x in Ti,Zr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='vCrMnFeNiH11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' (a, b) XRD profiles of hydrogen-free HEAs TixZr2-xCrMnFeNi where (b) shows magnified view of (110) peak of C14 Laves phase, (c) XRD profiles of hydrogen-free HEA Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi and corresponding high-entropy hydride, and (d) lattice parameters of hydrogen-free HEAs TixZr2- xCrMnFeNi and corresponding high-entropy hydrides determined from XRD profiles using Rietveld method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Data for Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0CrMnFeNi in (d) were taken from literature [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Lattice parameters of C14 Laves phase obtained experimentally using XRD analysis for HEAs TixZr2-xCrMnFeNi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Alloy a (Å) c (Å) Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='982 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='008 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='148 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='020 Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2CrMnFeNi 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='945 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='014 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='093 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='030 Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8CrMnFeNi 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='908 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='010 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='002 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='020 Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4CrMnFeNi 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='871 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='011 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='942 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='020 a TixZr2-xCrMnFeNi b TixZr2-xCrMnFeNi ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='C14 Normalized Intensity C14 (110) X=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6·: = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 Normalized X=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 ·^α 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' X= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 X=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8- X=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 ^ X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 30 40 50 60 70 80 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 Diffraction Angle, 20 (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=') Diffraction Angle, 20 (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=') c d 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='7 Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr16CrMnFeNi O-- Hydride C14 Alloy A 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 Hydride 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 --Hydride 卜 Alloy 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 A Alloy a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 30 40 50 60 70 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 Diffraction Angle, 20 (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=') x in Ti,Zr2-x CrMnFeNi(H6)12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Unit cell volume of C14 Laves phase obtained experimentally using XRD analysis and calculated using ab initio without (fixed) and with relaxation of atomic positions for (a) HEAs TixZr2- xCrMnFeNi and (b) corresponding high-entropy hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' It is now possible to compare the unit cell volumes obtained experimentally and theoretically to validate the models used for the first-principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 5 compares the equilibrium volumes obtained by DFT with those obtained experimentally for the (a) alloys and (b) hydrides with different fractions of titanium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The experimental and theoretical data show similar trends and the unit cell volume decreases with increasing the titanium content for both alloys and hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The linear change in the unit cell volume is qualitatively consistent with Vegard’s law by considering the difference in the atomic radius of titanium and zirconium [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The ab initio calculated volumes are slightly smaller than those obtained in experiments: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5% for the alloys and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5% for the hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Such an underestimation of volume using GGA (like PW91 [62,63] or PBE [54]) is found often in systems containing magnetic 3d elements such as pure bcc iron [64-66] and the HEA CrMnFeCoNi [67,68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The small differences between the experimental and theoretical cell volumes confirm the validity of the models used for the simulation of the HEAs and hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' a 180 TixZr2-xCrMnFeNi Unit Cell Volume (A) 175 170 165 Experiment 160 DFT(Fixed) DFT (Relaxed) 155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 x in TixZr2-xCrMnFeNi b 215 TixZr2-xCrMnFeNiH6 Unit Cell Volume (A") 210 205 200 Experiment 195 DFT (Fixed) DFT (Relaxed) 190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 x in Ti,Zr2-xCrMnFeNiH613 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Distribution of elements and their atomic fractions in HEA Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi examined by EDS elemental mappings using (a) SEM and (b) STEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Microstructure of alloys The microstructure of Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi, taken as a representative alloy with the lowest hydrogen binding energy, is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 6 shows (a) SEM-EDS and (b) STEM-EDS analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The distribution of elements is reasonably uniform at the micrometer and nanometer levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Moreover, the fraction of elements is reasonably consistent with the nominal composition within the detection limits of EDS analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The EDS analysis confirms that arc melting can be successfully employed to synthesize the HEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' An SEM micrograph and corresponding EBSD crystal orientation and phase mappings are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 7a-c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 7b illustrates that the HEA includes mainly large and elongated grains with sizes of several hundred micrometers and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 7c confirms that the alloy contains mainly a C14 Laves phase in good agreement with the XRD analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 7d-f show the high-resolution TEM images of the HEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The C14 Laves phase was the only phase that could be detected by high-resolution TEM images in good agreement with the XRD and EBSD analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Most of the examined regions such as the one in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 7d are free from defects, while high- angle grain boundaries and dislocations are visible in some regions, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 7e and 7f, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' These microstructural features are rather similar to the microstructures of other Laves phases synthesized by arc melting [15,16,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' a Element at% b Element at% Ti 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 Ti 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 Zr 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 Zr 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 Cr 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Cr 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='9 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 Mn 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Mn 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 10μm Fe 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 Fe 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 SEM 500 nm STEM Ni 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Ni 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='3 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 Ti Zr Ti Zr Mn Cr Mn Fe Fe Ni14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' (a) SEM micrograph and corresponding (b) crystal orientation and (c) phase mappings achieved by EBSD with beam step size of 1 μm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' and (d-f) high-resolution TEM lattice images for (d) C14 Laves phase lattice, (e) high-angle grain boundary and (f) dislocation for HEA Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Room temperature hydrogen storage Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 8a-d show the PCT absorption/desorption isotherms for three cycles at 298 K for the TixZr2-xCrMnFeNi alloys with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 8a indicates that the Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi alloy reversibly absorbs and desorbs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='7 wt% of hydrogen with good cycling performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The maximum hydrogen-to-metal (H/M) ratio for the alloy reaches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='06, confirming that the composition of the hydride can be reasonably considered as Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNiH6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' PCT isotherms for the Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2CrMnFeNi alloy in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 8b show a similar trend as for Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Further, the alloy can store 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 wt% of hydrogen with a hydrogen-to-metal ratio of 1, corresponding to the Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2CrMnFeNiH6 hydride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The main difference between Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 8a and 8b is that the alloy with less titanium exhibits a lower plateau pressure which should be due to its stronger hydrogen binding energy, as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 3 using ab initio simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 8c shows the PCT isotherms for the Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8CrMnFeNi sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' This HEA absorbs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 wt% hydrogen with a hydrogen- to-metal ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Since the plateau pressure for this alloy is high, its complete hydrogenation does not occur by increasing the pressure to 9 MPa (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=', the upper limit of pressure in the authors’ gas absorption facility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 8d, the Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4CrMnFeNi alloy absorbs a minor amount of a C C14 1010 100μm 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='um 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='μm SEM C14 00012110 [112]] [121 C14 [311] [122] [014] [112] C14 2 nm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='nm C14 [641] 311 12115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 wt% hydrogen, suggesting that the plateau pressure of this alloy is higher than 9 MPa because of its weak hydrogen binding energy, as discussed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' PCT absorption/desorption isotherms at room temperature for HEAs (a) Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi, (b) Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2CrMnFeNi, (c) Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8CrMnFeNi and (d) Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4CrMnFeNi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' and (e) comparison of third cycle of PCT absorption isotherms for HEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Data for Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0CrMnFeNi in (e) were taken from literature [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 8e provides a clear comparison between the PCT isotherms in the third absorption cycle for the TixZr2-xCrMnFeNi alloys with x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6, where the data for the a Hydrogen to Metal Ratio, H/M b HydrogentoMetalRatio,H/M 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='CrMnFeNi Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8Zr12CrMnFeNi 10 10 (MPa) Pressure (MPa) 1 Pressure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 H2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='01 1st Cycle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='01 O- 1st Cycle — 2nd Cycle 2nd Cycle 3rd Cycle 3rd Cycle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='001 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 Hydrogen Content (wt%) Hydrogen Content (wt%) c Hydrogen to Metal Ratio, H/M d Hydrogen to Metal Ratio, H/M 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2Zro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' :CrMnFeNi Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6Zro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4CrMnFeNi Pa) 10 (MPa) 10 Pressure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='01 O- 1st Cycle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='01 O- 1st Cycle H2 2nd Cycle —2nd Cycle 3rdCycle 3rd Cycle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='001 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 Hydrogen Content (wt%) Hydrogen Content (wt%) e 3rd Absorption Cycle at 298 K 10 (MPa) Pressure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6Zro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4CrMnFeNi Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2Zro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8CrMnFeNi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='01 Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='oZr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='oCrMnFeNi Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2CrMnFeNi O- Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='001 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 Hydrogen Content (wt%)16 Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0CrMnFeNi alloy were taken from the literature [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The plateau pressure systematically increases by increasing the fraction of titanium, while the Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi and Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2CrMnFeNi alloys have plateau pressures close to ambient pressure which renders them appropriate for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Such an increase in the plateau pressure by increasing the titanium content can be well explained by the influence of titanium on the hydrogen binding energy, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 9 demonstrates the kinetic measurements of hydrogen storage under an initial hydrogen pressure of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='7 MPa at room temperature for the HEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The alloys absorb hydrogen very fast within almost 30 seconds, although the total amount of hydrogen decreases with increasing the fraction of titanium, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=', with increasing the plateau pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The amount of stored hydrogen in Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi in both kinetic and PCT measurements is higher than the capacity of commercial room-temperature hydrogen storage materials [1-6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Hydrogenation kinetic curves at room temperature for HEAs TixZr2-xCrMnFeNi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 10 summarizes the cyclic hydrogen storage measurements for the Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi for up to 1000 cycles, where (a) shows the amount of hydrogen absorbed in each cycle within 10 min under an initial hydrogen pressure of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='7 MPa at 298 K and (b) shows the corresponding PCT absorption/desorption isotherms for cycles 4, 30, 100 and 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The amount of absorbed hydrogen remains almost constant within 1000 cycles, and the PCT isotherm also does not show any significant change after 1000 hydrogenation-dehydrogenation cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The current results confirm the excellent cycling performance and stability of this HEA, which is a critical issue in the commercialization of hydrogen storage materials [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Discussion HEAs are the most recent alloys that have been explored for hydrogen storage [8-25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The presence of multi-principal elements in the lattice of these alloys makes it possible to tune their electronic structures, crystal structures and physical properties in a much more straightforward way as compared to conventional alloys and intermetallics [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Since the capability to store hydrogen strongly depends on the electronic structure [1-3], the flexibility in controlling the electronic structure of HEAs introduces them as potential materials for hydrogen storage applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' However, the investigations on high-entropy hydrogen storage materials are still in their infancy, and only a few attempts have been pursued to combine theoretical calculations and experiments to develop new Hydrogen Content (wt%) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 TixZr2-xCrMnFeNi x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 PH2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='7 MPa 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 TH2 = 298 K x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 X = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Time (min)17 HEAs with appropriate electronic structures for room-temperature hydrogen storage [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The current study addresses this challenge and combines the existing empirical knowledge on the development of HEAs for low-temperature hydrogen storage with first-principles calculations to design alloys that can reversibly store hydrogen at ambient temperature under pressures close to atmospheric pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The empirical knowledge suggests that HEAs with AB2-type Laves phase structure and a VEC close to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 have a high potential for low-temperature hydrogen storage [8,15,16,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Previous first- principles calculations on conventional Mg-based alloys also suggest that a hydrogen binding energy of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 eV or slightly more negative can lead to room-temperature hydrogen storage [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The TixZr1- xCrMnFeNi alloys were therefore selected for the present study because they satisfy all empirical requirements and because their theoretical hydrogen binding energy can be tuned to negative values smaller than -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 eV by changing the fraction of titanium and zirconium atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' (a) Hydrogenation cycling tests for 1000 cycles at room temperature and (b) corresponding PCT absorption/desorption isotherms for cycles 4, 30, 100 and 1000 for HEA Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The designed alloys reversibly adsorb and desorb hydrogen at room temperature, while the equilibrium pressure can be easily tuned to appropriately low values by reducing the hydrogen binding energy to more negative values via increasing the fraction of zirconium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The kinetics of hydrogen storage is quite fast in these alloys due to their Laves phase structure [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Moreover, such fast storage kinetics occur without any thermal activation or catalyst addition, in contrast to many other storage materials including TiFe which suffer from the activation problem at room temperature a Hydrogen Content (wt%) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 PH2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='7 MPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 TH2 = 298 K 0 0 200 400 600 800 1000 Cycle Number b Hydrogen to Metal Ratio, H/M 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi 10 Pressure (MPa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 4th Cycle 30th Cycle H2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='01 100thCycle 1000thCycle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='001 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8 Hydrogen Content (wt%)18 [3,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Another advantage is that the good hydrogen storage performance of the present HEAs was achieved even though they were handled and crushed under an air atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Such a good air resistance cannot be achieved easily for other room-temperature hydrogen storage materials without conducting chemical modification [3] mechanochemical process [69] or mechanical treatment through severe plastic deformation [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Among the selected alloys in this study, the amount of stored hydrogen in Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi and Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='8Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='2CrMnFeNi alloys is higher than the capacity of commercial rare-earth-based LaNi5 for room-temperature hydrogen storage [4,71,72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The working pressure of these two alloys is also close to atmospheric pressure, in good agreement with their hydrogen binding energies, suggesting their high potential for stationary applications [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Therefore, a combination of first-principles calculations and experimental studies is an effective approach to design and synthesize new high-entropy hydrides for room-temperature hydrogen storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Such an approach can also contribute to the advancement of nickel-metal-hydride (Ni-MH) batteries because HEAs were recently reported to have high potential as anode materials of Ni-MH batteries [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Conclusion This study demonstrates the successful design and synthesis of high-entropy alloys having the capability to store hydrogen at room temperature and under pressures close to atmospheric pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The AB2-type Laves phase TixZr2-xCrMnFeNi (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6) alloys with a valence electron concentration of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4 and low hydrogen binding energies (negative values close to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='1 eV) were designed theoretically by using first-principles calculations and fabricated experimentally by the conventional arc melting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' These alloys reversibly absorb and desorb up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='7 wt% of hydrogen at room temperature (298 K) with fast kinetics, while their (de)hydrogenation pressure is systematically reduced by strengthening the hydrogen binding energy through increasing the zirconium fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' To the authors’ knowledge, this study is the first demonstration of adjusting the hydrogen storage temperature and pressure of high-entropy alloys to ambient conditions by employing the concept of binding-energy engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The concept introduced in the current study can be universally employed to discover many hydrogen storage materials for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Acknowledgments The authors thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fritz Körmann of Max-Planck-Institut für Eisenforschung GmbH, Germany, and Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Ricardo Floriano of the University of Campinas, Brazil, for fruitful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' This work is supported in part by Grants-in-Aid for Scientific Research on Innovative Areas from the MEXT, Japan (JP19H05176 & JP21H00150), in part by the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (grant agreement No 865855), in part by the State of Baden-Württemberg through bwHPC, and in part by the German Research Foundation (DFG) through grant number INST 40/467-1 FUGG (JUSTUS cluster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Appendix A Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' A shows the variations of magnetization versus magnetic field for (a) Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi and (b) Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4CrMnFeNi at 5 K and 300 K obtained using a superconducting quantum interference device (SQUID) magnetometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Both alloys are paramagnetic at cryogenic and ambient temperatures and do not show a clear ferromagnetic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The ab initio and experimental results may be comprehensively interpreted as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' in the real alloys, each metal atom has a small magnetic moment on average, but they are randomly orientated even at cryogenic temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 19 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Magnetization versus magnetic field examined using SQUID at 5 K and 300 K for HEAs (a) Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi and (b) Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4CrMnFeNi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Appendix B Atomic displacements are often considered to reflect the local lattice distortion in a HEA [74,75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The variation of atomic displacements and lattice distortion in a HEA is particularly correlated with the magnitude of solid solution strengthening [76,77], although such distortions can also influence the functional properties of HEAs such as their activity for hydrogen storage [7-24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Here, we quantify the atomic displacements from the ideal lattice sites for the TixZr2-xCrMnFeNi alloys and TixZr2-xCrMnFeNiH6 hydrides (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5) by ab initio DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' B shows the ab initio computed mean atomic displacements from the ideal lattice sites for each metal element as a function of unit cell volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Overall, the atomic displacements become substantially larger after hydriding which is likely due to the presence of hydrogen atoms at interstitial sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' For the B-type elements, chromium shows the largest atomic displacements, followed by manganese, iron, and nickel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' The magnitude of atomic displacements for the B-type elements is likely related to the binding energies to hydrogen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' This justification is consistent with the hydrogen binding energies in cubic C15 ZrX2 alloys (X = V, Cr, Mn, Fe, Co, Ni) reported in a previous ab initio study [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' For the A-type elements, the atomic displacements of titanium increase with increasing the unit a 3 Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6CrMnFeNi (emu/g) 2 0 300 K 1 2 5K 3 60 40 20 0 20 40 60 Magnetic Field (Oe) X 103 b 3 Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='6Zro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='4CrMnFeNi emul 2 0 300 K 1 5K 2 3 60 40 20 0 20 40 60 MagneticField(Oe) X 10320 cell volume more strongly than for zirconium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' This is likely because titanium has a smaller atomic radius than zirconium [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' That is, at a large volume, titanium atoms have a larger space than zirconium atoms and thus can deviate from the ideal lattice sites more easily than zirconium atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Ab initio calculated mean atomic displacements from ideal lattice sites for each metal element as a function of unit cell volumes after relaxation of atomic positions for (a, c, e) HEAs TixZr2- xCrMnFeNi and (b, d, f) corresponding high-entropy hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='20 Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5CrMnFeNi Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5CrMnFeNiH6 A Displacement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='10 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='05 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='05 OTiZr△CrMnVFeNi OTiZr △CrMnVFeNi 0 0 140 150 160 170 180 185 195 205 215 225 Unit Cell Volume (A") Unit Cell Volume (A3 c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='20 Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='Zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='oCrMnFeNi Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='oZr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='oCrMnFeNiHe A blacement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='10 Displ Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='05 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='05 OTi Zr△CrMnFe Ni OTiZr △CrMnFeNi 0 0 U 140 150 160 170 180 185 195 205 215 225 Unit Cell Volume (A3 Unit Cell Volume (A") e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='20 20 5CrMnFeNi id Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5Zro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='5CrMnFeNiH6 A lacement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='10 Displ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='05 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='05 OTi Zr△CrMnFeNi OTi Zr△CrMnVFe Ni 0 0 140 150 160 170 180 185 195 205 215 225 Unit Cell Volume (A21 References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' von Colbe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Ares, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Barale, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Baricco, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Buckley, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Capurso, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Gallandat, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Grant, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Guzik, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Jacob, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Jensen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Jensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Jepsen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Klassen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Lototskyy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Manickam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Montone, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Puszkiel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Sartori, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Sheppard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Stuart, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Walker, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Webb, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 2 (2021) 2524-2560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Cohen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' West, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Wernick, Degradation of LaNi5 by temperature-induced cycling, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Less- Common Met.' metadata={'source': 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Ultrastrong medium-entropy single-phase alloys designed via severe lattice distortion, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} +page_content=' 31 (2019) 1807142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQf-QJD/content/2301.02811v1.pdf'} diff --git a/x9FQT4oBgHgl3EQfAzVW/content/tmp_files/2301.13224v1.pdf.txt b/x9FQT4oBgHgl3EQfAzVW/content/tmp_files/2301.13224v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..218e83aa3f862cccbb81b3f1a52ffb45bc2d8a48 --- /dev/null +++ b/x9FQT4oBgHgl3EQfAzVW/content/tmp_files/2301.13224v1.pdf.txt @@ -0,0 +1,985 @@ +1 + +Near-perfect Reachability of Variational Quantum Search with Depth-1 Ansatz +Junpeng Zhan1* +Abstract +Grover’s search algorithm is renowned for its dramatic speedup in solving many important +scientific problems. The recently proposed Variational Quantum Search (VQS) algorithm +has shown an exponential advantage over Grover’s algorithm for up to 26 qubits. However, +its advantage for larger numbers of qubits has not yet been proven. Here we show that the +exponentially deep circuit required by Grover’s algorithm can be replaced by a multi- +controlled NOT gate together with either a single layer of Ry gates or two layers of circuits +consisting of Hadamard and NOT gates, which is valid for any number of qubits greater +than five. We prove that the VQS, with a single layer of Ry gates as its Ansatz, has near- +perfect reachability in finding the good element of an arbitrarily large unstructured data set, +and its reachability exponentially improves with the number of qubits, where the +reachability is defined to quantify the ability of a given Ansatz to generate an optimal +quantum state. Numerical studies further validate the excellent reachability of the VQS. +Proving the near-perfect reachability of the VQS, with a depth-1 Ansatz, for any number of +qubits completes an essential step in proving its exponential advantage over Grover’s +algorithm for any number of qubits, and the latter proving is significant as it means that the +VQS can efficiently solve NP-complete problems. + +1. Introduction +Quantum algorithms can be broadly categorized into two types, depending on whether they are +based on Grover’s search algorithm or quantum Fourier transformation1. Grover’s algorithm2,3 +provides a quadratic speedup in unstructured search and has numerous important applications1. +However, the depth of the quantum circuit required in Grover’s algorithm2,3 grows exponentially +with the number of qubits. To address this issue, a recently proposed algorithm called variational +quantum search (VQS)4 is capable of amplifying the total probability of the good element(s) to +nearly 1 using a shallow circuit that grows linearly with the number of qubits, as verified up to 26 +qubits due to the limitations of GPU memory. That is, the VQS shows an exponential advantage +over Grover's algorithm in terms of circuit depth. +The VQS is a variational quantum algorithm (VQA)5–7. There is a lot of research on VQAs +from different aspects, e.g., trainability8–11, expressibility12–14, reachability15,16. However, a similar +analysis for the VQS has not been seen but is essential to verify the advantage of the VQS for any +number of qubits, which is important as it determines whether the VQS can efficiently solve an +NP-complete problem17. +The reachability discusses the capability of a given Ansatz of VQA with parameters to +represent a quantum state that minimizes a given objective function8. Reachability for +QAOA15,16,18 and variational Grover search16,19 has been investigated. This paper focuses on the +reachability of the VQS for unstructured search and proves that the exponentially deep circuit in +Grover’s algorithm can be replaced by either a single layer of ������������������������(θ) gates or a two-layer circuit + +1 Department of Renewable Energy Engineering, Alfred University, Alfred, NY, USA. *E-mail: zhanj@alfred.edu + +2 + +consisting of Hadamard and X gates. Furthermore, our numerical studies have verified the +effectiveness of the VQS, with a single layer of ������������������������(θ) gates as its Ansatz, in solving the +unstructured search problem. The rest of the paper consists of the Method, Result, and Conclusion +sections. + +2. Method +The problem to be solved in this paper is to find the good element in an unstructured data set +ⅅ, which has only one good element and (2������������ − 1) bad elements. We use n qubits and n Hadamard +gates to generate an equal superposition of all 2������������ elements, as shown in the left-hand side of the +leftmost dashed red line in Fig. 1a. In the rest of this section, we propose three methods to solve +the problem: 1) a quantum circuit (see Fig. 1a,b) based on an HX layer, which is generated by +Algorithm 1 (Section 2.1), 2) a quantum circuit (see Fig. 1c,d) based on an ������������������������ layer, which is +generated by Algorithm 2 (Section 2.2), and 3) the VQS using the ������������������������ layer as the Ansatz (Section +2.4). Section 2.4 also discusses the reachability of the VQS. Section 2.3 clarifies the scalability of +Algorithms 1 and 2. + + +Fig. 1. The quantum circuit to generate data set ⅅ and amplify the probability of the only good +element in it to nearly 1. a,c, the circuits in compact form for n-qubit data using an HX layer and +an ������������������������ layer (the blue blocks in Fig. 1a and 1c), respectively, where the HX layer consists of +Hadamard and X gates and the ������������������������ layer consists of ������������������������(������������) gates. b,d, the detailed circuits for n=6 +a +c +d +b + +HX layer l +0, o) +0, o +12 +1 +laver +label +label +Ry(π) +0 +qubit +qubit +data +dat. +J(H,X +Xn +HOn +qubit. +qubit:0.bo> +1 +laver +0. o +HX layer +1 +2 +label +label +qubit +95 +qubit +95 = 10> +H +q5 = 10 +94 = [0> +H +0 = +93 = +93 = +H +data +data +qubits +qubits +q2 =/0) +92 二 +H +q1 = 10) +91 = +H +T += 0b +H +/)3 + +using the HX and ������������������������ layers, respectively. The yellow block, excluding the label qubit, generates +a state that is an equal superposition of all elements (a single good element and (2������������ − 1) bad +elements), i.e., all elements have the same initial probability. The red block (Oracle) provides label +|1〉 at the label qubit to the good element and |0〉 to all bad elements. The blue block amplifies the +probability of the good element to nearly 1. The label qubit is the highest one (the most significant +one). In panel a, the ������������(������������, ������������) represents a circuit consisting of Hadamard and X gates. In panels +b,d, ������������ = 6 and the index of the good element is 39 (its binary form is 100111). + +2.1 Design the HX-layer-based Circuit +This section consists of four parts. The first part details an algorithm, Algorithm 1, which +generates the HX layer (the blue block in Fig. 1a,b). The second part provides the vector forms of +the three key quantum states in the HX-layer-based circuit, shown in Fig. 1a,b. The third part +explains the design goal of the HX layer. The last part proves that Algorithm 1 can always achieve +the design goal of the HX layer for any number of qubits. +Algorithm 1. Here we propose Algorithm 1 to construct the so-called HX layer, i.e., a two-layer +circuit consisting of only Hadamard and X gates. The HX layer together with the Oracle (the red +blocks in Fig. 1) has the same function as Grover’s search algorithm, i.e., amplify the probability +of the good element to nearly 1. +Algorithm 1 | Pseudo code for generating the HX layer (the blue block in Fig. 1a). + + +Vector Forms of Three Quantum States. This sub-section details the vector forms of three +quantum states, i.e., |0, ������������0⟩, |������������1⟩, and |������������2⟩, which are respectively indicated in the three dashed +red lines in Fig. 1a. +The state, |0, ������������0⟩, can be written as |0〉⊗(|0〉+|1〉)⊗������������, which can be represented in the vector +form: +|0, ������������0⟩ = [������������0 +b, ������������1 +b, ⋯ , ������������������������−1 +b +, ������������������������ +g, ������������������������+1 +b +, ⋯ , ������������������������−1 +b +����������������������� +1st half: ������������ elements +, +0,0, ⋯ ,0 +�����]������������ +2nd half: ������������ elements + (1) +Input: the number of qubits n and the index of the good element in decimal form, k, ∀������������ ∈ +[0, 2������������ − 1] +Output: quantum gates in the HX layer (the blue block in Fig. 1a). +1 +Convert k into the binary form ������������������������−1������������������������−2 ⋯ ������������2������������1������������0. +2 +Add an X gate in the label qubit (the most significant qubit). +3 +Let m=n +4 +while ������������ ≥ 1 +5 +if ������������������������−1 = 1 +6 + Add a Hadamard gate followed by an X gate to qubit ������������������������−1. +7 +else +8 + Add a Hadamard gate to qubit ������������������������−1. +9 +m ← m−1 + + + + +4 + +where ������������ = 2������������, super scripts b and g indicate bad and good elements, respectively, and subscripts, +0~N−1, represent the index of an element in the vector. Note that the index always counts from 0 +in the rest of the paper. For example, ������������������������ +g means the kth element is a good element. +The relationship between |������������1⟩ and |0, ������������0⟩ can be represented as: +|������������1⟩ = ������������������������|0, ������������0⟩ = [������������0 +b, ������������1 +b, ⋯ , ������������������������−1 +b +, 0, ������������������������+1 +b +, ⋯ , ������������������������−1 +b +��������������������� +1st half: ������������ elements +, 0, ⋯ ,0, ������������������������+������������ +g +, 0, ⋯ ,0 +�������������]������������ +2nd half: ������������ elements + (2) +where oracle ������������������������ is implemented as the ������������������������(������������), which is an n-qubit-controlled X gate, as shown in +Fig. 1b. As shown in Eq. (2), the ������������������������(������������) changes the index of the good element from k to N+k. +The HX layer (the blue block in Fig. 1a) can be expressed as +������������ ⊗ ������������(������������, ������������) = � +������������ +������������(������������, ������������) +������������(������������, ������������) +������������ +� (3) +where ������������(������������, ������������) is an N by N matrix. For the convenience of expression, we call the matrix given +in (3) the HX-layer matrix. Then the relationship between |������������2⟩ and |������������1⟩ can be represented as: + |������������2⟩ = � +������������ +������������(������������, ������������) +������������(������������, ������������) +������������ +� |������������1⟩ += � +������������ +������������(������������, ������������) +������������(������������, ������������) +������������ +� [������������0 +b, ������������1 +b, ⋯ , ������������������������−1 +b +, 0, ������������������������+1 +b +, ⋯ , ������������������������−1 +b +��������������������� +1st half: ������������ elements +, 0, ⋯ ,0, ������������������������+������������ +g +, 0, ⋯ ,0 +�������������]������������ +2nd half: ������������ elements + (4) +For the convenience of analysis, we write |������������2〉 as a vector form: + |������������2〉 = [������������0, ������������1, ⋯ , ������������������������−1, ������������������������, ⋯ , ������������2������������−1]������������ (5) +where +∑ +|������������������������|2 +2������������−1 +������������=0 += 1 (6) +Design Goal of the HX Layer. The goal of designing the ������������(������������, ������������) is to let each element in its row +k be 1 √������������ +⁄ + such that ������������������������+������������, the (������������ + ������������)th element of |������������2⟩, can be calculated as: +������������������������+������������ = �∑ +������������������������ +b +������������−1 +������������=0 ++ ∑ +������������������������ +b +������������−1 +������������=������������+1 +� √������������ +⁄ + (7) +where the right-hand side is obtained from Eq. (4), i.e., multiply the (������������ + ������������)th row of the HX- +layer matrix with the vector form of |������������1⟩, and the integer ������������ ∈ [0, 2������������ − 1]. +From here on, we assume all elements have the same initial magnitude. That is, ������������������������ +b = ������������������������ +g = +������������������������+������������ +g += 1 √������������ +⁄ +, ∀������������ ∈ [0, ������������ − 1]. Then, (7) can be reformed as: +������������������������+������������ = �1 √������������ +⁄ +�(������������ − 1) √������������ +⁄ += 1 − 1 ������������ +⁄ += 1 − 1 2������������ +⁄ + (8) +The probability of obtaining the good element is equal to ������������������������+������������ +2 += (1 − 1 2������������ +⁄ +)2, which is equal +to 0.25, 0.5625, 0.7656, 0.8789, 0.9386, 0.9690, 0.9844, 0.9922, and 0.9961 for n=1~9, +respectively. That is, the probability of finding the good element is larger than 0.95 and 0.99 for n +being larger than 5 and 7, respectively. + +5 + +Position Index of the All-1 Row for the HX Layer. In the rest of this sub-section, we answer the +question of why the HX layer generated by Algorithm 1 can realize the design goal specified in +the previous sub-section. +The result of tensor product �������������������������,0 +������������������������,1� ⊗ �������������������������−1,0 +������������������������−1,1� ⊗ ⋯ ⊗ �������������2,0 +������������2,1� ⊗ �������������1,0 +������������1,1� ⊗ �������������0,0 +������������0,1� is a column +vector with 2������������+1 elements. The position index of the element ������������������������,������������������������������������������������−1,������������������������−1 ⋯ ������������2,������������2������������1,������������1������������0,������������0 in the +column vector is equal to the decimal value of a binary form ������������������������������������������������−1 ⋯ ������������2������������1������������0 , where ������������������������ ∈ +{0,1}, ∀ ������������ ∈ [0, ������������]. To better understand this, we provide two examples: the position index of the +element ������������3,0������������2,0������������1,0������������0,1 in the column vector associated with �������������3,0 +������������3,1� ⊗ �������������2,0 +������������2,1� ⊗ �������������1,0 +������������1,1� ⊗ �������������0,0 +������������0,1� is +1 (its binary form is 0001) and the position index of ������������3,1������������2,0������������1,0������������0,1 in the same vector is 9 (its +binary form is 1001). +We can express the tensor product ������������������������−1 ⊗ ������������������������−2 ⊗ ⋯ ⊗ ������������2 ⊗ ������������1 ⊗ ������������0 as +1 +√2������������ ������������, where ������������ +represents a 2������������ by 2������������ matrix and ������������������������, ∀ ������������ ∈ [0, ������������ − 1], represents either ������������������������ or ������������. Note that ������������ = +�0 1 +1 0� , ������������ = +1 +√2 �1 1 +1 − 1� , ������������������������ = +1 +√2 �1 − 1 +1 1�. It can be easily verified that M has only one all-1 +row (i.e., each element in the row is 1) while each of all the other rows consists of −1 and 1. The +question then becomes, where is the all-1 row, which will be answered in the next paragraph. +This paragraph explains how to find the all-1 row for the tensor product given in the previous +paragraph, and conversely, for any given number k, how to construct a quantum circuit such that +row k of the matrix associated with the circuit is an all-1 row. Given that each ������������������������ has exactly one +row of [1 1]/√2, the all-1 row is generated from the tensor product of the [1 1]/√2 row of each +������������������������. In other words, if the [1 −1]/√2 row is involved, the corresponding tensor product result will +not be all 1's. Let ������������������������ = 0 represent that the [1 1]/√2 is located in row 0 of ������������������������ (i.e., ������������������������ = ������������) and +������������������������ = 1 represent that the [1 1]/√2 is located in row 1 of ������������������������ (i.e., ������������������������ = ������������������������). According to the +paragraph before the previous one, we can know that the all-1 row is located at row k where k is +the decimal value of its binary form ������������������������−1������������������������−2 ⋯ ������������2������������1������������0. Conversely, for any given k∈ [0, 2������������ − 1], +if we want to generate a matrix that represents the tensor product ������������������������−1 ⊗ ������������������������−2 ⊗ ⋯ ⊗ ������������2 ⊗ ������������1 ⊗ +������������0, such that each element in row k of the matrix is 1 (ignoring the coefficient 1 √2 +⁄ + of each ������������������������), +we can convert k into its binary form ������������������������−1������������������������−2 ⋯ ������������2������������1������������0, set ������������������������ to be ������������������������ if ������������������������ = 1, and set ������������������������ to be +������������ if ������������������������ = 0. This is exactly what Algorithm 1 does. To better understand this, we provide three +examples for k=5, 8, and 39 in the next three paragraphs. +For k=5, its binary form is ������������2������������1������������0 =101. Then we have ������������2 ⊗ ������������1 ⊗ ������������0. By replacing ������������2 and ������������0 +by XH as ������������2 = ������������0 = 1, and replacing ������������1 by H as ������������1 = 0, we can obtain: + ������������������������ ⊗ ������������ ⊗ ������������������������ = +1 +√2 �1 − 1 +1 1� ⊗ +1 +√2 �1 1 +1 − 1� ⊗ +1 +√2 �1 − 1 +1 1� + +6 + += +1 +2√2 +⎣ +⎢ +⎢ +⎢ +⎢ +⎢ +⎡1 +−1 +1 +1 +1 +−1 +1 +1 +1 +−1 +1 +1 +−1 +1 +−1 +−1 +−1 +1 +−1 +−1 +−1 +1 +−1 +−1 +−1 +1 +−1 +−1 +1 +−1 +1 +1 +1 +−1 +1 +1 +1 +−1 +1 +1 +1 +−1 +1 +1 +−1 +1 +−1 +−1 +1 +−1 +1 +1 +1 +−1 +1 +1 +1 +−1 +1 +1 +−1 +1 +−1 +−1 ⎦ +⎥ +⎥ +⎥ +⎥ +⎥ +⎤ + (9) +which shows that row 5 is the only all-1 row (ignoring the coefficient +1 +2√2). It is important to note +that in this paper, the row index count always starts at 0, i.e., the first row is row 0. +For k=8, its binary form is 1000. Then we have +������������������������ ⊗ ������������ ⊗ ������������ ⊗ ������������ = +1 +√2 �1 − 1 +1 1� ⊗ +1 +√2 �1 1 +1 − 1� ⊗ +1 +√2 �1 1 +1 − 1� ⊗ +1 +√2 �1 1 +1 − 1� (10) +We can easily verify that the tensor product shown in Eq. (10) has only one all-1 row (ignoring +the coefficient ¼) located at row 8. +For k=39, its binary form is 100111. Then we can use ������������������������ ⊗ ������������ ⊗ ������������ ⊗ ������������������������ ⊗ ������������������������ ⊗ ������������������������ to +obtain a matrix whose only all-1 row is located at row 39, as shown in Fig. 1b. +In summary, by using a ������������������������(������������) gate and an HX layer, we can amplify the probability of the +good element from 1 2������������ +⁄ + to a value larger than 0.95 and 0.99 for n values greater than 5 and 7, +respectively, where we already know the position index of the good element. +2.2 Design the ������������������������-layer-based Circuit +This section consists of four parts. The first part details an algorithm, Algorithm 2, to generate +the ������������������������ layer (the blue block in Fig. 1c,d). The second part provides the vector form of the quantum +state |������������2⟩ in the ������������������������-layer-based circuit, shown in Fig. 1c,d. The third part explains the ������������������������ layer is +equivalent to the HX layer in solving the problem of finding the good element. +Algorithm 2. Here we design an ������������������������ layer (the blue block in Fig. 1c) to replace the HX layer (the +blue block in Fig. 1a), i.e., use a single layer of Ry gates to replace the two layers consisting of X, +H, and XH gates. Given that ������������������������(������������) = � +cos( +������������ +2) +−sin( +������������ +2) +sin( +������������ +2) +cos( +������������ +2) +�, we have ������������������������ � +������������ +2� = +1 +√2 �1 +−1 +1 +1 � which +is exactly equivalent to XH, ������������������������(������������) = �0 +−1 +1 +0 � which can be used to replace the X gate despite the +phase difference, and ������������������������(3������������ 2 +⁄ ) = +1 +√2 �−1 +−1 +1 +−1� which can be used to replace the H gate despite +the phase difference. The impact of the phase difference can be ignored with the reason given in +the rest of this section. Algorithm 2 details the procedure of how to generate the ������������������������ layer. + + + + +7 + +Algorithm 2 | Pseudo code for generating the ������������������������ layer (the blue block in Fig. 1c). + + +Vector Forms of Quantum State |������������2⟩. The states |0, ������������0⟩ and |������������1⟩ in Fig. 1c are the same as +those in Fig. 1a. Therefore, only state |������������2⟩ in Fig. 1c is provided here. The ������������������������ layer obtained from +Algorithm 2, as shown in the blue blocks in Fig. 1c,d, can be expressed as +������������������������(������������) ⊗ ������������������������(������������)⊗������������ = �0 +−1 +1 +0 � ⊗ ������������������������(������������)⊗������������ = � +������������ +−������������������������(������������)⊗������������ +������������������������(������������)⊗������������ +������������ +� (11) +For the convenience of expression, we refer to the matrix given in Eq. (11) as the ������������������������-layer matrix. +Like Eq. (4), based on Eq. (11), states |������������2⟩ and |������������1⟩ in Fig. 1c have the following relationship: + |������������2⟩ = � +������������ +−������������������������(������������)⊗������������ +������������������������(������������)⊗������������ +������������ +� |������������1⟩ += � +������������ +−������������������������(������������)⊗������������ +������������������������(������������)⊗������������ +������������ +� [������������0 +b, ������������1 +b, ⋯ , ������������������������−1 +b +, 0, ������������������������+1 +b +, ⋯ , ������������������������−1 +b +��������������������� +1st half: ������������ elements +, 0, ⋯ ,0, ������������������������+������������ +g +, 0, ⋯ ,0 +�������������]������������ +2nd half: ������������ elements + (12) +where ������������������������(������������)⊗������������ is an N by N matrix. +Relation Between the ������������������������ Layer and the HX Layer. This sub-section will explain that the ������������������������ +layer is equivalent to the HX layer in solving the problem of finding the good element from an +unstructured data set. +Similar to the analysis given in (4)-(7), the calculation of ������������������������+������������, defined in Eq. (5), involves +only the bottom left part, i.e., ������������������������(������������)⊗������������, of the ������������������������-layer matrix but not its top right part, i.e., +−������������������������(������������)⊗������������. Therefore, The phase difference between ������������������������(������������) and X does not have any impact on +the value of ������������������������+������������. In other words, we can use ������������������������(������������) to replace X gate without any impact on the +probability of finding the good element. +Input: the number of qubits n and the position index of the good element in decimal form, +k, ∀������������ ∈ [0, 2������������ − 1] +Output: quantum gates in the ������������������������ layer (the blue block in Fig. 1c). +1 +Convert k into the binary form ������������������������−1������������������������−2 ⋯ ������������2������������1������������0. +2 +Add an ������������������������(������������) gate in the label qubit (the most significant qubit). +3 +Let m=n +4 +while ������������ ≥ 1 +5 +if ������������������������−1 = 1 +6 + Add an ������������������������(������������ 2 +⁄ ) gate to qubit ������������������������−1. +7 +else +8 + Add an ������������������������(3������������ 2 +⁄ ) gate to qubit ������������������������−1. +9 +m ← m−1 + + +8 + +We can write ������������������������(������������)⊗������������ as a tensor product ������������������������−1 ⊗ ⋯ ⊗ ������������2 ⊗ ������������1 ⊗ ������������0 , where ������������������������, ∀������������ ∈ +[0, ������������ − 1], represents either ������������������������(������������ 2 +⁄ ) or ������������������������(3������������ 2 +⁄ ). Then ������������������������(������������)⊗������������ will have only one all-1 (all- +negative-1) row if the number of ������������������������(3������������ 2 +⁄ ) gates is even (odd), where each element in the all- +negative-1 row is −1, ignoring the coefficient 1/√2������������. When the number of ������������������������(3������������ 2 +⁄ ) gates is +even, the calculation given in (7) and (8) is also valid for Fig. 1c. When the number of ������������������������(3������������ 2 +⁄ ) +gates is odd, each element in row N+k is −1, ignoring the coefficient 1/√2������������, then we have +������������������������+������������ = − �∑ +������������������������ +b +������������−1 +������������=0 ++ ∑ +������������������������ +b +������������−1 +������������=������������+1 +� √������������ +⁄ + (13) +������������������������+������������ = − �1 √������������ +⁄ +�(������������ − 1) √������������ +⁄ += −1 + 1 ������������ +⁄ += −1 + 1 2������������ +⁄ + (14) +Since the probability is the square of magnitude ������������������������+������������, the magnitude of the good element +given in (8) and that given in (14) result in the same probability. Therefore, no matter whether the +number of ������������������������(3������������ 2 +⁄ ) gates is even or odd, the probability of finding the good element from the +output of Fig. 1c, |������������2〉, is larger than 0.95 and 0.99 for n values greater than 5 and 7, respectively, +which is the same as the analysis for Fig. 1a given in the previous section (see the paragraph below +Eq. (8)). Therefore, we can use the ������������������������ layer generated by Algorithm 2 to replace the HX layer +generated by Algorithm 1. +2.3 Scalability of Algorithms 1 and 2 +In the analysis for Figs. 1a and 1c above, there is neither restriction nor assumption associated +with the number of qubits. That is, the analysis given in Sections 2.1 and 2.2 is valid for any +number of qubits. Specifically, Algorithm 1 (Algorithm 2) each can generate a quantum circuit +whose corresponding matrix form has only one all-1 (all-1 or all-negative-1) row located at row k, +k∈ [0, 2������������ − 1], for any value of n, where n is the number of qubits. +In Algorithms 1 and 2, the location index, k, of the good element must be known in advance. +This means that they are not useful for finding the good element in a data set if the location index +is unknown. However, the analysis given above has proved that either the HX layer (generated by +Algorithm 1) or the ������������������������ layer (generated by Algorithm 2), together with a ������������������������(������������) gate (the red and +blue blocks in Fig. 1a,c), can amplify the probability of the good element from 1 2������������ +⁄ + to nearly 1 if +n is larger than 5. Note that for the same task, Grover’s search algorithm requires a quantum circuit +whose depth increases exponentially with the number of qubits, which manifests the exponential +advantage of the circuits generated by Algorithms 1 and 2, in terms of circuit depth. +2.4 Variational Quantum Search and its Reachability +This section briefs the VQS and then details the definition of reachability for the VQS. The +VQS4 is a VQA, which involves the interaction between classical and quantum computers. In the +classical part, an optimizer is used to update the parameter ������������ of the Ansatz based on the objective +function ������������(������������): +������������(������������) = −0.5⟨������������1|������������2⟩ + 0.5⟨������������1|������������ ⊗ ������������⊗������������|������������2⟩ (15) +where ������������1 and ������������2 are the states before and after the ansatz, respectively, as shown in Fig. 1a,c, Z +and I are Pauli Z and identity matrix, respectively, and n is the number of qubits. The first and +second terms in the objective function can be respectively obtained from measuring two quantum +circuits based on the Hadamard test, as detailed in paper4. Papar4 has validated two types of shallow + +9 + +and effective Ansatzs for the VQS. However, the analysis given above implies that the VQS with +only a single ������������������������ layer as the Ansatz, which is shallower than those two types of Ansatzs used in +paper4, is sufficient to find the only good element in an unknown data set. Note that unlike +Algorithms 1 and 2, the VQS does not need to know the position index, k, of the good element +beforehand. To quantify the capability of how best the Ansatz can generate a quantum state that +minimizes a given objective function, we propose a definition of reachability for the VQS in the +rest of this section. +The objective function given in Eq. (15) can be reformed as +������������(������������) = ⟨������������1|������������|������������2⟩ (16) +where +������������ = 0.5�−������������⊗������������+1 + ������������ ⊗ ������������⊗������������� = diag([ 0,0, ⋯ ,0 +����� +������������ elements +, −1, −1, ⋯ , −1 +��������� +������������ elements +]) (17) +where diag(w) represents a square diagonal matrix with elements of vector w on the main diagonal. +Then we can define a reachability15,16 for the VQS: +ℜ = +min +������������ ⟨������������1|������������|������������(������������)⟩− min +|������������⟩∈ℋ⟨������������1|������������|������������⟩ +− min +|������������⟩∈ℋ⟨������������1|������������|������������⟩ + (18) +where |������������1⟩ denotes the state before the ansatz (see Fig. 1c), the min������������ term, used to refer to +min +������������ ⟨������������1|������������|������������(������������)⟩ for the convenience of expression, is the minimum over all reachable states +generated by the Ansatz of the VQS, the ������������(������������) in the min������������ term represents the |������������2⟩ in Fig. 1c, and +the minH term, used to refer to min +|������������⟩∈ℋ⟨������������1|������������|������������⟩ for the convenience of expression, represents the +minimum over all states |������������⟩ of the Hilbert space ℋ. +This paragraph discusses the possible value range of the numerator in (18). Note that |������������1⟩ has +only a single non-zero element which is equal to 1 √������������ +⁄ + in this paper and located in its second half +and that ⟨������������|������������|������������⟩ is equal to the tensor product of the second half of −|������������⟩ and the second half of +|������������⟩, where the second half of a state represents the second half elements of the vector form of the +state. We can then know that, in the numerator in (18), the value range of ⟨������������1|������������|������������(������������)⟩ is a subset +of [− 1 √������������ +⁄ +, 1 √������������ +⁄ +] and the value range of ⟨������������1|������������|������������⟩ is [− 1 √������������ +⁄ +, 1 √������������ +⁄ +]. Therefore, the range +of ⟨������������1|������������|������������(������������)⟩ − ⟨������������1|������������|������������⟩ is a subset of [− 2 √������������ +⁄ +, 2 √������������ +⁄ +], which is an extremely small range +when n is large as both the upper and lower bounds converge exponentially towards zero. +Given that a well-defined metric should not consistently reside within a minuscule range +around zero, using the numerator in (18) as a measure of reachability is not appropriate as it is +extremely small when the number of qubits is large and converges exponentially towards zero as +the increase of the number of qubits, regardless of the value of ������������. To address the issue, we use the +minH term in the denominator as explained in the next paragraph. +According to the analysis given in paper4, the minimum value of the minH term, i.e., − 1 √������������ +⁄ +, +is achieved when |������������⟩ is equal to the computational state |������������ + ������������⟩, which is represented by a vector + +10 + +with a single element of 1 at index N+k and all other elements being 0's, where ������������ = 2������������. Given that +dividing the range [− 2 √������������ +⁄ +, 2 √������������ +⁄ +] by the minH term results in a range of [−2, 2], we use the +minH term as the denominator such that the range of ℜ defined in (18) becomes a subset of [0, 2] +instead of the extremely small range mentioned above. A range of [0, 2] is more suitable for the +reachability metric, as it is not confined to a minuscule range around zero, and this is the reason +why we introduce the denominator in the definition of reachability. +Now we explain why we add the absolute sign into each term, as shown in Eq. (19). As per +Eqs. (8) and (14), the value of ������������������������+������������ being equal to either 1 − 1 ������������ +⁄ or −1 + 1 ������������ +⁄ results in the +same probability of finding the good element, i.e., both solutions are equally good, where ������������ = 2������������. +However, putting these two values of ������������������������+������������ into |������������2⟩, which replaces |������������(������������)⟩, results in different +values of ⟨������������1|������������|������������(������������)⟩ and thereby leads to different values of reachability, but they should +correspond to the same reachability. To address this issue, we use absolute terms, i.e., +|⟨������������1|������������|������������(������������)⟩| and |⟨������������1|������������|������������⟩|, to replace ⟨������������1|������������|������������(������������)⟩ and ⟨������������1|������������|������������⟩, respectively, such that two +amplitudes with different signs lead to the same reachability. +This paragraph discusses a better definition of reachability, as given in Eq. (19). Note that +max +|������������⟩∈ℋ|⟨������������1|������������|������������⟩| reaches its maximum when ⟨������������1|������������|������������⟩ equals 1 √������������ +⁄ + or − 1 √������������ +⁄ +, corresponding +to finding the good element with a probability of 1. On the other hand, min +|������������⟩∈ℋ|⟨������������1|������������|������������⟩| reaches +its minimum when ⟨������������1|������������|������������⟩ equals 0, corresponding to finding the good element with a +probability of 0. Thus, we use ‘max’ rather than ‘min’ when using the absolute terms such that we +can find the good element with a high probability. According to the discussion given in the +previous paragraph, two amplitudes with different signs lead to different values of reachability if +we use (18) as the definition of reachability, which could be addressed if we use Eq. (19). +Therefore, we recommend using Eq. (19) as the definition of reachability instead of (18). +ℜ = +max +������������ |⟨������������1|������������|������������(������������)⟩|− max +|������������⟩∈ℋ|⟨������������1|������������|������������⟩| +− max +|������������⟩∈ℋ|⟨������������1|������������|������������⟩| + (19) +The smaller the value of ℜ, the better the reachability. When ℜ = 0, the VQS has perfect +reachability. The use of denominator and absolute terms distinguishes our definition of reachability +from that presented in papers15,16. +For the circuit depicted in Fig. 1c where the ������������������������ layer is generated by Algorithm 2, the (N+k)th +element of the output state |������������2⟩ is either equal to 1 − 1 ������������ +⁄ or −1 + 1 ������������ +⁄ , as shown in Eqs. (8) and +(14), respectively. If we replace the max +������������ |⟨������������1|������������|������������(������������)⟩| with |⟨������������1|������������|������������2⟩| in Eq. (19), we have +|⟨������������1|������������|������������2⟩|− max +|������������⟩∈ℋ|⟨������������1|������������|������������⟩| +− max +|������������⟩∈ℋ|⟨������������1|������������|������������⟩| += +1−1 ������������ +⁄ +√������������ − 1 +√������������ +− 1 +√������������ += +1 +������������ = +1 +2������������ (20) +As |������������2⟩ is generated by Algorithm 2, it is the |������������(������������)⟩ with the ������������ specified in Algorithm 2. Then +we know max +������������ |⟨������������1|������������|������������(������������)⟩| ≥ |⟨������������1|������������|������������2⟩|. Therefore, we have + +11 + +ℜ = +max +������������ |⟨������������1|������������|������������(������������)⟩|− max +|������������⟩∈ℋ|⟨������������1|������������|������������⟩| +− max +|������������⟩∈ℋ|⟨������������1|������������|������������⟩| +≤ +|⟨������������1|������������|������������2⟩|− max +|������������⟩∈ℋ|⟨������������1|������������|������������⟩| +− max +|������������⟩∈ℋ|⟨������������1|������������|������������⟩| += +1 +2������������ (21) +where the two sides of the ≤ come from Eqs. (19) and (20), respectively. Eq. (20) provides a +specific circuit, i.e., using the ������������������������ layer generated by Algorithm2 as the Ansatz of the VQS, whose +reachability value is equal to 1 2������������ +⁄ +, which is an extremely small value when n is large and +converges exponentially to zero as the increase of n. That is, this circuit already achieves near- +perfect reachability. +In conclusion, the VQS has at least near-perfect reachability, and the more qubits, the better +the reachability of the VQS. +3. Result +The results provided in this paper are obtained from quantum simulators using Pennylane’s +devices20. Results related to 2-, 8-, and 14-qubit (20- and 26-qubit) input states are obtained using +Pennylane’s default.qubit (lightning.gpu) device on an Intel i5-6500 CPU (A40x4 48-GB GPU). +The initial values of ������������ in the Ansatz are randomly sampled from a uniform distribution between 0 +and 2π. Same as paper4, two termination criteria for the iterative process are used in the VQS. The +first one is that the number of iterations reaches a given number (set to 300 in this paper). The +second one is that a small-change event occurs consecutively for a given number of times (set to +5 in this paper), where the small-change event is defined as: the absolute value of the relative +change of objective functions in two consecutive iterations is smaller than a given small value (set +to 1 × 10−4 in this paper). When one of the criteria is met, whichever comes first, the iterative +process of VQS terminates. +Here we numerically verify that the VQS with only one layer of ������������������������(������������) gates can efficiently +find the good element from an unstructured data set without knowing the position index, k, of the +good element beforehand, which is related to the trainability of the VQS. We apply the VQS, +where the Ansatz is an ������������������������ layer, to find the good element from 2-, 8-, 14-, 20-, and 26-qubit data +sets. +The results are shown in Fig. 2, which show that the VQS indeed can find the good element as +the amplified probability is very close to 1 for most runs out of 100 runs for n=8, 14, 20, and 26, +and that in some runs (22, 16, 16, and 16 for the four cases) out of 100 runs, the amplified +probability is close to 0. Compared to the results shown in paper4, although the VQS using the ������������������������ +layer in the Ansatz is less stable than the VQS using the two types of Ansatzs shown in paper4, the +former can still amplify the probability to nearly 1 in most runs, excluding the 2-qubit case. The +relatively poor performance of the 2-qubit case is not a concern, but actually further validates our +analysis as explained below. When n equals 2 and if we use the ������������������������ layer as the Ansatz of the VQS, +according to Eq. (8), the probability of finding the good element is (1 − 1 22 +⁄ +)2 = 0.5625, which +is roughly in the middle of the box result for the 2-qubit case (the leftmost one in Fig. 2a). In other +words, the results in Fig. 2a validate the conclusion obtained from the reachability analysis, i.e., +the more the number of qubits, the better the reachability of the VQS. + +12 + +The importance of using VQS with the ������������������������ layer is that it has been proved above that the ������������������������ +layer, together with the ������������������������(������������), can amplify the probability of the good element from 1 2������������ +⁄ + to +nearly 1 for any number of qubits being larger than 5. That is, the reachability of the VQS using +the ������������������������ layer is guaranteed for any number of qubits being larger than 5, which means we do can +scale the VQS to any large number of qubits while keeping the circuit depth to be 2 (i.e., one ������������������������ +layer and one ������������������������(������������) layer), which is an exponential advantage over Grover’s algorithm. + + +Fig. 2. Box plot results from 100 runs of VQS using the ������������������������ layer as the Ansatz for an n-qubit +input state. a, the amplified probability of good element. b, the number of iterations used when a +termination criterion is met. +4. Conclusion +We have proposed two algorithms to construct a depth-2 (the HX layer) and a depth-1 (the ������������������������ +layer) circuits, both of which can replace the exponentially deep circuit required by Grover’s +algorithm, if the position index of the good element is known beforehand. We have proved that +the VQS, with the ������������������������ layer as the Ansatz, has near-perfect reachability for any number of qubits +greater than five and its reachability exponentially improves with the number of qubits, which has +been further verified by numerical experiments. In these experiments, we use the VQS, with the +������������������������ layer as Ansatz, to search for the only good solution in unstructured data sets, represented by +8, 14, 20, and 26 qubits, and successfully find the good element with a probability of nearly 1 in +78 to 84 out of 100 independent runs. The experiments also indicate the good trainability of the +VQS for up to 26 qubits. We will further investigate the trainability of the VQS for more qubits in +the future. + +References +1. +Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information: 10th Anniversary +Edition. +Cambridge +University +Press +(Cambridge +University +Press, +2011). +doi:10.1017/CBO9780511976667. +2. +Grover, L. K. Quantum mechanics helps in searching for a needle in a haystack. Phys Rev Lett 79, +(1997). +3. +Grover, L. K. A fast quantum mechanical algorithm for database search. in Proceedings of the +Annual ACM Symposium on Theory of Computing vol. Part F129452 (1996). +a +b + +13 + +4. +Zhan, J. Variational Quantum Search with Exponential Speedup. (2022). +5. +Cerezo, M. et al. Variational quantum algorithms. Nature Reviews Physics vol. 3 625–644 Preprint +at https://doi.org/10.1038/s42254-021-00348-9 (2021). +6. +McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid +quantum-classical algorithms. New J Phys 18, (2016). +7. +Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat Commun +5, (2014). +8. +Bharti, K. et al. Noisy intermediate-scale quantum (NISQ) algorithms. arXiv:2101.08448 Preprint at +(2021). +9. +Bittel, L. & Kliesch, M. Training Variational Quantum Algorithms Is NP-Hard. Phys Rev Lett 127, +120502 (2021). +10. +McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R. & Neven, H. Barren plateaus in quantum +neural network training landscapes. Nat Commun 9, (2018). +11. +Cerezo, M., Sone, A., Volkoff, T., Cincio, L. & Coles, P. J. Cost function dependent barren plateaus +in shallow parametrized quantum circuits. Nat Commun 12, (2021). +12. +Liao, Y. & Zhan, J. Expressibility-Enhancing Strategies for Quantum Neural Networks. (2022). +13. +Haug, T., Bharti, K. & Kim, M. S. Capacity and quantum geometry of parametrized quantum circuits. +PRX Quantum 2, (2021). +14. +Sim, S., Johnson, P. D. & Aspuru-Guzik, A. Expressibility and Entangling Capability of Parameterized +Quantum Circuits for Hybrid Quantum-Classical Algorithms. Adv Quantum Technol 2, 1900070 +(2019). +15. +Akshay, V., Philathong, H., Zacharov, I. & Biamonte, J. Reachability Deficits in Quantum +Approximate Optimization of Graph Problems. Quantum 5, (2020). +16. +Akshay, V., Philathong, H., Morales, M. E. S. & Biamonte, J. D. Reachability Deficits in Quantum +Approximate Optimization. Phys Rev Lett 124, 090504 (2020). +17. +Zhan, J. Quantum Feasibility Labeling for NP-complete Vertex Coloring Problem. (2023) +doi:10.48550/arxiv.2301.01589. +18. +Bharti, K. et al. Noisy intermediate-scale quantum algorithms. Rev Mod Phys 94, 015004 (2022). +19. +Morales, M. E. S., Tlyachev, T. & Biamonte, J. Variationally Learning Grover’s Quantum Search +Algorithm. Phys Rev A (Coll Park) 98, (2018). +20. +Bergholm, V. et al. PennyLane: Automatic differentiation of hybrid quantum-classical +computations. Preprint at https://doi.org/10.48550/ARXIV.1811.04968 (2018). + +Acknowledgements +This research was partially supported by the NSF ERI program, under award number 2138702. +This work used the Delta system at the National Center for Supercomputing Applications through +allocation CIS220136 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services +& Support (ACCESS) program, which is supported by National Science Foundation grants +#2138259, #2138286, #2138307, #2137603, and #2138296. We acknowledge the use of IBM +Quantum services for this work. The views expressed are those of the authors, and do not reflect +the official policy or position of IBM or the IBM Quantum team. + + + diff --git a/x9FQT4oBgHgl3EQfAzVW/content/tmp_files/load_file.txt b/x9FQT4oBgHgl3EQfAzVW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f4a5763e1448df70dd1228d6cb5c6553d25119fb --- /dev/null +++ b/x9FQT4oBgHgl3EQfAzVW/content/tmp_files/load_file.txt @@ -0,0 +1,687 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf,len=686 +page_content='1 Near-perfect Reachability of Variational Quantum Search with Depth-1 Ansatz Junpeng Zhan1* Abstract Grover’s search algorithm is renowned for its dramatic speedup in solving many important scientific problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The recently proposed Variational Quantum Search (VQS) algorithm has shown an exponential advantage over Grover’s algorithm for up to 26 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' However, its advantage for larger numbers of qubits has not yet been proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Here we show that the exponentially deep circuit required by Grover’s algorithm can be replaced by a multi- controlled NOT gate together with either a single layer of Ry gates or two layers of circuits consisting of Hadamard and NOT gates, which is valid for any number of qubits greater than five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' We prove that the VQS, with a single layer of Ry gates as its Ansatz, has near- perfect reachability in finding the good element of an arbitrarily large unstructured data set, and its reachability exponentially improves with the number of qubits, where the reachability is defined to quantify the ability of a given Ansatz to generate an optimal quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Numerical studies further validate the excellent reachability of the VQS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Proving the near-perfect reachability of the VQS, with a depth-1 Ansatz, for any number of qubits completes an essential step in proving its exponential advantage over Grover’s algorithm for any number of qubits, and the latter proving is significant as it means that the VQS can efficiently solve NP-complete problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Introduction Quantum algorithms can be broadly categorized into two types, depending on whether they are based on Grover’s search algorithm or quantum Fourier transformation1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Grover’s algorithm2,3 provides a quadratic speedup in unstructured search and has numerous important applications1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' However, the depth of the quantum circuit required in Grover’s algorithm2,3 grows exponentially with the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' To address this issue, a recently proposed algorithm called variational quantum search (VQS)4 is capable of amplifying the total probability of the good element(s) to nearly 1 using a shallow circuit that grows linearly with the number of qubits, as verified up to 26 qubits due to the limitations of GPU memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=" That is, the VQS shows an exponential advantage over Grover's algorithm in terms of circuit depth." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The VQS is a variational quantum algorithm (VQA)5–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' There is a lot of research on VQAs from different aspects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', trainability8–11, expressibility12–14, reachability15,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' However, a similar analysis for the VQS has not been seen but is essential to verify the advantage of the VQS for any number of qubits, which is important as it determines whether the VQS can efficiently solve an NP-complete problem17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The reachability discusses the capability of a given Ansatz of VQA with parameters to represent a quantum state that minimizes a given objective function8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Reachability for QAOA15,16,18 and variational Grover search16,19 has been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' This paper focuses on the reachability of the VQS for unstructured search and proves that the exponentially deep circuit in Grover’s algorithm can be replaced by either a single layer of ������������������������(θ) gates or a two-layer circuit 1 Department of Renewable Energy Engineering, Alfred University, Alfred, NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' *E-mail: zhanj@alfred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='edu 2 consisting of Hadamard and X gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Furthermore, our numerical studies have verified the effectiveness of the VQS, with a single layer of ������������������������(θ) gates as its Ansatz, in solving the unstructured search problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The rest of the paper consists of the Method, Result, and Conclusion sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Method The problem to be solved in this paper is to find the good element in an unstructured data set ⅅ, which has only one good element and (2������������ − 1) bad elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' We use n qubits and n Hadamard gates to generate an equal superposition of all 2������������ elements, as shown in the left-hand side of the leftmost dashed red line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' In the rest of this section, we propose three methods to solve the problem: 1) a quantum circuit (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1a,b) based on an HX layer, which is generated by Algorithm 1 (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1), 2) a quantum circuit (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1c,d) based on an ������������������������ layer, which is generated by Algorithm 2 (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='2), and 3) the VQS using the ������������������������ layer as the Ansatz (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='4 also discusses the reachability of the VQS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='3 clarifies the scalability of Algorithms 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The quantum circuit to generate data set ⅅ and amplify the probability of the only good element in it to nearly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' a,c, the circuits in compact form for n-qubit data using an HX layer and an ������������������������ layer (the blue blocks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1a and 1c), respectively, where the HX layer consists of Hadamard and X gates and the ������������������������ layer consists of ������������������������(������������) gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' b,d, the detailed circuits for n=6 a c d b HX layer l 0, o) 0, o 12 1 laver label label Ry(π) 0 qubit qubit data dat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' J(H,X Xn HOn qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' qubit:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='bo> 1 laver 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' o HX layer 1 2 label label qubit 95 qubit 95 = 10> H q5 = 10 94 = [0> H 0 = 93 = 93 = H data data qubits qubits q2 =/0) 92 二 H q1 = 10) 91 = H T = 0b H /)3 using the HX and ������������������������ layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The yellow block, excluding the label qubit, generates a state that is an equal superposition of all elements (a single good element and (2������������ − 1) bad elements), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', all elements have the same initial probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The red block (Oracle) provides label |1〉 at the label qubit to the good element and |0〉 to all bad elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The blue block amplifies the probability of the good element to nearly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The label qubit is the highest one (the most significant one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' In panel a, the ������������(������������, ������������) represents a circuit consisting of Hadamard and X gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' In panels b,d, ������������ = 6 and the index of the good element is 39 (its binary form is 100111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1 Design the HX-layer-based Circuit This section consists of four parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The first part details an algorithm, Algorithm 1, which generates the HX layer (the blue block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The second part provides the vector forms of the three key quantum states in the HX-layer-based circuit, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1a,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The third part explains the design goal of the HX layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The last part proves that Algorithm 1 can always achieve the design goal of the HX layer for any number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Here we propose Algorithm 1 to construct the so-called HX layer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', a two-layer circuit consisting of only Hadamard and X gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The HX layer together with the Oracle (the red blocks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1) has the same function as Grover’s search algorithm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', amplify the probability of the good element to nearly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Algorithm 1 | Pseudo code for generating the HX layer (the blue block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Vector Forms of Three Quantum States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' This sub-section details the vector forms of three quantum states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', |0, ������������0⟩, |������������1⟩, and |������������2⟩, which are respectively indicated in the three dashed red lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' |0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������0⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' can be written as |0〉⊗(|0〉+|1〉)⊗������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' which can be represented in the vector form: |0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������0⟩ = [������������0 b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������1 b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������−1 b ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������+1 b ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������−1 b ����������������������� 1st half: ������������ elements ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0 �����]������������ 2nd half: ������������ elements (1) Input: the number of qubits n and the index of the good element in decimal form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ∀������������ ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 2������������ − 1] Output: quantum gates in the HX layer (the blue block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1 Convert k into the binary form ������������������������−1������������������������−2 ⋯ ������������2������������1������������0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 2 Add an X gate in the label qubit (the most significant qubit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 3 Let m=n 4 while ������������ ≥ 1 5 if ������������������������−1 = 1 6 Add a Hadamard gate followed by an X gate to qubit ������������������������−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 7 else 8 Add a Hadamard gate to qubit ������������������������−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 9 m ← m−1 4 where ������������ = 2������������, super scripts b and g indicate bad and good elements, respectively, and subscripts, 0~N−1, represent the index of an element in the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Note that the index always counts from 0 in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' For example, ������������������������ g means the kth element is a good element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The relationship between |������������1⟩ and |0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������0⟩ can be represented as: |������������1⟩ = ������������������������|0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������0⟩ = [������������0 b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������1 b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������−1 b ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������+1 b ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������−1 b ��������������������� 1st half: ������������ elements ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������+������������ g ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0 �������������]������������ 2nd half: ������������ elements (2) where oracle ������������������������ is implemented as the ������������������������(������������),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' which is an n-qubit-controlled X gate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' As shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (2), the ������������������������(������������) changes the index of the good element from k to N+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The HX layer (the blue block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1a) can be expressed as ������������ ⊗ ������������(������������, ������������) = � ������������ ������������(������������, ������������) ������������(������������, ������������) ������������ � (3) where ������������(������������, ������������) is an N by N matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' For the convenience of expression, we call the matrix given in (3) the HX-layer matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Then the relationship between |������������2⟩ and |������������1⟩ can be represented as: |������������2⟩ = � ������������ ������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������) ������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������) ������������ � |������������1⟩ = � ������������ ������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������) ������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������) ������������ � [������������0 b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������1 b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������−1 b ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������+1 b ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������−1 b ��������������������� 1st half: ������������ elements ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������+������������ g ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0 �������������]������������ 2nd half: ������������ elements (4) For the convenience of analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' we write |������������2〉 as a vector form: |������������2〉 = [������������0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������2������������−1]������������ (5) where ∑ |������������������������|2 2������������−1 ������������=0 = 1 (6) Design Goal of the HX Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The goal of designing the ������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������) is to let each element in its row k be 1 √������������ ⁄ such that ������������������������+������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' the (������������ + ������������)th element of |������������2⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' can be calculated as: ������������������������+������������ = �∑ ������������������������ b ������������−1 ������������=0 + ∑ ������������������������ b ������������−1 ������������=������������+1 � √������������ ⁄ (7) where the right-hand side is obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', multiply the (������������ + ������������)th row of the HX- layer matrix with the vector form of |������������1⟩, and the integer ������������ ∈ [0, 2������������ − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' From here on, we assume all elements have the same initial magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' That is, ������������������������ b = ������������������������ g = ������������������������+������������ g = 1 √������������ ⁄ , ∀������������ ∈ [0, ������������ − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Then, (7) can be reformed as: ������������������������+������������ = �1 √������������ ⁄ �(������������ − 1) √������������ ⁄ = 1 − 1 ������������ ⁄ = 1 − 1 2������������ ⁄ (8) The probability of obtaining the good element is equal to ������������������������+������������ 2 = (1 − 1 2������������ ⁄ )2, which is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='5625, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='7656, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='8789, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='9386, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='9690, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='9844, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='9922, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='9961 for n=1~9, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' That is, the probability of finding the good element is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='95 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='99 for n being larger than 5 and 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 5 Position Index of the All-1 Row for the HX Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' In the rest of this sub-section, we answer the question of why the HX layer generated by Algorithm 1 can realize the design goal specified in the previous sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The result of tensor product �������������������������,0 ������������������������,1� ⊗ �������������������������−1,0 ������������������������−1,1� ⊗ ⋯ ⊗ �������������2,0 ������������2,1� ⊗ �������������1,0 ������������1,1� ⊗ �������������0,0 ������������0,1� is a column vector with 2������������+1 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The position index of the element ������������������������,������������������������������������������������−1,������������������������−1 ⋯ ������������2,������������2������������1,������������1������������0,������������0 in the column vector is equal to the decimal value of a binary form ������������������������������������������������−1 ⋯ ������������2������������1������������0 , where ������������������������ ∈ {0,1}, ∀ ������������ ∈ [0, ������������].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' To better understand this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' we provide two examples: the position index of the element ������������3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0������������2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0������������0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1 in the column vector associated with �������������3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0 ������������3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1� ⊗ �������������2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0 ������������2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1� ⊗ �������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0 ������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1� ⊗ �������������0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0 ������������0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1� is 1 (its binary form is 0001) and the position index of ������������3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1������������2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0������������0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1 in the same vector is 9 (its binary form is 1001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' We can express the tensor product ������������������������−1 ⊗ ������������������������−2 ⊗ ⋯ ⊗ ������������2 ⊗ ������������1 ⊗ ������������0 as 1 √2������������ ������������, where ������������ represents a 2������������ by 2������������ matrix and ������������������������, ∀ ������������ ∈ [0, ������������ − 1], represents either ������������������������ or ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Note that ������������ = �0 1 1 0� , ������������ = 1 √2 �1 1 1 − 1� , ������������������������ = 1 √2 �1 − 1 1 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' It can be easily verified that M has only one all-1 row (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', each element in the row is 1) while each of all the other rows consists of −1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The question then becomes, where is the all-1 row, which will be answered in the next paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' This paragraph explains how to find the all-1 row for the tensor product given in the previous paragraph, and conversely, for any given number k, how to construct a quantum circuit such that row k of the matrix associated with the circuit is an all-1 row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Given that each ������������������������ has exactly one row of [1 1]/√2, the all-1 row is generated from the tensor product of the [1 1]/√2 row of each ������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=" In other words, if the [1 −1]/√2 row is involved, the corresponding tensor product result will not be all 1's." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Let ������������������������ = 0 represent that the [1 1]/√2 is located in row 0 of ������������������������ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', ������������������������ = ������������) and ������������������������ = 1 represent that the [1 1]/√2 is located in row 1 of ������������������������ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', ������������������������ = ������������������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' According to the paragraph before the previous one, we can know that the all-1 row is located at row k where k is the decimal value of its binary form ������������������������−1������������������������−2 ⋯ ������������2������������1������������0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Conversely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' for any given k∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 2������������ − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' if we want to generate a matrix that represents the tensor product ������������������������−1 ⊗ ������������������������−2 ⊗ ⋯ ⊗ ������������2 ⊗ ������������1 ⊗ ������������0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' such that each element in row k of the matrix is 1 (ignoring the coefficient 1 √2 ⁄ of each ������������������������),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' we can convert k into its binary form ������������������������−1������������������������−2 ⋯ ������������2������������1������������0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' set ������������������������ to be ������������������������ if ������������������������ = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' and set ������������������������ to be ������������ if ������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' This is exactly what Algorithm 1 does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' To better understand this, we provide three examples for k=5, 8, and 39 in the next three paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' For k=5, its binary form is ������������2������������1������������0 =101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Then we have ������������2 ⊗ ������������1 ⊗ ������������0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' By replacing ������������2 and ������������0 by XH as ������������2 = ������������0 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' and replacing ������������1 by H as ������������1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' we can obtain: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='������������������������ ⊗ ������������ ⊗ ������������������������ = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='√2 �1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1� ⊗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='√2 �1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='− 1� ⊗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='√2 �1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='2√2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='⎣ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='⎢ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='⎢ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='⎢ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='⎥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='⎤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='(9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='which shows that row 5 is the only all-1 row (ignoring the coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='2√2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' It is important to note that in this paper, the row index count always starts at 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', the first row is row 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' For k=8, its binary form is 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Then we have ������������������������ ⊗ ������������ ⊗ ������������ ⊗ ������������ = 1 √2 �1 − 1 1 1� ⊗ 1 √2 �1 1 1 − 1� ⊗ 1 √2 �1 1 1 − 1� ⊗ 1 √2 �1 1 1 − 1� (10) We can easily verify that the tensor product shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (10) has only one all-1 row (ignoring the coefficient ¼) located at row 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' For k=39, its binary form is 100111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Then we can use ������������������������ ⊗ ������������ ⊗ ������������ ⊗ ������������������������ ⊗ ������������������������ ⊗ ������������������������ to obtain a matrix whose only all-1 row is located at row 39, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' In summary, by using a ������������������������(������������) gate and an HX layer, we can amplify the probability of the good element from 1 2������������ ⁄ to a value larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='95 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='99 for n values greater than 5 and 7, respectively, where we already know the position index of the good element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='2 Design the ������������������������-layer-based Circuit This section consists of four parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The first part details an algorithm, Algorithm 2, to generate the ������������������������ layer (the blue block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The second part provides the vector form of the quantum state |������������2⟩ in the ������������������������-layer-based circuit, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1c,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The third part explains the ������������������������ layer is equivalent to the HX layer in solving the problem of finding the good element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Here we design an ������������������������ layer (the blue block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1c) to replace the HX layer (the blue block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1a), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', use a single layer of Ry gates to replace the two layers consisting of X, H, and XH gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Given that ������������������������(������������) = � cos( ������������ 2) −sin( ������������ 2) sin( ������������ 2) cos( ������������ 2) �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' we have ������������������������ � ������������ 2� = 1 √2 �1 −1 1 1 � which is exactly equivalent to XH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������(������������) = �0 −1 1 0 � which can be used to replace the X gate despite the phase difference,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' and ������������������������(3������������ 2 ⁄ ) = 1 √2 �−1 −1 1 −1� which can be used to replace the H gate despite the phase difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The impact of the phase difference can be ignored with the reason given in the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Algorithm 2 details the procedure of how to generate the ������������������������ layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 7 Algorithm 2 | Pseudo code for generating the ������������������������ layer (the blue block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Vector Forms of Quantum State |������������2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The states |0, ������������0⟩ and |������������1⟩ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1c are the same as those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Therefore, only state |������������2⟩ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1c is provided here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The ������������������������ layer obtained from Algorithm 2, as shown in the blue blocks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1c,d, can be expressed as ������������������������(������������) ⊗ ������������������������(������������)⊗������������ = �0 −1 1 0 � ⊗ ������������������������(������������)⊗������������ = � ������������ −������������������������(������������)⊗������������ ������������������������(������������)⊗������������ ������������ � (11) For the convenience of expression, we refer to the matrix given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (11) as the ������������������������-layer matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Like Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (4), based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (11), states |������������2⟩ and |������������1⟩ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1c have the following relationship: |������������2⟩ = � ������������ −������������������������(������������)⊗������������ ������������������������(������������)⊗������������ ������������ � |������������1⟩ = � ������������ −������������������������(������������)⊗������������ ������������������������(������������)⊗������������ ������������ � [������������0 b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������1 b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������−1 b ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������+1 b ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������−1 b ��������������������� 1st half: ������������ elements ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ������������������������+������������ g ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='0 �������������]������������ 2nd half: ������������ elements (12) where ������������������������(������������)⊗������������ is an N by N matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Relation Between the ������������������������ Layer and the HX Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' This sub-section will explain that the ������������������������ layer is equivalent to the HX layer in solving the problem of finding the good element from an unstructured data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Similar to the analysis given in (4)-(7), the calculation of ������������������������+������������, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (5), involves only the bottom left part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', ������������������������(������������)⊗������������, of the ������������������������-layer matrix but not its top right part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', −������������������������(������������)⊗������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Therefore, The phase difference between ������������������������(������������) and X does not have any impact on the value of ������������������������+������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' In other words, we can use ������������������������(������������) to replace X gate without any impact on the probability of finding the good element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Input: the number of qubits n and the position index of the good element in decimal form, k, ∀������������ ∈ [0, 2������������ − 1] Output: quantum gates in the ������������������������ layer (the blue block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1 Convert k into the binary form ������������������������−1������������������������−2 ⋯ ������������2������������1������������0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 2 Add an ������������������������(������������) gate in the label qubit (the most significant qubit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 3 Let m=n 4 while ������������ ≥ 1 5 if ������������������������−1 = 1 6 Add an ������������������������(������������ 2 ⁄ ) gate to qubit ������������������������−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 7 else 8 Add an ������������������������(3������������ 2 ⁄ ) gate to qubit ������������������������−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 9 m ← m−1 8 We can write ������������������������(������������)⊗������������ as a tensor product ������������������������−1 ⊗ ⋯ ⊗ ������������2 ⊗ ������������1 ⊗ ������������0 , where ������������������������, ∀������������ ∈ [0, ������������ − 1], represents either ������������������������(������������ 2 ⁄ ) or ������������������������(3������������ 2 ⁄ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Then ������������������������(������������)⊗������������ will have only one all-1 (all- negative-1) row if the number of ������������������������(3������������ 2 ⁄ ) gates is even (odd), where each element in the all- negative-1 row is −1, ignoring the coefficient 1/√2������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' When the number of ������������������������(3������������ 2 ⁄ ) gates is even, the calculation given in (7) and (8) is also valid for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' When the number of ������������������������(3������������ 2 ⁄ ) gates is odd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' each element in row N+k is −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ignoring the coefficient 1/√2������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' then we have ������������������������+������������ = − �∑ ������������������������ b ������������−1 ������������=0 + ∑ ������������������������ b ������������−1 ������������=������������+1 � √������������ ⁄ (13) ������������������������+������������ = − �1 √������������ ⁄ �(������������ − 1) √������������ ⁄ = −1 + 1 ������������ ⁄ = −1 + 1 2������������ ⁄ (14) Since the probability is the square of magnitude ������������������������+������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' the magnitude of the good element given in (8) and that given in (14) result in the same probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Therefore, no matter whether the number of ������������������������(3������������ 2 ⁄ ) gates is even or odd, the probability of finding the good element from the output of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1c, |������������2〉, is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='95 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='99 for n values greater than 5 and 7, respectively, which is the same as the analysis for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1a given in the previous section (see the paragraph below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Therefore, we can use the ������������������������ layer generated by Algorithm 2 to replace the HX layer generated by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='3 Scalability of Algorithms 1 and 2 In the analysis for Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1a and 1c above, there is neither restriction nor assumption associated with the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' That is, the analysis given in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='2 is valid for any number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Specifically, Algorithm 1 (Algorithm 2) each can generate a quantum circuit whose corresponding matrix form has only one all-1 (all-1 or all-negative-1) row located at row k, k∈ [0, 2������������ − 1], for any value of n, where n is the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' In Algorithms 1 and 2, the location index, k, of the good element must be known in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' This means that they are not useful for finding the good element in a data set if the location index is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' However, the analysis given above has proved that either the HX layer (generated by Algorithm 1) or the ������������������������ layer (generated by Algorithm 2), together with a ������������������������(������������) gate (the red and blue blocks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1a,c), can amplify the probability of the good element from 1 2������������ ⁄ to nearly 1 if n is larger than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Note that for the same task, Grover’s search algorithm requires a quantum circuit whose depth increases exponentially with the number of qubits, which manifests the exponential advantage of the circuits generated by Algorithms 1 and 2, in terms of circuit depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='4 Variational Quantum Search and its Reachability This section briefs the VQS and then details the definition of reachability for the VQS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The VQS4 is a VQA, which involves the interaction between classical and quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' In the classical part, an optimizer is used to update the parameter ������������ of the Ansatz based on the objective function ������������(������������): ������������(������������) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='5⟨������������1|������������2⟩ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='5⟨������������1|������������ ⊗ ������������⊗������������|������������2⟩ (15) where ������������1 and ������������2 are the states before and after the ansatz, respectively, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1a,c, Z and I are Pauli Z and identity matrix, respectively, and n is the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The first and second terms in the objective function can be respectively obtained from measuring two quantum circuits based on the Hadamard test, as detailed in paper4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Papar4 has validated two types of shallow 9 and effective Ansatzs for the VQS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' However, the analysis given above implies that the VQS with only a single ������������������������ layer as the Ansatz, which is shallower than those two types of Ansatzs used in paper4, is sufficient to find the only good element in an unknown data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Note that unlike Algorithms 1 and 2, the VQS does not need to know the position index, k, of the good element beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' To quantify the capability of how best the Ansatz can generate a quantum state that minimizes a given objective function, we propose a definition of reachability for the VQS in the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The objective function given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (15) can be reformed as ������������(������������) = ⟨������������1|������������|������������2⟩ (16) where ������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='5�−������������⊗������������+1 + ������������ ⊗ ������������⊗������������� = diag([ 0,0, ⋯ ,0 ����� ������������ elements , −1, −1, ⋯ , −1 ��������� ������������ elements ]) (17) where diag(w) represents a square diagonal matrix with elements of vector w on the main diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Then we can define a reachability15,16 for the VQS: ℜ = min ������������ ⟨������������1|������������|������������(������������)⟩− min |������������⟩∈ℋ⟨������������1|������������|������������⟩ − min |������������⟩∈ℋ⟨������������1|������������|������������⟩ (18) where |������������1⟩ denotes the state before the ansatz (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1c), the min������������ term, used to refer to min ������������ ⟨������������1|������������|������������(������������)⟩ for the convenience of expression, is the minimum over all reachable states generated by the Ansatz of the VQS, the ������������(������������) in the min������������ term represents the |������������2⟩ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1c, and the minH term, used to refer to min |������������⟩∈ℋ⟨������������1|������������|������������⟩ for the convenience of expression, represents the minimum over all states |������������⟩ of the Hilbert space ℋ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' This paragraph discusses the possible value range of the numerator in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Note that |������������1⟩ has only a single non-zero element which is equal to 1 √������������ ⁄ in this paper and located in its second half and that ⟨������������|������������|������������⟩ is equal to the tensor product of the second half of −|������������⟩ and the second half of |������������⟩, where the second half of a state represents the second half elements of the vector form of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' We can then know that, in the numerator in (18), the value range of ⟨������������1|������������|������������(������������)⟩ is a subset of [− 1 √������������ ⁄ , 1 √������������ ⁄ ] and the value range of ⟨������������1|������������|������������⟩ is [− 1 √������������ ⁄ , 1 √������������ ⁄ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Therefore, the range of ⟨������������1|������������|������������(������������)⟩ − ⟨������������1|������������|������������⟩ is a subset of [− 2 √������������ ⁄ , 2 √������������ ⁄ ], which is an extremely small range when n is large as both the upper and lower bounds converge exponentially towards zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Given that a well-defined metric should not consistently reside within a minuscule range around zero, using the numerator in (18) as a measure of reachability is not appropriate as it is extremely small when the number of qubits is large and converges exponentially towards zero as the increase of the number of qubits, regardless of the value of ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' To address the issue, we use the minH term in the denominator as explained in the next paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' According to the analysis given in paper4, the minimum value of the minH term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=", − 1 √������������ ⁄ , is achieved when |������������⟩ is equal to the computational state |������������ + ������������⟩, which is represented by a vector 10 with a single element of 1 at index N+k and all other elements being 0's, where ������������ = 2������������." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Given that dividing the range [− 2 √������������ ⁄ , 2 √������������ ⁄ ] by the minH term results in a range of [−2, 2], we use the minH term as the denominator such that the range of ℜ defined in (18) becomes a subset of [0, 2] instead of the extremely small range mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' A range of [0, 2] is more suitable for the reachability metric, as it is not confined to a minuscule range around zero, and this is the reason why we introduce the denominator in the definition of reachability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Now we explain why we add the absolute sign into each term, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' As per Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (8) and (14), the value of ������������������������+������������ being equal to either 1 − 1 ������������ ⁄ or −1 + 1 ������������ ⁄ results in the same probability of finding the good element, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', both solutions are equally good, where ������������ = 2������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' However, putting these two values of ������������������������+������������ into |������������2⟩, which replaces |������������(������������)⟩, results in different values of ⟨������������1|������������|������������(������������)⟩ and thereby leads to different values of reachability, but they should correspond to the same reachability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' To address this issue, we use absolute terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', |⟨������������1|������������|������������(������������)⟩| and |⟨������������1|������������|������������⟩|, to replace ⟨������������1|������������|������������(������������)⟩ and ⟨������������1|������������|������������⟩, respectively, such that two amplitudes with different signs lead to the same reachability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' This paragraph discusses a better definition of reachability, as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Note that max |������������⟩∈ℋ|⟨������������1|������������|������������⟩| reaches its maximum when ⟨������������1|������������|������������⟩ equals 1 √������������ ⁄ or − 1 √������������ ⁄ , corresponding to finding the good element with a probability of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' On the other hand, min |������������⟩∈ℋ|⟨������������1|������������|������������⟩| reaches its minimum when ⟨������������1|������������|������������⟩ equals 0, corresponding to finding the good element with a probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Thus, we use ‘max’ rather than ‘min’ when using the absolute terms such that we can find the good element with a high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' According to the discussion given in the previous paragraph, two amplitudes with different signs lead to different values of reachability if we use (18) as the definition of reachability, which could be addressed if we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Therefore, we recommend using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (19) as the definition of reachability instead of (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' ℜ = max ������������ |⟨������������1|������������|������������(������������)⟩|− max |������������⟩∈ℋ|⟨������������1|������������|������������⟩| − max |������������⟩∈ℋ|⟨������������1|������������|������������⟩| (19) The smaller the value of ℜ, the better the reachability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' When ℜ = 0, the VQS has perfect reachability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The use of denominator and absolute terms distinguishes our definition of reachability from that presented in papers15,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' For the circuit depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 1c where the ������������������������ layer is generated by Algorithm 2, the (N+k)th element of the output state |������������2⟩ is either equal to 1 − 1 ������������ ⁄ or −1 + 1 ������������ ⁄ , as shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (8) and (14), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' If we replace the max ������������ |⟨������������1|������������|������������(������������)⟩| with |⟨������������1|������������|������������2⟩| in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (19), we have |⟨������������1|������������|������������2⟩|− max |������������⟩∈ℋ|⟨������������1|������������|������������⟩| − max |������������⟩∈ℋ|⟨������������1|������������|������������⟩| = 1−1 ������������ ⁄ √������������ − 1 √������������ − 1 √������������ = 1 ������������ = 1 2������������ (20) As |������������2⟩ is generated by Algorithm 2, it is the |������������(������������)⟩ with the ������������ specified in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Then we know max ������������ |⟨������������1|������������|������������(������������)⟩| ≥ |⟨������������1|������������|������������2⟩|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' we have 11 ℜ = max ������������ |⟨������������1|������������|������������(������������)⟩|− max |������������⟩∈ℋ|⟨������������1|������������|������������⟩| − max |������������⟩∈ℋ|⟨������������1|������������|������������⟩| ≤ |⟨������������1|������������|������������2⟩|− max |������������⟩∈ℋ|⟨������������1|������������|������������⟩| − max |������������⟩∈ℋ|⟨������������1|������������|������������⟩| = 1 2������������ (21) where the two sides of the ≤ come from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (19) and (20), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (20) provides a specific circuit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', using the ������������������������ layer generated by Algorithm2 as the Ansatz of the VQS, whose reachability value is equal to 1 2������������ ⁄ , which is an extremely small value when n is large and converges exponentially to zero as the increase of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' That is, this circuit already achieves near- perfect reachability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' In conclusion, the VQS has at least near-perfect reachability, and the more qubits, the better the reachability of the VQS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Result The results provided in this paper are obtained from quantum simulators using Pennylane’s devices20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Results related to 2-, 8-, and 14-qubit (20- and 26-qubit) input states are obtained using Pennylane’s default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='qubit (lightning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='gpu) device on an Intel i5-6500 CPU (A40x4 48-GB GPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The initial values of ������������ in the Ansatz are randomly sampled from a uniform distribution between 0 and 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Same as paper4, two termination criteria for the iterative process are used in the VQS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The first one is that the number of iterations reaches a given number (set to 300 in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The second one is that a small-change event occurs consecutively for a given number of times (set to 5 in this paper), where the small-change event is defined as: the absolute value of the relative change of objective functions in two consecutive iterations is smaller than a given small value (set to 1 × 10−4 in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' When one of the criteria is met, whichever comes first, the iterative process of VQS terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Here we numerically verify that the VQS with only one layer of ������������������������(������������) gates can efficiently find the good element from an unstructured data set without knowing the position index, k, of the good element beforehand, which is related to the trainability of the VQS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' We apply the VQS, where the Ansatz is an ������������������������ layer, to find the good element from 2-, 8-, 14-, 20-, and 26-qubit data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 2, which show that the VQS indeed can find the good element as the amplified probability is very close to 1 for most runs out of 100 runs for n=8, 14, 20, and 26, and that in some runs (22, 16, 16, and 16 for the four cases) out of 100 runs, the amplified probability is close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Compared to the results shown in paper4, although the VQS using the ������������������������ layer in the Ansatz is less stable than the VQS using the two types of Ansatzs shown in paper4, the former can still amplify the probability to nearly 1 in most runs, excluding the 2-qubit case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The relatively poor performance of the 2-qubit case is not a concern, but actually further validates our analysis as explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' When n equals 2 and if we use the ������������������������ layer as the Ansatz of the VQS, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' (8), the probability of finding the good element is (1 − 1 22 ⁄ )2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='5625, which is roughly in the middle of the box result for the 2-qubit case (the leftmost one in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' In other words, the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 2a validate the conclusion obtained from the reachability analysis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', the more the number of qubits, the better the reachability of the VQS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 12 The importance of using VQS with the ������������������������ layer is that it has been proved above that the ������������������������ layer, together with the ������������������������(������������), can amplify the probability of the good element from 1 2������������ ⁄ to nearly 1 for any number of qubits being larger than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' That is, the reachability of the VQS using the ������������������������ layer is guaranteed for any number of qubits being larger than 5, which means we do can scale the VQS to any large number of qubits while keeping the circuit depth to be 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=', one ������������������������ layer and one ������������������������(������������) layer), which is an exponential advantage over Grover’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Box plot results from 100 runs of VQS using the ������������������������ layer as the Ansatz for an n-qubit input state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' a, the amplified probability of good element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' b, the number of iterations used when a termination criterion is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Conclusion We have proposed two algorithms to construct a depth-2 (the HX layer) and a depth-1 (the ������������������������ layer) circuits, both of which can replace the exponentially deep circuit required by Grover’s algorithm, if the position index of the good element is known beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' We have proved that the VQS, with the ������������������������ layer as the Ansatz, has near-perfect reachability for any number of qubits greater than five and its reachability exponentially improves with the number of qubits, which has been further verified by numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' In these experiments, we use the VQS, with the ������������������������ layer as Ansatz, to search for the only good solution in unstructured data sets, represented by 8, 14, 20, and 26 qubits, and successfully find the good element with a probability of nearly 1 in 78 to 84 out of 100 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The experiments also indicate the good trainability of the VQS for up to 26 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' We will further investigate the trainability of the VQS for more qubits in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Nielsen, M.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content='04968 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' Acknowledgements This research was partially supported by the NSF ERI program, under award number 2138702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' This work used the Delta system at the National Center for Supercomputing Applications through allocation CIS220136 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by National Science Foundation grants #2138259, #2138286, #2138307, #2137603, and #2138296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' We acknowledge the use of IBM Quantum services for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} +page_content=' The views expressed are those of the authors, and do not reflect the official policy or position of IBM or the IBM Quantum team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfAzVW/content/2301.13224v1.pdf'} diff --git a/zdFRT4oBgHgl3EQfjzd-/content/tmp_files/2301.13592v1.pdf.txt b/zdFRT4oBgHgl3EQfjzd-/content/tmp_files/2301.13592v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3d5ba9b363cc008a53b27734f6215ab193c7f84b --- /dev/null +++ b/zdFRT4oBgHgl3EQfjzd-/content/tmp_files/2301.13592v1.pdf.txt @@ -0,0 +1,634 @@ +Priors are Powerful: Improving a Transformer for +Multi-camera 3D Detection with 2D Priors +Di Feng∗, Francesco Ferroni∗ +Abstract— Transfomer-based approaches advance the recent +development of multi-camera 3D detection both in academia +and industry. In a vanilla transformer architecture, queries +are randomly initialised and optimised for the whole dataset, +without considering the differences among input frames. In this +work, we propose to leverage the predictions from an image +backbone, which is often highly optimised for 2D tasks, as +priors to the transformer part of a 3D detection network. The +method works by (1). augmenting image feature maps with +2D priors, (2). sampling query locations via ray-casting along +2D box centroids, as well as (3). initialising query features with +object-level image features. Experimental results shows that 2D +priors not only help the model converge faster, but also largely +improve the baseline approach by up to 12% in terms of average +precision. +I. INTRODUCTION +Towards 360-degree 3D perception, self-driving vehicles +are usually equipped with multiple monocular cameras, and +reliable and accurate multi-camera 3D detection has become +an important research challenge and industrial effort. The +traditional approach takes advantage of the convolution neu- +ral nets (convnets) that are highly-optimised for 2D tasks, +by performing 2D scene understanding on images, followed +by a 2D to 3D projection. Recent advancements, in contrast, +propose to project 2D images into 3D space, before running +3D tasks on a bird’s eye view (BEV) representation [1]. This +new paradigm not only provides a generic scene represen- +tation for multi-modal perception, mapping, and prediction, +but also achieves improved accuracy with the help of the +transformer architecture [2]. +A typical pipeline for multi-camera 3D detection with +transformers is shown in Fig. 1. First, multi-level feature +maps from multiple camera images are extracted from a +backbone network, commonly a convnet. Afterwards, a trans- +former decoder iteratively processes queries and interacts +with image feature maps via cross-attention [3]. Finally, each +updated query is fed into a detection head to categorize +objects and regress their cuboid parameters (such as centroid +locations, cuboid extents, and orientations). In a vanilla +transformer architecture, such as from the seminal work +detr3d [4], query features and their location information are +randomly initialised and optimised for the whole dataset, +without considering the heterogeneity of inputs from dif- +ferent frames. We find that such query design suffers from +slow training convergence and strong smearing effect with +∗Work done while both authors were at Argo AI (https://www. +argo.ai/). Di Feng is now with Apple, and Francesco Ferrorni is with +Nvidia. Correspondence: fengdi1015@gmail.com +Query + ref. point +Transformer +decoder +… +Feature maps +Backbone +… +Cross +attn. +Updated query +2D Priors +Fig. 1: The pipeline for multi-camera 3D detector with trans- +formers. First, a backbone network, commonly a convolution +neural network (convnet), extracts multi-level feature maps +from multi-camera inputs. Afterwards, a transformer decoder +iteratively processes queries and interacts with feature maps. +Finally, each query is fed into a detection head to predict +object classes and cuboid parameters. In this work, we pro- +pose to leverage 2D predictions from the convnet backbone +as priors to the transformer decoder for 3D detection. Those +priors are incorporated into feature maps, as well as each +query and reference point. +erroneous depth estimation (shown in Fig. 2(a) and will be +discussed in Sec. II). +Several methods extend detr3d [4] with improved query +design. For example, PETR [5] generates 3D position em- +bedding to image features as the input of a transformer +decoder. SpatialDETR [6] encodes camera intrinsics and +extrinsics features to keys and queries. Graph-detr3d [7] +replaces self-attention with a graph neural network for better +query interaction. Finally, BEVFormer [8] discretises the 3D +world with bird’s eye view grids, and considers each grid as +a query location. +Since convnets are often highly optimised for 2D tasks, +why not reusing those 2D predictions as priors to the +transformer part of 3D detection? In this work, we verify +this idea in the detr3d pipeline, and incorporate 2D object +detection, semantic segmentation, and depth estimation from +a image backbone to the transformer decoder. We propose +three simple strategies to use 2D priors: augmenting image +feature maps for cross attention, sampling query locations +via ray-casting along 2D box centroids, as well as initialising +query features with object-level image features. Experimental +results on an internal dataset shows that our methods largely +improve the vanilla detr3d by up to 12% in terms of +average precision, and make the model converge faster during +training. +In parallel to our work, MV2D [9] also proposes to +arXiv:2301.13592v1 [cs.CV] 31 Jan 2023 + +Feature +embedding +MLP +Positional +embedding +Random +init. +Random +init. +Query +Ref. point +(a) Vanilla detr3d +Feature +embedding +Cropped features +MLP +MLP +Positional +embedding +Ref. point +Query +[x, y, z] +(b) Ours +Fig. 2: A comparison of query generation strategies between the vanilla detr3d [4] and our proposed methods with 2D priors. +(a). The vanilla detr3d randomly initialises the positional embedding and the feature embedding vectors. Reference points are +predicted by a small multi-layer perceptron (MLP). (b). Our proposed approach leverages 2D detection and semantic maps +as 2D priors. We sample reference points via ray-casting along 2D box centroids, which generates the positional embedding +vector through a MLP. Besides, we initialize the feature embedding vector with the object-level features, weighted by +semantic scores. +Classifications scores +x [m] +y [m] +(a) Predictions from detr3d +x [m] +y [m] +(b) Query sampling results +Fig. 3: (a). Raw predictions from the vanilla detr3d model on +the Bird’s Eye View (BEV). Each circle represents a query +prediction. Strong smearing effect can be observed along the +ray. (b). Our proposed query sampling strategy. Blue dots are +generated reference points, and orange dots are ground truth +centroids. All ground truth centroids can be associated with +nearby reference points, though with some errors. +leverage 2D detections as priors for the transformer part of +a multi-camera 3D detector. Unlike our approaches which +generate multiple reference points from a 2D box centroid +and employs multiple 2D cues (2D boxes, semantic maps, +and depth maps), MV2D only studies how to exploit 2D +detections, and how to predict one reference point for each +2D box via a dynamic object query generator. Their ex- +perimental results demonstrate higher recall rates compared +to the vanilla transformer model, especially for small and +distant objects. +In the sequel, Sec. II reviews the transformer decoder part +of the detr3d model, with a focus on the query generation +process. Sec. III introduces our proposed three methods to +improve the detr3d network. Sec. IV shows the experimental +results, followed by a summary and discussion in Sec. V. +II. VANILLA DETR3D REVISITED +The architecture of detr3d [4] has been summarised in the +previous section and depicted in Fig. 1. In this section, let +us take a closer look at the transformer decoder. +A +decoder +is +built +from +six +standard +transformer +blocks [3]. In each block, queries are interacted with each +other via self-attention, and fused with multi-camera multi- +level feature maps via cross-attention. Unlike the common +“global” cross-attention mechanism [3], detr3d only asso- +ciates a query to image features that correspond to its query +location (also called reference point). To do this, a position +p = [x,y,z] in the 3D coordinate system is computed for +each query. The position is projected onto image planes +given camera intrinsics and extrinsics parameters. The image +features from the projected pixels are weighted and averaged +over all feature levels and cameras for updating a query in +a “local” cross-attention manner. +Denote d as the embedding dimension for a query, +Fig. 2(a) illustrates how a query q ∈ Rd and its reference +point p ∈ R3 are built. First, a position embedding vector +qpos ∈ Rd and a feature embedding vector qfeat ∈ Rd are +randomly initialized (following a uniform or a normal dis- +tribution). Afterwards, qpos is mapped to the reference point +p via a multi-layer perceptron (MLP), and added to qfeat to +generate the final positional-aware query features q. Through +training the network with the standard Hungarian assignment +and the set prediction loss [3], both qpos and qfeat learn to +encode the object statistics for the whole dataset, which can +be considered as pre-defined “anchors” in common object +detection pipeline, such as Faster-RCNN [10]. +Though simple and straightforward, the network design +in [4] lacks prior knowledge in the query and reference point +generation process, resulting in slow training convergence +and ambiguity in prediction. Fig. 3(a) illustrates a typical +detr3d output on the bird’s eye view (BEV) without post- + +top preds +gtloc priors +40 +ground truth +20 +D +-20 +-40 +40 +-20 +0 +20 +40processing. All queries are marked by circles, and those +with high classification scores are further demonstrated with +red bounding boxes. We observe strong smearing effect in +object detection, i.e. queries are converged along the ray from +detections on 2D images, bringing many false positives. This +is due to erroneous depth estimation and ambiguous target +assignment during training1. +III. THREE WAYS OF ADDING PRIORS +We propose three methods to improve the detr3d network, +by incorporating 2D priors to the transformer decoder, illus- +trated in Fig. 2(b). To do this, we select 2D object detection, +semantic segmentation, and depth estimation predicted by +our convnet backbone as priors, as they are common, well- +optimized 2D tasks for autonomous driving (e.g. HydraNet +from Tesla [1]). The depth estimation is represented as a +single-channel depth map rescaled to [0,1]. The semantic +segmentation is represented by a semantic map with C +channels, where each channel shows pixel-wise classification +scores of a category. +A. Feature Map Priors +We simply concatenate semantic and depth maps with the +multi-camera feature maps at different scales. In this way, +semantic and depth priors are added to queries in a cross- +attention operation. +B. Location Priors +We generate reference points p ∈ R3 only along the rays +from the centroids of 2D box predictions. For each ray, a +simple uniform sampling with 5 meters interval is performed. +In this way, the search space for objects can be narrowed +efficiently, which helps reduce false positives, limit the +number of queries, and accelerate model convergence. When +performing cuboid prediction, the detection head regresses +the offset to its reference point, denoted by ∆x,∆y,∆z, as +the cuboid centroids. It also regresses the cuboid’s height, +length, width, and yaw angle. +The reference points may not accurately overlap with the +cuboid centroids, because the centers of 2D boxes are differ- +ent from those from the projected cuboids, and the 5-meters +sampling interval is rough compared to the common discreti- +sation thresholds in many well-known detection networks +(e.g 0.5-meters interval in Lift-spalt-shoot [11], 0.16-meters +in CaDNN [12], and 0.2-meters in Pointpillar [13]). However, +we find such a simple point generation strategy provides +rough-and-ready estimates to the actual object locations, as +illustrated in Fig. 3(b). Besides, the location errors can be +compensated by the iterative query refinement in transformer +blocks. We expect more accurate reference point generation, +when introducing a centerness head for projected cuboid +centers (similar to CenterNet [14]), or sampling points only +around predicted depth (similar to CramNet [15]). +1Imagine multiple queries generate reference points, which are close to +each other and are projected to the same object on images. Due to the +Hungarian assignment [3], only one query is labelled “positive”, punishing +other positive queries with “negative” signals. +Inspired by Anchor-DETR [16], we further incorporate +location priors to queries, by projecting a reference point +p ∈ R3 to a position embedding vector qpos ∈ Rd via a small +MLP. Interestingly, this is a reversed procedure compared to +the vanilla detr3d, which maps a position embedding vector +to its reference point. +C. Query Priors +All queries generated from a ray come from the same 2D +object. Therefore, we propose to incorporate the same object- +level 2D priors to those queries, and further distinguish +among them with positional information. We follow five +steps: First, the semantic map, the depth map, and the multi- +level multi-camera feature maps are cropped based on the +2D box estimates. Then, the channel of the cropped semantic +map, which corresponds to the predicted object class, is used +to weight the cropped depth map and feature maps by a +pixel-wise dot production. Afterwards, a channel-wise global +average-pooling operation is used to generate a 1D vector for +each query prior, inspired from the squeeze-and-excitation +operation from SENet [17]. Furthermore, the query prior +vector, appended with an object class index, an objectness +score, and the 2D bounding box parameters, is fed into a +small MLP to generate query embedding features. Finally, +the positional embedding features are added to the query +embedding features as the final query features, so that the +queries from the same ray are distinguishable. +IV. EXPERIMENTAL RESULTS +Based +on +a +pre-trained +convnet +backbone, +we +re- +implement the detr3d transformer decoder, and experiment +its detection performance with different 2D priors. Following +the original detr3d [4], we set the initial learning rate to +be 2 ∗ 10−4 with a weight decay of 10−5. The AdamW +optimiser with a consine decay is used. Unlike [4], we do +not use any data augmentation tricks, and find that training +with more epochs improve the model performance. All +models, unless mentioned otherwise, are trained with a tiny +subset of our internal dataset, with approx. 60k training, 10k +validation, and 4k testing samples. The data was recorded +in various locations in the US and Europe, with different +lightning conditions (daytime, nighttime, rainy, sunny etc.) +and scenarios (cities, rural areas, etc.). We report the Average +Precision (AP) scores at the IoU=0.1 threshold on the bird’s +eye view (BEV) for the VEHICLE and HUMAN classes, +and only consider detections within the 50 meters range. +A. Main Results +Tab. I compares the AP scores between the vanilla detr3d +model with its variants with different 2D priors. The model +“+ feat prior ” only adds feature map priors, “+ feat, loc +priors” additionally uses location priors, and “+ feat, loc, +query priors” exploits all three priors. Fig. 4(a) and Fig. 4(b) +show the precision recall curves for VEHICLE and HUMAN +classes, respectively. We observe that all 2D priors improve +the vanilla detr3d model with higher AP scores up to nearly +12%. The largest performance gain comes from location + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Recall +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Precision +VEHICLE +vanilla detr3d ++ feat prior ++ feat,loc priors ++ feat,loc,query priors +(a) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Recall +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Precision +HUMAN +vanilla detr3d ++ feat prior ++ feat,loc priors ++ feat,loc,query priors +(b) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Recall +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Precision +Comparison with Lidar points ++ feat,loc priors (VEHICLE) ++ feat,lidar priors (VEHICLE) ++ feat,loc priors (HUMAN) ++ feat,lidar priors (HUMAN) +(c) +Fig. 4: Precision recall curves. (a). A comparison among the vanilla detr3d model and its variants with 2D priors on the +VEHICLE class. (b). A comparison on the HUMAN class. (c). Ablation study by replacing reference points with lidar +observations. +priors, verifying the effectiveness of our design choice for +reference point generation. +Models +AP VEHICLE (%) +AP HUMAN (%) +vanilla detr3d +70.57 +6.36 ++ feat prior +72.40 (+1.83) +9.16 (+1.80) ++ feat, loc priors +79.93 (+9.36) +13.52 (+7.16) ++ feat, loc, query priors +82.01 (+11.44) +14.68 (+8.32) +TABLE I: A comparison of Average Precision (AP) scores +at the IoU=0.1 threshold. +Models +AP VEHICLE (%) +AP HUMAN (%) +vanilla detr3d +75.84 +24.79 ++ feat, loc, query priors +86.52 (+10.68) +58.06 (+33.27) +TABLE II: A comparison of Average Precision (AP) scores +at 4 meters centroid distance threshold. +In addition, Tab. II reports AP scores at the 4 meters +threshold, which are commonly used in the Nuscenes met- +rics [18]. This threshold is less strict than the IoU=0.1 +threshold when evaluating location errors, thus resulting in +higher AP scores when evaluating on the same model. In this +setting, we observe that the model with 2D priors largely +improves the HUMAN detection by more than 30%. +B. Using Lidar Points as Location Priors +We conduct a simple ablation study by replacing the +location priors with Lidar point clouds. To do that, we train a +model called “+ feat, lidar priors”, which uses the uniformly +sub-sampled lidar observations as reference points. Fig. 4(c) +shows that our camera-only model “+ feat, loc priors” +achieves similar performance with its camera-lidar fusion +counterpart when detecting the VEHICLE class, but performs +much worse for the HUMAN class. The result indicates that +localization errors are still the bottleneck for the camera- +only detection pipeline, especially for small objects. Similar +findings are also reported in [2]. +Models +AP VEHICLE (%) +AP HUMAN (%) +Single-camera baseline +77.78 +12.86 +Ours +83.48 +16.60 +TABLE III: Comparing the proposed model (Ours) with a +single-camera baseline at the IoU=0.1 threshold. +C. Training with Larger Data +We experiment our model with ×20 more data, and +compare it with a single-camera baseline model, which +runs detection on each monocular camera separately, and +aggregates results from all cameras as the final multi- +camera detection outputs (with non-maximum-suppression). +The baseline model follows a network architecture similar to +FCOS3D [19], which regresses cuboid parameters directly +from 2D images. The baseline and our proposed models +use the same pre-trained image backbone. Tab. III shows +the inference results on the same test subset in Sec. IV- +A. Our model (detr3d + feat, loc, query priors) outperforms +the baseline model by 5.70% and 3.74% AP for VEHICLE +and HUMAN classes, respectively. Besides, larger training +data brings approx. 1.5% performance gain, when comparing +results from the small training data shown in Tab. I. This +marginal AP improvement suggests that the 2D priors from +the image backbone might compensate the benefits from +large dataset, saving the training cost. +D. Training convergence +We show the learning curves in Fig. 5, by overfitting a +small dataset with approx. 300 data frames. Compared to +the vanilla detr3d model, the model with 2D priors reaches + +Epoch loss +Epoch loss +Epochs +Epochs +vanilla detr3d +detr3d with 2D priors +Fig. 5: The learning curves for overfitting a small dataset. +the same epoch loss with much fewer epochs, implying the +benefits of 2D priors for faster training convergence. +V. SUMMARY +Transformer-based methods advance the recent develop- +ment of mulit-camera 3D detection. The vanilla transformer +architecture randomly initializes queries, without considering +the heterogeneity of inputs from different frame. We argue +that this approach is sub-optimal in query generation. In this +regard, we propose to leverage multiple predictions from +an image backbone network as 2D priors to improve the +transformer part of the network, including 2D detections, +semantic maps, and depth maps. The method works by +augmenting image feature maps with 2D priors, sampling +query locations via ray-casting along 2D box centroids, as +well as initialising query features with object-level image +features. Experiments results show that 2D priors can be +used to largely improve the detection accuracy in terms of +average precision, and to accelerate the model convergence. +In the future, we intend to add more 2D priors, such as scene +flow and instance masks, and extend the framework into a +multi-modal fusion setting (e.g. combining cameras, lidars, +and radars) [20]–[22]. +ACKNOWLEDGEMENT +The authors would like to thank the full detection team +at Argo AI for the technical discussions and the ML infra +support. Special thanks to Jan Martin and Ahsan Iqbal for +making this publication possible. +REFERENCES +[1] “Tesla +AI +Day +2021,” +https://www.youtube.com/watch?v= +j0z4FweCy4M, August 2021. +[2] H. Li, C. Sima, J. Dai, W. Wang, L. Lu, H. Wang, E. Xie, Z. Li, +H. Deng, H. Tian et al., “Delving into the devils of bird’s-eye- +view perception: A review, evaluation and recipe,” arXiv preprint +arXiv:2209.05324, 2022. +[3] N. Carion, F. Massa, G. Synnaeve, N. Usunier, A. 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Gl¨aser, +“Deepfusion: A robust and modular 3d object detector for lidars, cam- +eras and radars,” in IEEE/RSJ International Conference on Intelligent +Robots and Systems (IROS), 2022. +[22] X. Chen, T. Zhang, Y. Wang, Y. Wang, and H. Zhao, “Futr3d: A +unified sensor fusion framework for 3d detection,” arXiv preprint +arXiv:2203.10642, 2022. + +epoch_loss +r7 +LJ +3 +2 +1 +0 +20 +40 +60 +80 +RunValue +Step Time +Relative +0.430977 +10/5/22.6:32PM2.745hr +epochepoch_loss +5 +4 +3 +2 +0 +100 +200 +300 +400 +RunValueStepTime +Relative +1.79 +381 +9/23/228:39AM7.364hr +epe \ No newline at end of file diff --git a/zdFRT4oBgHgl3EQfjzd-/content/tmp_files/load_file.txt b/zdFRT4oBgHgl3EQfjzd-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d92b4203ba0eda6e6d8131f83d189015557bc9b6 --- /dev/null +++ b/zdFRT4oBgHgl3EQfjzd-/content/tmp_files/load_file.txt @@ -0,0 +1,459 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf,len=458 +page_content='Priors are Powerful: Improving a Transformer for Multi-camera 3D Detection with 2D Priors Di Feng∗, Francesco Ferroni∗ Abstract— Transfomer-based approaches advance the recent development of multi-camera 3D detection both in academia and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' In a vanilla transformer architecture, queries are randomly initialised and optimised for the whole dataset, without considering the differences among input frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' In this work, we propose to leverage the predictions from an image backbone, which is often highly optimised for 2D tasks, as priors to the transformer part of a 3D detection network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The method works by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' augmenting image feature maps with 2D priors, (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' sampling query locations via ray-casting along 2D box centroids, as well as (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' initialising query features with object-level image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Experimental results shows that 2D priors not only help the model converge faster, but also largely improve the baseline approach by up to 12% in terms of average precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' INTRODUCTION Towards 360-degree 3D perception, self-driving vehicles are usually equipped with multiple monocular cameras, and reliable and accurate multi-camera 3D detection has become an important research challenge and industrial effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The traditional approach takes advantage of the convolution neu- ral nets (convnets) that are highly-optimised for 2D tasks, by performing 2D scene understanding on images, followed by a 2D to 3D projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Recent advancements, in contrast, propose to project 2D images into 3D space, before running 3D tasks on a bird’s eye view (BEV) representation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' This new paradigm not only provides a generic scene represen- tation for multi-modal perception, mapping, and prediction, but also achieves improved accuracy with the help of the transformer architecture [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' A typical pipeline for multi-camera 3D detection with transformers is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' First, multi-level feature maps from multiple camera images are extracted from a backbone network, commonly a convnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Afterwards, a trans- former decoder iteratively processes queries and interacts with image feature maps via cross-attention [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Finally, each updated query is fed into a detection head to categorize objects and regress their cuboid parameters (such as centroid locations, cuboid extents, and orientations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' In a vanilla transformer architecture, such as from the seminal work detr3d [4], query features and their location information are randomly initialised and optimised for the whole dataset, without considering the heterogeneity of inputs from dif- ferent frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' We find that such query design suffers from slow training convergence and strong smearing effect with ∗Work done while both authors were at Argo AI (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' argo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='ai/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Di Feng is now with Apple, and Francesco Ferrorni is with Nvidia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Correspondence: fengdi1015@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='com Query + ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' point Transformer decoder … Feature maps Backbone … Cross attn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Updated query 2D Priors Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 1: The pipeline for multi-camera 3D detector with trans- formers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' First, a backbone network, commonly a convolution neural network (convnet), extracts multi-level feature maps from multi-camera inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Afterwards, a transformer decoder iteratively processes queries and interacts with feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Finally, each query is fed into a detection head to predict object classes and cuboid parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' In this work, we pro- pose to leverage 2D predictions from the convnet backbone as priors to the transformer decoder for 3D detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Those priors are incorporated into feature maps, as well as each query and reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' erroneous depth estimation (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 2(a) and will be discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Several methods extend detr3d [4] with improved query design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' For example, PETR [5] generates 3D position em- bedding to image features as the input of a transformer decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' SpatialDETR [6] encodes camera intrinsics and extrinsics features to keys and queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Graph-detr3d [7] replaces self-attention with a graph neural network for better query interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Finally, BEVFormer [8] discretises the 3D world with bird’s eye view grids, and considers each grid as a query location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Since convnets are often highly optimised for 2D tasks, why not reusing those 2D predictions as priors to the transformer part of 3D detection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' In this work, we verify this idea in the detr3d pipeline, and incorporate 2D object detection, semantic segmentation, and depth estimation from a image backbone to the transformer decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' We propose three simple strategies to use 2D priors: augmenting image feature maps for cross attention, sampling query locations via ray-casting along 2D box centroids, as well as initialising query features with object-level image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Experimental results on an internal dataset shows that our methods largely improve the vanilla detr3d by up to 12% in terms of average precision, and make the model converge faster during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' In parallel to our work, MV2D [9] also proposes to arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='13592v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='CV] 31 Jan 2023 Feature embedding MLP Positional embedding Random init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Random init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Query Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' point (a) Vanilla detr3d Feature embedding Cropped features MLP MLP Positional embedding Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' point Query [x, y, z] (b) Ours Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 2: A comparison of query generation strategies between the vanilla detr3d [4] and our proposed methods with 2D priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The vanilla detr3d randomly initialises the positional embedding and the feature embedding vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Reference points are predicted by a small multi-layer perceptron (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Our proposed approach leverages 2D detection and semantic maps as 2D priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' We sample reference points via ray-casting along 2D box centroids, which generates the positional embedding vector through a MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Besides, we initialize the feature embedding vector with the object-level features, weighted by semantic scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Classifications scores x [m] y [m] (a) Predictions from detr3d x [m] y [m] (b) Query sampling results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 3: (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Raw predictions from the vanilla detr3d model on the Bird’s Eye View (BEV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Each circle represents a query prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Strong smearing effect can be observed along the ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Our proposed query sampling strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Blue dots are generated reference points, and orange dots are ground truth centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' All ground truth centroids can be associated with nearby reference points, though with some errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' leverage 2D detections as priors for the transformer part of a multi-camera 3D detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Unlike our approaches which generate multiple reference points from a 2D box centroid and employs multiple 2D cues (2D boxes, semantic maps, and depth maps), MV2D only studies how to exploit 2D detections, and how to predict one reference point for each 2D box via a dynamic object query generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Their ex- perimental results demonstrate higher recall rates compared to the vanilla transformer model, especially for small and distant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' In the sequel, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' II reviews the transformer decoder part of the detr3d model, with a focus on the query generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' III introduces our proposed three methods to improve the detr3d network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' IV shows the experimental results, followed by a summary and discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' VANILLA DETR3D REVISITED The architecture of detr3d [4] has been summarised in the previous section and depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' In this section, let us take a closer look at the transformer decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' A decoder is built from six standard transformer blocks [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' In each block, queries are interacted with each other via self-attention, and fused with multi-camera multi- level feature maps via cross-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Unlike the common “global” cross-attention mechanism [3], detr3d only asso- ciates a query to image features that correspond to its query location (also called reference point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' To do this, a position p = [x,y,z] in the 3D coordinate system is computed for each query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The position is projected onto image planes given camera intrinsics and extrinsics parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The image features from the projected pixels are weighted and averaged over all feature levels and cameras for updating a query in a “local” cross-attention manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Denote d as the embedding dimension for a query, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 2(a) illustrates how a query q ∈ Rd and its reference point p ∈ R3 are built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' First, a position embedding vector qpos ∈ Rd and a feature embedding vector qfeat ∈ Rd are randomly initialized (following a uniform or a normal dis- tribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Afterwards, qpos is mapped to the reference point p via a multi-layer perceptron (MLP), and added to qfeat to generate the final positional-aware query features q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Through training the network with the standard Hungarian assignment and the set prediction loss [3], both qpos and qfeat learn to encode the object statistics for the whole dataset, which can be considered as pre-defined “anchors” in common object detection pipeline, such as Faster-RCNN [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Though simple and straightforward, the network design in [4] lacks prior knowledge in the query and reference point generation process, resulting in slow training convergence and ambiguity in prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 3(a) illustrates a typical detr3d output on the bird’s eye view (BEV) without post- top preds gtloc priors 40 ground truth 20 D 20 40 40 20 0 20 40processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' All queries are marked by circles, and those with high classification scores are further demonstrated with red bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' We observe strong smearing effect in object detection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' queries are converged along the ray from detections on 2D images, bringing many false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' This is due to erroneous depth estimation and ambiguous target assignment during training1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' THREE WAYS OF ADDING PRIORS We propose three methods to improve the detr3d network, by incorporating 2D priors to the transformer decoder, illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' To do this, we select 2D object detection, semantic segmentation, and depth estimation predicted by our convnet backbone as priors, as they are common, well- optimized 2D tasks for autonomous driving (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' HydraNet from Tesla [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The depth estimation is represented as a single-channel depth map rescaled to [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The semantic segmentation is represented by a semantic map with C channels, where each channel shows pixel-wise classification scores of a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Feature Map Priors We simply concatenate semantic and depth maps with the multi-camera feature maps at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' In this way, semantic and depth priors are added to queries in a cross- attention operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Location Priors We generate reference points p ∈ R3 only along the rays from the centroids of 2D box predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' For each ray, a simple uniform sampling with 5 meters interval is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' In this way, the search space for objects can be narrowed efficiently, which helps reduce false positives, limit the number of queries, and accelerate model convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' When performing cuboid prediction, the detection head regresses the offset to its reference point, denoted by ∆x,∆y,∆z, as the cuboid centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' It also regresses the cuboid’s height, length, width, and yaw angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The reference points may not accurately overlap with the cuboid centroids, because the centers of 2D boxes are differ- ent from those from the projected cuboids, and the 5-meters sampling interval is rough compared to the common discreti- sation thresholds in many well-known detection networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='5-meters interval in Lift-spalt-shoot [11], 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='16-meters in CaDNN [12], and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='2-meters in Pointpillar [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' However, we find such a simple point generation strategy provides rough-and-ready estimates to the actual object locations, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Besides, the location errors can be compensated by the iterative query refinement in transformer blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' We expect more accurate reference point generation, when introducing a centerness head for projected cuboid centers (similar to CenterNet [14]), or sampling points only around predicted depth (similar to CramNet [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 1Imagine multiple queries generate reference points, which are close to each other and are projected to the same object on images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Due to the Hungarian assignment [3], only one query is labelled “positive”, punishing other positive queries with “negative” signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Inspired by Anchor-DETR [16], we further incorporate location priors to queries, by projecting a reference point p ∈ R3 to a position embedding vector qpos ∈ Rd via a small MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Interestingly, this is a reversed procedure compared to the vanilla detr3d, which maps a position embedding vector to its reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Query Priors All queries generated from a ray come from the same 2D object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Therefore, we propose to incorporate the same object- level 2D priors to those queries, and further distinguish among them with positional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' We follow five steps: First, the semantic map, the depth map, and the multi- level multi-camera feature maps are cropped based on the 2D box estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Then, the channel of the cropped semantic map, which corresponds to the predicted object class, is used to weight the cropped depth map and feature maps by a pixel-wise dot production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Afterwards, a channel-wise global average-pooling operation is used to generate a 1D vector for each query prior, inspired from the squeeze-and-excitation operation from SENet [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Furthermore, the query prior vector, appended with an object class index, an objectness score, and the 2D bounding box parameters, is fed into a small MLP to generate query embedding features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Finally, the positional embedding features are added to the query embedding features as the final query features, so that the queries from the same ray are distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' EXPERIMENTAL RESULTS Based on a pre-trained convnet backbone, we re- implement the detr3d transformer decoder, and experiment its detection performance with different 2D priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Following the original detr3d [4], we set the initial learning rate to be 2 ∗ 10−4 with a weight decay of 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The AdamW optimiser with a consine decay is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Unlike [4], we do not use any data augmentation tricks, and find that training with more epochs improve the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' All models, unless mentioned otherwise, are trained with a tiny subset of our internal dataset, with approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 60k training, 10k validation, and 4k testing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The data was recorded in various locations in the US and Europe, with different lightning conditions (daytime, nighttime, rainy, sunny etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=') and scenarios (cities, rural areas, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' We report the Average Precision (AP) scores at the IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='1 threshold on the bird’s eye view (BEV) for the VEHICLE and HUMAN classes, and only consider detections within the 50 meters range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Main Results Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' I compares the AP scores between the vanilla detr3d model with its variants with different 2D priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The model “+ feat prior ” only adds feature map priors, “+ feat, loc priors” additionally uses location priors, and “+ feat, loc, query priors” exploits all three priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 4(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 4(b) show the precision recall curves for VEHICLE and HUMAN classes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' We observe that all 2D priors improve the vanilla detr3d model with higher AP scores up to nearly 12%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The largest performance gain comes from location 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='0 Recall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='0 Precision VEHICLE vanilla detr3d + feat prior + feat,loc priors + feat,loc,query priors (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='5 Recall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='8 Precision HUMAN vanilla detr3d + feat prior + feat,loc priors + feat,loc,query priors (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='0 Recall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='0 Precision Comparison with Lidar points + feat,loc priors (VEHICLE) + feat,lidar priors (VEHICLE) + feat,loc priors (HUMAN) + feat,lidar priors (HUMAN) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 4: Precision recall curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' A comparison among the vanilla detr3d model and its variants with 2D priors on the VEHICLE class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' A comparison on the HUMAN class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Ablation study by replacing reference points with lidar observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' priors, verifying the effectiveness of our design choice for reference point generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Models AP VEHICLE (%) AP HUMAN (%) vanilla detr3d 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='57 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='36 + feat prior 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='40 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='83) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='16 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='80) + feat, loc priors 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='93 (+9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='36) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='52 (+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='16) + feat, loc, query priors 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='01 (+11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='44) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='68 (+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='32) TABLE I: A comparison of Average Precision (AP) scores at the IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='1 threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Models AP VEHICLE (%) AP HUMAN (%) vanilla detr3d 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='84 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='79 + feat, loc, query priors 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='52 (+10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='68) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='06 (+33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='27) TABLE II: A comparison of Average Precision (AP) scores at 4 meters centroid distance threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' In addition, Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' II reports AP scores at the 4 meters threshold, which are commonly used in the Nuscenes met- rics [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' This threshold is less strict than the IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='1 threshold when evaluating location errors, thus resulting in higher AP scores when evaluating on the same model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' In this setting, we observe that the model with 2D priors largely improves the HUMAN detection by more than 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Using Lidar Points as Location Priors We conduct a simple ablation study by replacing the location priors with Lidar point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' To do that, we train a model called “+ feat, lidar priors”, which uses the uniformly sub-sampled lidar observations as reference points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 4(c) shows that our camera-only model “+ feat, loc priors” achieves similar performance with its camera-lidar fusion counterpart when detecting the VEHICLE class, but performs much worse for the HUMAN class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The result indicates that localization errors are still the bottleneck for the camera- only detection pipeline, especially for small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Similar findings are also reported in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Models AP VEHICLE (%) AP HUMAN (%) Single-camera baseline 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='78 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='86 Ours 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='48 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='60 TABLE III: Comparing the proposed model (Ours) with a single-camera baseline at the IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='1 threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Training with Larger Data We experiment our model with ×20 more data, and compare it with a single-camera baseline model, which runs detection on each monocular camera separately, and aggregates results from all cameras as the final multi- camera detection outputs (with non-maximum-suppression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The baseline model follows a network architecture similar to FCOS3D [19], which regresses cuboid parameters directly from 2D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The baseline and our proposed models use the same pre-trained image backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' III shows the inference results on the same test subset in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' IV- A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Our model (detr3d + feat, loc, query priors) outperforms the baseline model by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='70% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='74% AP for VEHICLE and HUMAN classes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Besides, larger training data brings approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='5% performance gain, when comparing results from the small training data shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' This marginal AP improvement suggests that the 2D priors from the image backbone might compensate the benefits from large dataset, saving the training cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Training convergence We show the learning curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 5, by overfitting a small dataset with approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 300 data frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Compared to the vanilla detr3d model, the model with 2D priors reaches Epoch loss Epoch loss Epochs Epochs vanilla detr3d detr3d with 2D priors Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' 5: The learning curves for overfitting a small dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' the same epoch loss with much fewer epochs, implying the benefits of 2D priors for faster training convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' SUMMARY Transformer-based methods advance the recent develop- ment of mulit-camera 3D detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The vanilla transformer architecture randomly initializes queries, without considering the heterogeneity of inputs from different frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' We argue that this approach is sub-optimal in query generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' In this regard, we propose to leverage multiple predictions from an image backbone network as 2D priors to improve the transformer part of the network, including 2D detections, semantic maps, and depth maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' The method works by augmenting image feature maps with 2D priors, sampling query locations via ray-casting along 2D box centroids, as well as initialising query features with object-level image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Experiments results show that 2D priors can be used to largely improve the detection accuracy in terms of average precision, and to accelerate the model convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' In the future, we intend to add more 2D priors, such as scene flow and instance masks, and extend the framework into a multi-modal fusion setting (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' combining cameras, lidars, and radars) [20]–[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' ACKNOWLEDGEMENT The authors would like to thank the full detection team at Argo AI for the technical discussions and the ML infra support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' Special thanks to Jan Martin and Ahsan Iqbal for making this publication possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content=' REFERENCES [1] “Tesla AI Day 2021,” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFRT4oBgHgl3EQfjzd-/content/2301.13592v1.pdf'} +page_content='youtube.' metadata={'source': 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PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Abstract. We introduce the notion of a zebra structure on a surface, which is a more general geometric +structure than a translation structure or a dilation structure that still gives a directional foliation of every +slope. We are concerned with the question of when a free homotopy class of loops (or a homotopy class +of arcs relative to endpoints) has a canonical representative or family of representatives, either as closed +leaves or chains of leaves joining singularities. We prove that such representations exist if the surface has a +triangulation with edges joining singularities (in the zebra structure sense). Our results hold for both closed +surfaces and non-compact surfaces. +1. Introduction +A quadratic differential q on a Riemann surface X, possibly with simple poles, naturally endows that +surface with a half-translation structure via coordinate charts obtained by integration. Associated to q is its +divisor, which may be interpreted as a function +(1) +α ∶ X → Z≥−1 = {−1,0,1,...} +whose support is discrete and closed. We have α(x) = −1 if x is a simple pole, and α(x) = k ≥ 1 if x is a zero +of order k. The half-translation structure gives charts from a neighborhood of each x ∈ X to the α(x)+2-fold +branched cover of C/⟨z ↦ −z⟩ branched over the origin, and transition maps are given by translations or +180○ rotations in local coordinates. Geometrically x ∈ X has the local structure of a Euclidean cone point +with cone angle π(α(x)+2). Thus the support Σ of α is the collection of singularities of the half-translation +structure. Throughout this paper, we allow our surfaces to be non-compact, though the closed surface case +is of special interest and we prove new results in this setting as well. +Given a half-translation structure on an oriented topological surface S, for each slope m ∈ ˆR = R ∪ {∞}, +the foliation of the plane by lines of slope m pulls back under charts to give a singular foliation Fm of the +surface by leaves of slope m. Formally Fm is a foliation of S ∖ Σ whose local behavior near a point p is +governed by the value α(p). In particular, each Fm has α(p) + 2 prongs at each point p ∈ S. +More generally, a singular foliation F of a topological surface S is a foliation of S ∖ Σ, where Σ ⊂ S is +a closed discrete subset and such that there is a singular data function α ∶ S → Z≥−1 whose support is Σ +such that F is locally homeomorphic at p ∈ Σ to the horizontal foliation of a half-translation surface in a +neighborhood of a cone point with cone angle π(α(p) + 2). Note that F determines both Σ and α: The +subsurface S ∖ Σ is the union of leaves, and α can be determine by the number of prongs at a point. +Singular foliations need not come from quadratic differentials. Indeed, those that come from quadratic +differentials carry the additional structure of a measured foliation, which appears because half-translation +surfaces have a natural path metric obtained by pulling back the Euclidean metric on the plane. In this +paper, we investigate what happens when these additional structures are not required, but where we still +have a family of singular foliations that fit together nicely. +Definition 1.1. Let S be an oriented topological surface. Consider a family {Fm ∶ +m ∈ ˆR} of singular +foliations on S indexed by slope that determine the same singular set Σ and the same singular data function +α. We say such a family is stellar if each p ∈ S has a neighborhood U such that U ∖ {p} is foliated by +segments of leaves containing p in their closure in a manner homeomorphic to the standard half-translation +model associated to α(p) as depicted in Figure 1. If {Fm} is stellar, we say it induces a stellar foliation +structure or a zebra1 structure on (S,α). We call a pair (S,{Fm}m∈ˆR) where {Fm} is stellar a zebra surface. +Date: January 11, 2023. +1The name zebra surface was inspired by the Zebra Slot Canyon in Grand Staircase-Escalante National Monument. +1 +arXiv:2301.03727v1 [math.GT] 10 Jan 2023 + +2 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Figure 1. The local structure of leaves through p, with α(p) ∈ {−1,0,1,2} from left to right. +Here the bold leaves are of slope zero and slopes cyclically increase in the counterclockwise +direction. +See Section 2 for a more detailed and formal definition. Observe that half-translation structures induce +zebra structures. +Another geometric structure on a surface that induces a zebra structure is a dilation +structure; see Section 1.2. This paper originated from our interest in dilation structures on surfaces. These +structures have recently generated a lot of interest, see for example [ABW22,BGT21,DFG19,Wan21]. We +discuss some relevant results in Sections 1.2.3, 1.2.4, and 10. In Section 5.6, we show that there are zebra +structures that do not arise from dilation or translation structures. A broader discussion about foliations on +surfaces can be found in [Nik01]. +It is a fundamental observation in Teichm¨uller theory that the bundle Qg of quadratic differentials (without +poles) over the moduli space of closed Riemann surfaces of genus g is naturally identified with the cotangent +bundle of that space. The geodesic flow in the Teichm¨uller metric is conjugate via the identification of +quadratic differentials with half-translation structures to the diagonal action of PSL(2,R) on the space +of half-translation surfaces, acting affine-linearly by simultaneous post-composition with coordinate charts. +This action of PSL(2,R) on Qg is an active area of research [Wri15]. This action extends to an action +of PSL(2,R) on zebra surfaces: In fact, the PSL(2,R)-action extends to a Homeo+(ˆR)-action given by +reindexing the foliations; see Section 2.8. +Our initial motivation when writing this paper was to extend fundamental facts about length-minimizing +representatives of curves on translation surfaces to the more general context of dilation surfaces, where there +is no natural notion of length. One motivation for studying this question is Thurston’s theory of simple +closed curves and their relation with the classification of surface homeomorphisms. +When looking into +this, we realized that surprisingly little is known about distinguished representatives of curves for related +structures (such as dilation structures with cone-type singularities and translation structures on non-compact +surfaces) and an answer can be given in the very general context of zebra surfaces. Working in this more +general context makes some things challenging, but the limited tools available lead to a natural and general +approach to the problems under consideration. +1.1. Main results. We briefly introduce some important definitions so that we can state our results. +Let (S,{Fm}) be a zebra surface, with S any oriented topological surface. +As indicated above, this +information determines a singular set Σ and a singular data function α ∶ S → Z≥−1 whose support is Σ. +A leaf is a leaf of any of the foliations Fm. Leaves are contained in S∖Σ and so do not contain singularities. +A leaf is closed if it is homeomorphic to a circle. If a leaf is not closed, then it is homeomorphic to an open +interval, and such a leaf can have singular endpoints in its closure. A saddle connection is a leaf together with +two singular endpoints. A leaf triangulation of S is a triangulation of S whose edges are saddle connections. +We require that triangles to meet edge-to-edge and that the union of the triangles be all of S. In such a +triangulation, only finitely many triangles can meet at each vertex. +A trail is a maximal bi-infinite parameterized path that follows a sequence of leaves, transitioning between +leaves only at singularities in such a way that the two angles made at the singular transitions are at least +π. (Angles made between leaves meeting at a point can be measured using the stellar neighborhood of the +point.) We call the angles made at singular transitions bending angles. Each bending angle appears either +on the right or left side of the trail. We require trails to “bounce off” poles, returning along the leaf through + +ZEBRA SURFACES +3 +which it arrived. (This “bouncing off” is not allowed at other singularities.) A trail is closed if it can be +reparameterized to be periodic. +A zebra plane is a zebra structure (Z,{ ˜Fm}) where Z is an open disk and the singular data function ˜α +is non-negative. (Examples of simply connected zebra surfaces which are not zebra planes can be found in +[Pan09].) If (S,{Fm}) is any connected zebra surface, and α is non-negative, then the structure lifts to the +universal cover to give a zebra plane. If S has poles, then there is a larger cover which is a zebra plane, where +we require double branching over poles. We call this the pole-resolved universal cover (the PRU cover), see +Section 3.1. Note that because of the double branching, preimages of poles are non-singular points in the +zebra plane. The PRU cover coincides with the universal cover if S has no poles. +A basic question in the geometry of metric spaces is whether any two points can be joined by a geodesic, +and if so, whether this geodesic is unique. For example, the Cartan–Hadamard theorem guarantees that +any two points can be connected by a unique geodesic segment in a complete non-positively curved simply- +connected metric space [BH13]. In our setting, we observe that distinct points on a zebra plane can be joined +by at most one arc of a trail (Proposition 3.6). We say a subset of a zebra plane is convex if any two distinct +points can be joined by an arc of a trail contained in that subset. We prove: +Theorem 1.2. If a zebra plane Z has a leaf triangulation, then Z is convex. +The primary case of interest is when the zebra plane is the PRU cover of a closed surface. As a consequence +if we can “triangulate” the closed surface in a manner that lifts to a leaf triangulation on the cover, then +that cover is convex. Figure 2 shows a surface with no leaf triangulation whose PRU cover is not convex. +Figure 2. A dilation surface: edges are glued by dilations and translations. The orange, +pink, and blue points are dilation singularities, which are not singularities in the induced +zebra structure. The foliated region is a full zebra cylinder (see Section 1.2.4), and so no +closed trail can cross this cylinder by Proposition 1.9. +Of course, it is only possible for a zebra plane to have a leaf triangulation when α takes positive values, +(that is, when there are singularities). We also provide a criterion for convexity of the PRU cover of a closed +surface when α is non-positive; see Corollary 7.5. Some examples, like R2 which covers the square torus are +convex, but others like the universal cover of a Hopf torus is not; see Section 1.2.3. +We will need to extend the notion of homotopy of paths, to the case of zebra surfaces with poles. See +Section 9.1 for details on this construction. We call this extended notion of homotopy pole-resolved homotopy +or PR homotopy, and it coincides with the usual notion if there are no poles. The PR free homotopy classes +of loops are in natural bijective correspondence with conjugacy classes in the deck group of the PRU covering. +We say a PR free homotopy class of closed curves is polar if there is a simple closed curve in the class that +bounds a disk whose only interior singularity is a pole. We say that a PR free homotopy class ⟦γ⟧ of closed +curves is a power if there is another PR free homotopy class ⟦β⟧ and a k ≥ 2 such that ⟦γ⟧ is homotopic to +the k-fold cover of ⟦β⟧. +The closed standard cylinder is C = [−1,1] × S1 where S1 = R/Z. This cylinder comes equipped with its +vertical foliation by fibers of the projection C → [−1,1], and has two boundary components ∂±C = {±1}×S1. +Its interior is the subset C○ = (−1,1) × S1. We split C○ into two halves: +H○ +− = (−1,0] × S1 +and +H○ ++ = [0,1) × S1. + +4 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +We have: +Theorem 1.3 (Closed trails). Let (S,{Fm}) be a zebra surface, where S is any connected oriented topological +surface. Fix a PR free homotopy class of closed curves ⟦γ⟧ which is nontrivial, non-polar, and not a power. +Then one of the following mutually exclusive statements holds: +(NR) (Non-realization case) There is no closed trail in ⟦γ⟧. +(TF) (Toral foliation case) The surface S is the torus, the zebra structure has no singularities, and the +collection of all closed trails in ⟦γ⟧ is a collection of simple closed leaves that foliate S. +(Cyl) (Cylinder case) There is an embedding ϵ ∶ C○ → S such that the closed leaves of S in ⟦γ⟧ are precisely +the images under the embedding of the vertical closed leaves of C○. +(UT) (Unique trail case) There is a unique closed trail in ⟦γ⟧ and this closed trail has at least one bending +angle greater than π on each side. +Furthermore, +(1) If the PRU cover of S is convex, then case (NR) cannot occur. +(2) In case (Cyl), define σ to be the collection of signs s ∈ {±} such that ϵ(H○ +s) has compact closure. Set +¨C = C○ ∪ ⋃ +s∈σ +∂sC. +Then there is a map ¨ϵ ∶ ¨C → S whose restriction ¨ϵ∣C○ satisfies (Cyl) such that for each s ∈ σ, the +curve ¨ϵ∣∂sC is a closed trail in ⟦γ⟧ passing through a non-empty collection of singularities, and every +bending angle made when passing through such a singularity on the side of ¨ϵ(C○) has measure π. +Furthermore, all closed trails in ⟦γ⟧ are obtained as restrictions of ¨ϵ to vertical circles in ¨C. +In order to prove this theorem, we prove a criterion for existence of a closed trail that does not require +convexity of the PRU cover; see Theorem 9.7. +In the context of closed zebra surfaces with singularities and convex PRU covers, Theorem 1.3 specializes +to the following: +Corollary 1.4. Suppose that S is a closed surface and {Fm} is a zebra structure on S with a non-empty +singular set Σ. Suppose also that the PRU cover of S is convex. Then, if ⟦γ⟧ is a PR free homotopy class of +closed curves that is nontrivial, non-polar, and not a power, then either there is a unique closed trail in ⟦γ⟧ +as in case (UT) or there is a continuous map from the closed standard cylinder ¨ϵ ∶ C → S whose restriction +to C○ is an embedding as in case (Cyl) and whose restriction to each boundary component is a closed trail +as described in (2), and such that all closed trails in the homotopy class are given by restriction as in (2). +However, there certainly are some closed surfaces for which there is a ⟦γ⟧ that falls into case (NR). The +PRU cover of such a surface is not convex. An example of this situation is depicted in Figure 2. The surface +is constructed by starting with a polygonal annulus in R2 and making boundary identifications (which can +be chosen to give the surface a dilation structure). The homotopy class of a homotopically non-trivial loop +traveling around the interior of this annulus gives an example of a homotopy class satisfying (NR). See the +caption of the figure for more details. +1.2. Contexts. Here we consider our main theorems in specific contexts moving roughly from more specific +structures to more general structures. The chart below depicts the various geometric structures on surfaces +that we consider, together with arrows from one structure to another to indicate that a surface with the first +structure is also a surface with the second structure. (E.g., a translation surface atlas is also a half-translation +surface atlas.) +Translation +Dilation with +cone singularities +Dilation with +dilation singularities +Euclidean +cone +Half-translation +Half-dilation with +cone singularities +Half-dilation with +dilation singulartites +Zebra + +ZEBRA SURFACES +5 +1.2.1. Closed translation surfaces and cone surfaces. A translation surface is an oriented surface with an +atlas of charts to the plane whose transition functions are translations, where we allow cone points with cone +angles that are integer multiples of 2π (so, in our notation, α takes even values). We briefly explain why our +main results are true in the case of a closed translation surface and in a related case. +A (Euclidean) cone surface is an oriented surface with an atlas of charts to the plane where transition +functions are in the orientation-preserving isometry group, and where we allow cone singularities with any +positive real cone angle. Thus a translation surface is a special case of a cone surface. +If S is a closed translation surface, its universal cover is a Hadamard space, i.e., a complete metric space +that is non-positively curved in the CAT(0) sense. More generally, we could consider a closed cone surface +all of whose cone singularities have cone angles greater than 2π. The universal cover is again a Hadamard +space. For details see [BL18, §2.1]. This also works for closed half-translation surfaces, but if the surface has +poles, then we have to replace the universal cover with the PRU cover described in this article. +From local considerations, a curve on such a surface is a geodesic if and only if it satisfies our definition of +a trail. (Indeed, this is the motivation for our definition.) Hadamard spaces are well known to be geodesically +convex, so this gives Theorem 1.2 in this context. +The strategy for deducing Theorem 1.3 in this context is to use facts about isometries of Hadamard spaces. +Isometries of metric spaces can categorized based on their attained translation lengths. Here the translation +length of a point p ∈ X under an isometry ϕ ∶ X → X is TL(x) = d(x,ϕ(x)). The isometry ϕ is elliptic if it +has a fixed point, hyperbolic if TL attains a strictly positive minimum, and parabolic if the infimum of values +of TL is not attained. An isometry of a locally CAT(0) space translates along some geodesic if and only if +the isometry is hyperbolic [BH13, II, Thm 6.8]. +Given a PR homotopy class ⟦γ⟧ on a closed surface S as above, we get a deck transformation ∆γ ∶ ˜S → ˜S +where ˜S is the Hadamard cover described above, which by hypothesis is not elliptic (because ⟦γ⟧ is nontrivial +and non-polar). A cocompact group of isometries acting on a Hadamard space cannot contain parabolic +isometries [BH13, II, Prop 6.10]. Therefore, ∆γ is hyperbolic and translates along a geodesic. It is not hard +to move from this point to the description in Theorem 1.3 using elementary facts about Euclidean cone +surfaces, though in the cone surface case cylinders are immersed rather than embedded. Also the closed +trails in a cylinder in this case are parallel (globally in the translation surface case and locally in the cone +surface case). +1.2.2. Non-compact translation surfaces and cone surfaces. We will briefly explain how the argument from +Section 1.2.1 proving special cases of our main results breaks when we consider non-compact translation +surfaces and Euclidean cone surfaces whose cone angles are larger than 2π. +There are two difficulties. First, there are non-compact translation surfaces, all of whose singularities +are finite cone singularities, that admit leaf triangulations (i.e., triangulations by saddle connections joining +singularities) but whose universal covers are not complete; see Figure 3 for an example. Second, since the +surface is not compact, it is unclear how to rule out parabolic isometries in the deck group. +Figure 3. Part of an infinite triangulation of a connected open subset of the plane is +depicted. Let S be the smallest double cover of this disk with double branching over the +vertices and no other branching. The surface S is naturally a translation surface with a leaf +triangulation, but is not complete as a metric space. Theorem 1.2 and Theorem 1.3 apply +to S, with the latter giving a closed trail in every nontrivial free homotopy class. + +6 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +It is interesting to note that the open unit disk in R2 has a complete metric for which geodesics are +straight lines: The Klein disk model of the hyperbolic plane. We wonder: +Question 1.5. Which zebra surfaces have a complete CAT(0) metric whose geodesics are the trails? +Translation structures are special cases of zebra structures, so our results hold in this setting. We have +the following consequence: +Corollary 1.6. If S is a non-compact translation surface whose universal cover ˜S is geodesically convex (or +has a leaf triangulation), then every nontrivial deck transformation is a hyperbolic isometry of ˜S. +We will explain that Figure 4 illustrates a failure of our conclusions to hold in the context of non-compact +cone surfaces with all singularities having cone angles larger than 2π. Note that this surface is not a zebra +surface, because it has cone singularities with cone angle strictly between 2π and 3π. The figure illustrates +a Euclidean cone structure on the annulus, but with no geodesic core curve, because the annulus continues +to get thinner as we move towards one boundary. +This surface is depicted with a decomposition into +quadrilaterals, which when cut along their diagonals gives a leaf triangulation, in the sense that edges are +saddle connections and vertices are singularities with cone angle larger than 2π. It follows that one of the +following two implications must be incorrect in this context: A leaf triangulation implies convexity of the +universal cover, or convexity of the cover implies the existence of a geodesic representative in every nontrivial +free homotopy class of loops. However, we conjecture: +Conjecture 1.7. Fix ϵ > 0. Let S be non-compact Euclidean cone surface such that all cone singularities have +angle at least 2π + ϵ. Suppose that S admits a triangulation by saddle connections. Then the universal cover +˜S is convex and every nontrivial free homotopy class of closed curves in S contains a geodesic representative. +Figure 4. A Euclidean cone structure on the annulus built using infinitely many trapezoids. +1.2.3. Dilation surfaces with cone singularities. A half-dilation surface with cone singularities is a surface +with an atlas of charts to the plane and finite covers of C/⟨z ↦ −z⟩ branched over the origin such that +transition maps are in the group generated by translations, dilations, and rotations by π. A dilation surface +with cone singularities is the same, only covers should be of the plane branched over the origin and the +transition maps are in the group generated by translations and dilations. +The (dilational) holonomy around an oriented loop in such a surface is the ratio of lengths of a segment +parallel translated around a loop and the original segment, measured in a fixed local coordinate chart and +interpreted as an element of R+ [Wan21]. Because our singularities are cone singularities, the holonomy +around any contractible loop is trivial and the notion of holonomy around a loop gives rise to the holonomy +homomorphism π1(S) → (R+,×). + +ZEBRA SURFACES +7 +Earlier we mentioned the Hopf tori, given by C∗/⟨z ↦ λz⟩ where C∗ = C ∖ {0} and λ is a positive real +number, because these surfaces have non-convex universal covers. The fibers of the map arg ∶ C∗ → R/2πZ +give a foliation of a Hopf torus. To see a Hopf torus has non-convex universal cover, observe that the torus +has two closed leaves of every slope, and if p and q are points from distinct closed leaves of the same slope then +there is no trail connecting p with q. See [DFG19] for background on Hopf tori and related constructions. +Consider an immersion of the cylinder C = [−1,1] × S1 into a Hopf torus T that sends vertical closed +leaves of C to the foliation of T given by fibers of arg. The pullback of the dilation structure to C is called a +dilation (or affine) cylinder. The universal cover of C with this structure can be seen to be isomorphic to a +sector in a branched cover of the plane, and we call the angle of this sector the angle of the dilation cylinder. +It is not hard to see that a dilation surface (or half-dilation surface) containing an affine cylinder with angle +π or more cannot be convex, and no closed curve crossing such a cylinder can have a closed trail representing +it. See Proposition 1.9.We note that dilation surfaces can also contain flat cylinders, isomorphic to rotations +of [0,w] × R/cZ for c and w positive. +We were unable to find a simple argument for proving Theorem 1.3 in the context of dilation surfaces +with cone singularities that bypasses the technique we use for zebra surfaces. +1.2.4. Dilation surfaces with dilation singularities. A dilation singularity is more general than a cone singu- +larity: We allow a loop around a dilation singularity to have nontrivial dilational holonomy. +The dilation singularities must locally look like certain natural models. Let U denote the closed upper +half-plane, {z ∈ C ∶ Iz ≥ 0}. One example of such a model is given by Mλ = U/ ∼ with λ > 0, where ∼ is +the finest equivalence relation on U where for every positive x ∈ R, −x ∼ λx. Here the singularity of Mλ +is at the origin, and the dilational holonomy of a counterclockwise loop around the origin is λ. In general, +the model singularities are given by branched covers M n +λ of Mλ of degree n ≥ 1 branched over the origin 0. +The dilational holonomy around the singularity in M n +λ is λn and the angle at the singularity is nπ. These +models have natural local coordinate maps from M n +λ ∖ {0} to C whose transition functions are in the group +generated by translations, dilations, and rotations by π. If n is even, we can specify local coordinate maps +to C where the transition functions are in the group generated by translations and dilations. +A half-dilation surface with dilation singularities is a surface together with an atlas of charts to the plane +and the spaces M n +λ , where the transition maps are in the group generated by translations, dilations, and +rotations by π in local coordinates. +A dilation surface with dilation singularities is the same, only the +transition maps are in the group generated by translations and dilations, and the model singularities are of +the form M 2n +λ . +Note that the universal cover of a dilation surface with dilation type singularities has no natural metric, +because now there is dilational holonomy around loops in the cover. This makes metric methods to deduce +convexity and existence of closed trails seem unlikely to work. It is conceivable there is a different metric +worth considering. See Question 1.5. +Interestingly, some papers in the field of dilation surfaces only allow cone singularities, while others allow +dilation singularities. Veech was probably the first to consider dilation surfaces, though he worked in the +more general context of (singular) complex affine structures on surfaces [Vee93] [Vee97]. +Veech proved +fundamental results on the moduli spaces of these structures. This understanding was recently improved +in [ABW22] which specifically considers moduli spaces of dilation surfaces allowing dilation singularities, +and proving (among other results) that the moduli space of dilation surfaces with singular data fixed is an +orbifold covering of the usual moduli space of the corresponding punctured surface. +The directional foliations on the spaces Mλ constructed above are isomorphic to the directional foliations of +C∗/⟨z ↦ −z⟩, and the foliations on n-fold branched covers M n +λ are isomorphic to those on the n-fold branched +covers of C∗/⟨z ↦ −z⟩. Thus, dilation surfaces with dilation singularities still induce zebra structures on the +surface. +Veech proved that a dilation surface has a triangulation by saddle connections if and only if it contains no +dilation cylinders with angle π or more [DFG19, Appendix]. Note however that dilation singularities with +angle 2π are singularities in the dilation surface sense but are non-singular on the induced zebra structure. +Therefore a triangulation by saddle connections may not be a leaf triangulation. Nonetheless, by combining +Veech’s result with Theorem 1.2 and Corollary 1.4 we obtain: +Theorem 1.8. Suppose S is a closed surface with a dilation structure and at least one singularity, but +without dilation singularities with angle 2π. Then, the following are equivalent: + +8 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +● S has a triangulation by saddle connections. +● The universal cover ˜S is convex. +● Every nontrivial free homotopy class ⟦γ⟧ of closed curves is realized by either a unique closed trail +or a cylinder. +● S contains no dilation cylinder with angle π or more. +We will observe however that Theorem 1.8 does not hold if one allows dilation singularities with angle 2π. +For this, we need the notion of a zebra cylinder in a zebra surface S, which we take to be the union of all +closed trails in a PR free homotopy class of closed curves ⟦γ⟧ that contains at least two such closed trails. +By Theorem 1.3, these closed trails must have the structure of a cylinder, and each closed trail has constant +slope (though these slopes vary with the closed trail). A zebra cylinder is full if every slope in ˆR is the slope +of one of the closed trails in the cylinder. +When dilation singularities with angle 2π are allowed in a dilation surface, there can be multiple dilation +and flat cylinders with homotopic core curves that join together to form one cylinder in the zebra sense. +(Dilation singularities with angle 2π are not considered to be singularities on the zebra surface.) An example +of a full zebra cylinder made from two dilation cylinders and one flat cylinder is shown in Figure 2. +The reason we cannot extend Theorem 1.8 to allow dilation singularities with angle 2π is the following: +Proposition 1.9. Suppose S is a zebra surface and that S has a PR free homotopy class of closed curves +⟦γ⟧ whose closed trails constitute a full zebra cylinder. If ⟦β⟧ is any PR free homotopy class of closed curves +whose geometric intersection number with ⟦γ⟧ is non-zero, then ⟦β⟧ contains no closed trails. Furthermore +if such a ⟦β⟧ exists, then the PRU cover of S is not convex. +Proof. Suppose ⟦γ⟧ and ⟦β⟧ are as stated. Let ¨ε ∶ ¨C → S be the full zebra cylinder containing the closed +trails in ⟦γ⟧. Suppose to the contrary that β is a closed trail in ⟦β⟧. Because of the intersection number +condition, there must be an arc β′ ⊂ ¨ε( ¨C) of β that crosses every closed trail in the cylinder. Since ¨ε(C○) +contains no singularities, the slope of β′ must be constant. Let m denote this slope. Let ℓ ⊂ ¨C be a vertical +closed leaf whose image ¨ε(ℓ) is a closed trail whose constant slope is also m. We will derive a contradiction +from the fact that β′ must both contain points in ¨ε(ℓ) and not in ¨ε(ℓ). This is clearly impossible when ℓ is +contained in the interior of C○, because in this case both ¨ε(ℓ) and β′ are leaves of the same foliation of slope +m restricted to the cylinder. If ℓ is one of the two boundary curves of C, then ¨ε(ℓ) is a trail of constant +slope m and the bending angles along ¨ε(ℓ) on the side of ¨ε(C○) are all π. Because of these bending angles +and the local structure at the singular points, no leaf of slope m emanating from a singularity on ¨ε(ℓ) enters +¨ε(C○), again contradicting the existence of β′. This completes the proof that ⟦β⟧ contains no closed trail. +The last statement follows directly from statement (1) of Theorem 1.3. +□ +1.2.5. Zebra surfaces. Based on Theorem 1.8, we state: +Conjecture 1.10. Suppose S is a closed surface with a zebra structure and at least one singularity. Then, +the following are equivalent: +(a) S has a leaf triangulation. +(b) The PRU cover ˜S is convex. +(c) Every PR free homotopy class ⟦γ⟧ of closed curves that is nontrivial and non-polar either contains a +unique closed trail or contains closed leaves (as described in Corollary 1.4). In particular, case (NR) +of Theorem 1.3 does not occur. +(d) S contains no full cylinders. +The implication (a) implies (b) is Theorem 1.2. The implication (b) implies (c) is Corollary 1.4. The +implication (c) implies (d) follows directly from Proposition 1.9. It remains to prove that (d) implies (a), +i.e., if S has no full cylinders, then it has a leaf triangulation. +1.3. Outline of paper. We will now explain what is done in this paper. Because the paper proves state- +ments about zebra surfaces but some reader’s interests will only include translation surfaces or dilation +surfaces, we try to point out what can be skipped for such a reader. +In Section 2, we carefully define what a zebra surface is and establish basic terminology. All results here +are well known for translation and dilation surfaces. We prove that the Gauss-Bonnet theorem holds for +zebra surfaces and subsurfaces with polygonal boundaries. + +ZEBRA SURFACES +9 +In Section 3, we formally define the pole-resolved universal cover of a zebra surface. This is also natural +for half-translation surfaces with poles, but we have not seen it in the literature. We consider basic geometric +objects on zebra planes (such as PRU covers) such as polygons and trails. We prove basic results about these +objects, which are all obvious when working with half-translation and half-dilation surfaces. For example, we +construct rectangles, prove arcs of trails have maximal extensions as trails, and show trails on zebra planes +are proper maps. +In Section 4, we define the notion of a zebra structure on a surface with boundary. Our definition allows +for a polygonal boundary, generalizing the natural idea of a dilation surface with piecewise-linear boundary. +This is important for laying a rigorous foundation for the next section. +In Section 5, we consider surgical constructions on zebra surfaces, building new zebra surfaces from +subsurfaces of others with polygonal boundary. Because of the flexibility of the zebra structure, we allow +gluing subsurfaces together by homeomorphism of edges in the boundary. This is useful for simplifying +several arguments appearing later in the paper. In Section 5.6, we use surgery to show that there are zebra +surfaces that do not arise from half-dilation structures. +Section 6 focuses on producing foliations of polygons in zebra planes by combining leaves from the foliations +Fm with m varying. These foliations form the foundation of our later arguments, and some such foliations +seem interesting even in the Euclidean plane (though proofs would be easier in this context where analytic +methods are available). In Section 6.2, we show that a triangle in a zebra surface can be foliated by leaves +emanating from a vertex, and that slopes of these leaves vary monotonically. In Section 6.3, we prove that +a polygon can be foliated by leaves passing through one edge, where the slopes of leaves passing through a +given point on that edge are given by a monotone function (subject to obvious constraints). This second +result has a slick proof using surgery on zebra surfaces. +In Section 7, we investigate the behavior of trail rays emanating from a point in a zebra plane Z, and also +arcs of trails joining two points. This work is fundamental for our convexity arguments later in the paper. +Considering all the trail rays emanating from a point in Z leads to a foliation of an open subset of Z with a +different singular structure. We use this structure to prove that polygonal regions in Z all of whose exterior +angles are at least π are convex. This is clear from CAT(0) arguments when the polygon is in a translation +surface, but seems unclear for polygons in dilation surfaces with dilation-type singularities in the interior of +the polygon. This result allows us to prove a criterion for convexity of PRU covers of closed zebra surfaces +where α is non-positive; see Corollary 7.5. This statement applies to dilation tori, all of whose singularities +are dilation-type with angle 2π. In Section 7.3, we prove a continuity statement for the map sending a pair +of points to the arc of a trail between the two points. +In Section 8, we prove Theorem 1.2, which says that a zebra plane Z with a leaf triangulation is convex. +We choose an arbitrary point p ∈ Z and consider all trail rays emanating from p. We inductively show that +these rays cover every triangle in our triangulation. Given the results from the prior section, the proof is +largely combinatorial. This argument is likely of interest to anyone interested in non-compact translation +surfaces or in dilation structures. +In Section 9, we consider the question of finding closed trails in a zebra surface. We begin with a discussion +of PR free homotopy classes of curves and a pole-resolved version of the fundamental group in Section 9.1. +In Section 9.3, we prove a theorem that guarantees the existence of a closed trail. This result, Theorem 9.7, +is of interest to those thinking about dilation surfaces. Later subsections are concerned with developing the +remainder of the structure described in Theorem 1.3. Arguments should be readable to experts interested +in the contexts of translation or dilation surfaces. We prove Corollary 1.6 in Section 9.8. +Section 10 provides a list of open questions. We know very little about zebra surfaces. +2. Formal definitions +2.1. Surfaces and structures. For us a surface is a second countable Hausdorff space that is locally +homeomorphic to R2. +Throughout this paper, all surfaces are oriented. +A closed surface is a compact +connected surface without boundary. +Let X be a topological space and S be a surface. An atlas of charts from S to X is a collection of charts +of the form φ ∶ U → X whose domains are open and cover S and such that each chart φ ∶ U → X has an +open image φ(U), and is a homeomorphism from U to its image. A transition map between two charts with +intersecting domains φ1 ∶ U1 → X and φ2 ∶ U2 → X is the restriction of φ2 ○ φ−1 +1 +to φ1(U1 ∩ U2). + +10 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +In general, geometric structures are specified by defining a pseudogroup of homeomorphisms between the +open sets of X, and insisting that transition maps lie in the pseudogroup. We call X the model space. We +refer the uninitiated reader to Chapter 3 of [Thu14]. Foliations can be considered to be a particular case of +a geometric structure, see [Thu14, Example 3.1.9]. +2.2. Foliated surfaces. The horizontal foliation H of R2 is the collection of all horizontal lines in the plane. +If U ⊂ R2 is open, we say that two points (x1,y1) and (x2,y2) are horizontally equivalent in U if y1 = y2 +and the horizontal line segment between the points is contained in U. The horizontal foliation of U is the +collection H∣U of horizontal equivalence classes. We call these equivalence classes leaves. +The horizontal foliation pseudogroup of R2 is the collection of homeomorphisms h ∶ U → V between open +subsets of R2 that induces a bijection from the leaves of H∣U to the leaves of H∣V . +A foliation atlas on a surface S is an atlas of charts to R2 whose transition functions lie in the horizontal +foliation pseudogroup. A foliation atlas determines a foliation equivalence relation on S, namely the finest +one such that given any chart φ ∶ U → R2, preimages of points in the same leaf of H∣φ(U) are equivalent. A +foliation of S is the collection of foliation equivalence classes obtained from a foliation atlas. We call the +equivalence classes leaves. +If S is a surface with a foliation F, and A ⊂ S is a subsurface (possibly with boundary), then the restricted +foliation on A is the collection F∣A of connected components of intersections A ∩ ℓ, where ℓ varies over the +leaves of F. If A is an open set, then it is a surface and the restricted foliation is a foliation on A, because a +foliation atlas for A can be obtained by restricting each chart φ ∶ U → R2 in the atlas for F to the function +φ∣A∩U ∶ A ∩ U → R2. +Definition 2.1 (Leaf topology). Let S be a topological surface, perhaps with boundary, and let ℓ ⊂ S be +a subset. The leaf topology on ℓ is the coarsest topology such that for each open U ⊂ S, each connected +component of U ∩ ℓ is open. If ℓ ∈ F is a leaf of a foliated surface without boundary, then each point of ℓ has +a neighborhood homeomorphic to an open interval. This gives ℓ the structure of a connected 1-manifold. +A local homeomorphism f ∶ S0 → S1 is a map such that for every point p ∈ S0, there is an open neighborhood +U of p such that f(U) is open in S1 and f∣U ∶ U → f(U) is a homeomorphism. Suppose S1 is a space with a +foliation F1. Let S0 be another topological space and suppose f ∶ S0 → S1 is a local homeomorphism. Then +there is a natural pullback equivalence relation, namely the finest equivalence relation on S0 such that the +points p and q of S0 are equivalent when there is an open set U ⊂ S0 containing p and q such that f(U) is +open, f∣U ∶ U → f(U) is a homeomorphism, and f(p) and f(q) lie on the same leaf of F1∣f(U). The pullback +foliation f ∗(F1) is the collection of equivalence classes of the pullback equivalence relation. If (S0,F0) and +(S1,F1) are two foliated spaces and f ∶ S0 → S1 is a homeomorphism such that F0 = f ∗(F1), then we say +that f is an isomorphism. That is, f must induce a bijection from F0 to F1. +Given a collection {(Si,Fi)} of foliated spaces as above, the disjoint union ⊔i Fi is a foliation on X = ⊔i Si. +In any of the foliated spaces (X,F) constructed as above, the foliation pseudogroup consists of all isomor- +phisms between open subsets of X endowed with restricted foliations. +Remark 2.2. One can define a foliated surface (S,F) to be the geometric structure determined by a foliation +atlas. But, treating a foliation as its collection of leaves seems more natural, and from the collection of leaves +derived from such an atlas we can recover an atlas defining the structure. The charts can be taken to be the +collection of all φ ∶ U → R2 where U ⊂ S is open and φ is an isomorphism from (U,F∣U) to (f(U),H∣f(U)). +2.3. Standard singularities. Note that the action of multiplication by −I on R2 preserves the horizontal +foliation. Let Π−1 denote R2/−I, which has a cone point with cone angle π at the image of the origin. We call +this cone point the origin 0 ∈ Π−1 and write Π∗ +−1 = Π−1 ∖{0}. Note that a collection of inverses of restrictions +of the covering map R2 ∖ {0} → Π∗ +−1 gives a foliation atlas on Π∗ +−1. We call the foliation associated to this +atlas the horizontal foliation H−1 of Π∗ +−1. +For each integer n ≥ 0, we define Πn to be the branched cover of Π−1 of degree n + 2 branched over the +origin. In all these spaces, we use 0 to denote the unique preimage of 0 ∈ Π−1, and call 0 the origin. Note +that geometrically 0 ∈ Πn is a cone singularity with cone angle (n + 2)π. We define Π∗ +n = Πn ∖ {0}. The +pullback of the horizontal foliation on Π∗ +−1 under the covering map Π∗ +n → Π∗ +−1 is the horizontal foliation Hn +of Π∗ +n. +Note that Π0 is naturally homeomorphic to R2, and the foliation H0 of Π0 is carried by this homeomor- +phism to the horizontal foliation of R2 ∖ {0}. + +ZEBRA SURFACES +11 +We further define Π−2 to be the plane R2 equipped with the foliation of R2 ∖{0} by circles with center 0. +For convenience we call this foliation the horizontal foliation of Π∗ +−2 = Π−2 ∖ {0} and denote it by H−2. This +is locally homeomorphic to one of the straight-line foliations that arises near a double pole of a quadratic +differential. +Singularities of this form show up in some of our arguments, but are not allowed in zebra +surfaces. +A prong of Πn is a leaf of the horizontal foliation with 0 as an endpoint. There are n + 2 prongs in Πn. +2.4. Singular foliations. Consider the model space +(2) +X = Π−1 ⊔ ⊔ +n≥1 +Πn, +and let +X∗ = Π∗ +−1 ⊔ ⊔ +n≥1 +Π∗ +n ⊂ X, +which comes equipped with its horizontal foliation HX = ⊔Hn. +Let h ∶ U → V be an orientation-preserving homeomorphism between open subsets of X. We say h is in +the horizontal pseudogroup (of X) if : +(1) We have h(U ∩ X∗) = h(U) ∩ X∗. +(2) The restriction h∣U∩X∗ is in the foliation pseudogroup of (X∗,HX). +Observe that if Ui ⊂ U is a connected component then Ui ⊂ Πm for some m = m(i) and h(Ui) ⊂ Πn for +some n = n(i). Note that statement (1) implies that 0 ∈ Ui if and only if 0 ∈ h(Ui) and that h(0) = 0 if +0 ∈ Ui. +A singular foliation atlas on a surface S is an atlas of charts to X whose transition functions lie in the +horizontal pseudogroup. We say a point p ∈ S is a singularity if there is a chart φ ∶ U → Πn such that +φ(p) = 0. Because elements of the pseudogroup send origins to origins, the notion of being a singularity is +independent of the chart. The singular set Σ ⊂ S is the collection of all singularities. Every point p has a +neighborhood U such that U ∖ {p} contains no singularities, so Σ is a closed discrete subset. +Observe that a singular foliation atlas determines a foliation on S ∖ Σ. We call this foliation a singular +foliation on S. The singular data of the atlas consists of Σ and the function α ∶ S → Z≥−1 whose support is +Σ and which sends p ∈ Σ to the n such that there is a chart from a neighborhood of p to a neighborhood of +0 ∈ Πn. Observe that α is well-defined, because a single chart tells you that there are n+2 prongs emanating +from p. Here a prong emanating from a point p ∈ S is the germ of an injective path γ ∶ (0,1) → ℓ into a +leaf ℓ with limt→0+ γ(t) = p, i.e., an equivalence class of such paths where two such paths γ1 ∶ (0,1) → ℓ +and γ2 ∶ (0,1) → ℓ are equivalent if there are constants ϵ1,ϵ2 ∈ (0,1) such that the restrictions γ1∣(0,ϵ1) and +γ2∣(0,ϵ2) are the same up to orientation-preserving reparameterization. (Our definition of prong is slightly +non-standard in that typically prongs are only defined at singular points, but we define them at all points.) +If α(p) = −1, the point p is called a pole. +Remark 2.3 (Removable singularities). Since allowing removable singularities would make statements of +our main results more technical and since we don’t have much use for them in this paper, we have purposely +not allowed removable singularities in our zebra surfaces. However, if the model space X is altered to include +Π0 with its horizontal foliation, then a point mapping to 0 ∈ Π0 would be a removable singularity or marked +point. A marked point in a singular foliation can be removed by replacing a chart to (Π0,H0) with a chart +to (R2,H) or an isomorphic subset of X. So, removable singularities are much the same as regular points. +However, treating a regular point as a singularity creates some problems, because including a regular point +in Σ alters the space being foliated. Thus, this change alters the notion of what a leaf is. In particular, if we +allow removable singularities, the statements of Theorem 1.2 and Theorem 1.3 or the definitions they depend +on need to be suitable altered to make the results still true. +Given a prong contained in a leaf ℓ emanating from a singular point p, a parameterization of the leaf +(0,1) → ℓ can be extended to include p. In such a case we call the singularity an endpoint of the leaf. If a +leaf has two endpoints, then we call the union of the leaf with its endpoints a saddle connection. If the leaf +has one endpoint, then we call this union a separatrix. A leaf with no endpoints is called bi-infinite. A leaf +that is homeomorphic to a circle is said to be closed. +Sometimes we will allow double poles to appear in our singular foliations, though we do not allow these +more general singular foliations in our zebra surfaces. A generalized singular foliation atlas on a surface is +defined as above but including Π−2 in the model space with its horizontal foliation F−2. In this case α takes +values in Z≥−2 and we call a point p where α(p) = −2 a double pole. Note that a singular foliation is also a +generalized singular foliation. + +12 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Proposition 2.4 (Euler–Poincar´e Formula). Let F be a generalized singular foliation on a closed surface S +with singular set Σ and singular data α. Then, +∑ +p∈Σ +α(p) = −2χ(S), +where χ(S) denotes the Euler characteristic of S. +This result follows from the Poincar´e-Hopf theorem, which gives the Euler characteristic in terms of the +sums of indices of a zeros of a vector field with isolated zeros. For a proof see Proposition 5.1 of [FLP21]. +The main idea is to pass to a double cover, so the foliation becomes oriented and gives rise to a vector field. +Note that here we allow α to take the values −2 and −1 while [FLP21] does not (though the proof goes +through in this case). +2.5. The extended real numbers. Let ˆR = R ∪ {∞}, which is homeomorphic to a circle. +The usual +increasing order on R extends to a cyclic order on ˆR. +We use interval notation to denote subsets of ˆR. If m0 ≠ m1, we use interval notation such as (m0,m1) +to denote the set of slopes m ∈ ˆR for which the triple (m0,m,m1) is in strictly increasing cyclic order. Then +[m0,m1] will denote the associated closed interval, (m0,m1) ∪ {m0,m1}. +2.6. The stellar functions. The stellar function on R2 ∖ {0} is the function which sends a point p to the +slope of the line joining p to 0: +(3) +ρ ∶ R2 ∖ {0} → ˆR; +(x,y) ↦ y +x. +Observe that the value of ρ is preserved by the action of −I, so ρ descends to a well-defined map ρ−1 ∶ Π∗ +−1 → ˆR. +Then for n ≥ 0, we can obtain functions ρn ∶ Π∗ +n → ˆR by composing the covering map Π∗ +n → Π∗ +−1 with ρ−1. +We call ρn the stellar function on Π∗ +n. +A ray in Πn of slope m is a connected component of ρ−1 +n (m). +2.7. Definition of zebra structure. Let {Fm ∶ m ∈ ˆR} be a collection of singular foliations on a connected +surface S with the same singular set Σ and the same singularity data function α ∶ S → Z≥−1. +A stellar neighborhood of a point p ∈ S is an open neighborhood U of p such that there is an integer n ≥ −1 +and a homeomorphism h ∶ U → Πn such that h(p) = 0 and the following statements hold for each slope m ∈ ˆR. +(1) For each ray r ⊂ Π∗ +n of slope m, h−1(r) is contained in a leaf of Fm. +(2) For each prong of Fm emanating from p, there is a ray r ⊂ Π∗ +n of slope m such that for any path +γ ∶ (0,1) → r with limt→0+ γ(t) = 0, the preimage h−1 ○ γ represents the prong. +The above two statements guarantee that for each m, h induces a bijection from prongs of Fm emanating +from p and rays of slope m in Πn. We call h a stellar homeomorphism. It follows by counting prongs and +rays that n = α(p). +We say that the collection of singular foliations {Fm}m∈ˆR is a stellar foliation structure or a zebra structure +on S if the foliations have the same singular sets, the same singular data, and every p ∈ S has a stellar +neighborhood. We call a surface together with a zebra structure a zebra surface. +2.8. The action by homeomorphisms of the circle. Our foliations are parameterized by the topological +circle ˆR. There is a natural action of the group Homeo+(ˆR) of all orientation-preserving homeomorphisms +of ˆR on zebra surfaces defined as follows. Suppose {Fm} defines a zebra structure on S. If ϕ ∈ Homeo+(ˆR), +then for each m ∈ ˆR we can define F′ +m = Fϕ−1(m), and {F′ +m}m∈ˆR will define another zebra structure on S. +Note the original structure and the new structure have the same singular data. +Let Homeo−(ˆR) denote the collection of all orientation-reversing homeomorphisms of ˆR. Then Homeo+(ˆR)∪ +Homeo−(ˆR) forms the full homeomorphism group of ˆR. For ϕ ∈ Homeo−(ˆR), we define +ϕ(S,{Fm}) = ( ¯S,{Fϕ−1(m)}) +where ¯S denotes S with its opposite orientation. + +ZEBRA SURFACES +13 +2.9. Leaves on zebra surfaces. As in the introduction, a leaf of a zebra surface is a leaf of one of the +foliations Fm. The slope of a leaf on S is the m for which the leaf belongs to Fm. The word horizontal +means slope zero, and vertical means slope ∞. Terms like saddle connection, separatrix and bi-infinite leaf +all make sense on S. +Observe: +Proposition 2.5 (Transversality). If leaves ℓ1 and ℓ2 have distinct slopes and intersect at a non-singular +point, then they cross transversely in the sense that there is an open disk containing the intersection point +such that there is only one intersection between ℓ1 and ℓ2 in this disk and the disk is cut in two by ℓ1 with +points in ℓ2 in both halves. +Proof. Since we are on a zebra surface, the intersection point has a stellar neighborhood that gives the +desired properties. +□ +2.10. Angles and Gauss-Bonnet. Zebra surfaces have a natural notion of angle. Let pq and qr be two +segments of leaves on a zebra surface, where we allow any of these three points to be singular. Let U be +a stellar neighborhood of q and h ∶ U → Πα(q) be the corresponding stellar homeomorphism. Then ∡pqr +indicates the counterclockwise angle measured at the origin of Πα(q) from h(pq ∩ U) to h(qr ∩ U). +We +normalize this measurement so +(4) +0 ≤ ∡pqr < (α(q) + 2)π. +Suppose S′ is a subsurface of a zebra surface. Let q ∈ ∂S′, and suppose that in a neighborhood of q, +∂S′ = rq ∪ qp where rq and qp are segments of leaves and S′ is on the left as we move from r to q to p along +this boundary curve. Then the interior angle of S′ at q is ∡pqr. +Theorem 2.6 (The Gauss-Bonnet Theorem for zebra surfaces). Let K be a compact subsurface of a zebra +surface with a boundary consisting of a union of disjoint simple closed curves that are piecewise given by +segments of leaves (with finitely many pieces). Let α be the singular data function on the surface containing +K. For a boundary point q, let θq denote the interior angle at q. Then, +(5) +∑ +q∈∂K +(π − θq) − +∑ +p∈Σ∩K○ πα(p) = 2πχ(K). +Note that in (5), there are only finitely many points in each sum for which the contribution to the sum is +non-zero. +Proof. Consider K as equipped with a foliation of slope m which does not coincide with the slope of any of +the finitely many leaves in the boundary of K. Double K across its boundary to obtain a closed surface X to +which we can apply the Euler–Poincar´e Formula. The doubled surface satisfies χ(X) = 2χ(K). The surface +X inherits a foliation from the two copies of K, where we allow our leaves to pass between the copies of K +through the boundary. We will see that this foliation of slope m of K lifts to a generalized singular foliation +of X. We allow our leaves to pass between copies of K through the boundary, so this will be a foliation of +the complement of the singular points in the interior of K and endpoints of boundary edges. Let ˜α denote +the singularity data on X for the lifted foliation. (Checking that ˜α is well defined will prove that the lifted +foliation to X is a generalized singular foliation.) Each singular point p ∈ K○ has two lifts ˜p1, ˜p2 ∈ X, and +we have ˜α(˜p1) = ˜α(˜p2) = α(p). For q ∈ ∂K an endpoint of a boundary edge, we have only one lift ˜q and by +considering a stellar neighborhood of q we see that ˜α(˜q) = 2vq − 2 where vq is the number of prongs in K +terminating at q. Then by the Euler–Poincar´e Formula, we have +∑ +q∈∂K +(2 − 2vq) − +∑ +p∈Σ∩K○ 2α(p) = 2χ(X) = 4χ(K). +Dividing by 2 and multiplying by π yields: +(6) +∑ +q∈∂K +π(1 − vq) − +∑ +p∈Σ∩K○ πα(p) = 2πχ(K). +Now consider a single boundary component γ of K whose vertices are qi for i ∈ Z/nZ written in increasing +cyclic order as we travel around γ with the region K on the left. Given any i, choose a stellar homeomorphism +hi ∶ Ui → Πα(qi). By possibly shrinking Ui, we can assume that h(K ∩ Ui) is a sector σi ⊂ Πα(qi). Then +the angle of this sector coincides with the interior angle θqi, and the starting ray of the sector has the same + +14 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +slope as qiqi+1 and the ending ray has the same slope as qi−1qi. Consider the union of these sectors σi with +the starting ray of σi glued to the ending ray of σi+1. Note that the glued rays have the same slopes, so this +gluing of sectors produces a copy of Πk where ∑θqi = (k + 2)π. It follows that the total number of prongs +∑vqi = k + 2 and so we have +(7) +n−1 +∑ +i=0 +θqi = +n−1 +∑ +i=0 +πvqi. +Substituting (7) into (6) (for each boundary component) yields (5). +□ +It is useful to observe the following result for future Gauss-Bonnet calculations: +Proposition 2.7. Let γ be a closed curve in a zebra surface that is piecewise given by segments of leaves. Let +{θi ∶ i = 1,...,n} denote the measures of angles at the transitions between the segments of leaves, measured +uniformly on one side of γ. Then the sum ∑n +i=1 θi is an integer multiple of π. +Proof. Let qi denote the endpoints of the segments of leaves making up γ, with the angle θi being measured +at qi. Orient γ, and consider a vector traveling around γ in the direction of the orientation and pointed +along the curve on the interiors of the arcs making up γ, and turning at each qi between the arcs. Then, the +unit vector turns by a signed angle equivalent to π ± θi modulo πZ at qi, where the sign only depends on +the side of γ where measurements were made. Since when the vector travels completely around γ, it ends +pointing in a direction with the same slope as its start, we have that ∑i(π ± θi) is an integer multiple of π. +Since the signs are uniform, the conclusion follows. +□ +3. Basic observations, definitions, and results +3.1. The pole-resolved universal cover. Let S, Σ, and α be as in Section 2, but add the hypothesis +that S is connected. Define Σ−1 = α−1(−1) to be the set of poles. The points in Σ−1 are our only source +of “positive curvature.” The goal here is to define a variant of the universal cover but with no “positive +curvature.” The cover described here was also considered in [Fra18, §3]. +Let S+ = S ∖ Σ−1. Similar to language used in the introduction, call a loop in S+ polar if it is freely +homotopic in S+ to a simple loop bounding a disk in S containing exactly one point in Σ−1. Choose a +basepoint p0 ∈ S+ and define +(8) +N = ⟨γ2 ∶ γ is polar⟩ ⊂ π1(S+,p0). +Then, N is a normal subgroup of π1(S+,p0), because being polar is a conjugacy invariant. +Proposition 3.1. There is a largest branched cover ˜S of S which is at most doubly branched over each point +in Σ−1 and is unbranched over other points. The restriction of the covering π ∶ ˜S → S to π−1(S+) → S+ is the +normal cover of S+ associated to N ⊂ π1(S+,p0). The surface ˜S is homeomorphic to a disk, and the covering +π is doubly branched over every pole, i.e., the covering map is locally 2 − 1 near any preimage of a pole. +We define ˜S to be the pole-resolved universal cover (PRU cover) of S. It follows from the result above +that ˜S is a branched cover of the usual universal cover. If Σ−1 = ∅, then the PRU cover coincides with the +universal cover. +Proof of Proposition 3.1. Let ˜S+ denote the cover of S+ associated to the subgroup N, as in covering space +theory. We claim that ˜S+ is ˜S with the preimages of points in Σ−1 removed. +Suppose ˜T is some other branched cover of S which is only branched over points in Σ−1 and at most +doubly branched over these points. Puncturing ˜T at preimages of Σ−1, we obtain a covering ˜T + of S+, which +is associated to a subgroup G ⊂ π1(S+,p0). To show that ˜S+ covers ˜T +, it suffices to prove that N ⊂ G. To +this end, let γ ∶ [0,1] → S+ be a polar loop with γ(0) = γ(1) = p0. Then there is a homotopy hs ∶ [0,1] → S+ +such that h0 = γ and h1 is a loop in S bounding a disk enclosing exactly one point in Σ−1. Let η ∈ π1(S+,p0) +be the loop which follows s ↦ hs(0) for s ∈ [0,1], then follows t ↦ h1(t) for t ∈ [0,1] and returns to the +basepoint following s ↦ hs(0) parameterized backward from s = 1 to s = 0. It is not hard to show that η is +homotopic rel endpoints to γ in S+, thus determining the same element of π1(S+,p0). Since ˜T is at most +doubly branched over points in Σ−1, the square of the element of π1(S+,p0) associated to the common class +of γ and η lies in G. Since γ was an arbitrary polar curve N ⊂ G as claimed, proving that ˜S+ is the largest + +ZEBRA SURFACES +15 +such cover. This argument also shows that ˜S+ is locally at most a double cover in neighborhoods of Σ−1, +and we can fill in these points to form the branched cover ˜S. +It remains to show that the cover ˜S is actually doubly branched over points in Σ−1 and that ˜S is a +topological disk. To see that ˜S → S is doubly branched over some point p ∈ Σ−1, it suffices to find a branched +cover ˜T as above which is doubly branched over p. To see that ˜S is a disk, it suffices to find a ˜T whose +universal cover is a disk. (If ˜T satisfies the double branching condition, the so does its universal cover.) If S +has positive genus, this is clear since its universal cover is a disk and given any p ∈ Σ−1, we can find a linear +map H1(S+;Z/2Z) → Z/2Z sending a loop wrapping once around p to 1. Covering space theory associates +this linear map to a double cover of S+ which is doubly branched over p. If S is a sphere, then by the +Gauss-Bonnet Theorem there are at least four points in Σ−1. Choosing four points including our favorite +point p, we can puncture only at those four points and define a linear map as before such that the homology +classes of the loops around each of these points are sent to one. The associated double cover is a torus which +is doubly branched over these four points as desired. Since this torus has a disk as its universal cover, this +also proves that ˜S is a disk in this case. +□ +Now suppose {Fm}m∈ˆR is a family of foliations determining a zebra structure on S. Let ˜Fm be the singular +foliation of ˜S whose leaves are lifts of leaves of Fm. We call { ˜Fm}m∈ˆR the lifted family of foliations. These +foliations have common singularity data ˜α, where if ˜p ∈ ˜S projects to p ∈ S, we have +˜α(˜p) = α(p) if α(p) ≥ 0 +and +˜α(˜p) = 0 if α(p) = −1. +3.2. Zebra planes. A zebra plane is a zebra structure on the open topological disk such that the singular +data function α is non-negative. From the discussion above, the PRU cover of a zebra surface is always a +zebra plane. +3.3. Polygons. A polygon p0p1 ...pn−1 in a zebra plane is a topological disk bounded by a simple closed +curve of the form p0p1 ∪p1p2 ∪...∪pn−2pn−1 ∪pn−1p0, where each edge pipi+1 with i ∈ Z/nZ is a segment of a +leaf from a directional foliation. The Jordan Curve Theorem guarantees that this curve bounds a topological +disk, and we’ll use the counterclockwise ordering when describing polygons so that the polygon is on the left +as we move from pi to pi+1 along pipi+1. We call the pi vertices. The interior angle of P at pi is ∡pi+1pipi−1. +The external angle is ∡pi−1pipi+1. The sum of the interior and exterior angles at pi is α(pi)π + 2π, the total +angle at pi. +We’ll call a vertex straight if the interior angle equals π. As we are typically interested in the internal +geometry of a polygon, we will typically ignore straight vertices. So, a k-gon is a polygon with k vertices +that are not straight. A triangle is a 3-gon, and we’ll use other similarly obvious terminology coming from +plane figures to describe objects in ˜S. +Suppose S is a zebra surface and ˜S is its PRU cover. Let π ∶ ˜S → S denote the covering map. If ˜P is +a polygon in ˜S such that the restriction π∣ ˜ +P ∶ ˜P → S is injective on the interior of ˜P, then we’ll call the +restriction π∣ ˜ +P a polygon P in S. These are maps rather than subsets of S, because it gives the right notion +of the interior of P (the image of the interior) and boundary (the further restriction to the boundary of ˜P). +These notions are confusing even when considering the square in the center of the usual square torus (in +that the closed square is the whole torus, but you still want it to have boundary for instance). +The next proposition shows that interior angles and slopes of edges of a zebra triangle behave as they do +in plane geometry. +Proposition 3.2. Triangles contain no singularities in their interiors and the sum of the interior angles of +a triangle in a zebra plane is always π. Let (m0,m1,m2) ∈ ˆR3 be a triple of slopes of edges of a triangle, +listed in counterclockwise order as we travel around the boundary of the triangle. Then the triple of slopes is +distinct and appear in decreasing cyclic order on ˆR. +Proof. Let θi for i = 0,...,2 be the interior angles of a triangle T. Since the curve is simple, we have θi > 0 +for all i. By the Gauss-Bonnet Theorem, we have +θ0 + θ1 + θ2 = π − ∑ +p∈T + πα(p) > 0, +where the sum is taken over all interior singularities of T. Since α(p) ≥ 1 at all singularities in a zebra plane, +there must be no singularities in T + and thus θ0 + θ1 + θ2 = π as desired. It follows the slopes of the sides of + +16 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +a triangle must be the same as the slopes of the sides of a Euclidean triangle. Therefore, the slopes appear +in decreasing cyclic order. +□ +Proposition 3.3. The sum of the interior angles of a quadrilateral Q in a zebra plane is either π or 2π. If +the sum is 2π, then Q contains no singularities in its interior. If the sum is π, then Q contains a singularity +q in its interior with α(q) = 1. +Proof. From the Gauss-Bonnet Theorem, the interior angles satisfy +θ0 + θ1 + θ2 + θ3 = 2π − ∑ +p∈T + πα(p) > 0, +immediately giving the result. +□ +We also have the following more general result: +Proposition 3.4. Any closed n-gon P in a zebra plane must have at least three interior angles whose +measure is less than π. +Proof. Again by the Gauss-Bonnet Theorem, +∑ +q∈∂P +(π − θq) = 2π + ∑ +p∈K○ πα(p) ≥ 2π. +Since for each q, we have π − θq < π, there must be at least three q ∈ ∂P for which π − θq > 0. +□ +3.4. Arcs of trails. Let Z be a zebra plane and let { ˜Fm} denote the associated foliations. +Let p ∈ Z be a singularity and pq and pr be segments of leaves. We say these arcs satisfy the angle +condition at p if +∡qpr ≥ π +and +∡rpq ≥ π. +A parameterized arc of a trail on Z is a parameterized curve γ ∶ I → Z, where I ⊂ R is a non-degenerate +interval, such that if t is any point in the interior of I then the following statements are satisfied: +● If γ(t) is not singular, then in a neighborhood of t, γ(t) moves injectively along a leaf. +● If γ(t) is singular, then the two arcs made at γ(t) formed by increasing and decreasing t satisfy the +angle condition. That is, the angle made by γ at γ(t) is at least π on both sides. +Suppose γ ∶ I → Z and η ∶ J → Z are parameterized arcs of trails. We say that γ is a subarc of η if there is +an orientation-preserving continuous injective map φ ∶ I → J such that γ = η ○ φ. We say γ is a proper subarc +of η if the map φ is not surjective. +The condition that two arcs of trails are each subarcs of the other (i.e., reparameterizations of one another) +is an equivalence relation, and we’ll call an equivalence class an arc of a trail. The notion of subarc induces +a well-defined partial ordering on arcs of trails. +An arc of a trail on S is the image of an arc of a trail on the PRU cover ˜S under the covering ˜S → S. We +likewise use the cover to define the other notions above. +Proposition 3.5 (No monogons). If τ ∶ I → Z is a parameterized arc of a trail in a zebra plane, then τ is +injective. +Proof. Suppose τ is not injective. +Then we can assume without loss of generality that I = [a,b] and +τ(a) = τ(b). Observe that because I is closed and bounded and τ is continuous and locally injective, the set +J of all t ∈ [a,b] for which there is a t′ ∈ [a,t) such that τ(t) = τ(t′) is closed. Let x = inf J. Then there is an +x′ ∈ [a,x) such that τ(x) = τ(x′). Restricting τ to [x′,x] yields a simple closed curve, which by the Jordan +Curve Theorem bounds a disk D. Observe that D must be a polygon, and since τ is an arc of a trail, all its +interior angles are larger than π except possibly at τ(x′) = τ(x). This violates Proposition 3.4. +□ +Proposition 3.6 (No bigons). Suppose τ1 and τ2 are arcs of trails in a zebra plane Z. Then τ1 ∩ τ2 is the +empty set, is a single point, or is a common subarc (possibly with different induced orientations). +Proof. Assume τ1 and τ2 intersect. Consider each τi to be parameterized by functions with the same name, +τi ∶ Ii → Z. +The statement can be observed to be true as long as τ −1 +1 (τ2) is connected. +If τ −1 +1 (τ2) is +disconnected, we can let J1 ⊂ I1 ∖ τ −1 +1 (τ2) be a bounded open interval such that both boundary points lie +in τ −1 +1 (τ2). Since τ2 is injective, there is also a unique interval J2 ⊂ I2 such that τ2(∂J2) = τ1(∂J1). By +construction τ1(J1)∪τ2(J2) forms a simple closed curve, which again by the Jordan Curve Theorem bounds + +ZEBRA SURFACES +17 +a disk D. This time the only possible interior angles less than π are the two points in τ1(∂J1), again violating +Proposition 3.4. +□ +Proposition 3.7. Let P be a polygon in a zebra plane all of whose exterior angles are at least π. Then +there is no parameterized arc of a trail τ ∶ [0,1] → Z such that τ(0),τ(1) ∈ ∂P and τ((0,1)) ∩ P = ∅. +Proof. If this were the case, the union of τ and an arc of P bound a polygon Q whose interior is in the +complement of P. Points in ∂Q ∩ τ have interior angles at least π since τ is an arc of a trail, and points +on ∂Q ∩ P that are not endpoints of τ have interior angles for Q which are the same as the exterior angles +for P. Thus, Q can have at most two interior angles less than π (namely, the endpoints of τ). Again this +violates Proposition 3.4. +□ +3.5. Trapezoids. We speak of two segments of leaves in a zebra plane Z as being parallel if they are segments +of leaves coming from the same foliation ˜Fm. A trapezoid in Z is a 4-gon with a pair of opposite edges that +are parallel. A parallelogram is a 4-gon such that both opposite pairs of edges are parallel. +Proposition 3.8. Let T be a trapezoid in Z. Then the angles of T add to 2π and there are no singularities +in the interior of T. +Proof. Suppose our trapezoid is pqrs, with vertices ordered counterclockwise and with pq parallel to rs. +Then ∡rqp + ∡srq = π. The other pair of angles add to π as well, so the sum of all the angles is 2π. Then +Proposition 3.3 tells us that T has no singular points in its interior. +□ +We say a trail has constant slope if there is an m such that every segment of a leaf contained in the trail +has slope m. +Proposition 3.9. Let pq,rs ⊂ Z be arcs of trails of the same constant slope m. Suppose that qr and sp are +disjoint segments of leaves whose slopes are not m. Then the curve pq ∪ qr ∪ rs ∪ sp is a 4-gon (trapezoid) +whose vertices are p, q, r and s. In particular, all interior angles at singularities in the interior of segments +pq and rs are π, and the interior angles at p, q, r and s are each less than π. +Proof. The curve pq ∪ qr ∪ rs ∪ sp is a closed curve. If we can show it is simple, then by the Jordan curve +theorem it bounds a disk, which is our polygon. Suppose it has n sides. As in Euclidean geometry, the +Gauss-Bonnet Theorem guarantees that the sum of the interior angles is (n − 2)π. Because pq and rs are +parallel, we have +∡q + ∡r = aπ +and +∡s + ∡p = bπ +for some integers a,b ≥ 1. +Suppose t1,...,tn−4 are the singularities in the interiors of pq and rs. Then ∡ti = kiπ for some integer ki ≥ 1, +because both these arcs of trains have constant slope. Thus the sum of the interior angles is (a+b+∑n−4 +i=1 ki)π. +The smallest this sum can be is (n − 2)π and so we must have a = b = k1 = ... = kn−4 = 1. +Now we will argue that pq ∪ qr ∪ rs ∪ sp is simple. We know that the arcs pq, qr, rs, and sp are simple +by Proposition 3.5. Using the slope conditions, we see that Proposition 3.6 guarantees that the intersection +of adjacent edges (e.g., pq ∩ qr) consists only of the common vertex. Therefore, the only way that the curve +can fail to be simple is if opposite edges intersect in their interiors. By hypothesis rq ∩ ps = ∅. We claim +that pq ∩ rs = ∅. Suppose to the contrary that pq ∩ rs ≠ ∅. Then by Proposition 3.6, they intersect in either +a single point or a common compact subarc. Let x ∈ pq be the point closest to q in pq ∩ rs (where “closest” +is measured in a parameterization of pq). Then the path γ = xq ∪ qr ∪ rx is simple. Observe that γ has at +most two points at which there are angles whose measure is less than π (on either side of the curve), namely +the points q and r. (It could be that x coincides with either q or r, but otherwise because the arcs of γ on +both sides of x are parallel, the angle at x must be at least π.) This contradicts Proposition 3.4, proving our +claim and completing the proof. +□ +Lemma 3.10 (Trapezoid construction lemma). Let pq ⊂ Z be an arc of a trail, where the angle measured on +the left side as we move from p to q at any singularities in the interior of pq is π. Let U ⊂ Z be an open set +containing pq. Let ps and qr be additional segments such that 0 < ∡qps < π and 0 < ∡rqp < π. Then there +exist s′ ∈ ps ∖ {p} and r′ ∈ qr ∖ {q} and a segment s′r′ parallel to pq forming a trapezoid pqr′s′ contained in +U. Furthermore, we can construct the trapezoid in such a way so that leaves parallel to pq passing through +the interior of the trapezoid pass through the interiors of ps′ and qr′. + +18 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Figure 5. Depiction of the trapezoid produced by Lemma 3.10. +See Figure 5 for an illustration of the trapezoid construction. +We briefly discuss the idea of the proof before giving a detailed proof. By compactness, we can produce +a finite covering of pq by foliation charts for the foliation parallel to pq. We can find stellar neighborhoods +of p and q contained in the charts containing p and q. The angle conditions at p and q and the fact that +there are only finitely many charts can be used to guarantee that we can find a leaf parallel to pq that joins +a point of ps in the stellar neighborhood of p to a point of qr in the stellar neighborhood of q. A technical +point is that the charts might contain a singularity. +Proof. Assume without loss of generality that pq is horizontal. For each point x ∈ pq there is a foliation +chart from a neighborhood of x to (−1,1) × (−1,1) (or if x is singular, a singular chart to Πn carrying x +to 0) for the horizontal foliation that intersects pq in an interval. By possibly shrinking the chart, we can +assume that this neighborhood of x is contained in U. For this proof, all our foliation charts lie in U. We +can normalize these charts so that the portion of pq in the domain maps to a subset of (−1,1) × {0} and so +that the portion of the chart to the left of pq when moving from p to q is mapped into (−1,1) × (0,1). In +case x is singular, since the angle made on the left side of pq at x is π, the portion of U in the chart to the +left of pq is mapped into a half-plane of Πn. We can put coordinates on this closed half-plane of the form +R × [0,+∞), and like in the non-singular case we can restrict the chart and rescale it so that the portion +of pq in the domain can be mapped to (−1,1) × {0} and the portion to the left of pq in the domain can be +mapped to (−1,1) × (0,1) in these coordinates. Using compactness of pq we can produce a minimal finite +subcovering of pq. Restrict these charts to the points whose images have non-negative y-coordinates. We +can order this collection of restricted charts φi ∶ Bi → (−1,1) × [0,1) for i = 0,...,n such that Bi ∩ pq gives a +sequence of open subintervals of pq such that p ∈ B0, q ∈ Bn and pq ∩ Bi ∩ Bj ≠ ∅ if and only if ∣i − j∣ ≤ 1. +We claim that we can define intervals Ji ⊂ [0,1) which are open as subsets of [0,1) together with continuous +strictly increasing functions ψi ∶ Ji → [0,1) such that for all y ∈ Ji, the leaf φ−1 +0 ((−1,1)×{y}) can be continued +across B1, B2, . . . , Bi as +(9) +i +⋃ +k=0 +φ−1 +k ((−1,1) × {ψk(y)}). +We do this by induction. Define J0 = [0,1) and φ0 ∶ J0 → [0,1) to be the identity map. Now assuming Ji +and φi are defined, we can let Ui be the connected component of Bi ∩ Bi+1 containing pq ∩ Bi ∩ Bi+1 and +there is a continuous strictly increasing function +hi ∶ πy ○ φi(Ui) → πy ○ φi+1(Ui), +where +πy(x,y) = y +coming from the transition between the charts such that φ−1 +i ((−1,1)×{y}) continues across Ui as φ−1 +i+1((−1,1)× +{hi(y)}). Then by defining Ji+1 = Ji ∩ ψ−1 +i (Ui) and ψi+1 = hi ○ ψi we see that (9) is satisfied, completing the +induction and giving definitions for Jn and ψ0,...,ψn satisfying (9). This proves the claim. +Let N be a stellar neighborhood at p and h ∶ N → Πn′ be the stellar homeomorphism. Then the connected +component of N ∩ pq containing p maps under h to the closure of a horizontal ray r0 ⊂ Πn. Let H ⊂ Πn be +the closed half-space consisting of 0 and all rays r ⊂ Πn′ such that the counterclockwise angle from r0 to +r lies in [0,π]. Observe that p ∈ B0. By restricting to a smaller neighborhood N and rescaling the stellar +homeomorphism h, we can assume that h−1(H) ⊂ B0. Let ℓ be the connected component of (ps ∖ {s}) ∩ N + +ZEBRA SURFACES +19 +containing p. Then h(ℓ) is a ray contained in H. Since h−1(H) ⊂ B0, we see that ℓ ⊂ B0. Since ℓ is not +horizontal and contains p, the horizontal leaves in B0 meet ℓ transversely and so πy ○φ0(ℓ) = [0,b0) for some +b0 > 0. Similarly, there is a half-open arc ℓ′ ⊂ qr ∩ Bn containing q such that interval πy ○ φn(ℓ′) = [0,bn) +for some bn > 0. Since Jn is an open subset of [0,1) containing 0 and ψn is strictly increasing and preserves +zero, we can find y ∈ (0,b0) so that ψn(y) ∈ (0,bn). Then we have a segment of a leaf that cuts across B0, +. . . , Bn as described in (9). Let s′ be the place this leaf crosses ps and r′ be the place this leaf crosses qr. +We conclude that pqr′s′ is a trapezoid using Proposition 3.9. Furthermore, if 0 < y′ < y, then the leaf as +constructed in (9) (with y′ replacing y) cuts across this trapezoid passing through the interiors of edges ps′ +and qr′ as desired. +□ +Corollary 3.11 (Generalized rectangles). Let p ∈ Z and let U ⊂ Z be an open set containing p. Then there +is a 2α(p) + 4-gon P ⊂ U with alternating horizontal and vertical sides and all interior angles of π +2 such +that p ∈ P ○. Furthermore, there is a bijection between the horizontal prongs at p and the vertical edges of P +such that each horizontal prong has a realization as a horizontal path joining the corresponding edge to p. +These α(p) + 2 paths cut P into α(p) + 2 rectangles, each of which has p in the interior of an edge formed by +two of the prong realizations. Every horizontal leaf that enters the interior of P either crosses through the +interior of one of the rectangles joining opposite vertical sides of the rectangle, or follows one of the α(p)+2 +horizontal paths and terminates at p. +We call the polygon P a generalized rectangle because of the alternating horizontal and vertical sides and +angles of π +2 . An example is depicted in Figure 6. +Figure 6. A generalized rectangle surrounding a point p where α(p) = 1. +Proof. Construct horizontal segments of leaves with endpoints at p realizing every prong. +By possibly +shortening them, we can assume they are pairwise disjoint. There are n = α(p) + 2 such arcs. Denote them +{βi ∶ i ∈ Z/nZ} and order them counterclockwise. Then the counterclockwise angle from βi to βj at p is π +if and only if j = i + 1. Let ei denote the path βi ∪ {p} ∪ βi+1. Using Lemma 3.10, for each i we can produce +a rectangle Ri one of whose edges is ei such that the counterclockwise angle from βi to βi+1 at p is interior +to Ri. +Define Q = (⊔Ri)/ ∼ where ∼ identifies the corresponding points in the common subarcs {p}∪βi ⊂ Ri−1∩Ri. +Observe that Q is homeomorphic to a closed disk. Let π ∶ Q → Z be the natural map induced by the inclusions +of Ri into Z. Observe that π restricted to each Ri is injective. By considering horizontal foliation charts +at p and at points of each βi, we can see that π is locally injective. If we knew π were globally injective, +then P = π(Q) will be a polygon satisfying the statements in the corollary, with the statement about the +horizontal leaves following from Lemma 3.10. +Now assume π ∶ Q → Z is not injective. We will alter the construction to produce a smaller Q′ ⊂ Q with +the same properties such that π restricted to Q′ is injective. This will complete the proof. +We will actually construct a sequence of subsets Qn playing the role of Q′. For each i, construct a sequence +of rectangles Rn +i ⊂ ˜Ri such that are nested ( ˜Rn+1 +i +⊂ ˜Rn +i for all n) and satisfy ⋂∞ +n=0 Rn +i = ei. These Ri +n can be +produced from Ri by cutting along a horizontal leaf through the interior. We define Qn ⊂ Q to be the union +over i of the Rn +i . If the restriction of π to Qn is injective, then we can define P = π(Qn) to be our generalized +rectangle, completing the proof. Now suppose to the contrary that the restriction of π to Qn is not injective +for any n. Then for each n, we can find distinct points xn,yn ∈ Qn such that π(xn) = π(yn). By passing to a + +20 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +subsequence, we can assume that limxn = x and limyn = y both exist in Q. Then π(x) = π(y) by continuity +of π. Observe that ⋂n Qn = {p} ∪ ⋃i βi, and each Qn is closed, so we have x,y ∈ {p} ∪ ⋃i βi. But π restricted +to {p} ∪ ⋃i βi is injective because the βi were constructed to be pairwise disjoint realizations of prongs of +p. Thus we must actually have x = y. But then the facts that xn ≠ yn, π(xn) = π(yn) and limxn = limyn +violates the local injectivity of π. +□ +3.6. Properness of leaves. Let ℓ be a separatrix. Then it has a parameterization of the form ℓ ∶ [0,+∞) → +Z where ℓ(0) is its singular endpoint. Bi-infinite leaves have a parameterizations of the form ℓ ∶ R → Z. +Recall that a proper map between topological spaces is one for which preimages of compact subsets are +compact. +Proposition 3.12. A separatrix or bi-infinite leaf in a zebra plane that is parameterized as above is a proper +map. +Remark 3.13. Singular foliations of the disk where α is allowed to take the value −1 can have separatrices +and bi-infinite leaves that fail to be proper in the above sense even if there are only finitely many singularities +[Ros83]. This phenomenon also occurs with periodically arranged singularities where α = −1 [Pan09]. Avoid- +ing this phenomenon is a reason for requiring α to be non-negative in a zebra plane and for our definition +of the PRU cover. +Proof of Proposition 3.12. Let ℓ be as above, and assume without loss of generality that its slope is horizontal. +If ℓ is not proper, there is a compact set K ⊂ Z whose preimage is not compact. Since its preimage is +necessarily closed, the preimage must not be bounded. So, we can find a sequence tn in the domain of ℓ such +that each ℓ(tn) ∈ K and ∣tn∣ → +∞. Then by passing to a subsequence, we can assume that ℓ(tn) converges +to some point p ∈ K. Using Corollary 3.11, we can produce a generalized rectangle P containing p in its +interior. Therefore there are infinitely many ℓ(tn) ∈ P. Fix such an n. Since ℓ is horizontal and doesn’t +approach p as ∣t∣ → +∞, Corollary 3.11 guarantees that the portion of the horizontal leaf through ℓ(tn) must +enter, cut across one of the rectangles making up P and then exit. In particular the connected component +of ℓ−1(P) containing tn is a closed and bounded interval I. But then by hypothesis there is a tm in our +sequence such that ℓ(tm) ∈ P and tm /∈ I. This contradicts Proposition 3.7, which tells us that ℓ cannot later +return to P. +□ +Corollary 3.14. Let T = pqrs be a trapezoid with pq and rs parallel and of slope m. Then the restriction +of Fm to the interior of T consists of segments of leaves joining ps to qr. +Proof. Consider a parameterized leaf ℓ ∶ (a−,a+) → Z of slope m that intersects the interior T ○. We claim +that ℓ(t) must exit T or approach a point in ∂T as t approaches either endpoint. If s is a sign and limt→as ℓ(t) +is a singularity, this follows from the fact that T ○ has no singularities by Proposition 3.8. On the other hand, +if limt→as ℓ(t) is not a singularity, then ℓ must be a bi-infinite leaf or must be a separatrix with the limit to +the other endpoint limt→a−s ℓ(t) a singularity. Then, Proposition 3.12 guarantees that this parameterization +can be made proper. Setting ts to be the closest element of (a−,a+) to as such that ℓ(ts) ∈ T, we see ℓ exits +T at ℓ(ts) and never returns. +Now consider a maximal segment of a leaf of Fm in T ○. From the previous paragraph, traversing such a +maximal segment in either direction approaches a point in ∂T. The segment can’t approach a point in pq +or in rs because these are trails of slope m with interior angles of π; see Proposition 3.9. This means there +are no prongs of slope m approaching points in pq or rs. Now observe that the two points approached in +the two different directions can’t lie on the same edge by Proposition 3.6. Thus each maximal segment must +join ps to qr as claimed. +□ +3.7. Trails. A trail is an arc of a trail which is maximal with respect to the subarc partial order. The +following result tells us that trails exist, and every arc of a trail can be extended to a trail: +Theorem 3.15. If γ is an arc of a trail in a zebra surface, then γ is a subarc of a trail. +Let I ⊂ R be an interval, ¯I ⊂ R∪{±∞} be its closure, and γ ∶ I → Z be a parameterized arc of a trail. Let ℓ +be a leaf which, since Z is a zebra surface, is not closed and thus is homeomorphic to an open interval. If γ(a) +is in ℓ then ℓ∖γ(a) has two connected components. We’ll say that γ finishes a leaf ℓ in the positive direction +if there are a ∈ I and b ∈ ¯I with a < b such that γ(a) ∈ ℓ and ℓ∖γ([a,b)) has one connected component. That +is, there is a c ∈ (a,b] such that γ((a,c)) is one of the connected components of ℓ ∖ γ(a). Then, any further + +ZEBRA SURFACES +21 +extension of γ in the positive direction will require adding points not in ℓ (e.g., a singularity and a portion +of a new leaf). We make a similar definition of finishing a leaf in the negative direction. The following is the +main ingredient in the proof of this theorem: +Lemma 3.16. Let Z be a zebra plane, let I ⊂ R be an interval with endpoints −1 and 1, and let γ ∶ I → Z +be a parameterized arc of a trail. +(1) (Right limit) If limt→1− γ(t) exists, then there is a parameterized arc of a trail η ∶ I ∪ [1,2) → Z +extending γ such that η finishes a leaf in the positive direction that γ does not. +(2) (Left limit) If limt→−1+ γ(t) exists, then γ can be similarly extended to left as an arc of a trail. +Conversely if neither limit exists, then γ is a trail. +Proof. We will prove statement (1). Statement (2) will follow by symmetry. Let γ(1) denote the limit +limt→1− γ(t) (regardless of whether 1 is formally in the domain of γ). If γ(1) is not singular, let m be the +slope of γ at γ(1). Then we can extend γ by following the leaf of Fm through γ(1) until it finishes the leaf. +Now suppose γ(1) is singular. Let L be a leaf with an endpoint at γ(1) which satisfies the angle condition +at γ(1). (One can see by inspection of the angle condition that such a leaf always exists.) Then γ can be +extended to finish L. +The final statement can be proved by showing the contrapositive. Suppose γ ∶ I → Z is a proper subarc +of an arc of a trail η ∶ J → Z. By a change of coordinates, we may assume that I has endpoints −1 and 1. +Let φ ∶ I → J be the continuous orientation-preserving map satisfying γ = η ○ φ. Since φ is not surjective, we +may assume without loss of generality that the limit limt→1− φ(t) exists in J. Then we have +lim +t→1− γ(t) = lim +t→1− η ○ φ(t) = η( lim +t→1− φ(t)) +by continuity of η, so the limit in statement (1) exists. +□ +Proof of Theorem 3.15. Observe that it suffices to prove the statement for zebra planes, because arcs of trails +on zebra surfaces are defined to be images of arcs of trails on their PRU cover. So, throughout this proof, +we will only consider trails in a zebra plane Z. +Let γ1 ∶ J1 → Z be an arc of a trail where J1 ⊂ R is a bounded interval. We produce k ∈ N ∪ {+∞} and +a finite or infinite sequence of parameterized arcs of trails {γi ∶ Ji → Z}1≤i 0 such that γ((m,+∞)) is contained in N. Observe that γ((m,+∞)) contains infinitely +many complete leaves, giving us our desired contradiction to the statement that limt→+∞ γ(t) = p. +There are two other cases. The case where the left limit always exists and the forward limit eventually +stops existing is symmetric. The case where both limits always exists is similar: we may simultaneously +handle the right and left limits say by defining Ji = (−i,i). +□ +We record the following basic consequence of the results above: +Corollary 3.17. Every trail on Z has a parameterization γ ∶ R → Z. + +22 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Proof. Let γ ∶ I → Z be a parameterization of a trail. By Lemma 3.16, limt→inf I γ(t) and limt→sup I γ(t) do +not exist, so I must be an open interval. By possibly reparameterizing, we may assume that I = R. +□ +This allows us to strengthen Proposition 3.7: +Corollary 3.18. Let P ⊂ Z be a polygon all of whose exterior angles are at least π. Then given any trail +τ ∶ R → Z, the preimage τ −1(P) is either empty or a (possibly degenerate) closed and bounded interval. +Proof. Since τ is continuous and P is a closed set, τ −1(P) is closed. By Proposition 3.7, the preimage τ −1(P) +is an interval. It remains to show that τ −1(P) is bounded. +Recall that a trail follows leaves, transitioning between leaves at singularities. By Proposition 3.12, if +τ∣I ∶ I → Z parameterizes a bi-infinite leaf or a separatrix including its singular endpoint, the preimage +(τ∣I)−1(P) is compact. Because transitions only happen at singularities, restrictions of τ can parameterize +at most one bi-infinite leaf and at most two separatrices. +Now consider the collection of all parameterized saddle connections of the form τ∣J that intersect P. Such +an interval J must be bounded by Lemma 3.16. By Proposition 3.7, τ can’t leave P and later return, so +there are at most two parameterized saddle connections that intersect P but are not contained in P. By +Proposition 3.5, trails are simple curves and therefore each singularity in P is the endpoint of at most two +parameterized saddle connections of the form τ∣J. Since P is compact and the singularities are isolated, there +are at most finitely many singularities in P. It follows that there are at most finitely many parameterized +saddle connections of the form τ∣J that are contained in P. +Putting it all together, we have shown that τ −1(P) is contained in the union of at most two compact +subsets of intervals I such that τ∣I parameterizes a bi-infinite leaf or a separatrix and finitely many bounded +intervals J parameterizing saddle connections intersecting P. Thus τ −1(P) is bounded. +□ +3.8. Properness of trails. +Theorem 3.19. If τ ∶ R → Z is parameterized trail, then τ is a proper map. +Proof. Let τ ∶ R → Z be a parameterized trail. Suppose to the contrary that τ is not proper. Then there is a +compact subset K ⊂ Z such that τ −1(K) is not compact. Since τ is continuous, τ −1(K) is closed. Therefore, +τ −1(K) must be unbounded. Without loss of generality, we may assume that there is a sequence tn with +limtn → +∞ and τ(tn) ∈ K for all n. Since K is compact, by passing to a subsequence we may assume that +limτ(tn) = p ∈ K. Use Corollary 3.11 to construct a generalized rectangle P containing p in its interior. +Then there are infinitely many n such that τ(tn) ∈ P, so τ −1(P) is unbounded violating Corollary 3.18. +□ +Corollary 3.20. If τ is a trail in Z, then Z ∖ τ has two connected components, both homeomorphic to an +open disk. +Proof. Let X be the one-point compactification of Z with x∞ denoting the point added. Since Z is an +open topological disk, X is homeomorphic to the 2-sphere. Our trail can be parameterized by an injective +proper map from R, so it extends continuously to a simple closed curve ¯τ ∶ ˆR → τ ∪ {x∞} with ¯τ(∞) = x∞. +By the Jordan Curve Theorem, X ∖ ¯τ consists of two components, each homeomorphic to a disk. We have +Z ∖ τ = X ∖ ¯τ. +□ +Corollary 3.21. Let τ ∶ R → Z be a parameterized trail and let P ⊂ Z be a polygon all of whose interior +angles are less than or equal to π. If τ passes through the interior of P, then I = τ −1(P) is a non-degenerate +closed and bounded interval and τ −1(∂P) = ∂I. +Proof. Since P is closed and τ is continuous, I = τ −1(P) is closed. Since τ is proper, I is bounded. If I were +not an interval, there would be a bounded open interval J that is a component of R ∖ I, and the restriction +of τ to the closure ¯J would lead to a contradiction to Proposition 3.7. Observe that I is non-degenerate +because U = τ −1(P ○) is open, non-empty and contained in I. +Observe that τ(t) ∈ ∂P if and only if t ∈ I ∖ U. Thus, it remains to show that U is the interior of I. +Let U0 ⊂ U be a connected component of U. Observe that τ(U0) is an arc of τ in the interior of P whose +endpoints τ(∂U0) lie in ∂P. Let p ∈ τ(∂U0) be one of those endpoints. The interior angle at p is at most π, +so any path ℓ ∶ [0,1) → P with ℓ(0) = p such that ℓ((0,1)) is contained in a leaf must make an angle that is +strictly less than π with τ(U0) at p. But by the angle condition of trails, the continuation of τ outside of U0 +cannot make such an angle with τ(U0) at p, so after τ passes through p it immediately leaves P. Therefore + +ZEBRA SURFACES +23 +U only has one connected component, whose endpoints are also endpoints of I. In particular U is the interior +of I. +□ +4. Zebra surfaces with boundary +Surgeries are useful but require some preparations, namely a clear definition of zebra structure on a surface +with boundary. There are no surprises but we were unable to find work on topological singular foliations on +surfaces with boundary that was sufficiently detailed for our needs. +4.1. Surfaces with boundary. Let U = {(x,y) ∈ R2 ∶ +y ≥ 0} be the closed upper half-plane, whose +boundary is ∂U = {0} × R ⊂ U. A surface with boundary S is a second countable Hausdorff space that is +locally homeomorphic to U. That is, for each point p ∈ S, there is an open neighborhood N and an open +subset U ⊂ U together with a homeomorphism h ∶ N → U. A point p is said to be in the boundary of +S if h(p) ∈ ∂U, and the set of points in the boundary is denoted ∂S. It is a standard observation that +this definition is well-defined, that ∂S is a 1-manifold, and that ∂S is a closed subset of S. The points in +S○ = S ∖ ∂S are said to be interior points of S. +4.2. Sectors. Recall the objects constructed in Section 2.3: The spaces Πn which are n-fold covers of +Π−1 = R2/ − I branched over the origin, the stellar functions ρn ∶ Πn ∖ {0} → ˆR, and the stellar foliations of +Πn ∖ {0}. As in Section 2.6, a ray in Πn is a connected component of ρ−1 +n (m) ⊂ Πn ∖ {0}, where m ∈ ˆR is +some slope. +If r1 and r2 are distinct rays in some Πn, then r1 ∪ {0} ∪ r2 is a simple curve that divides Πn into two +connected components. A sector σ ⊂ Πn is the union of r1 ∪ {0} ∪ r2 and one of the connected components. +This is an example of a surface with boundary, with ∂σ = r1 ∪ {0} ∪ r2. As we move outward along one of +the boundary rays, σ is on the left. We call this ray the initial ray. As we move out along the other ray, the +sector σ is on the right. We call this second ray the terminal ray. A sector also has an interior angle defined +in Section 2.10, which measures the angle from the initial ray to the terminal ray. A half-plane sector is a +sector whose interior angle is π. We write σ∗ for σ ∖ {0}. +4.3. Horizontal foliations of sectors. Recall that Hn denotes the horizontal foliation of Πn. If σ ⊂ Πn is +a sector, its horizontal foliation is Hσ = Hn∣σ∗, which is really a foliation of σ∗. Note that the boundary rays +of a sector are either horizontal leaves or everywhere transverse to the horizontal foliation. See Figure 7 for +some examples. +Figure 7. Some sectors in Π0 with their horizontal foliations. +We say two sectors σ1 ⊂ Πm and σ2 ⊂ Πn have isomorphic horizontal foliations if there is an orientation- +preserving homeomorphism h ∶ σ1 → σ2 such that h(0) = 0 and h∣σ∗ +1 ∶ σ∗ +1 → σ∗ +2 is an isomorphism between +the horizontal foliations of these sectors. Observe: +Proposition 4.1. Two sectors σ1 and σ2 have isomorphic horizontal foliations if and only if the two state- +ments “the initial ray of σi is horizontal” and “the terminal ray of σi is horizontal” each have a truth value +independent of the choice i ∈ {1,2} and the number of horizontal rays in the two sectors is the same. +Proof. The listed conditions are clearly necessary for an isomorphism to exist. To see the converse, we need +to break into cases. Recall that the M¨obius action of PSL(2,R) on ˆR (obtained from the action of lines in +R2 through the origin) acts transitively on counterclockwise ordered triples in ˆR. So, if σ1,σ2 ⊂ Π−1 are two +sectors that do not contain the horizontal ray in Π−1, there is an affine map of Π−1 which carries σ1 to σ2 +and preserves the horizontal foliation. Every sector σ ⊂ Πn containing no horizontal rays is isomorphic to + +24 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +such a sector in Π−1 under an isomorphism obtained by restricting the covering map Πn → Π−1 to σ, so this +handles the case when there are no horizontal rays in the sectors. +Now suppose that the same number of horizontal rays of σ1 ⊂ Πm and σ2 ⊂ Πn exist. By lifting to a +common cover, we can assume that m = n and the complementary angles of the sectors are at least 2π. By +rotation of Πn, we can assume that if the initial rays are both horizontal, then they are the same, and if they +are not horizontal then they lie in the same half-plane Hi with horizontal boundary. We can act affinely +on Hi as before while preserving the horizontal foliation to make the portions of the sectors in Hi coincide. +Then because the sectors contain the same number of horizontal rays, the terminal rays coincide if they are +horizontal or lie in the same half-plane Hr with horizontal boundary if not. Again, we can act affinely to +make the sectors coincide. In this case, there is a piecewise affine map between the sectors that gives an +isomorphism of the foliations. +□ +4.4. The model space. Let S be a set of sectors, with one from each horizontal foliation isomorphism +class. Consider the model space +X∂ = Π−1 ⊔ +∞ +⊔ +n=1 +Πn ⊔ ⊔ +σ∈S +σ +with its horizontal foliation +H∂ = H−1 ⊔ +∞ +⊔ +n=1 +Hn ⊔ ⊔ +σ∈S +Hσ. +We let X∗ +∂ denote the space X∂ with the origin removed from each Πn and each sector in S. Then, H∂ is a +foliation of X∗ +∂. +Imitating the definition in Section 2.4, we define the pseudogroup G∂ to consist of all orientation-preserving +homeomorphisms h ∶ U → V between open subsets U,V ⊂ X∂ such that the following two statements hold: +(1) h(U ∩ X∗ +∂) = h(U) ∩ X∗ +∂. +(2) The restriction h∣U∩X∗ +∂ is in the foliation pseudogroup of (X∗ +∂,H∂). +Note that elements of G∂ send boundary points of sectors to boundary points of sectors. As in Section 2.4, +statement (1) guarantees that origins are sent to origins. In particular, if the origin in a sector is in the +domain of an element of G∂, then it must be sent to the origin of another sector. +It will be convenient to notice that (X∂,H∂) looks fairly uniform, with points having standard neighbor- +hood. We will use this to simplify our gluing arguments. Recall that U denotes the closed upper half-plane. +Let V = {(x,y) ∈ R2 ∶ x ≥ 0} be the closed right half-plane. Both half-planes are surfaces with boundary +that come with horizontal foliations obtained by restricting the horizontal foliation H on R2. +Proposition 4.2. Let U ⊂ X∂ be open and p ∈ U. Then: +(1) If p lies in the interior of X∂ and is not the origin of a Πn, then there is an open neighborhood +V ⊂ U ∩ Π∗ +n of p such that (V,H∂∣V ) is isomorphic to (R2,H) under a homeomorphism carrying p +to 0. +(2) If p ∈ ∂σ ∖ {0} for some sector σ, then there is an open interval I ⊂ U ∩ ∂σ ∖ {0} containing p such +that for any subinterval J ⊂ I containing p there is an open neighborhood of p, V ⊂ U, such that +V ∩∂σ = J and (V,H∂∣V ) is isomorphic to (U,H∣U) or (V,H∣V) and such that p ↦ 0 under this map. +(See Figure 8 for an example.) +(3) If p = 0 ∈ σ for some sector σ, then there is an open neighborhood V ⊂ U of p such that (V,H∂∣V ) is +isomorphic to (σ,Hσ). Furthermore, if σ contains at least one horizontal ray, then there is an open +interval I ⊂ U ∩ ∂σ containing 0 such that for any open subinterval J ⊂ I containing 0 there is an +open neighborhood V ⊂ U of p such that V ∩ ∂σ = J and (V,H∂∣V ) is isomorphic to (σ,Hσ). +(4) If p = 0 ∈ Πn, then there is an open neighborhood V ⊂ U of p such that (V,H∂∣V ) is isomorphic to +(Πn,Hn). +Observe that in particular, this means that we can think of (R2,H), (U,H∣U) and (V,H∣V) as part of our +model space Xσ: These spaces are all realizable up to isomorphism by subsets of X∂. +Proof. To see (1), we choose V ⊂ U to be an open rectangle in X∗ +∂ with center p. Identify V with (− π +2 , π +2 )2 +affinely and in these coordinates define the map to R2 by +(10) +(x,y) ↦ (tanx,tany). +Observe that this map is an isomorphism as desired. +To see (2), observe that using Proposition 4.1, we can assume that the boundary ray of σ containing p +is either vertical or horizontal. Then we can find a rectangle R ⊂ U containing p that intersects ∂σ in an + +ZEBRA SURFACES +25 +Figure 8. Left: The situation of statement (2) of Proposition 4.2. The parallelogram has +a foliation isomorphic to (V,H∣V). +Right: A sector containing subspaces isomorphic to +(R2,H), (U,H∣U) and (V,H∣V). +interval I ⊂ ∂σ ∖ {0}. Then for any J ⊂ I containing p, we can define V ⊂ R to be a smaller rectangle +with boundary J. Place coordinates on V of the form (− π +2 , π +2 ) × [0, π +2 ) or [0, π +2 ) × (− π +2 , π +2 ) where p is given +coordinates of (0,0). Then (10) defines a map to U or V, respectively that is an isomorphism as claimed. +Now consider the first statement of (3) in the case when σ has no horizontal rays. In this case σ has +isomorphic horizontal foliations to the sector depicted on the left side of Figure 7. So, assume that all σ ⊂ Π0 +which we identify with R2. Rays in σ are uniquely determined by their slope m, and the horizontal leaves are +uniquely determined by their y-coordinate in R. We can assume by possibly applying a 180○ rotation that +y ≥ 0 on σ. Then (m,y) is a coordinate system on σ. Given U containing p = 0 ∈ σ, there is a y0 > 0 such that +the triangle V = {(m,y) ∈ σ ∶ y < y0} ⊂ U. Then the homeomorphism V → σ given by (m,y) ↦ (m,tan πy +2y0 ) +is an isomorphism from (V,Hσ∣V ) to (σ,Hσ) as desired. +In case σ has a horizontal ray, the second statement of (3) implies the first. To see the second statement +holds, suppose p = 0 ∈ σ where σ has a horizontal ray. Then by Proposition 4.1, we can assume that σ has +horizontal and vertical boundary rays. The interior angle of σ is nπ +2 +for some n ≥ 1. Then we can find a +neighborhood N ⊂ U of 0 that is a union of n rectangles meeting edge-to-edge, with each rectangle in a +subsector whose interior angle is π +2 with horizontal and vertical sides. See Figure 9. We let I = N ∩∂σ which +is an interval. For any open subinterval J ⊂ I containing 0, we can construct a neighborhood V ⊂ N of 0 +from a union of n rectangles meeting edge-to-edge such hat V ∩ ∂σ = J. Then each rectangle R of V can be +mapped homeomorphically onto the subsector of σ containing the rectangle via a map such as (10). Observe +that the maps agree along the common edges of rectangles, and so together they define a homeomorphism +V → σ that gives the desired isomorphism. +Figure 9. A neighborhoods V and N of 0 built from rectangles in an open subset U of a +sector with horizontal and vertical sides and interior angle 3π +2 . +A similar argument proves (4), because a neighborhood V ⊂ U of 0 ∈ Πn can be constructed from a union +of 2n rectangles with 0 as a vertex meeting edge-to-edge. +□ + +26 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Let Y be a surface possibly with boundary and F be a foliation of a subset Y ∗ ⊂ Y . +(We are only +interested in the cases when Y = Πn, Y is a sector, or Y ∈ {U,V}.) Note that we can recover Y ∗ from F as it +is the union of all leaves in F. We’ll say that an automorphism of (Y,F) is a homeomorphism h ∶ Y → Y such +that h(Y ∗) = Y ∗ and h∣Y ∗ is an isomorphism from (Y ∗,F) to itself. The collection of orientation-preserving +automorphisms forms the group Aut+(Y,F). +Proposition 4.3. +(1) If (Y,F) ∈ {(U,H∣U),(V,H∣V)} then Y ∗ = Y and any orientation-preserving +homeomorphism h ∶ ∂Y → ∂Y can be continuously extended to an element of Aut+(Y,F). +(2) If σ is a sector with at least one horizontal ray, then any orientation-preserving homeomorphism +h ∶ ∂σ → ∂σ such that h(0) = 0 can be continuously extended to an element of Aut+(σ,Hσ). +(3) If σ is a sector that contains no horizontal rays, then there is a unique involution ι ∶ ∂σ ∖ {0} → +∂σ ∖ {0}, swapping the two components, such that for all p ∈ ∂σ ∖ {0}, the leaf of Hσ through p +also passes through ι(p). If h ∶ ∂σ → ∂σ is any orientation-preserving homeomorphism such that +h(0) = 0 and such that h∣∂σ∖{0} commutes with ι, then h can be continuously extended to an element +of Aut+(σ,Hσ). +Proof. In statement (1), Y is a subset of R2 and is naturally a product space Y = ∂Y × [0,+∞) where leaves +are fibers of one of the coordinate projections. Observe that if f ∶ [0,+∞) → [0,+∞) is any homeomorphism +then h × f ∈ Aut+(Y,F). +Consider the case of (2) when σ has one horizontal boundary ray and one vertical boundary ray and an +interior angle of π +2 . Then we can naturally identify σ with ¯r1 × ¯r2 where r1 and r2 are the boundary rays +whose closures ¯ri include 0. The foliation H∂ consists of the fibers of the perpendicular projection from +σ∗ to the vertical boundary ray. Given h as in statement (2), the map h restricts to two homeomorphisms +h1 ∶ ¯r1 → ¯r1 and h2 ∶ ¯r2 → ¯r2. The product h1 × h2 lies in Aut+(σ,Hσ). +Now consider the general case of a sector σ with at least one horizontal ray. By Proposition 4.1, we may +assume that the boundary rays of σ are either horizontal or vertical. Then the interior angle is nπ +2 for some +integer n ≥ 1, and there are a total of n + 1 horizontal and vertical rays, say r0,...,rn in counterclockwise +order. As in the previous part, an h as in (2) determines homeomorphisms h0 ∶ ¯r0 → ¯r0 and hn ∶ ¯rn → ¯rn. +Choose arbitrary homeomorphisms hi ∶ ¯ri → ¯ri for the other rays. Observe that the horizontal and vertical +rays partition σ into n-subsectors σ1,...,σn each bounded by ¯ri−1 and ¯ri, and as in the previous part we +can think of each σi as the product ¯ri−1 × ¯ri. Then the map which restricts to the product hi−1 × hi on each +σi lies in Aut+(σ,Hσ). This completes the proof of (2). +Now let σ be a sector that has no horizontal rays and consider (3). Then up to isomorphism, we may +think of σ as lying in the upper half-plane of Π0 which we can identify with R2. The leaves in Hσ are then +fibers of the restriction to σ∗ of the projection πy ∶ R2 → R. Here πy(σ∗) = (0,+∞) while πy(0) = 0. Note +that each y ∈ (0,+∞) has two preimages in ∂σ ∖ {0} and ι must swap them. This shows that ι exists and is +unique as claimed. If h is as in statement (3), then h induces a homeomorphism hy ∶ [0,+∞) → [0,+∞) from +its action on leaves. Now consider the stellar function ρ0 ∶ Π∗ +0 → ˆR. Observe that ρ0(σ∗) is a closed interval +I with 0 /∈ I, and +(11) +(πy × ρ0)∣σ∗ ∶ σ∗ → (0,+∞) × I +is a homeomorphism. If f ∶ I → I is any orientation-preserving homeomorphism, then the homeomorphism +of σ whose restriction to σ∗ is conjugate under (11) to hy × f is in Aut+(σ,Hσ). +□ +4.5. Definition of singular foliation on surfaces with boundary. Let S be a surface with boundary. +A singular foliation atlas on S is an atlas of charts to X∂ such that transition functions lie in G∂. Observe +that such an atlas induces a singular foliation on the interior S○ in the sense of Section 2.4, because the +restriction of a chart φ ∶ U → σ induces a chart from U ∩ S○ → Πn obtained by i ○ φ∣U∩S○ where i ∶ σ○ → Πn +denotes the inclusion of the interior of a sector into the Πn containing it. Thus we get a singular set Σ ⊂ S○ +that is discrete and closed and a singular data function α ∶ S○ → Z≥−1 supported on Σ as before. Observe +that Σ is closed as a subset of S, because each point p ∈ S has a neighborhood N such that N ∖{p} contains +no singular points. +Given S with a singular foliation atlas, a point v ∈ ∂S is called a vertex if there is a chart φ ∶ U → X∂ +such that v ∈ U and φ(v) is the origin in a sector. Again, each point p ∈ S has a neighborhood N such that +N ∖ {p} contains no vertices, so the collection V ⊂ ∂S of all vertices is both closed and discrete. An edge +of S is a connected component of ∂S ∖ V. Recalling that ∂S is a 1-manifold, we see that each edge is also + +ZEBRA SURFACES +27 +a 1-manifold. It is important to note that edges can be homeomorphic to an open interval or to a circle +(because a component of ∂S might be homeomorphic to a circle and not contain any vertices). +An edge e will be said to be incident to a vertex v if v ∈ ¯e. Each v has two incident edges (counting with +multiplicity, since there could be an edge e and a vertex v such that e∪{v} is homeomorphic to a circle). We +will name these two edges to match the terminology used for sectors: As we travel around ∂S with S○ on the +left, we move from the terminal incident edge of v, to v, and then to the initial incident edge to e. Because +these two edges may be counted with multiplicity, it is useful to distinguish them with an orientation: We +orient them away from v. +Given a singular foliation atlas on a surface S with boundary, we get a foliation equivalence relation on +S ∖ (Σ ∪ V) as before: the coarsest one for which two points are equivalent if they can be connected by an +arc whose image under a chart is contained in a leaf of the foliation H∂ of X∂. Leaves are again equivalence +classes, and the foliation associated to the atlas is the collection of equivalence classes. Leaves get a topology +by restricting the charts to connected components of intersections with charts as before. Observe that a +leaf of (σ,Hσ) that is transverse to a boundary ray has boundary, so leaves ℓ of our singular foliation are +1-manifolds some of which have non-empty boundary ∂ℓ. +Let F be a singular foliation. We’ll say F is transverse to an edge e of F if for any p ∈ e, the point p is in +the boundary of the leaf containing p. +Proposition 4.4. For each edge e, either e is a leaf or the singular foliation is transverse to e. If a leaf ℓ +is not an edge, then its interior ℓ○ is contained in S○ and every point in ∂ℓ lies in an edge that is transverse +to the singular foliation. +As a consequence we see that each edge e is either a leaf edge meaning it is also a leaf, or a transverse +edge meaning that the singular foliation is transverse to e. +Proof. Consider the first statement. Let p ∈ e and φ ∶ U → X∂ be a chart with p ∈ U. By (2) of Proposition 4.2, +there is a neighborhood Np ⊂ U containing p such that (Np,F∣Np) is isomorphic to either (U,H∣U) or (V,H∣V). +Doing this for every point in e, gives local isomorphisms to these spaces covering e. But observe that whenever +two neighborhoods Np and Nq contain a common point of e, they must have the same type (either isomorphic +to U or V) because in (U,H∣U) the boundary is a leaf and in (V,H∣V) leaves hit the boundary transversely. +Since e is connected, it follows that all points have neighborhoods of the same type. If each Np is isomorphic +to (U,H∣U), then e is a leaf edge and otherwise it is a transverse edge. +Now consider the last statement. Let ℓ ∈ F be any leaf and suppose there is a point p ∈ ℓ ∩ ∂S. Then p +must lie in some edge e. If e is a leaf edge, then we must have ℓ = e. Otherwise e is a transverse edge, and +so p ∈ ∂ℓ. Thus if ℓ is not an edge, every point of ∂ℓ is contained in a transverse edge and ℓ○ ⊂ S○. +□ +Now we give a more detailed description of how a singular foliation on a surface with boundary gives rise +to a singular foliation of its interior, explaining exactly how the leaves of the two foliations are related: +Corollary 4.5. If F is a singular foliation of a surface S with boundary, then +F○ = {ℓ○ ∶ ℓ ∈ F and ℓ /⊂ ∂S} +is a singular foliation of S○. +Proof. By Proposition 4.4, each point p ∈ S○ ∖ Σ lies in the interior of a leaf of F. Thus F○ is a partition of +S○ ∖ Σ. To see F○ is a singular foliation, we need to see that it arises from a singular foliation atlas on S○. +We’ll obtain this atlas by altering the charts of the atlas determining F. If φ ∶ U → X∂ is a chart for the +structure on S and Ui ⊂ U is a connected component, then φ∣Ui∩S○ is either a map to Πn or to the interior of +a sector σ○ ⊂ Πn for some n. In the latter case, we obtain a new chart by post-composing φ∣Ui∩S○ with the +inclusion σ○ → Πn. To see the corresponding singular foliation F○ of S○ is as claimed, suppose that ℓ ∈ F +is a leaf with ℓ /⊂ ∂S. Then Proposition 4.4 tells us that that ℓ○ ⊂ S○. Now observe that any interval I ⊂ ℓ○ +that is contained in the domain of a chart φ ∶ U → X∂ will also be contained in the domain of altered charts +defined as above. Thus, ℓ○ ∈ F○ as desired. +□ +4.6. Constructing singular foliations. Formally a singular foliation F is a partition of a subset of a +surface that is determined from a singular foliation atlas. We offer a recipe for proving that a partition F is +a singular foliation: + +28 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Lemma 4.6. Let S be a surface with boundary. Let Σ ⊂ S○ and V ⊂ ∂S be sets such that Σ∪V is closed and +discrete. Suppose that F is a partition of S ∖(Σ∪V) into path connected subsets. Suppose further that there +is an atlas A of charts φ ∶ U → X∂ from S to X∂ such that each chart satisfies the two conditions: +(1) We have φ(U ∗) = φ(U) ∩ X∗ +∂, where U ∗ = U ∖ (Σ ∪ V). +(2) The partition of U ∗ given by connected components of intersections of elements ℓ ∈ F with U ∗ is the +same as the pullback of H∂∣φ(U ∗) under φ. +Then, A is a singular foliation atlas on S and F is the singular foliation determined by A. +Proof. We claim that A is a singular foliation atlas. To this end, let φ1 ∶ U1 → X∂ and φ2 ∶ U1 → X∂ be charts. +Let V ⊂ U1 ∩ U2 be a connected component of the intersection. We must show that the associated transition +function φ2 ○ φ−1 +1 +lies in G∂. But, since G∂ is a pseudogroup, it suffices to show that h = φ2 ○ φ−1 +1 ∣φ1(V ) is in +G∂. +Since V is connected, the images φ1(V ) and φ2(V ) must lie either in some Πn or in some sector σ ∈ S. +Then statement (1) of the lemma guarantees that there can be at most one point of Σ ∪ V in V , since there +is only one point in each Πn ∖ X∗ +∂ and each σ ∖ X∗ +∂. Furthermore, if such a point exists, it must be sent to +the origin in the containing Πn or σ. Thus h satisfies part (1) of the definition of G∂. +To see that part (2) of the definition of Gδ is satisfied by h, set V ∗ = V ∖ (Σ ∪ V). Observe that h must +be an isomorphism from (φ1(V ),H∂∣φ1(V ∗)) to (φ2(V ),H∂∣φ2(V ∗)) because both foliations pull back to the +same partition of V ∗ by statement (2) of the Lemma. Thus h∣φ1(V ∗) is in G∂ as desired. +Let F′ be the singular foliation determined by the atlas A. We need to show that F = F′. Let ℓ ∈ F. +Then given any path γ ∶ [0,1] → ℓ, γ stays within the same leaf of F′ locally because given any p ∈ ℓ there is +an open set U and a chart φ ∶ U → X∂ such that connected components of ℓ∩U are obtained by pullback just +as they are in the definition of F′. By hypothesis, elements of F are path connected, so this local agreement +guarantees that ℓ ⊂ ℓ′ for some ℓ′ ∈ F′. Now suppose that ℓ was a proper subset of ℓ′. Then, we can choose +a point p ∈ ℓ and a point q ∈ ℓ′ ∖ ℓ. Since ℓ′ is a connected 1-manifold, we can join p to q by a path η. Then +we can cover η by connected components of Ui ∩ η where each Ui comes from a chart φi ∶ Ui → X∂ in A. By +compactness, we can pass to a finite subcover by these connected components Ci ⊂ Ui∩η where we now index +by i ∈ {1,...,n}. Observe that if p and q are in different elements of F, then some component Ci contains +points both in ℓ and in ℓ′ ∖ℓ. But this would violate statement (2) of the lemma for the corresponding chart. +Thus in fact ℓ = ℓ′. +We’ve shown that each element of F is an element of F′. But, since both F and F′ are partitions of +S ∖ (Σ ∪ V), they must coincide. +□ +We have the following consequence, which indicates that every singular foliation has a nice atlas: +Proposition 4.7. Let F be a singular foliation on a surface S with boundary. Then there is a singular +foliation atlas for F such that every chart is a homeomorphism from an open set U to one of the spaces in +{R2,U,V} ∪ {Πn ∶ n = −1 or n ≥ 1} ∪ {σ ∶ σ ∈ S} +such that restriction to U ∗ = U ∖ (Σ ∪ V) is an isomorphism to the respective foliated space in +{(R2,H),(U,H∣U),(V,H∣V)} ∪ {(Π∗ +n,Hn) ∶ n = −1 or n ≥ 1} ∪ {(σ∗,Hσ) ∶ σ ∈ S}. +Furthermore, if D ⊂ S is any closed discrete subset, then such an atlas can be produced where each point of +D appears in the domain of exactly one chart, and such that each p ∈ (D ∩ S○) ∖ Σ is sent to the origin R2, +and each p ∈ (D ∩ ∂S) ∖ V is sent to the origin in U or V. +We remark that above we are thinking of R2, U and V as subspaces of X∂, see Figure 8. +Proof. By definition F is determined by an atlas A. We will define a new atlas A′ with maps as described +determining the same foliation. +Suppose a closed discrete set D has been provided. Fix a p ∈ D and a chart φ ∶ U → X∂ in A with +p ∈ U. By Proposition 4.2, for each chart φ ∶ U → X∂ in A and each p ∈ U, we can define a new chart +ψφ,p from a neighborhood Vφ,p ⊂ U ∖ (D ∖ {p}) of p to one of the spaces described and whose restriction to +V ∗ +φ,p = Vφ,p ∖(Σ∪V) is an isomorphism to the corresponding foliated space. Observe that these isomorphisms +will satisfy the last statement. Each of these will be included in the atlas A′. + +ZEBRA SURFACES +29 +For every point p ∈ X∂ ∖ D, define a chart ψφ,p from a neighborhood of p, Vφ,p ⊂ U ∖ D, in the same way. +Add these charts to A′. Now the domains of charts in A′ cover S, and there is only one chart containing +each point in D. +Observe that statements (1) and (2) of Lemma 4.6 are satisfied by A′ since the atlas came from restrictions +of charts determining F. Thus, Lemma 4.6 guarantees that F is the foliation determined by the atlas A′ as +desired. +□ +4.7. Removable vertices. By Proposition 4.1, there are only two half-plane sectors up to isomorphism. +These isomorphism classes are represented by the upper and right half-planes in Π0. Given a singular foliation +F on a surface S with boundary, a vertex v ∈ V is removable if there is a chart φ ∶ U → X∂ such that v ∈ U +and the connected component C ⊂ U containing v is mapped into a sector that is isomorphic to a half-plane +sector. The edges on either side of a removable vertex v are either both leaf edges or both transverse edges. +In removing v from the list of vertices, we either join these two leaf edges or add an endpoint to the single +leaf with v as an endpoint in the transverse case. +Lemma 4.6 can be used to prove the following result: +Proposition 4.8. Let F be a singular foliation on a surface S with boundary. Let Σ and V be the singularity +and vertex sets, respectively. If V′ ⊂ V is any collection of removable vertices, then there is a unique singular +foliation F′ on S whose vertex set is V ∖ V′ such that every leaf of F is contained in a leaf of F′. +Proof. By Proposition 4.7, there is an atlas A for F such that every point of Σ ∪ V appears in exactly one +chart, and such that charts are as described in that proposition. We will alter A to define a new atlas A′. +The only charts we alter are those whose domain contains a point in V′. Each v ∈ V′ appears in the domain +of a chart to a half-plane sector, which is isomorphic to either the upper or right half-plane of Π0. We replace +this chart with one to U or V obtained by isomorphism to the half-plane in Π0 followed by the natural map +to R2. Observe that A′ is a singular foliation atlas that determines a foliation F′ as described. +Uniqueness follows from properties of the singular foliations. If v ∈ V′, then because v has a neighborhood +isomorphic to a half-plane sector, the two edges meeting at v must be either both leaf edges or both transverse +edges. If two leaf edges meet at v, these two leaf edges must be part of the same leaf of F′. If the two edges +are transverse edges, then there is exactly one leaf whose closure contains v and the union of {v} and this +leaf is contained in a leaf of F′. These conditions determine F′. +□ +We also have the reverse construction: +Proposition 4.9. Let F be a singular foliation on a surface S with boundary. Let Σ and V be the singularity +and vertex sets, respectively. If V′ ⊂ ∂S ∖ V is any collections of points such that V ∪ V′ is a closed discrete +subset of ∂S, then there is a unique singular foliation F′ of S whose vertex set is V ∪ V′ such that every leaf +of F′ is contained in a leaf of F. +Proof. Again by Proposition 4.7, we can produce an atlas A for F as described there such that each point +of D = Σ ∪ V ∪ V′ appears in exactly one chart. Then a point p ∈ V′ must be mapped to the origin in either +U or V. Recall that there are sectors σU,σV ∈ S isomorphic to (U,H∣U∖{0}) and (V,H∣V∖{0}), respectively. +We form a new atlas A′ by post-composing the chart associated to each p ∈ V′ with the isomorphism from +the codomain (U or V) to σU or σV . This new atlas determines F′ as described. Uniqueness is clear: Leaves +of F′ must be the connected components of ℓ ∖ V′ taken over all ℓ ∈ F. +□ +5. Surgery on zebra surfaces +5.1. Gluing singular foliations. Let S be an oriented topological surface with boundary and F be a +singular foliation on S. We do not require S to be connected, and we could therefore obtain S as the disjoint +union of surfaces with singular foliations. We will explain how to glue edges of S to obtain a new surface +(perhaps with a smaller boundary) equipped with a singular foliation. +Recall that ∂S is a 1-manifold and edges are connected components of ∂S ∖ V where V ⊂ ∂S is a closed +discrete subset. Let E denote the collection of edges of S . Then ∂S = V ∪ ⋃e∈E e. The boundary of an edge +∂e consists of any vertices in the closure ¯e of e viewed as a subset of ∂S. Thus ∂e can consist of zero, one, +or two vertices. + +30 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +We will construct the quotient by gluing together closures of edges. Edges have a natural orientation: +If we move in the direction of the orientation, the interior S○ should be on the left. An edge gluing is an +orientation-reversing homeomorphism between the closures of two edges ¯e1 → ¯e2. +An edge identification scheme for (S,F) consists of a collection of edges E ⊂ E, a fixed-point free involution +ε ∶ E → E, and a choice of an edge gluing for each e ∈ E of the form: +ge ∶ ¯e → ε(e) +such that ge ○ gε(e) is the identity on ¯e for each e ∈ E. +An edge e ∈ E will be called glued if e ∈ E and unglued if e /∈ E. +An edge identification scheme determines an equivalence relation ∼ on ⋃e∈E ¯e: The finest equivalence +relation such that each p ∈ ¯e is equivalent to ge(p) ∈ ε(e). We extend ∼ to an equivalence relation on all of S +by defining ∼-equivalence classes of a point p ∈ S ∖ ⋃e∈E ¯e to be [p] = {p}. Let π ∶ S → S/ ∼ be the quotient +projection. +Observe that edge gluings send vertices to vertices, because these are endpoints of edges, so the ∼- +equivalence class of a vertex [v] consists only of vertices. Let V/ ∼ denote the equivalence classes in S/ ∼ +containing vertices. We introduce several notions related to identified vertices. We say [v] is finite if it is +finite as an equivalence class of vertices of V. We say [v] is completely glued if for each e ∈ E containing a +point of [v] in its closure is glued. Finally, we let Pr([v]) ∈ Z≥0 ∪ {∞} denote the total number of prongs of +F taken over all v ∈ V , with each pair of glued leaf edges counting only once. +We will explain how an edge identification scheme satisfying certain requirements leads to a singular +foliation on the quotient surface S′. These requirements are: +(a) Each [v] ∈ V/ ∼ is finite. +(b) If [v] ∈ V/ ∼ is completely glued, then Pr([v]) ≥ 1. +(c) For each e ∈ E, the edge e is a leaf edge if and only if ε(e) is a leaf edge. +Note that condition (c) says that glued edges must have the same type, because there are only two edge +types: leaf edges and transverse edges. +Theorem 5.1 (Surgery on singular foliations). Let S and F be as above and suppose (E,ε,{ge ∶ e ∈ E}) +is an edge identification scheme satisfying conditions (a)-(c). Then S′ = S/ ∼ is a surface with boundary +∂S′ = V′ ∪ ⋃e′∈E′ e′, where +V′ = {[v] ∈ V/ ∼ ∶ [v] is not completely glued} +and +E′ = {π(e) ∶ e ∈ E ∖ E}. +Also define +Σ′ = π(Σ) ∪ {[v] ∈ V/ ∼ ∶ [v] is completely glued and Pr([v]) ≠ 2}. +Let F′ be the finest partition of S′ ∖ (Σ′ ∪ V′) such that the following two statements hold: +(1) Images of leaves of F are contained in leaves of F′, i.e., for each ℓ ∈ F, we have π(ℓ) ⊂ ℓ′ for some +ℓ′ ∈ F′. +(2) If v ∈ V is a vertex such that [v] is completely glued and Pr([v]) = 2, and the leaf ℓ ∈ F contains a +prong emanating from v, then we have that [v] is contained in the same leaf of F′ as π(ℓ). +Then F′ is a singular foliation on S′, with singular set Σ′, vertex set V′, edge set E′ and singular data +function α′ ∶ (S′)○ → Z≥−1 whose only non-zero values are given by α′([p]) = α(p) if [p] ∈ π(Σ) and +α′([v]) = Pr([v]) − 2 if [v] ∈ Σ′ ∖ π(Σ). +Proof. Let A be a singular foliation atlas for (S,F). We will use A to produce an atlas of charts A′ from +S′ to X∂ satisfying Lemma 4.6. Existence of this atlas will prove that S′ is a surface with boundary and +Lemma 4.6 will guarantee that the atlas is a singular foliation atlas whose singular foliation is the partition +F′ as described above. Since the connected components of the codomain of each chart consists of spaces +homeomorphic to R2 or U, it will follow that S is a surface with boundary. Attention to the definition of +the charts will verify that ∂S′ is as claimed. +In order for this to work, we need to check that the conditions of Lemma 4.6 are satisfied. One condition +that needs checking is that the elements of F′ are path connected. We verify this now. For ℓ ∈ F, let ℓ● +denote the union of ℓ and any v ∈ V with [v] completely glued and Pr([v]) = 2 such that there is a prong +emanating from v and contained in ℓ. Observe that ℓ● is homeomorphic to an interval in R, since the added +points would have to be endpoints of an open end of ℓ. Observe that F′ (as defined in the statement of +the Theorem) can be seen to be the finest partition of S′ ∖ (Σ′ ∪ V′) such that for each ℓ ∈ F, the image +π(ℓ●) is contained in a partition element of F′. Then by definition, for any two points [p],[q] ∈ S′ that lie + +ZEBRA SURFACES +31 +in the same partition element of F′, there is a finite sequence of leaves ℓ1,...,ℓn ∈ F such that [p] ∈ π(ℓ● +1), +[q] ∈ π(ℓ● +n), and π(ℓ● +i) ∩ π(ℓ● +i+1) ≠ ∅ for i = 1,...,n − 1. We can then explicitly describe a path from [p] to +[q] in S′ by moving along π(ℓ● +1) from [p] to a point of intersection with π(ℓ● +2) then along π(ℓ● +2) to a point +of intersection with π(ℓ● +3), et cetera. +We will also need to check that the atlas A′ we produce satisfies statements (1) and (2) of Lemma 4.6. +But these conditions pertain only to individual charts. So these conditions will be checked as we define our +charts for S′. We will define charts covering any point not taking part in the gluing, then charts handling +each point on a glued edge, and finally charts including any vertex taking part in the gluing. +The set of points that take part in the gluing construction is G = ⋃e∈E ¯e. We claim that G is closed. Since +G ⊂ ∂S and ∂S is closed, we know that ¯G ⊂ ∂S. Now recall that every point of ∂S is either in an edge or is +a vertex. But edges are open in ∂S, and a vertex v ∈ V has a neighborhood in ∂S consisting of at most two +edges e with v ∈ ∂e. Thus, ¯G ∖ G contains neither edges nor vertices, and therefore ¯G = G as desired. +Because no gluing is taking place in the complement of G, the map π induces an isomorphism between +(S ∖ G,F∣S∖G) and (π(S ∖ G),F′∣π(S∖G)). (By definition F′∣π(S∖G) consists of the connected components of +ℓ′ ∖π(G) taken over ℓ′ ∈ F′.) Given our atlas A, the collection of restricted charts φ∣U∖G ∶ U ∖G → X∂ taken +over all φ ∶ U → X∂ in A gives a singular foliation atlas for F∣S∖G. Thus pre-composing these charts with +π−1 gives a singular foliation atlas for F′∣π(S∖G), namely the collection of charts +φ ○ π∣−1 +S∖G ∶ π(U ∖ G) → X∂ +taken over all φ ∶ U → X∂ in A. We include each such chart in A′. Since A was a singular foliation atlas +for (S,F), these charts cover π(S ∖ G) and form a singular foliation atlas for F′∣π(S∖G). Since they are a +singular foliation atlas, each individual chart satisfies statements (1) and (2) of Lemma 4.6. +We will now define a chart of A′ for each pair (e1,p1) consisting of an edge e1 ∈ E and a point p1 ∈ e1. +Then [p1] = {p1,p2} where p2 = ge1(p1). Define e2 = ε(e1). Statement (2) of Proposition 4.2 guarantees we +can restrict charts of A and post-compose these restricted charts with an isomorphism to obtain new charts +for the original structure of the form +ψi ∶ Vi → H +for i ∈ {1,2}, +where V1 and V2 are disjoint neighborhoods of p1 and p2 respectively, H ∈ {U,V} is independent of i, and +ge1(e1 ∩V1) = e2 ∩V2. Then using statement (1) of Proposition 4.3 we can alter ψ2 by post-composition with +an automorphism of (H,H∣H) to ensure +(12) +ψ2 ○ ge(q) = −ψ1(q) +for any q ∈ e1 ∩ V1. +Then we can define the glued neighborhood and corresponding chart +W = π(V1 ∪ V2) +and +χ ∶ W → R2; +χ ○ π(q) = +⎧⎪⎪⎨⎪⎪⎩ +ψ1(q) +if q ∈ V1, +−ψ2(q) +if q ∈ V2. +This map χ is a well-defined homeomorphism since V1 and V2 are disjoint, and the only gluing taking place +is according to ge and our charts were built to satisfy (12). We include χ in our atlas A′ by identifying +R2 with a subset of X∂ as in Proposition 4.2. The domain W contains no singularities so statement (1) +of Lemma 4.6 is automatically satisfied. Also observe that statement (2) of Lemma 4.6 is satisfied, i.e., χ +induces a bijection between the collection of connected components of ℓ′ ∩ (W ∖ {[v]}) taken over all ℓ′ ∈ F′ +and the elements of the horizontal foliation H of R2, because leaves of F′ pass through the glued edges +according to the edge gluing. +Now suppose [v] is an equivalence class of vertices that is not completely glued. For each such [v] we will +define a single chart on a domain containing [v]. Condition (a) tells us that [v] is finite. Because our edge +gluings are required to be orientation-reversing, whenever two vertices v and v′ are identified by a gluing +map ge, the initial incident edge of one of the vertices must be glued to the terminal incident edge of the +other vertex. Say v′ ∈ [v] is the successor of v if the terminal incident edge of v is glued to the initial incident +edge of v′. From the preceding remarks, we can enumerate [v] = {v1,...,vn} such that vi+1 is the successor +of vi for i = 1,...,n − 1. Let e− +i denote the initial incident edge to vi and let e+ +i denote the terminal incident +edge. Let gi ∶ e+ +i → e− +i+1 denote the gluing maps. For each i, fix a chart φi ∶ Ui → X∂ such that vi ∈ Ui. By +restricting these charts to smaller open subsets we may assume the Ui are pairwise disjoint. By statement +(3) of Proposition 4.2, for each i, we can choose open intervals I− +i ⊂ e− +i and I+ +i ⊂ e+ +i with vi ∈ ∂I± +i such that: + +32 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +(1) We have gi(I+ +i ) = I− +i+1 for i = 1,...,n − 1, +(2) For each i ∈ {1,...,n − 1}, there is an open subset Vi ⊂ Ui such that Vi ∩ ∂S = I− +i ∪ {vi} ∪ I+ +i and +such that there exists a sector σi and a homeomorphisms ψi ∶ Vi → σi that is an isomorphism from +(Vi,F∣Vi) to (σi,Hσi). +Observe that condition (c) guarantees that for i ∈ {1,...,n − 1}, the terminal ray of σi is horizontal if and +only if the initial ray of σi+1 is horizontal. Let σ′ be the a sector such that the initial ray of σ′ is horizontal if +and only if the initial ray of σ1 is horizontal, such that the terminal ray of σ′ is horizontal if and only if the +terminal ray of σn is horizontal, and such that σ′ has the same number of horizontal rays as the total number +of horizontal rays in {σ1,...,σn} with glued rays counting only once. Using Proposition 4.1, observe that σ′ +can be be decomposed by cutting along rays into sectors σ′ +1,...,σ′ +n arranged in counterclockwise order such +that each σi has a horizontal foliation isomorphic to that of σ′ +i via a map hi ∶ σi → σ′ +i. We will define a chart +χ ∶ W → σ′ +where +W = π ( +n +⋃ +i=1 +Vi) +by defining each restriction χ∣π(Vi) ∶ Vi → σ′ +i. We define +(13) +χ∣π(V1) = h1 ○ ψ1 ○ π∣−1 +V1. +Then proceeding by induction, assuming i ∈ {1,...,n − 1} and χ∣π(Vi) has been defined, we define +(14) +χ∣π(Vi+1) = αi+1 ○ hi+1 ○ ψ1 ○ π∣−1 +V1, +where αi+1 ∶ σ′ +i+1 → σ′ +i+1 is an automorphism chosen so that for all q ∈ e− +i+1, we have +(15) +χ∣π(Vi+1)(q) = χ∣π(Vi) ○ ge− +i+1(q). +Such an automorphism αi+1 exists by statement (2) or (3) of Proposition 4.3. Equation (15) guarantees that +χ is well defined. We include χ in our atlas A′. The only singular point in the domain is [v] which is sent +to 0 ∈ σ′, so χ satisfies statement (1) of Lemma 4.6. The identifications made by π on V1 ∪ ⋅⋅⋅ ∪ Vn glue [v] +to one point and each q ∈ e− +i+1 to ge− +i+1(q) ∈ e+ +i , just as the map χ does. By construction, the restriction χ∣Vi +sends leaves of F∣Vi to leaves of Hσ′ +i. Since the only gluings happening on π−1(W) are between the edges +already discussed, χ satisfies statement (2) of Lemma 4.6. +Now suppose [v] is an equivalence class of vertices that is completely glued. Again we will define a single +chart for A′ on a domain containing [v]. Much of the structure is the same, so we maintain as much of the +notation as possible. Here because [v] is completely glued, every v ∈ [v] has a successor. Thus it is natural +to enumerate [v] as +[v] = {vi ∶ i ∈ Z/nZ} +where n is the cardinality of [v] +so that for all i, vi+1 is the successor of vi. We define the charts φi ∶ Ui → X∂, intervals I− +i ⊂ e− +i and I+ +i ⊂ e+ +i , +gluing maps gi, open subsets Vi ⊂ Ui, sectors σi, and homeomorphisms ψi ∶ Vi → σi as before except that +now we require any statement relating an object with index i to an object with index i + 1 to be true for all +i ∈ Z/nZ. Let m = Pr([v]) − 2. Recall that Condition (b) guarantees that m ≥ −1. Then there is a partition +of Πm into sectors σ′ +1,...,σ′ +n cyclically ordered such that σi and σ′ +i always have an isomorphic horizontal +foliation. Cyclically shift the indexing so that vn has a horizontal prong. Then σn and σ′ +n have horizontal +rays. Now we will define a chart +χ ∶ W → Πm +where +W = π ⎛ +⎝ ⋃ +i∈Z/nZ +Vi +⎞ +⎠ +inductively. We first define χ∣π(V1) to σ′ +1 as in (13). Then we inductively define χ∣π(Vi+1) to σ′ +i+1 for i ∈ +{1,...,n − 2} as in (14) with each αi+1 defined so that (15) holds as before. It remains to define χ∣π(Vn) ∶ +π(Vn) → σ′ +n. Recalling that σ′ +n has a horizontal ray, we can define χ∣π(Vn) as in (14) where this time +αn ∶ σ′ +n → σ′ +n is chosen to satisfy the stronger condition that for each q− ∈ e− +n and each q+ ∈ e+ +n we have +(16) +χ∣π(Vn)(q−) = χ∣π(Vn−1) ○ ge−n(q−) +and +χ∣π(Vn)(q+) = χ∣π(V1) ○ ge+n(q+). +To find such an αn, we need that σ′ +n has a horizontal ray so that we can use the stronger statement (2) of +Proposition 4.3. Again χ is a well-defined homeomorphism because it respects all gluing maps. Assuming +m ≠ 0, we add χ to the atlas A′. In this case, [v] is a singular point α′([v]) = m as in the Theorem. Observe +that statements (1) and (2) of Lemma 4.6 are satisfied. If m = 0, then we alter the chart slightly by defining + +ZEBRA SURFACES +33 +χ′ = ι ○ χ, where ι ∶ Π0 → R2 is a homeomorphism that restricts to an isomorphism from (Π0 ∖ {0},H0) +to (R2 ∖ {0},H∣R2∖{0}). We include χ′ ∶ W → R2 into A′ by identifying R2 with a subset of X∂ as in the +edge gluing case. In this case statement (1) of Lemma 4.6 is satisfied because W contains no singularities. +Statement (2) is satisfied because χ′ respects the edge identifications and because χ′ joins the two leaves +containing prongs of points in [v]. +Since we have covered S′ = S/ ∼ by charts, this is a surface with boundary consisting only of vertices +that were not completely glued and the union of edges not taking part in the gluing. Furthermore, since +statements statements (1) and (2) of Lemma 4.6 are satisfied for all charts, A′ is a singular foliation atlas +whose singular foliation is F′. +□ +5.2. Stellar functions. Let σ ⊂ Πn be a sector. +Recall the definition of the stellar function on Πn in +Section 2.6. The stellar function on σ is the restriction +ρσ = ρn∣σ ∶ σ∗ → ˆR. +This function maps each point in a ray of σ to the slope of the ray. +Two sectors σ1 and σ2 are stellar equivalent if there is an orientation-preserving homeomorphism h ∶ σ1 → +σ2 such that h(0) = 0 and ρσ1 = ρσ2 ○ h on σ∗ +1. Observe: +Proposition 5.2. Two sectors are stellar equivalent if and only if their initial rays have the same slope and +their interior angles are the equal. +Proof. If such an equivalence h exists, then it must send the initial ray to the initial ray and so the slopes +of the initial rays are the same. Also the internal angle of the sector can be determined by the number of +rays of each slope. To see the converse assume σ1 ⊂ Πm and σ2 ⊂ Πn are two sectors with initial rays of the +same slope and the same internal angles. Let πm ∶ Πm → Π−1 and πn ∶ Πn → Π−1 denote the covering maps. +There is an h ∶ σ1 → Πn satisfying πm = πn ○ h obtained by lifting the restriction of πm to σ1. Namely, both +initial rays have the same image in Π−1, so we can define h on the initial ray of σ1 so it sends this ray to +the initial ray of σ2. Then because σ1 is simply connected, we can extend to all of σ1 in a unique way. The +image h(σ1) must be a sector with the same initial ray as σ2 and have the same angle, so σ2 = h(σ1) and h +can be seen to be a stellar equivalence. +□ +Corollary 5.3. Stellar equivalence is a finer equivalence relation than the relation of being horizontal foli- +ation isomorphic. +Proof. Suppose σ1 and σ2 are stellar equivalent. Then by Proposition 5.2, the initial rays have the same slope +and they have the same interior angles. It also therefore follows that the terminal rays have the same slope. +Thus by statement (1) of Proposition 4.1, the sectors σ1 and σ2 have isomorphic horizontal foliations. +□ +Later, we will need to glue sectors together in a manner respecting their stellar functions. Let σ be a +sector. We’ll say that a stellar epimorphism ψ ∶ U → σ is a homeomorphism whose domain U is an open +subset of σ containing 0 such that ψ(0) = 0 and such that for any ray r ⊂ σ∗, ψ(r∩U) = r. It then follows that +ρσ ○ ψ(p) = ρσ(p) for p ∈ σ∗. The following three propositions ensure we can construct stellar epimorphisms +that are useful for our gluing constructions. None of these proofs are difficult, and we’ll leave all but the last +to the reader. +Proposition 5.4. If σ is a sector and C ⊂ σ is a closed subset that does not contain 0, then there is a stellar +epimorphism ψ ∶ U → σ with U ∩ C = ∅. +Proposition 5.5. If σ is a sector and I ⊂ ∂σ is an open interval containing 0, then there is a stellar +epimorphism ψ ∶ U → σ with U ∩ ∂σ = I. +Proposition 5.6. If σ is a sector and ψ0 ∶ ∂σ → ∂σ is an orientation-preserving homeomorphism such that +ψ0(0) = 0, then there is a homeomorphism ψ ∶ σ → σ extending ψ0 such that ψ is also a stellar epimorphism. +Proof. Let {rt ∶ t ∈ [0,1]} denote the collection of rays of σ ordered counterclockwise, so r0 is the initial +ray and r1 is the terminal ray. Since ψ0 is orientation-preserving and ψ0(0) = 0, we know ψ0(r0) = r0 and +ψ0(r1) = r1. For t ∈ {0,1}, define ht ∶ [0,+∞) → [0,+∞) such that ht(0) = 0 and we have +d(0,ψ0(p)) = ht(d(0,p)) +if p ∈ rt. + +34 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Then since ψ0 is a homeomorphism, both h0 and h1 are homeomorphisms. We extend our definition of ht +to include t ∈ (0,1) by defining +ht(x) = (1 − t)h0(x) + th1(x) +for all x ∈ [0,+∞). +Then each ht is a homeomorphism and the map (t,x) ↦ ht(x) is continuous. Observe that the function +ψ ∶ σ → σ defined by ψ(0) = 0 and ψ(p) = q if p ∈ rt where q ∈ rt is the point such that d(0,q) = ht(d(0,p)) +is a stellar epimorphism such that the restriction to ∂σ is ψ0. +□ +5.3. Zebra structures on surfaces with boundary. Let S be a surface with boundary and let {Fm}m∈ˆR +be a collection of singular foliations on S with the same singular set Σ, the same singular data α ∶ S → Z≥−1, +and the same vertex set V. +A stellar neighborhood of a point p in the interior S○ is defined exactly as in Section 2.7. If p ∈ ∂S, a +stellar neighborhood of p is an open neighborhood U of p such that there is a sector σ and a homeomorphism +h ∶ U → σ such that h(p) = 0 and the following statements hold for each slope m ∈ ˆR: +(1) For each ray r ⊂ σ of slope m, h−1(r) is contained in a leaf of Fm. +(2) For each prong of Fm emanating from p, there is a ray r ⊂ σ∗ of slope m such that for any path +γ ∶ (0,1) → r with limt→0+ γ(t) = 0, the preimage h−1 ○ γ represents the prong. +This homeomorphism h is called a stellar homeomorphism. We remark that statement (1) guarantees that if +U is a stellar neighborhood U of a point p, then there can be at most one singularity or vertex in U, namely +p itself. +We say {Fm} is a stellar foliation structure or a zebra structure on a surface S with boundary if the +singular foliations all have the same singular set, same singular data, and same vertex set, and each point +p ∈ S has a stellar neighborhood. The pair (S,{Fm}) will be called a zebra surface with boundary. +Again a leaf in a zebra surface is a leaf of any of the foliations Fm. Because the foliations all have the +same singular set and the same vertices, they also have the same edges. Each edge is a leaf: +Proposition 5.7. Let {Fm} be a zebra structure on a surface S with boundary. Then: +(1) For each edge e, there is a unique m such that e is a leaf of Fm. +(2) If p is a point of an edge e, then the stellar homeomorphism associated to p is a homeomorphism to +a half-plane sector. +The m appearing in statement (1) is the slope of e. +Proof. Let e be an edge. First we will establish that if e is both a leaf of Fm1 and of Fm2 then m1 = m2. +Choose any p ∈ e. Then the two ways of approaching p within e determine two prongs at p of each of the two +singular foliations Fm1 and Fm2. Let h ∶ U → σ be a stellar homeomorphism associated to p. Each prong +must arise from a ray, so we find there are two rays of slope m1 and two rays of slope m2 that give rise to +the prongs. But observe that these rays must be contained in ∂σ because h is a homeomorphism and their +preimages under h are contained in e ⊂ ∂S. We conclude that both pairs of rays are the boundary rays of σ. +Thus, m1 = m2 and boundary rays of ∂σ have the same slope. This proves the uniqueness part of (1) and +that the interior angle of σ is an integer multiple of π (since the boundary rays have the same slope). +We still need to show that e is a leaf of some Fm. Again let h ∶ U → σ be a stellar homeomorphism +associated to p ∈ e. Then h(p) = 0. Let m be the slope of the initial ray r of σ. From statement (1) of the +stellar neighborhood definition, h−1(r) is a subset of a leaf ℓ ∈ Fm. Because h is a homeomorphism from an +open subset of S to σ, we have h−1(r) ⊂ ∂S. Then Proposition 4.4 guarantees that ℓ is a leaf edge. Observe +that both e and ℓ are edges containing h−1(r). Edges are pairwise disjoint, so we conclude that e = ℓ. Thus +e is a leaf of Fm proving (1). +It remains to prove that if p ∈ e and h ∶ U → σ is a stellar homeomorphism, then σ is a half-plane. That +is, we need to show that the interior angle of σ is π. Let m be such that e is a leaf of Fm. From the first +paragraph, the interior angle of σ is (n − 1)π where n is the number of prongs of Fm at p. But any point in +a leaf ℓ in a singular foliation has a single prong if the point lies in ∂ℓ and has two prongs if the point lies +in ℓ○. Since edges are homeomorphic to open intervals, we see that n = 2 and so the interior angle of σ is π, +proving (2). +□ + +ZEBRA SURFACES +35 +Now let (S,{Fm}) be a zebra surface with boundary and let v ∈ V be a vertex. Since v ∈ ∂S, the stellar +homeomorphism associated to v must have the form h ∶ U → σ with h(v) = 0. The interior angle of S at v +is the same as the interior angle of σ at 0. This notion is well-defined: +Proposition 5.8. For each v ∈ V, if h1 ∶ U1 → σ1 and h2 ∶ U2 → σ2 are two stellar homeomorphisms with +hi(v) = 0, then σ1 and σ2 are stellar equivalent. In particular, the interior angle of S at v is well-defined. +Proof. Fix v ∈ V. Recall from Section 4.5 that v is isolated within V. Thus, as we travel around ∂S through +v with S○ on the left, we pass through an edge e1 then through v and finally through an edge e2. By +Proposition 5.7, each of the these edges are leaves of some slope. Let m1 and m2 denote the slopes of e1 and +e2, respectively. Now let hi ∶ Ui → σi be two stellar homeomorphisms with hi(v) = 0 as in the statement. +Then for each i ∈ {1,2}, the preimage of the terminal ray of σi is contained in e1 and the preimage of the +initial ray is contained in e2. The boundary slopes of σi are therefore determined. Also, the interior angles +of each σi can be determined from knowing the number of rays of slope m1 in σi. But by definition of stellar +neighborhood this number of rays is the same of the number of prongs of Fm1 at v. Since the boundary +slopes are the same and the interior angles are the same, Proposition 5.2 tells us that σ1 and σ2 are stellar +equivalent. +□ +Proposition 5.9. If (S,{Fm}) is a zebra surface with boundary, then {F○ +m} is a zebra structure on S○. +This is a direct consequence of Corollary 4.5 and the definitions. Details are left to the reader. +5.4. Removable vertices of zebra structures. Let S be a zebra surface with boundary and let v ∈ V be +a vertex. We say v is removable if its internal angle is π. This is equivalent to the stellar homeomorphism +for v being a map to a half-plane sector. It then follows from Corollary 5.3 that a removable vertex for the +zebra structure is also a removable vertex for each singular foliation Fm in the structure. +Proposition 5.10 (Removing marked points). Let (S,{Fm}) be a zebra surface with boundary. Let Σ and +V be the singularity and vertex sets, respectively. Suppose V′ ⊂ V is any collection of removable vertices. For +each m ∈ ˆR, let F′ +m be the singular foliation obtained by removing the removable vertices as in Proposition 4.8. +Then (S,{F′ +m}) is a zebra surface with boundary. +Proof. The stellar neighborhoods and homeomorphisms for (S,{Fm}) still work for (S,{F′ +m}). +□ +Proposition 5.11 (Adding marked points). Let (S,{Fm}) be a zebra surface with boundary. Let Σ and +V be the singularity and vertex sets, respectively. Suppose V′ ⊂ ∂S ∖ V is a collections of points such that +Σ ∪ V ∪ V′ is a closed discrete subset of S. For each m ∈ ˆR, let F′ +m be the singular foliation obtained by +adding each point in V′ to the collection of vertices as in Proposition 4.9. Then (S,{F′ +m}) is a zebra surface +with boundary with singular set Σ and vertex set V ∪ V′. Furthermore, each v ∈ V′ is a removable vertex in +(S,{F′ +m}). +Proof. We must find a stellar neighborhood of each point p ∈ S for the new structure. Let h ∶ U → Y be +a stellar homeomorphism for p for the original structure, where Y is either some Πn or a sector. Since +V′ is closed and discrete, there is an r > 0 such that there are no points of h(U ∩ (V′ ∖ {p})) in the open +ball B consisting of all points whose Euclidean distance from 0 in Y is less than r. Let f ∶ B → Y be +a homeomorphism such that f(0) = 0 and for each ray r of Y , f(r ∩ B) = r. Then h−1(B) is a stellar +neighborhood of p and f ○ h∣B is a stellar homeomorphism. Observe that if p ∈ V′ then p is a removable +vertex because p lies in an edge of the original structure and so Y is a half-plane sector by Proposition 5.7. +□ +5.5. Surgery on zebra surfaces. Let {Fm} be a zebra structure on a surface S with boundary, possibly +with multiple connected components. Let Σ denote the singular set and V denote the vertex set. We will +explain how to glue edges of S to make a new zebra surface. +As in Section 5.1, an edge gluing is an orientation-reversing homeomorphism between closures of distinct +edges. As before an edge identification scheme is a collection (E,ε,{ge}e∈E), where E is a collection of +edges, ε is a fixed-point free involution that describes which edges are to be glued, and {ge} is a choice +orientation-reversing gluing homeomorphisms ge ∶ ¯e → ε(e) such that g−1 +e += gε(e) for all e ∈ E. +Again there is an equivalence relation ∼ on S determined by the edge identification scheme. Let [p] ∈ S/ ∼ +denote the ∼-equivalence class of p ∈ S. As in Section 5.1, vertices can only be identified with other vertices. +The notions of [v] ∈ V/ ∼ being finite and being complete glued are defined as before. The total angle, + +36 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Ang([v]), of [v] is the sum of the interior angles at v over all v ∈ [v]. In order for our construction to +produce a zebra surface, we require that: +(d) Each [v] ∈ V/ ∼ is finite. +(e) For each edge e ∈ E, the slope of e is the same as the slope of ε(e). +As part of the proof of the result below, we will show that statements (b) and (c) from Section 5.1, which +were required to glue singular foliations, follow from statements (d) and (e) above. +Theorem 5.12 (Surgery on zebra surfaces). Let (S,{Fm}) be a zebra surface with boundary as above and +let (E,ε,{ge}) be an edge identification scheme satisfying statements (d) and (e) above. Define +Σ′ = π(Σ) ∪ {[v] ∈ V/ ∼ ∶ [v] is completely glued and Ang([v]) ≠ 2π} +and +V′ = {[v] ∈ V/ ∼ ∶ [v] is not completely glued}. +Then S′ = S/ ∼ is a surface with boundary ∂S′ = V′ ∪⋃e∈E∖E π(e), and for each m ∈ ˆR, the edge identification +scheme satisfies the hypotheses of Theorem 5.1 for (S,Fm) and F′ +m = Fm/ ∼ is a singular foliation F′ +m on +S′ = S/ ∼ with singular set Σ′, vertex set V′, and singular data function α′ ∶ (S′)○ → Z≥−1 whose only non-zero +values are given by +α′([p]) = α(p) +if [p] ∈ π(Σ) +and +α′([v]) = 1 +π Ang([v]) − 2 +if [v] ∈ Σ′ ∩ V/ ∼ . +Furthermore {F′ +m ∶ +m ∈ ˆR} is a zebra structure on S′, and if [v] ∈ V′, then the interior angle at [v] is +Ang([v]). +Proof. We begin by checking the hypotheses of Theorem 5.1 for (S,Fm). Condition (a) of Theorem 5.1 +is the same as condition (d) here. To check condition (b) of Theorem 5.1, suppose that [v] is completely +glued. For each vi ∈ [v], there is a stellar neighborhood Ui and a stellar homeomorphism hi ∶ Ui → σi for +some sector σi. Assume that [v] is indexed {vi ∶ i ∈ Z/nZ} where the terminal incident edge of vi is glued +to the initial incident edge to vi+1 for all i. Our condition (e) tells us that these pairs of edges have the +same slopes. Then the terminal ray of σi has the same slope as the initial ray of σi+1 for all i. Using stellar +neighborhoods at each vi, we see that each vi has prongs of varying slopes parameterized by a closed interval: +these prongs are identified with rays in the corresponding sector. The collection of all prongs of all slopes +at all vi with boundary prongs identified according to the edge gluings has a natural cyclic ordering coming +from counterclockwise rotation of rays in the corresponding sectors, and the map from this collection to +slopes is a covering map of ˆR. It follows that Ang([v]) is a positive integer multiple of π. Then by definition +of stellar neighborhood, regardless of slope Pr([v]) = 1 +πAng([v]) ≥ 1, verifying Condition (b). Condition (c) +of Theorem 5.1 follows trivially from condition (e) here: An edge e is a leaf edge for Fm if and only if e has +slope m, and we only allow gluing edges of the same slope. +We have shown that F′ +m is well-defined for all m. It remains to show that (S′,{F′ +m}) is a zebra surface. +This is a property that must be checked for every point on the surface. +Observe that the collection of points taking part in the gluing is G = ⋃e∈E ¯e ⊂ ∂S. For points p ∈ S ∖ G a +stellar neighborhood U of p for the zebra surface (S,{Fm}) can be produced that is disjoint from G. Then +π(U) is a stellar neighborhood for π(p) on (S′,{F′ +m}). +Now let p1 be a point in an edge e1 ∈ E. Let e2 = ε(e1) and let g ∶ e1 → e2 denote the gluing map. +Then [p1] = {p1,p2} where p2 = g(p1) ∈ e2. +Choose stellar neighborhoods Ui of pi for each i ∈ {1,2}. +From (e) we know that e1 and e2 have the same slope. Then by Proposition 5.7 we know that the stellar +homeomorphisms hi ∶ Ui → σi are both maps to half-plane sectors with the same boundary slopes. We +can assume that U1 ∩ U2 = ∅. +(Otherwise, we can find a closed set C ⊂ S ∖ {p1,p2} such that p1 and +p2 lie in different components of S ∖ C, and find new stellar neighborhoods in Ui ∖ C with new stellar +homeomorphisms defined by restricting hi to the smaller neighborhoods and post-composing with a stellar +epimorphism obtained from Proposition 5.4. As the new neighborhoods are connected, they are necessarily +disjoint.) Since the σi are half-planes with the same slope, we can assume that σ1 and σ2 are complementary +half-planes in Π0. Thus, Π0 = σ1 ∪ σ2. Now choose an open interval I1 ⊂ e1 containing p1 such that I1 ⊂ U1 +and I2 = ge1(I1) ⊂ U2. Using Proposition 5.5, we can find an open Vi ⊂ Ui such that Vi ∩ ∂S = Ii and a +stellar epimorphism ψi ∶ hi(Vi) → σi. Observe that the two maps ψi ○ hi send Ii to the common boundary + +ZEBRA SURFACES +37 +∂σi by homeomorphism, but they do not yet respect the gluing map. By Proposition 5.6, there is a stellar +epimorphism χ2 ∶ σ2 → σ2 such that +(17) +χ2 ○ ψ2 ○ h2 ○ g(q1) = ψ1 ○ h1(q1) +for each q1 ∈ I1. +Define W = π(V1 ∪V2). Then W is an open neighborhood of [p1] in S′ since ∼ identifies V1 and V2 according +to g∣I1 ∶ I1 → I2. We define +h ∶ W → Π0; +π(q) ↦ +⎧⎪⎪⎨⎪⎪⎩ +ψ1 ○ h1(q) +if q ∈ V1, +χ2 ○ ψ2 ○ h2(q) +if q ∈ V2. +Then (17) guarantees that h is well-defined on W and is a homeomorphism. We need to check that h is a +stellar homeomorphism. Observe that +h ○ π∣V1 = ψ1 ○ h1∣V1 +and +h ○ π∣V2 = χ2 ○ ψ2 ○ h2∣V2. +Since the original maps hi were stellar, for each m ∈ ˆR, it induces a bijection between prongs of Fm at pi and +rays of slope m in σi. Since h○π∣Vi is obtained from hi by post-composition with a stellar epimorphisms, these +bijections persist for h between prongs of F′ +m at [p1] approaching [p1] within π(Vi). The only identifications +between the Vi made by π occur in the identification of I1 with I2, so these bijections induce a bijection +between prongs of F′ +m at [p1] and rays of Π0 = σ1 ∪ σ2. Thus h is stellar as desired. +Now let v ∈ V and assume that [v] is not completely glued. We can order [v] = {v1,...,vn} such that the +terminal incident edge e+ +i of vi is glued to the initial incident edge e− +i+1 of vi+1 for i ∈ {1,...,n − 1} under +the gluing map gi ∶ e+ +i → e− +i+1. As above, we can choose pairwise disjoint stellar neighborhoods Ui of vi and +a stellar homeomorphism hi ∶ Ui → σi. For each i, choose open intervals I− +i ⊂ e− +i and I+ +i ⊂ e+ +i with vi as an +endpoint such that gi(I+ +i ) = I− +i+1 for all i ∈ {1,...,n − 1}. As above, for each i we can find an open subset +Vi ⊂ Ui such that Vi ∩∂S = I+ +i ∪{0}∪I+ +i and a stellar epimorphism ψi ∶ hi(Vi) → σi. Let σ′ be a sector whose +initial ray has the same slope as the initial ray of σ1, whose terminal ray has the same slope as the terminal +ray of σn, and whose interior angle is the sum of the interior angles of the σi. Then by cutting σ′ along rays, +we can partition σ′ into subsectors with the same internal angles as the σi in counterclockwise order. Then +each σi is stellar equivalent to the corresponding subsector of σ′. Therefore, we redefine σi so that it is this +subsector of σ′. We have that +ψi ○ hi(I+ +i ) = ψi+1 ○ hi+1(I− +i+1) +is the common boundary ray of σi and σi+1, but the maps to do not yet respect the gluing maps gi∣I+ +i ∶ I+ +i → +I− +i+1. Define W = π(⋃i Vi). Then W is an open neighborhood of [v]. We will define h ∶ W → σ′ by induction. +For q ∈ V1, we define h○π(q) = ψ1 ○h1(q). Also define χ1 ∶ σ1 → σ1 to be the identity map. Now assume that +h has been defined on π(Vi). Using Proposition 5.6, we see there is a stellar epimorphism χi+1 ∶ σi+1 → σi+1 +such that +(18) +χi+1 ○ ψi+1 ○ hi+1 ○ gi(qi) = χi ○ ψi ○ hi(qi) +for each qi ∈ I+ +i . +For q ∈ Vi+1 we define h ○ π(q) = χi+1 ○ ψi+1 ○ hi+1(q). Equation (18) guarantees that h is a well-defined +homeomorphism. Again h was defined from the hi by post-composing with stellar epimorphisms, so h is a +stellar homeomorphism for [v]. +The case when [v] is completely glued is similar. We will highlight the differences with the previous case. +This time we write [v] = {vi ∶ i ∈ Z/nZ} and write the elements of Z/nZ as 1,...,n to keep with the previous +paragraph. We get a gluing map gi ∶ e+ +i → e− +i+1 for all i and we require that gi(I+ +i ) = I− +i+1 hold for all i. As +noted in the first paragraph of the proof, in the completely glued case the sum of the interior angles of the +sectors is of the form (m + 2)π for some m ∈ Z≥−1. Thus, we can partition Πm into sectors that are stellar +equivalent to the σi. We think of σi ⊂ Πm and order these sectors cyclically counterclockwise. The maps +hi, subsets Vi ⊂ Ui, and stellar epimorphisms ψi can be defined as before. The set W is defined as above, +but we will define h ∶ W → Πm. We again proceed inductively. We define the base case of h on π(V1), +and the inductive step of h on π(Vi+1) when i ∈ {1,...,n − 2} as before, but things change in a minor way +when defining h in the last step. This time using Proposition 5.6 slightly differently, we define the stellar +epimorphism χn ∶ σn → σn such that +χn ○ ψn ○ hn ○ gn−1(qn−1) = χn−1 ○ ψn−1 ○ hn−1(qn−1) +for each qn−1 ∈ I+ +n−1, +and +χn ○ ψn ○ hn ○ g−1 +n (q1) = χ1 ○ ψ1 ○ h1(q1) +for each q1 ∈ I− +1 . + +38 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Then we define h on π(Vn) to be χn ○ ψn ○ hn. The maps again respect the gluings so h ∶ W → Πm is a +homeomorphism and it is a stellar homeomorphism for the same reason as before. +□ +5.6. Zebras not arising from dilation structures. On a closed half-dilation surface, in any direction, +there are at most finitely many isolated closed leaves. Indeed, this can be deduced from [DFG19] as follows. +It is enough to consider the case where the surface has a non-empty singular set or at least marked points. +Each isolated closed leaf of the singular foliation of slope m must lie in a dilation cylinder. Each dilation +cylinder contains only finitely many closed leaves of slope m. Furthermore, the dilation cylinders that contain +closed leaves of slope m must have disjoint interiors, and must be bounded by saddle connections. The total +of the interior angles of a dilation cylinder must be at least 2π, so this tells us that there can be only finitely +many dilation cylinders that contain closed leaves of slope m. The finiteness result follows. +The same holds for surfaces in the Homeo+(ˆR)-orbit of a zebra structure obtained by weakening a dilation +surface structure. However, there are zebra surfaces whose directional foliations have countably many isolated +closed leaves. Therefore: +Proposition 5.13. Let g ≥ 0 and let S be a closed and oriented topological surface of genus g. Suppose +α ∶ S → Z≥−1 has finite support and satisfies ∑p∈S α(p) = 4g − 4. Then there is a zebra structure {Fm} on S +with singular data α, such that for every ϕ ∈ Homeo+(ˆR), the zebra structure ϕ(S,{Fm}) is not isotopic to +a zebra structure obtained from a half-dilation surface. +Proof. There are square-tiled half-translation surfaces in each stratum determined by α. (In fact they are +dense; see [HS06, §1.5.2].) Let X be such a surface, and observe the horizontal foliation is periodic. Let I ⊂ X +be a vertical interval contained in the interior of a horizontal cylinder C ⊂ X. Slice I open, obtaining two +halves I+ and I−. Weakening the structure to a zebra structure, we see that the resulting object is a zebra +surface X′ with two boundary edges, I+ and I−. Let g ∶ I+ → I− be a gluing homeomorphism that is monotone +increasing in the y-coordinate such that the induced map I → I has countably many isolated fixed points. +Then the zebra surface X′′ obtained by gluing ∂X′ according to g has the same singular data α and has +countably many isolated closed leaves in the horizontal foliation F0. Every surface in the Homeo+(ˆR)-orbit +also has a directional foliation with countably many isolated closed leaves, and so cannot be isotopic to a +zebra structure arising from a half-dilation surface. +□ +6. Constructing new foliations +6.1. The Burp Lemma. Later we will be constructing foliations consisting of leaves from directional +foliations with varying slope. It is generally easy to prove that the proposed leaves do not cross, but it +is more difficult to prove that there are no gaps (bubbles) between the proposed leaves. We call the following +result the Burp Lemma because it is useful to rule out (burp) these bubbles: +Lemma 6.1 (The Burp Lemma). Let pq be an arc of a trail such that all bending angles on the left side, as +we move from p to q, are π. For every segment qr such that ∡rqp < π, there exists a point x ∈ qr ∖ {q} and +a segment of a leaf px. +Proof. We may assume by acting by an element of Homeo+(ˆR) that pq is horizontal and qr is vertical. Using +Lemma 3.10, we can construct a rectangle R whose vertices are p, q, a point r′ ∈ qr∖{q} and some additional +point s. The rectangle has no singularities in its interior by Proposition 3.8. Our task is to construct a +segment of a leaf px where x ∈ qr′ ∖ {q}. +Let γ ∶ [0,1] → pq be a parameterization of pq with γ(0) = p and γ(1) = q. Say that t ∈ [0,1] is bad, if +there is a vertical segment γ(t)β(t) with β(t) in the interior of R such that there is no segment of a leaf +joining p to a point on γ(t)β(t) ∖ {γ(t)}. +If the Burp Lemma is false, then there is a choice of points for which 1 is bad. Assume this is the case, +and we will derive a contradiction. Since there is a stellar neighborhood at p, every t for which γ(t) is in +this stellar neighborhood is not bad, because the neighborhood is foliated by leaves emanating from p. Thus, +setting t0 equal to the infimum of the bad values of t gives t0 > 0. +First we claim that t0 is not bad. Suppose to the contrary that t0 is bad. Consider β(t0), which is a +point on the vertical segment γ(t0)β(t0) below which no leaf from p can cross. Let ℓ denote the leaf through +β(t0) of slope 1. Moving leftward along ℓ, we must eventually leave R by Proposition 3.12. We will consider +where ℓ exits the rectangle. This situation is depicted on the left side of Figure 10. The leaf ℓ can’t exit + +ZEBRA SURFACES +39 +through the r′s ∖ {r′} because then the polygon formed whose vertices are γ(t0), q, r′, ℓ ∩ r′s, and β(t0) +would have interior angles adding to more than 3π, violating our Gauss-Bonnet Theorem which guarantees +that the interior angles of a pentagon with no interior singular points add up to 3π. Similarly, we can see +that ℓ cannot exit through qr′ ∖ {q} (by applying Gauss-Bonnet Theorem to the quadrilateral with vertices +γ(t0), q, qr′ ∩ ℓ and β(t0)) and cannot exit through γ(t0)q ∖ {γ(t0)} (by applying Gauss-Bonnet Theorem +to the triangle with vertices γ(t0), γ(t0)q ∩ ℓ and β(t0)). If ℓ exits through sp, then it can’t exit at p, or +else it would violate the definition of β(t0). If ℓ exits at a point of sp ∖ {p}, consider the segment τ of +slope 1 leaving p and entering R. The segment τ enters and so must eventually exit the quadrilateral with +vertices p, γ(t0), β(t0) and ℓ ∩ sp. It can’t exit through pq or ps because such an exit would create a bigon +contradicting Proposition 3.6 and cannot exit through ℓ because it is a leaf of the same foliation. So τ would +have to exit through γ(t0)β(t0) ∖ {γ(t0)}, which would violate the definition of β(t0). We’ve ruled out the +possibility that ℓ exits R through ps, and the only remaining segment of R that remains for ℓ to exit is +pγ(t0) ∖ {p}. If ℓ exited at γ(t0) it would create a bigon with the vertical segment γ(t0)β(t0). So, it must +exit through some point u ∈ pγ(t0)∖{p,γ(t0)}, forming a new triangle T = △uγ(t0)β(t0). Pick a t such that +γ(t) is in the interior of uγ(t0). Choose β(t) on the vertical leaf through γ(t) such that β(t) is in T. As +t < t0, the parameter t cannot be bad. Therefore, there is a point y ∈ γ(t)β(t) and a segment of a leaf py. +Since py enters the interior of T, it must enter through an edge. By definition of β(t0), it cannot be through +γ(t0)β(t0). It cannot be through uγ(t0), because this is part of the boundary of R, and no trail can exit R +and later reenter (Corollary 3.18). Therefore, it enters through ℓ = uβ(t0). But then it must exit T through +a different edge, or else it would create a bigon. But we’ve already showed that the leaf continuing py cannot +pass through any of the other edges. This contradicts our hypothesis that t0 is bad, proving that t0 is not +bad. +Figure 10. Configurations discussed in the proof of Lemma 6.1. +We have shown that t0 is not bad. We will also derive a contradiction from this. Construct a stellar +neighborhood N of γ(t0). Since t0 is the infimum of the bad values of t, there is a t > t0 such that t is bad +and γ(t) ∈ N. Let β(t) be as in the definition of bad. We can assume, by possibly moving β(t) closer to γ(t) +along γ(t)β(t) that β(t) ∈ N. Using the fact that N is stellar, we can construct γ(t0)β(t) forming a triangle +△γ(t0)γ(t)β(t). The slope of γ(t0)β(t) is some m > 0 by Proposition 3.2. Let m′ be a slope with 0 < m′ < m, +and construct the ray of slope m′ from β(t) moving leftward through R. This leaf cannot cross over β(t)γ(t0) +so it must cross the vertical leaf through γ(t0) at some point, call it x. (The vertical leaf through γ(t0) must +exit through sr′, and the ray cannot exit through γ(t)q ∪qr′ ∪r′s by repeating analysis done in the previous +paragraph.) Since t0 is not bad, there must be a segment py where y ∈ γ(t0)x ∖ {γ(t0)}. This leaf must +have positive slope, and any leaf of smaller positive slope through p must also intersect γ(t0)x ∖ {γ(t0)}, so +we can assume without loss of generality that the slope m′′ of py satisfies 0 < m′′ < m′. Then continuing +along py we enter the quadrilateral γ(t0)γ(t)β(t)x through the edge xγ(t0). The continuation cannot exit +through γ(t0)γ(t) or else it would create a bigon, and cannot exit through xβ(t) because it would create a +triangle whose counterclockwise slope triple is (∞,m′′,m′) which is increasing and violates Proposition 3.2. +Therefore, it must exit through γ(t)β(t), but this violates that t was bad. +We have shown that the infimum of bad points cannot exist. It follows that all elements of [0,1] are not +bad. In particular 1 is not bad, so there is a leaf joining p to qr′ ∖ {q}. +□ + +40 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +6.2. Foliating triangles. Let △pqr be a triangle in a zebra plane Z. For ∗ ∈ {p,q,r}, let m∗ denote the +slope of the edge opposite ∗. +Triangles inherit restricted foliations Fm of every slope m from inclusion in Z. For each vertex ∗, each +edge with ∗ as a vertex forms a section for the foliation of the triangle by leaves of slope m∗ (i.e., every +maximal segment of a leaf contained the triangle passes transversely through the edge exactly once), because +leaves of slope m∗ passing through the interior of the triangle cannot exit through the opposite edge. +Lemma 6.2. Let △pqr be a triangle in Z with vertices ordered counterclockwise. Then the collection of +leaves with slopes in [mr,mq] through p cover the triangle and are pairwise disjoint except for sharing the +common vertex p. Consider the function +(19) +h ∶ △pqr ∖ {p} → [mr,mq] × (pq ∖ {p}) +sending x to the pair consisting of the slope of px and the point on pq where the leaf of slope mp through x +exits the triangle. This function is a homeomorphism whose restriction to qr maps to [mr,mq] × {q}. +Equation 19 places a coordinate system on the triangle somewhat analogous to polar coordinates. +Proof. For m ∈ [mr,mq], let ℓm denote the arc of the leaf of slope m starting at p and entering the triangle +until it exits the triangle. (Exiting is guaranteed by Proposition 3.12.) Then ℓmr = pq and ℓmq = pr. In +a stellar neighborhood of p, it is easy to observe that the segments ℓm are disjoint except at p, and by +Proposition 3.6, these segments are disjoint except at p as subsets of △pqr. Thus if m ∈ (mr,mq), the arc of +the leaf ℓm exits the triangle through a point on qr. Thus for such m, △pqr∖ℓm consists of two components, +one containing q and the other containing r. We’ll say that ℓm runs through x if x ∈ ℓm, runs above x if x lies +in the component of △pqr ∖ℓm containing q, and runs below x if x lies in the component containing r. These +notions make sense for m ∈ [mr,mq]. We will first prove that for every x ∈ △pqr there is a m ∈ [mr,mq] +such that ℓm runs through x. +Observe that for m < m′ slopes in [mr,mq], the segments ℓm and ℓm′ together with a segment of qr +form a triangle. Thus by Proposition 3.2, the counterclockwise order for the edges of this triangle is ℓm, the +segment of qr, and finally ℓm′. In particular, if Am denotes the set of points that ℓm runs above and Bm +denote the set of points that ℓm runs below we have +(20) +Am ⊂ Am′ +and +Bm′ ⊂ Bm. +Now assume that there is a point x0 ∈ △pqr such that there is no m ∈ [mr,mq] for which ℓm runs through +x0. Define +mc = sup{m ∈ [mr,mq] ∶ ℓm runs below x0} +(where the supremum is taken using the increasing cyclic ordering on [mr,mq]). We may assume without +loss of generality (by possibly applying an orientation-reversing homeomorphism of the circle as described +in Section 2.8) that ℓmc runs below x0. Now define +X = Bmc ∩ +⋃ +m∈(mc,mq] +Am. +Observe that x0 ∈ X. By (20), for x ∈ X and m ∈ [mr,mc], m runs below x. +Consider the leaf β through x0 which is parallel to qr. (If x ∈ qr, we take β = qr.) Since β is a leaf of +the same foliation as qr, it cannot cross qr. Therefore, the leaf β must intersect both pq and qr and so must +cross each ℓm exactly once (or else it creates a bigon). Let y = ℓmc ∩ β and consider the segment x0y ⊂ β. +We claim that x0y ∖ {y} ⊂ X. We have x0y ∖ {y} ⊂ Bmc because ℓmc ∩ β = {y} and so x0y ∖ {y} lies in the +same component of △pqr ∖ ℓmc. Similarly if m ∈ (mc,mq], because x0 ∈ Am the leaf ℓm must intersect β +in the segment of β connecting pr to x0. Thus x0y ∖ {y} ⊂ Am because x0y lies in the same component of +△pqr ∖ ℓm as x0. +Observe that ∡x0yp < π but there is no leaf from p that intersects x0y ∖ {y}. This is a violation of +Lemma 6.1, proving that the ℓm cover △pqr. +Now we will show that the map h is a homeomorphism. To see it is continuous, fix an x in the triangle. +The topology on the codomain has a basis consisting of rectangles. Suppose h(x) is in the rectangle given +by the product of an interval of slopes with endpoints m1 and m2, and points y1,y2 ∈ pq. The preimage of +this set consists of the intersection of the triangle bounded by ℓm1, ℓm2 and a segment of qr and a trapezoid +consisting of the region between the leaves of slope mp through the points y1 and y2. Thus h is continuous. + +ZEBRA SURFACES +41 +It is onto since there must be an intersection between any ℓm and any leaf of slope mp passing through the +triangle (since the boundary of ℓm separates pq ∖ {p} from pr ∖ {r} in the boundary of the triangle). It is +one-to-one because of Proposition 3.6 and the fact that the two leaves being intersected have distinct slopes. +Thus h−1 is well-defined. To show h−1 is continuous, let U be an open subset of the triangle. Let x ∈ U +and suppose h(x) = (m,y). Let γ be the leaf of slope mp through x and y. Since x is in the interior of U, +we can find x1 ∈ U ∩ (xy ∖ {x}) and x2 ∈ U ∩ (γ ∖ xy), so that x1x2 ⊂ U. Let m1 and m2 be the slopes of +px1 and px2 respectively. Then Lemma 3.10 allows us to construct two trapezoids in U one on each side of +x1x2 with adjacent edges consisting of segments of ℓm1 and ℓm2. The union of these trapezoids is a larger +trapezoid with x in its interior. As we’ve already shown that the leaves ℓm vary monotonically, the image of +the interior of this trapezoid under h is a rectangle (m1,m2) × (y1,y2) with y1 and y2 being the places that +the edges of the trapezoid parallel to mq intersect pq. +The fact that qr maps to [mr,mq] × {q} is clear since qr is one of the leaves of slope mp. +□ +Corollary 6.3 (Vertex foliations). Let P = p0p1 ...pn−1 be a n-gon in a zebra plane with no interior singu- +larities all of whose interior angles are less than π. Let m0 be the slope of p0p1 and m1 be the slope of p0pn−1. +For m ∈ [m0,m1], let ℓm be the leaf of slope m entering P from p0. Then the leaves {ℓm ∶ m ∈ [m0,m1]} +cover P, foliate the interior of P, and for any m ∈ (m0,m1) the intersection ℓm ∩ ∂P consists of two points. +Figure 11. Corollary 6.3 and its proof. +Proof. Let Z denote the zebra plane containing P. Since P has no singularities in its interior and all of its +interior angles are less than π, there is a convex Euclidean n-gon Q = q0 ...qn−1 ⊂ R2 with the same boundary +slopes. Since Q is convex, there is an i ∈ {1,...,n − 2} such that the line ←��→ +qiqi+1 extending the side qiqi+1 +intersects the two rays ��→ +q0q1 and ���→ +q0qn−1. Define r1 = ��→ +q0q1 ∩ ←��→ +qiqi+1, and r2 = ���→ +q0qn−1 ∩ ←��→ +qiqi+1. Then the triangle +△q0r1r2 contains Q. Using surgery (Theorem 5.12), define Z′ = (R2 ∖ Q) ∪ P. Then applying Lemma 6.2, +we see that the leaves through p0 cover all of the triangle △q0r1r2 (which we are now viewing as in Z′), and +in particular these leaves cover P and foliate its interior. Finally, if m ∈ (m0,m1), we see by considering +a stellar neighborhood at p0 that ℓm immediately enters the interior of P after leaving p0. Thus, ℓm ∩ ∂P +contains only two points by Corollary 3.21. +□ +As a consequence, we see some uniformity to our stellar neighborhoods: +Corollary 6.4. Suppose P is polygon containing no singularities in its interior all of whose interior angles +are less than π. If x is in the interior of P, then the interior of P is a stellar neighborhood of x. The same +holds if P is a polygon satisfying the same properties, but with x being the only singularity in the interior of +P. +Proof. Consider the horizontal and vertical leaves emanating from x. These divide P into subpolygons all +of whose interior angles are less than π with no singularities in its interior. Then Corollary 6.3 allows us to +foliate all these subpolygons using leaves emanating from x. We can use this to construct leaves from x to +the vertices of P, cutting P into triangles. If v and w are consecutive vertices of P, then Lemma 6.2 gives a +homeomorphism from each triangle with x removed to rectangles [mv,mw] × (0,1], carrying the foliation of +the triangle by leaves through x to the vertical foliation of the rectangle. These rectangles can be stitched + +42 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +together (using bump functions to continuously adjust the homeomorphisms so that the vertical sides can +be identified by isometry), giving homeomorphism from P ∖ {x} to ˆRk × (0,1] where ˆRk denotes the k-fold +cover of ˆR and k = α(x) + 2. Then composing with a standard map from complex analysis gives the stellar +homeomorphism. +□ +6.3. Foliating polygons. In this subsection, we explain two methods of foliating a polygon by leaves passing +through an edge. +Lemma 6.5. Let P be an n-gon with vertices p0, . . . , pn−1 listed in counterclockwise order. Assume that +there are no singularities in the interior of P and interior angles at each pi are less than π. +For each +i ∈ Z/nZ, let mi denote the slope of pipi+1. Let γ ∶ [0,1] → p0pn−1 be a parameterization such that γ(0) = p0 +and γ(1) = pn−1. Suppose µ ∶ [0,1] → ˆR ∖ {mn−1} is continuous and, using the ordering on R ∖ {mn−1}, we +have that µ is strictly increasing and satisfies +µ(0) ≤ m0 +and +mn−2 ≤ µ(1). +Let ℓt denote the leaf of slope µ(t) through γ(t). Then the collection of leaves {ℓt ∶ t ∈ [0,1]} foliates P. +This foliation is transverse to p0p1 if µ(0) < m0, is transverse to pn−2pn−1 if mn−2 < µ(1), and is always +transverse to the other edges. +Figure 12. A foliation of a polygon as described by Lemma 6.5, depicted as constructed +in the proof. +Proof. By the Gauss Bonnet Theorem, the sum of the interior angles of P is (n − 2)π. Thus we can find +a strictly convex polygon P ′ = p′ +0 ...p′ +n−1 in the plane R2 with the same edge slopes and the same interior +angles as P. Let x ∈ R2 be the point at which the line of slope µ(0) through p′ +0 intersects the line of slope +µ(1) through p′ +n−1. We can extend the rays �→ +xp′ +0 and ���→ +xp′ +n−1 and join them by a segment forming a triangle +T ′ containing P ′ as depicted in Figure 12. +Consider the foliation of T ′ by rays emanating from x. Let η ∶ [0,1] → p′ +0p′ +n−1 be a parameterization +such that η(0) = p′ +0 and η(1) = p′ +n−1. For t ∈ [0,1], let ν(t) denote the slope of the segment xη(t). By a +standard Euclidean geometry exercise, ν is a function from [0,1] to [µ(0),µ(1)] which is strictly increasing +and satisfies ν(0) = µ(0) and ν(1) = µ(1). +We use surgery. Let X = (R2 ∖ P ′) ∪ P, where we reglue P in place of P ′ being sure to glue p0pn−1 to +segment p′ +0p′ +n−1 by the map +γ(t) ↦ η(ν−1 ○ µ(t)), + +ZEBRA SURFACES +43 +and gluing the other edges more freely by arbitrary homeomorphisms. The triangle T ′ remains a triangle in +X, and so by Lemma 6.2 there is a foliation G of T by leaves through x. By construction the leaf of slope +µ(t) enters P through the point γ(t) and so the restriction of G to P realizes the desired foliation of P. +To see the last statement, recall that on a zebra surface, any two foliations Fm and Fm′ with m ≠ m′ +are transverse. Since interiors of edges are segments of leaves, to show an edge is transverse to G, it suffices +to prove that the interior of the edge is not contained in a leaf of G. Observe that ℓm ∩ P is connected by +Proposition 3.7. First consider the leaf ℓµ(0) which passes through p0. By considering a stellar neighborhood +at p0 we see that ℓµ(0) ∩ P = {p0} if µ(0) < m0. Also if µ(0) = m0 then ℓµ(0) ∩ P = p0p1 because the interior +angles at p0 and p1 are both less than π. Thus the only edge that ℓµ(0) can contain is p0p1 and only if +µ(0) = m0. Similarly, the only edge ℓµ(1) can contain is pn−2pn−1 and only if mn−2 = µ(1). Now consider +a leaf ℓµ(t) with t ∈ (0,1). By construction ℓµ(t) intersects γ(t) and because µ(t) is distinct from mn−1, +it crosses pn−1p0 transversely and enters the interior of P. Then Corollary 3.21 guarantees that ℓµ(t) ∩ ∂P +contains only two points and so ℓµ(t) cannot contain any edges. We’ve shown that p0p1 and pn−2pn−1 are +the only edges that can fail to be transverse to G, and only in the circumstances allowed in the statement of +the lemma. +□ +7. Connecting points with trails +7.1. Trail rays. Let Z be a zebra plane. A trail ray (or simply ray) with initial point p ∈ Z is an arc of a +trail with a parameterization of the form γ ∶ [0,+∞) → Z with γ(0) = p which is maximal with respect to the +subarc partial order among all such arcs of trails with initial point p. Theorem 3.15 guarantees that any arc +of a trail starting at p can be extended to a ray (simply by extending to a trail and discarding the portion +before p). +Let Σ denote the set of singularities of Z, let γ ∶ [0,+∞) → Z be a trail ray, and let p = γ(0). Then +J = γ−1(Z ∖ (Σ ∪ {p})) is an open subset of R. For I a connected component of J, we call ℓ = γ(I) a leaf of +γ. It is either a leaf contained in the image of γ or a connected component of a leaf with p removed. Trail +rays and their leaves are oriented away from p. +Fix a point p and consider the collection Rp ⊂ Z of all trail rays leaving p. We will show: +Theorem 7.1. The union of all rays in Rp is an open subset Up ⊂ Z containing p. Let U ∗ +p = Up ∖ (Σ ∪ {p}). +Then the collection Rp of all leaves of rays leaving p is an oriented foliation of U ∗ +p . +We remark that if Z is convex, then Up will equal Z because any point can be connected to p by a trail. +The oriented foliation Rp of U ∗ +p should be considered to be an oriented singular foliation, but with a +singular structure which is different from that of the singular foliations defined in Section 2.4. The initial +point p is special: the leaves agree locally with the stellar neighborhood foliation with leaves oriented outward. +At the singularities s ∈ (Σ ∩ Up) ∖ {p}, the foliation has exactly one leaf ℓs oriented towards s. (If there were +two such leaves, we’d get a contradiction to Proposition 3.6.) Nearby leaves pass by the singularity, and +leaves emanating from s making angle with ℓs of at least π on each side are oriented away from s. See the +left side of Figure 13. +We’ll say a simple path α ∶ [0,1] → Up ∖ {p} is a leaf transversal to Rp if α((0,1)) is contained in a leaf +of some directional foliation Fm of Z but not contained in a leaf of Rp. +There is a natural slope map µ ∶ Up ∖{p} → ˆR that assigns to each point q ∈ U ∗ +p the slope of the leaf of Rp +through q. If s is a singularity in Up ∖ {p}, then we define µ(s) to be the slope of the unique leaf in Rp that +has s as an endpoint and is oriented towards s. We prove the foliation associated to Rp is monotonic in the +following sense. +Theorem 7.2. If α ∶ [0,1] → Up is a leaf transversal and is parameterized such that rays cross from the left +side of α to the right, then the function +[0,1] → ˆR; +t ↦ µ ○ α(t) +is continuous and (cyclically) strictly increasing. +To prove these theorems we will give an address to every ray in Rp. The directions of straight line paths +leaving p can be parameterized by an angle in R/(α(p) + 2)πZ. We fix an identification between leaves +emanating from p and this circle. Then, every ray must initially travel in one of these directions, call it +θ0. If the ray never hits a singularity, then θ0 is the complete address. Otherwise, it hits a singularity s1 + +44 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Figure 13. Left: The foliation near a singularity by rays as in Theorem 7.1. Here the +light blue curves represent rays passing near or hitting the singularity (bold), and the pink +curves are emanating from the singularity and can be viewed as trail continuations of the +ray hitting the singularity. Right: The correspondence between values of t1 ∈ [0,1] and +angles for continuations through a singularity s1. +with cone angle (α(s1) + 2)π. (Because s1 is a singularity in a zebra plane, α(s1) ≥ 1.) There is a non- +degenerate closed interval of directions in which the ray can continue, of total angle α(s1)π, because of the +angle condition. Apply an affine change of coordinates so this interval becomes [0,1]. That is, t1 ∈ [0,1] +represents continuing through s1 in a direction so that the counterclockwise angle from the initial leaf leaving +p and the continuation through s1 is π + t1α(s1)π; see the right side of Figure 13. Continuing along our ray, +we may encounter more singularities. We therefore have provided our ray with one of the following three +kinds of addresses: +(21) +(θ0;), +(θ0;t1,t2,...,tk), +or +(θ0;t1,t2,...). +In the first case, the ray never hits a singularity. In the second case, the ray hits a finite number of singularities +(namely, k) and then follows a separatrix. In the third case, the ray follows an infinite sequence of saddle +connections. +A truncated address is given by the choice of θ0 and a possibly empty finite sequence t1,t2,...,tj which +can be extended to a full address of a ray. The truncated addresses correspond to leaves of the directional +foliations making up a ray. Namely, we reach the leaf by following a ray under the instructions given by the +truncated address. So (θ0;t1,t2,...,tj) corresponds to the leaf at obtained by first moving in direction θ0 +away from p, we then hit a sequence of j singularities s1,...,sj and in each case move in the direction as +described by tj. The leaf followed after crossing sj is the leaf L(θ0;t1,t2,...,tj) referred to by the truncated +address. +Proof of Theorem 7.1 and Theorem 7.2. Let Up denote the union of rays in Rp. Since p has a stellar neigh- +borhood, p lies in the interior of Up. +Now consider a non-singular point q ∈ Up ∖{p}. Then q lies on one of the leaves L(θ0;t1,t2,...,tj), where +the truncated address can be extended to be the address of a ray. We consider several cases. Some arguments +are used more than once, so we highlight some ideas for later use in bold. See Figure 14 for illustrations of +these named arguments. +Argument 1. First, suppose the leaf through q has the form L(θ0;). We claim that the leaves of the +form L(θ;) foliate a neighborhood of q. To see this, fix a slope m distinct from the slope of pq, and let α +be the leaf of slope m through q. Then by two applications of Lemma 6.1 we can find points r and s on +α such that q ∈ rs and such that pr and ps are segments of leaves. Then △prs is a triangle. Since α was +a leaf, there are no singularities on rs and we can construct a quadrilateral rr′s′s such that rr′ extends pr +and ss′ extends ps using Lemma 3.10, forming a larger triangle △pr′s′ that contains q in its interior. Then +Lemma 6.2 guarantees that the leaves through p foliate this triangle, covering a neighborhood of q as desired. +It also follows by applying Lemma 6.2 to △prs that the function µ is continuous and monotonic on α as +described in Theorem 7.2 on rs. +Second, assume that the address of the leaf through q is L = L(θ0;t1,t2,...,tj) where tj /∈ {0,1}. The leaf +with this address starts at some singularity, call it sj−1, and the leaf L continues sj−1q. Repeating Argument +1 with sj−1 playing the role of p gives a foliation of a triangle containing q in its interior by leaves emanating + +ZEBRA SURFACES +45 +Figure 14. Figures of the arguments for the proof of Theorem 7.1. Top left: Argument 1; +Top right: Argument 3; Bottom: Argument 2 with k = 2. +from sj−1. The slopes of these leaves vary continuously and monotonically by Lemma 6.2, and so there is a +neighborhood of q in which points all lie on leaves with addresses L(θ0;t1,t2,...,t′ +j) with t′ +j near tj. +Argument 2. Third, assume that the address of the leaf through q is L = L(θ0;t1,t2,...,tj,1,1,...,1) +where there are exactly k ones at the end of the address. (It can be that j = 0, in which case the portion +of the address after θ0 is all ones.) Let p′ = p if j = 0 and let p′ = sj otherwise. Then the trail visits p′ +and then a sequence of k singularities, making an angle of π on the left as the trail moves through each of +these singularities. Let α be a leaf through q with a different slope than L. Repeating half of Argument 1 +for the portion of α reachable from sj+kq by making a right turn at q, gives a △sj+krq. We can extend this +triangle by a trapezoid through edge qr. This triangle is foliated by leaves through sj+k, and thus points +in the triangle have addresses of the form L = L(θ0;t1,t2,...,tj,1,1,...,t′ +j+k) for t′ +j+k in some interval of +the form [a,1]. This foliates a half-neighborhood of q. On the other side of α, because the angles at sj+c +are π for all c ∈ {1,...,k} we can apply Lemma 6.1 to construct △p′qs, where qs ⊂ α. We can extend the +triangle through qs, and foliate the triangle again. Points sufficiently close to q in this triangle lie on leaves +whose addresses have the form L(θ0;t1,t2,...,tj−1,t′ +j) where t′ +j > tj. This gives a foliation of the second half +of the neighborhood of q. Lemma 6.2 can again be applied to △p′qs and △p′rq to deduce continuity and +monotonicity of µ on α. +Fourth, if the address of the leaf through q is L = L(θ0;t1,t2,...,tj,0,0,...,0), then we can repeat a +symmetric version of Argument 2. (In fact, an orientation reversing map of the underlying surface has the +effect of changing each ti to 1 − ti.) +Argument 3. The above four paragraphs handle all possible addresses of regular points. A singular point +q ∈ Up appears as the endpoint of some leaf L = L(θ0;t1,t2,...,tj), i.e., L = sjq. The point q has a stellar +neighborhood N with total angle being nπ where n = α(q)+2. Say that the angle coordinate of a point x ∈ N +is the angle made with L: ∡xqsj. Arguing as in the previous paragraphs, we can foliate neighborhoods of +q with total angle π, namely points sufficiently close to q with angle coordinates in [0,π] or [(n − 1)π,nπ]. +Points in the stellar neighborhood whose angle coordinates lie in [π,(n − 1)π] lie on leaves with addresses +of the form L = L(θ0;t1,t2,...,tj,t′ +j+1) for some t′ +j+1 ∈ [0,1]. Since this is a stellar neighborhood, we have +foliated a neighborhood of q as desired. +□ +7.2. Polygonal convexity. The goal of this section is to prove: +Theorem 7.3. A polygon R in a zebra plane is convex if and only if all of the exterior angles of R are at +least π. + +46 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +We will prove the “if” part of this result first. The following complements Theorem 7.1, giving a subspace +in which the union of rays is closed. +Lemma 7.4. Let K be a compact topological disk in a zebra plane and suppose that p ∈ K. For a ray r ∈ Rp, +let gr denote the connected component of r ∩ K containing p, which is an arc of a trail from p to a point on +∂K or possibly just {p} if p ∈ ∂K. Then the union G = ⋃r∈Rp gr is closed. +Proof. Let qi ∈ G be a sequence approaching some q ∈ K. We will prove that q ∈ G. Since p ∈ G, we can +assume q ≠ p. Because singularities are isolated, we can assume (by passing to a subsequence) that each qi +is non-singular. +Let Li denote the leaf containing qi in a trail through p and qi. By compactness, there are only finitely +many singularities in K, so up to passing to a subsequence, we can assume that each Li either passes through +p or starts at the same singularity s ∈ K. In the first case, the truncated address of each Li has the form +(θi;). In the latter case, these addresses have the form (θ,t1,t2,...,tk,xi) for some xi ∈ [0,1], where θ and +t1,...,tk are fixed. We’ll continue the argument in the case of (θ,t1,t2,...,tk,xi), but the same argument +applies in the other case as well (taking s = p). +We may further assume by passing to a subsequence, that the sequence (xi) is monotonic. We may assume +without loss of generality that it is increasing. Let x = limi→∞ xi ∈ [0,1]. +Suppose first that x = xi for some i. Then the sequence (xi) would be eventually constant and therefore +for i large enough we’d have qi and q on the same leaf. In this case, we can fix gr that contains the qi that +lie on the same leaf as q, and since gr is closed we’d have q ∈ gr ⊂ G as well. +If (xi) is not eventually constant, we can pass to a subsequence such that the sequence (xi) is strictly +increasing and converging to x. Thus in particular x > 0. Since there are only finitely many singularities in +K, we can find an a ∈ [0,x) such that for any y ∈ (a,x), the leaf L(θ;t1,t2,...,tk,y) does not terminate in a +singularity in K. We will assume by passing to a further subsequence that each xi ∈ (a,x). +Consider the trail ray r∞ with address (θ;t1,t2,...,tk,x,0,0,0,...), where we add as many zeros as +possible. We claim that following r∞ we hit q before leaving K. To see this observe that all the arcs of +trails from p to qi follow the same path up to s as r∞, because of the common start to their coding. The +idea is that the segments sqi accumulate on a segment ending at q from the ray r∞, and since K is compact +it follows that this segment is in K and therefore the connected component of r∞ ∩ K that contains p also +contains q. See Figure 15. +s +q +qi +Figure 15. The end of the argument in Lemma 7.4. +The squares denote locations of +possible singularities along r∞. +To formalize the argument, observe that by Theorem 3.19, the trail ray r∞ must eventually exit the +compact set K. Since K is compact and the singularity set is discrete and closed, we can choose a point y in +the first open interval of r∞∖(K ∪Σ) after gr∞. Since K is compact, we can construct a stellar neighborhood +N of y that is disjoint from K. Let z be a point in N such that the angle at y from py ⊂ r∞ to yz is π +2 . Then +using Lemma 6.1 we can construct a triangle T = △sz′y where z′ ∈ yz ∖{y}. Using Lemma 6.2 we can foliate +T by leaves emanating from s. The leaf space is homeomorphic to the interval z′y, and the leaf st with +t ∈ z′y has slope that is continuous and strictly increasing as t moves from z′ to y. For i sufficiently large, +sqi extends to a leaf of this foliation of T, and because the xi from the addresses vary affine linearly with +slope, these leaves must approach the edge sy of T as i → ∞. Thus we must have that sq ⊂ K. (Every point +on sq is approached by a sequence of points from sqi with i → ∞.) It follows that sq ⊂ gr∞ and therefore +q ∈ G as desired. +□ + +ZEBRA SURFACES +47 +Proof of Theorem 7.3. Let P be a polygon with all exterior angles at least π. We will show that there is a +trail in P between any two points p,q ∈ P by showing that G = P, where as in Lemma 7.4, G is the union +over all rays r ∈ Rp of the connected component of r ∩ P containing p. By this lemma, G is closed. By +Corollary 3.18, we have gr = r ∩ P. Thus G coincides with the intersection of P with the union of rays, +which is open as a subset of P by Theorem 7.1. Since P is connected and G is non-empty and both open +and closed as a subset of P, we have that G = P. This proves the “if” part of the statement. +To prove the converse, suppose P is a polygon, and a, b, and c are consecutive vertices in counterclockwise +order such that the measure of the external angle ∡abc is less that π. Then we can find a slope m and an +open segment of a leaf or trail through b of constant slope m that is contained in the interior of P except for +touching the boundary at b. Considering a chart near b of the singular foliation Fm, we can find segments +of leaves of Fm that join ab to bc that enter the complement of P; see Figure 16. Let a′ ∈ ab and c′ ∈ bc be +the intersections of such a leaf with these two edges of P. We see that P cannot be convex, because there +can be at most one arc of a trail from a′ to c′ by Proposition 3.6, but by construction a′c′ exits P. +□ +Figure 16. The “only if” part of the proof of Theorem 7.3. +This theorem has a nice consequence for zebra tori without singularities: +Corollary 7.5. Suppose S is a closed surface with a zebra structure such that α is non-positive. Let S+ +denote S with its singularities removed. If S has two closed leaves which are not homotopic in S+, then the +PRU cover ˜S is convex. +Remark 7.6. The condition about the existence of non-homotopic closed leaves is necessary. Consider a +Hopf torus, C∗/⟨z ↦ λz⟩, as described in Section 1.2.3. The foliation by lines through the origin descends +to a foliation of a Hopf torus by homotopic closed leaves. The universal cover of such a torus is also the +universal cover of C∗, which is not convex, so these are the only closed leaves. +Proof of Corollary 7.5. By the Gauss-Bonnet Theorem, S is either a torus with α identically zero, or a +sphere with four poles. +Consider the case of the sphere. Each of our closed leaves is simple and so bounds a pair of disks, and by +the Gauss-Bonnet Theorem, each disk must contain two poles. If the two curves were disjoint, then they’d +have to bound an annulus and would therefore be homotopic. Thus, they must intersect. The sphere has +the torus as a double cover branched over the fours poles, and we can lift the zebra structure to the torus +and our closed leaves to intersecting closed curves on the torus. The foliations on the torus are orientable, so +the geometric intersection number coincides with the algebraic intersection number. The lifted closed leaves +on the torus cannot be homotopic, and thus it suffices to discuss the case of the torus. +Now let β and γ be the two non-homotopic closed leaves for a zebra structure on the torus without +singularities. Since these two curves are non-homotopic simple closed curves, they must intersect. It follows +that their slopes are distinct, and the intersection points are isolated. Let p0 ∈ β ∩ γ. Parameterize β ∶ +[0,1] → S and γ ∶ [0,1] → S such that both curves start and end at p0. Choose a lift ˜p0 ∈ ˜S, and select lifts +˜β and ˜γ starting at ˜p0. Let ∆β and ∆γ denote the deck transformations that carry ˜p0 to the endpoint of ˜β +and ˜γ, respectively. Consider the union of the four segments of leaves +(22) +˜β ∪ ∆β(˜γ) ∪ ∆γ(˜β) ∪ ˜γ. + +48 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +We claim that these four segments of leaves bound a parallelogram in ˜S, and the segments are edges listed in +cyclic order. Indeed consecutive segments above are not parallel and therefore, by Proposition 3.6, intersect +at exactly one point. Namely, ˜γ ∩ ˜β = {p0}, ˜β ∩ ∆β(˜γ) = {∆β(p0)}, +∆β(˜γ) ∩ ∆γ(˜β) = {∆β ○ ∆γ(˜p0) = ∆γ ○ ∆β(˜p0)}, +and +∆γ(˜β) ∩ ˜γ = {∆γ(p0)}. +Opposite edges must be disjoint: If they weren’t disjoint then because they are parallel, they’d have to lie +in the same leaf, but then one of the other edges would join a leaf to itself violating Proposition 3.6. Then +Proposition 3.9 guarantees that the four curves in (22) bound a parallelogram P1 ⊂ ˜S as claimed above. +Observe that ⟨∆β,∆γ⟩ is a finite index subgroup of the fundamental group of the torus, that ˜S/⟨∆β,∆γ⟩ +is a torus covering S, and that the parallelogram P1 constructed above is a fundamental domain for the +action of ⟨∆β,∆γ⟩ on ˜S. For each n ≥ 1, define +P2n+1 = +⋃ +−n≤i≤n +⋃ +−n≤j≤n +∆i +β ○ ∆j +γ(P1). +Each P2n+1 is a parallelogram formed from (2n+1)2 copies of P1 and bounded by four lifts of 2n+1-fold covers +of β and γ. Since P1 is a fundamental domain for the action of ⟨∆β,∆γ⟩ on ˜S, it follows that ⋃n P2n+1 = ˜S. +Each P2n+1 is convex by Theorem 7.3, and so ˜S is convex: If x,y ∈ ˜S, then there is an n such that x,y ∈ Pn +and so xy exists. +□ +7.3. Continuity of arcs of trails. If X is a topological space, then we use Cl(X) to denote the collection +of all closed subsets of X with its Fell topology, which has a subbase given by the sets, +V ∩ = {A ∈ Cl(X) ∶ A ∩ V ≠ ∅} +and +K /∩ = {A ∈ Cl(X) ∶ A ∩ K = ∅} +taken over all non-empty open sets V ⊂ X and all proper compact subsets K ⊂ X. If X is locally compact +second countable, then Cl(X) is compact and metrizable [Bee93, Theorem 5.1.5]. In particular, this holds +when X is a surface. +Consider the set +TA = {{p} ∶ p ∈ Z} ∪ {{x,y} ∶ x,y ∈ Z are distinct points that can be joined by a trail arc}. +If x and y are distinct and {x,y} ∈ TA, we use xy to denote the arc of a trail starting at x and ending at +y. A singleton {p} can also be written {p,p} and we define pp = {p}. In this way we have associated each +element of TA with a closed subset of Z. +Theorem 7.7. The set TA is an open subset of Z2 modulo permutation. The function +ta ∶ TA → Cl(Z); +{x,y} ↦ xy +is a homeomorphism onto its image. +Lemma 7.8. If xy is an arc of a trail in Z (possibly with x = y) then there is a decreasing sequence of +convex polygons Pn such that ⋂Pn = xy. In Cl(Z), we have limPn = xy. +Proof. In a locally compact and second countable space, a decreasing nested sequence of closed subsets +approaches its intersection in the Fell topology. Thus the second sentence follows from the first. +First consider the case of x = y. Since Z is a topological disk, x has a countable neighborhood base +which we can take to be nested, U1 ⊃ U2 ⊃ .... We produce our sequence of polygons by induction using +Corollary 3.11 to produce generalized rectangles. First define P1 to be a generalized rectangle contained in +U1 and containing x in its interior. Then assuming that Pk ⊂ Uk is generalized rectangle containing x in its +interior, define Pk+1 to be a generalized rectangle contained in Uk+1 ∩P ○ +k and containing x in its interior. To +see why the resulting sequence {Pn} approaches xy = {x}, first observe that {x} ∈ V ∩ implies x ∈ V and so +Pn ∈ V ∩ for all n. Second observe that {x} ∈ K /∩ implies there is a UN such that UN ∩ K = ∅ and therefore +Pn ∈ K /∩ for n ≥ N. +Now suppose that x ≠ y. We will cover a neighborhood of xy by polygons of three types. Observe that +that xy is the union of segments of leaves (or leaves) whose endpoints are in the union of {x,y} and the +collection of singularities in the interior of xy. We’ll call these segments edges in this proof. For each edge e +of xy, use Lemma 3.10 to construct two rectangles with an edge of e, one on each side of xy. These will be +called edge rectangles. Now fix a singularity z in the interior of xy. Since Z is a zebra plane, the cone angle +at z is at least 3π, and since xy is a trail, the angles made at z are at least π. So, the constructed right angles + +ZEBRA SURFACES +49 +of edge rectangles at z cannot overlap, but also can’t cover all of a neighborhood of z. The complement of +these right angles consists of one or two positive angles (one occurs if one side of xy makes an angle of π at +z), which we will call complementary angles. Choose finitely many segments of leaves emanating from z that +cut these complementary angles into smaller angles all of whose measures are less than π. Fix two adjacent +leaves emanating from z bounding such an angle, and construct a triangle with vertex z such that two edges +are arcs of these leaves, and the opposite angles are equal. We’ll call these equal angles the base angles and +these triangles vertex triangles. At endpoint x, the two edge rectangles with vertex x cover an angle at x of +measure π. Using Lemma 3.10, we can construct more rectangles to add to these two edge rectangles that +cover a neighborhood of x and have disjoint interiors. We call these rectangles end rectangles and also add +them around y. A picture of this situation is depicted in Figure 17. +Figure 17. Left: A neighborhood of a trail tiled by polygons as in the proof of Lemma 7.8. +Edge rectangles are colored blue, vertex triangles are red, and end rectangles are orange. +Right: Example foliations of the polygons, with the trail situated at the bottom of the +polygons. +We will foliate each of the polygons we have constructed, with the goal to enclose xy in a 1-parameter +family of boundaries of convex polygons that nest down to xy. Each edge rectangle has one edge on xy, and +by Corollary 3.14 we can foliate it by parallel leaves which run between the edges emanating from points +on xy. Each vertex triangle has T one vertex v that is a singularity on xy. Let m be the slope of the side +opposite v. Restricting Fm to T, we see from Proposition 3.6 that maximal segments of leaves through the +interior of T must join the edges of the triangle emanating from v. We foliate the end rectangles in a slightly +more technical way. These rectangles intersect xy in an endpoint (either x or y), and suppose we are given +a homeomorphism between the edges sharing this endpoint vertex, h ∶ e1 → e2 that preserves that endpoint +(i.e., h(x) = x or h(y) = y). A point p ∈ e1, then picks out two segments of leaves of the directional foliations: +the leaf through p parallel to e2 and the leaf through h(p) parallel to e1. By Corollary 3.14, these segments +of leaves join distinct pairs of opposite sides and so intersect. Therefore, we can join p to h(p) by a path +that follows these leaves, making the transition at the intersection point. (See the orange rectangle on the +right side of Figure 17 for examples of such paths.) +In order to achieve our goal, we need the leaves of the foliations around xy to close up. (If we attempted +to close the polygons using segments of say the boundaries of end rectangles, then the resulting curves might +not bound convex polygons.) To ensure closing, we need to take advantage of the flexibility built into the +foliations of the end rectangles. Pick one end rectangle with vertex x and call it R⋆, and choose foliations +as above, with arbitrary choices made for all end rectangles other than R⋆. Let e⋆ +1 and e⋆ +2 denote the edges +of R⋆ with endpoint x. Observe that there are closed intervals Ij ⊂ e⋆ +j with endpoint x for j ∈ {1,2} and +a homeomorphism h⋆ ∶ I1 → I2 such that h⋆(x) = x and for p ∈ I1 ∖ {x} there is a concatenation of leaves +joining p to h⋆(p). To foliate R⋆, extend h⋆ to a homeomorphism e⋆ +1 → e⋆ +2, and use this homeomorphism to +define the foliation as above. Observe that concatenations of leaves through our polygons passing through +I1 ∖ {x} all close up. +We claim that the closed leaves constructed above bound convex polygons. To see this we can check that +the interior angles all have measure less than π, by Theorem 7.3 and the fact that all points in zebra planes +have angle at least 2π. These interior angles only occur in the interiors of end rectangles, where all interior +angles are right angles, and in the transition between two vertex triangles or between a vertex triangle and an +edge rectangle. Observe that because our vertex triangles were constructed so that the base angles are equal, +these base angles always measure less than π +2 , while the contribution of edge rectangles to an interior angle +is always π +2 . So, all interior angles are less than π as claimed and so each bounded polygon is convex. +□ + +50 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Proof of Theorem 7.7. To see that TA is open, let {x,y} ∈ TA. Then xy exists, and Lemma 7.8 guarantees +that there is a convex polygon P1 containing xy in its interior, P ○ +1 . Observe that convexity guarantees that +P ○ +1 × P ○ +1 ⊂ TA, so TA is open. +To see ta is injective, let A ∈ ta(TA). Observe that if A consists of only one point, say p, then ta−1(A) +contains only {p}. Otherwise, A is an arc and ta−1(A) consists of the endpoints of A. (The endpoints of A +can be distinguished from the other points of A. If x is an endpoint of A, then for any neighborhood N of +x, there is a smaller neighborhood N ′ such that N ′ ∖ A is connected. On the other hand, if z ∈ A is not an +endpoint, then there is a neighborhood N of z such that N ∖A is homeomorphic to the disjoint union of two +open disks with z in the common boundary. Therefore, any smaller neighborhood N ′ must intersect both +these disks and so N ′ ∖ A is also disconnected.) +Now consider the continuity of ta. Fix xy ∈ ta(TA). Let Pn be the sequence of convex polygons approach- +ing xy guaranteed to exist by Lemma 7.8. A basis for Cl(Z) is given by sets of the form +U = V ∩ +1 ∩ V ∩ +2 ∩ ... ∩ V ∩ +n ∩ K /∩ +(since finite unions of compact sets are compact). +Suppose U contains xy. +We’ll produce an open set +containing {x,y} ∈ TA whose image is contained in U. +The case when x = y is simple: Here we have that x ∈ Vi for all i and since Pn → {x} in Cl(Z), for n large +enough, Pn that is disjoint from K. The open set consisting of pairs from P ○ +n ∩ ⋂n +i=1 Vi suffices. +Now suppose that x ≠ y. Then for any index i ∈ {1,...,n}, we can find a segment of a leaf γi ⊂ Vi such +that γi ∩xy consists of a single point pi that is distinct from x and y and that the crossing at pi is transverse, +in the sense that γi crosses from one side of xy to the other at pi. This situation is depicted in Figure 18. +Let E be the collection of endpoints of the collection of paths {γi ∶ i = 1,...,n}. Then there is a polygon +Pn that is disjoint from both K and E. Observe that by construction, x and y lie in distinct components of +P ○ +n ∖ γi. Denote these components by Xi and Yi respectively. Let X = ⋂n +i=1 Xi and Y = ⋂n +i=1 Yi. For x′ ∈ X +and y′ ∈ Y , both points lie in P ○ +n, so x′y′ exists. Since Pn is disjoint from K, so is x′y′. Furthermore, the +path x′y′ must intersect each of the γi and so must intersect each Vi. Thus x′y′ is in our basis element U as +desired. +Finally, we need to show that ta−1 ∶ ta(TS) → TA is continuous. Let U ⊂ TA be open and pick any +(x0,y0) ∈ U. We’ll find an open neighborhood of x0y0 in Cl(Z) such that whenever this open set contains +xy, we have {x,y} ∈ U. +We consider two cases separately. First suppose x0 = y0. Then there is an open subset U ′ ⊂ Z containing +x0 such that U ′ × U ′ ⊂ U. Using Lemma 7.8, we can produce a compact polygon P containing x0 in its +interior which does not intersect ∂U ′ and therefore is contained in U ′. Observe that the set (P ○)∩ ∩ (∂P)/∩ +is a neighborhood satisfying our condition. +Now assume that x0 ≠ y0. In this case we can find disjoint open neighborhoods Ux of x0 and Uy of y0 +such that the collection U ′ of pairs {x,y} with x ∈ Ux and y ∈ Uy satisfies U ′ ⊂ U. Applying Lemma 3.10, we +can produce produce a convex polygon Qx ⊂ Ux containing x0 such that x0y0 passes through the interior of +one of the edges of Qx. Define Kx to be the union of the edges that do not intersect x0y0. Similarly define +Qy ⊂ Uy and Ky. Consider the open subset of Cl(Z) defined by +(Q○ +x)∩ ∩ K /∩ +x ∩ (Q○ +y)∩ ∩ K /∩ +y . +Fix an xy from this set. Observe that since xy ∈ (Q○ +x)∩ it must contain points in Qx. Because Kx contains all +but one edge of Qx and xy ∈ K /∩ +x, if there was no endpoint in Qx, then xy must enter and exit Qx through the +same edge. This is impossible by Proposition 3.6, so Qx must contain either x or y. By the same analysis, +Qy must contain either x or y. Thus {x,y} ∈ U as desired. +□ +Corollary 7.9. If K ⊂ TA is compact, then so is the subset of Z given by ⋃{x,y}∈K xy. +Proof. Let Q = ⋃{x,y}∈K xy ⊂ Z. Since Z is metrizable, it suffices to prove that Q is sequentially compact, +so let qn ∈ Q be a sequence. Then for each n, we can find {xn,yn} ∈ K such that qn ∈ xnyn. Since K is +compact, there is a subsequence {xnk,ynk} that converges to some {x∞,y∞} ∈ K. By Lemma 7.8, we can +find a convex polygon P that contains x∞y∞ in its interior. Then the interior P ○ intersects x∞y∞ and the +boundary ∂P is disjoint from x∞y∞. By Theorem 7.7, we know that xnkynk converges to x∞y∞ in Cl(Z), +and so there is an index k′ such that k > k′ implies that xnkynk ∩ P ○ ≠ ∅ and xnkynk ∩ ∂P = ∅. Since xnkynk + +ZEBRA SURFACES +51 +Figure 18. Illustration of a configuration described in the proof of Theorem 7.7. +is connected, we conclude that xnkynk ⊂ P for k > k′. Then since P is compact, we know that qnk ∈ xnkynk +has a convergent subsequence. +□ +Theorem 7.10. The function µ that sends a pair of distinct points (x,y) ∈ Z2 with {x,y} ∈ TA and y +non-singular to the slope of xy measured at y is continuous. +Proof. Let I = (m0,m1) ⊂ ˆR be an open interval. Suppose µ(x0,y0) ∈ I. We need to find an open neighbor- +hood of (x0,y0) whose intersection with the domain maps into I. +Let α ⊂ x0y0 be an arc on the same leaf as the leaf of x0y0 containing y0, but such that α does not contain +y0. Using Lemma 3.10 twice, we can construct two rectangles with α as one edge, one on each side of x0y0. +The union of these two rectangles is a larger rectangle R such that α passes through the interior of R and +joins opposite sides. Recall that R is convex and contains no singularities in its interior. Choose a point +z from the interior of α, and through z construct segments ac and bd contained in R of slope m0 and m1, +respectively. Let Q denote the quadrilateral abcd. This construction is illustrated in Figure 19. Because +µ(x0,y0) ∈ I, direction considerations at z tell us that x0y0 passes through opposite sides of Q, and up to +swapping a and c, we can assume these sides are ab and cd. But x0y0 does not pass through bc or cd. So, +Lemma 7.8 guarantees we can find a convex polygon P containing x0y0 in its interior that does not intersect +bc∪ad. This lemma further allows us to ensure that P contains no singularities not found on x0y0, since the +singularities in any compact polygon form a compact set. Observe that x0 and y0 lie in distinct components +of P ○ ∖ Q, call these components X and Y . We claim the intersection of X × Y with the domain of µ maps +into I. Fix (x,y) ∈ X × Y . Then xy is forced to cross through edge ab into Q, then over both both ac +and bd, and exit through cd. Assuming xy does not pass through z, xy forms a triangle with ac and bd, +and Proposition 3.2 guarantees that the slope of xy measured in Q is in I. (The same holds if it does pass +through z because parallel trajectories form such a triangle.) The slope is the same as the slope measured +at y, because Y cannot contain any singularities, because by construction all singularities lie in X. Thus +µ(x,y) ∈ I as desired. +□ +Figure 19. Illustration for the proof of Theorem 7.10. +Theorem 7.11. Let K ⊂ Z be a compact subset of a zebra surface. Suppose {xn,yn} ∈ TA is a sequence of +pairs such that the limits x = limxn and y = limyn exist. If xnyn ⊂ K for all n, then {x,y} ∈ TA. +Proof. We offer a direct proof. Using Corollary 3.11, for each point p ∈ K, there is a generalized rectangle +Pp such that p ∈ P ○ +p and ∂Pp contains no singularities. Then {P ○ +p ∶ p ∈ K} is an open cover of K, so there is +a finite subcover. Let P denote the finite collection of polygons used in the finite subcover. +Let Q0 ∈ P denote a polygon containing x in its interior. If y ∈ Q0 then {x,y} ∈ TA by Theorem 7.3. +Now suppose y /∈ Q0. Then by discarding the first finitely many elements of our sequences, we may assume + +52 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +that xn ∈ Q○ +0 and yn /∈ Q0 for all n. Therefore Corollary 3.18 guarantees that for all n we have Q0 ∩ xnyn = +xnz1n for some point z1 +n ∈ ∂Q0. Since ∂Q0 is compact, by passing to a subsequence, we can assume that +z1 = limz1 +n ∈ ∂Q0 exists. Observe that xz1 exists as an arc of a trail by Theorem 7.3. +Since zn +1 ∈ xnyn for all n, we see that z1 ∈ K. Let Q1 ∈ P be such that z1 ∈ Q○ +1. If y ∈ Q1 then we can +construct the arc of a trail z1y by Theorem 7.3. In this case we claim that the concatenation xz1 ● z1y is a +trail. Since both paths in the concatenation are arcs of trails, we only need to check the path at z1 ∈ ∂Q0. +Since ∂Q0 contains no singularities, we know z1 is not singular. Similarly, the points z1 +n are not singular. +Since z1 +n ∈ xnyn, we have xnyn = xnz1n ● z1nyn. Let m1 +n denote the slope of xnyn measured at z1 +n. Applying +Theorem 7.10 to the sequence {xn,z1 +n} → {x,z1} tells us that m1 +n converges to the slope m1 of xz1 measured +at z1. Then applying the same result to {yn,z1 +n} → {y,z1} tells us that m1 is also the slope of z1y measured +at z1. Therefore, xz1 ● z1y is a trail, proving our claim. +Now assume that y /∈ Q1. By discarding the first finitely many elements of our sequences we may assume +that z1 +n ∈ Q○ +1 and yn /∈ Q1 for all n. Then Corollary 3.18 guarantees that for all n we have Q1∩z1nyn = z1nz2n for +some point z2 +n ∈ ∂Q1. By passing to a subsequence we can assume that z2 +n converges to some z2 ∈ ∂Q1. Then +Theorem 7.3 guarantees that we can construct z1z2. The argument from the previous paragraph guarantees +that xz1 ● z1z2 is an arc of a trail xz2. +The previous two paragraphs can be repeated inductively, producing a sequence of polygons Q0,...,Qk in +P. We also produce sequences zj +n ∈ xnyn ∩ ∂Qj−1 for j = 1,...,k that converge to zj ∈ ∂Qj−1 such that there +are arcs of trails xzj. At each such stage, either y ∈ Qk in which case we can construct a trail xy = xzk ●zky, +or we can extend the sequence one more step (passing to a subsequence as we do). Since P is finite, we must +either produce the trail xy at some point, or some polygon must appear twice in our sequence. To prove the +theorem, it suffices to show that no polygon can appear twice. Suppose to the contrary that Qj = Qk where +j < k. Temporarily fix n, which because of our passing to subsequences determines the points z1 +n,...,zk +n. In +our construction, we define zj+1 +n +to be the point such that Qj ∩ zj +nyn = zj +nzj+1 +n +. By construction we have the +sequence of proper subsets zj +nyn ⊃ zj+1 +n +yn ⊃ ... ⊃ zknyn. Thus zk +n ∈ zj +nyn ∖zj +nzj+1 +n +. Now recall that by definition +of Qk we have zk ∈ Q○ +k. Therefore for n large enough we have zk +n ∈ Q○ +k. But this is a contradiction: Since +Qj = Qk, for such an n we are supposed to have +zk +n ∈ Q○ +k, +Qk ∩ zj +nyn = zj +nzj+1 +n +, +and +zk +n ∈ zj +nyn ∖ zj +nzj+1 +n +. +□ +8. Convexity of triangulated zebra planes +In this section we prove Theorem 1.2, which says zebra planes with a leaf triangulation are convex. From +now on we assume our zebra plane Z has a leaf triangulation and we fix such a triangulation T . Since T +is a leaf triangulation, its vertices are singularities s ∈ Σ where α(s) > 0. (This guarantees that the angles +at the singular vertices of our triangle are all 3π or more. This point will be crucial for Lemma 8.2 below.) +Thus edges e of T are saddle connections, including their singular endpoints. As sets, a triangle T ∈ T is the +union of its three edges and the interior, and so is homeomorphic to a closed disk. +Proposition 8.1. In a leaf triangulation, only finitely many triangles meet at each vertex. +Proof. Let v be a vertex of a leaf triangulation. The zebra structure on the surface gives a bijection between +the prongs emanating from v and the circle R/α(v)πZ. Each triangle with vertex v corresponds to an interval +in R/α(v)πZ representing the prongs with representations that are entirely contained in the triangle.Since the +triangles are meeting edge-to-edge, these intervals of prongs meet endpoint-to-endpoint. Using compactness +of the circle, it is not hard to show that any collection of non-degenerate closed intervals in the circle that +have disjoint interiors, meet endpoint-to-endpoint, and cover the circle is necessarily a finite collection. +□ +To prove Theorem 1.2, we pick a point p ∈ Z. As above, let Rp denote the collection of all trail rays +emanating from p, and let Up denote the union of all these trail rays. By Theorem 7.1, Up is open. Our goal +is to prove that Up = Z, which indicates that any point q ∈ Z lies on a trail ray from p. Since p was arbitrary, +all pairs of points can be connected by an arc of a trail. + +ZEBRA SURFACES +53 +Our proof is an inductive argument showing that Rp covers larger and larger collections of triangles from +T whose union is Z. At the end of the day, the argument is largely combinatorial, so we begin by considering +how the rays in Rp can pass through a triangle T ∈ T . +Recall from Section 7.1 that Rp is a singular foliation of an open subset Up ⊂ Z by the leaves of rays in Rp +with singularities at p and at Σ. Let e be an edge of a triangle T not containing p. We’ll say e is transverse +to Rp if e ⊂ Up and as a leaf e is transverse to Rp. If e is an edge and T is a triangle with edge e, then rays +enter T through e if some rays intersect e before entering T or exit T through e if rays passing through e +have already passed through T. (There can be no transition between exiting and entering through e because +e is either everywhere transverse or coincides with a leaf.) +We say that an edge e of T is a leaf of Rp (or a leaf edge) if there is a trail ray containing e. In this case +we also have e ⊂ Up. Leaf edges get an orientation from inclusion in the ray containing them. +The following observations are the key to our induction. +Lemma 8.2. Fix p as above. Let T be a triangle that does not contain p and has an edge e0 that is transverse +to Rp. Assume that rays enter T through e0. Let e1 and e2 be the other two edges, labeled so that the list +of edges e0,e1,e2 is in counterclockwise order around ∂T. +Then T ⊂ Up and one of the following three +statements holds: +(1) The edges e1 and e2 are transverse to Rp and rays exit T through them. There is a ray in Rp that +passes through the interior of e0 and the opposite vertex of T. +(2) The edge e1 is transverse to Rp and rays exit T through e1, but e2 is a leaf of Rp whose orientation +agrees with the clockwise orientation on ∂T. +(3) The edge e2 is transverse to Rp and rays exit T through e2, but e1 is a leaf of Rp whose orientation +agrees with the counterclockwise orientation on ∂T. +Note that as a consequence of this lemma, no triangle has two edges through which rays enter T. +Proof. Normalize by a rotation so that edge e0 is vertical. +Parameterize e0 clockwise around T by γ ∶ +[0,1] → e0. Then by Theorem 7.2, there is a strictly increasing continuous function µ ∶ [0,1] → R such +that the ray crossing γ(t) has slope µ(t) at γ(t). Let m1 and m2 be the slopes of e1 and e2 respectively. +Then by Proposition 3.2, both are real and m2 < m1. Since µ(0) < µ(1), we have three mutually exclusive +possibilities: +(i) µ(0) < m1 and m2 < µ(1). +(ii) µ(0) < m1 and µ(1) ≤ m2. +(iii) m1 ≤ µ(0) and m2 < µ(1). +These possibilities are depicted in Figure 20. +In case (i), we can apply Lemma 6.5 to foliate T by the continuation of rays through e0. These rays enter +through e0 and therefore must exit transversely through e1 and e2. +In case (ii), there are two subcases. First, it could be that µ(1) = m2. In this case, there is a leaf of +Rp of slope µ(1) hitting e2, and the corresponding ray can be continued as a trail along e2, making an +angle of π counterclockwise from e2 to the ray hitting γ(1). Thus e2 is a leaf and is oriented clockwise. +Applying Lemma 6.5 again shows that rays hitting e0 exit transversely through e1. The second subcase +is when µ(1) < m2. In this case, we can continue the ray hitting γ(1) into the interior of the triangle T1, +making an angle of π at γ(1). The ray cannot exit through e2 or else it would violate the decreasing cyclic +order of slopes promised by Proposition 3.2. Therefore, it exits through e1 at some point, call it q. This +forms triangle T1 = △γ(0)γ(1)q, to which we can apply Lemma 6.5 again to foliate T1 by leaves passing +through e0 and exiting through γ(0)q ⊂ e1. Now consider the triangle T2 = T ∖ T1. This triangle has γ(1) as +one vertex and by Lemma 6.2 we can foliate T2 by leaves emanating from γ(1). The counterclockwise angle +from the ray hitting γ(1) to each leaf emanating from γ(1) into T2 is in the interval [π,2π) and since γ(1) +is a singularity with cone angle at least 3π, these are continuations of the trail hitting γ(1). The edge e2 is +one of these leaves emanating from γ(1), so e2 is a leaf oriented clockwise. Also, all of edge e1 is covered by +trail rays exiting T, so e1 is transverse to Rp and rays exit through e1. +Case (iii) is symmetric to case (ii) under an orientation reversing symmetry. +□ +We introduce some terminology associated to the triangles described in Lemma 8.2. We’ll call a triangle +T with an edge that is transverse to Rp such that rays enter T through the edge a transverse triangle. This + +54 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Figure 20. Cases (1)-(3) of Lemma 8.2 from left to right: fully transverse, clockwise leaf, +and counterclockwise leaf triangles. Edge e0 is always depicted at left, rays entering through +e0 are depicted in light blue with bold light blue arrows indicating µ(0) and µ(1). Pink +arrows indicate portions of rays emanating from a vertex. +includes any triangle covered by the lemma. We also introduce more specialized terminology for the three +cases. We’ll call T satisfying (1) a fully transverse triangle. A T satisfying (2) is a clockwise leaf triangle, +and a T satisfying (3) is a counterclockwise leaf triangle. Here “clockwise” and “counterclockwise” refer to +the orientation the (already oriented by Rp) leaf edges induce on ∂T. +There is a fourth type of triangle we need to understand. A double leaf triangle is one for which two +edges are leaves of Rp, and both are oriented away from their common vertex, and the third edge is a +transverse edge through which rays exit the triangle. These will show up as a consequence of the lemma +below. Combinatorial pictures of each triangle type are shown in Figure 21. +Figure 21. Combinatorial representations of the four triangle types (from left to right): +fully transverse, clockwise leaf, counterclockwise leaf, and double leaf. Green arrows indicate +leaf edges and their orientations. Blue arrows indicate transverse edges and the direction +rays cross. +We will now consider the local structure of the edges of T sharing a vertex v, as they relate to Rp. A flag +with vertex v is a pair (e,v) where e is an edge of T with endpoint v. Let F(v) denote the collection of all +flags with vertex v. Assuming e ⊂ Up, there are several possible relationships between the flag (v,e) and the +oriented foliation Rp. First, it could be that e is a leaf of Rp, in which case we call e a leaf edge and (v,e) +a leaf flag. Recalling that rays are oriented away from p, we see that leaf edges inherit orientations. Thus, a +leaf flag can either be oriented towards the vertex or away from the vertex. If e is not a leaf, then viewing v +as the center of a clock, we see that leaves of Rp either cross e in the clockwise or counterclockwise direction. +In these cases we call (v,e) a clockwise or counterclockwise transverse flag, respectively. See Figure 22. +Since singularities have cone angle at least 3π, there are at least four edges meeting at any vertex, and by +Proposition 8.1 there can be only finitely many edges meeting at a vertex. Thus, the flags in F(v) come with +a natural cyclic ordering. Given a flag f = (v,e) ∈ F(v), we’ll use f cc to denote the next flag counterclockwise +from f, and use f c to denote the next flag clockwise from f. We’ll say a collection of flags F ⊂ F(v) are +consecutive if with at most one exception, f ∈ F implies f cc ∈ F. We have: +Lemma 8.3. Let v ∈ Up ∖ {p} be a singularity. Suppose either: + +ZEBRA SURFACES +55 +Figure 22. Four examples of flags (v,e) drawn as a black point and segment. Portions of +the oriented foliation Rp are drawn in blue. From left to right: a leaf flag oriented towards +the vertex, a leaf flag oriented away from the vertex, a clockwise transverse flag, and a +counterclockwise transverse flag. +(a) There is a fully transverse triangle in Up such that v is vertex opposite the edge through which rays +enter the triangle. +(b) There is an edge e of the triangulation that is also a leaf of Rp oriented towards v, and the two +triangles sharing the edge e are contained in Up. +Then each triangle with vertex v is contained in Up and we have the following configuration of flags in F(v): +(1) In case (a) there are no leaf flags oriented towards v, and in case (b) there is exactly one leaf flag +oriented towards v. +(2) If f ∈ F(v) is a leaf edge oriented towards v, then f cc is a counterclockwise transverse flag and f c is +a clockwise transverse flag. +(3) The collections of clockwise transverse flags, counterclockwise transverse flags, and leaf flags oriented +away from v are all consecutive non-empty subsets of F(v). +Observe that the lemma forces clockwise and counterclockwise transverse flags to move outward from the +edges provided in the hypothesis (either a fully transverse triangle or leaf edge oriented towards v). At some +point, in each direction these leaves have to transition to (one or more) leaf edges oriented away from v. See +Figure 23 for examples. +Figure 23. Two examples of configurations by Lemma 8.3 are shown. An example satis- +fying hypothesis (a) is shown on the left and an example of (b) is on the right. Blue arrows +denote the direction Rp crosses a transverse edge, and green arrows are leaf edges indicating +their orientation. +Proof of Lemma 8.3. Note that if we have a leaf edge oriented towards v, we get an arc of a trail pv. Similarly, +in case (a), from Lemma 8.2(1) we get an arc of a trail pv. Statement (1) holds because there can be at most +one trail from p to v. +To see statement (2), suppose the leaf edge oriented towards v is e = wv ⊂ pv. Let T be the triangle to the +right of e as we move from w to v. Let x be the third vertex of T, making (v,vx) = (v,e)cc. Let ℓ ∶ [0,1] be +a parameterized segment of a leaf perpendicular to e such that ℓ(0) is in the interior of e and ℓ((0,1]) ⊂ T ○. +By Theorem 7.7, observe that for t sufficiently small, we have pℓ(t)∩vx = ∅ by definition of the Fell topology. +(Since pℓ(0) misses vx, so must pℓ(t) for t small.) Thus the ray extending such a pℓ(t) must exit through +vx, making (v,vx) a counterclockwise transverse flag. A symmetric argument handles the edge clockwise +from wv. +Either case (a) or (b) gives a counterclockwise transverse flag f = (v,e) in the initial triangle(s). Let T be +the triangle such that both the edges from f and f cc are edges of T. Then rays enter T through e, so we can +apply Lemma 8.2 to see that rays cover T and see that the f cc is either another counterclockwise transverse + +56 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +flag or a leaf flag edge oriented away from v. We can then repeat this inductively, forming a consecutive +collection of counterclockwise transverse flags. We observe that this process must terminate in a leaf edge +oriented away from v, because the trail pv can be continued as a trail by continuing the path through v such +that the counterclockwise angle from pv to the continuation is π. The next edge counterclockwise after this +continuation (or perhaps the edge that coincides with this continuation) is a leaf edge oriented away from v, +and prior edges are all counterclockwise transverse flags. +A similar analysis in the clockwise direction gives a consecutive collection of clockwise transverse flags +followed by a leaf edge oriented away from v. All triangles considered so far are contained in Up by Lemma 8.2. +If there are edges we haven’t seen yet, then they make an angle of more than π with pv on each side, so they +are leaf edges oriented away from v arranged in a consecutive collection as desired. Considering the remarks +above, we’ve proved (3). +It remains to show that triangles with two leaf edges oriented away from v are in Up. Let T be any such +triangle. Observe that the angles made between the leaf edges and pv are all larger than π. Therefore, any +leaf in T emanating from v is a trail continuation of pv. We conclude using Lemma 6.2 that T ⊂ Up. +□ +Lemma 8.3 provides us with an inductive step to prove Theorem 1.2. Assuming a vertex satisfies the +hypotheses of the lemma, it allows us to extend the known collection of triangles that are foliated by rays +to include all triangles with the given vertex. We apply this inductively to prove all of Z is covered. +To organize the induction used in the proof of Theorem 1.2, we borrow the idea of a queue from computer +programming [Knu97]. A queue Q is a data structure whose state is always a finite ordered list of elements +of a set V . We’ll write Qi to denote the state of the queue after i ≥ 0 operations have been performed. Thus +each Qi is an element of +V ∗ = +∞ +⋃ +n=0 +V n. +For simplicity our queue will always start by representing the empty list, which we denote by ε. Queues +support two operations. The enqueue operation E(w) ∶ V ∗ → V ∗ takes as input an element w ∈ V and +appends it to the list. Assuming this is the j-th operation performed, +Qj−1 = (w0,w1,...,wn−1) +implies +Qj = (w0,w1,...,wn−1,w). +The dequeue operation D ∶ V ∗ ∖ {ε} → V ∗ removes an element from the start of the list. +Thus if the +dequeue operation is applied to Qj−1 = (w0,w1,...,wn−1), we would remove w0 from the list and define +Qj = (w1,...,wn−1). A dequeue error is the attempt to apply the dequeue operation to ε. This is not defined +and must be avoided. Thus the set of operations we can perform is +O = {D} ∪ {E(w) ∶ w ∈ V }. +Suppose that we choose an infinite sequence of operations, {oj ∈ O}∞ +j=1. Then we can inductively update +the state of a queue Q by defining Q0 = ε and defining Qj to be operation oj applied to Qj−1 for each j ≥ 1. +The queuing sequence of Q is the sequence {vi ∈ V } in which elements are enqueued. That is, if {ji}∞ +i=0 is +the collection of j ≥ 1 such that oj is an enqueue operation and {ji} is enumerated in increasing order, we +define +{vi ∈ V }∞ +i=0 +where vi is such that +oji = E(vi). +Assuming no dequeue error occurs, the queuing sequence will be a well-defined infinite sequence. +We will use the following scheme to enumerate the vertices of an infinite triangulation of a connected +surface: +Lemma 8.4. Let G be an infinite connected graph all of whose vertices have finite degree. Let V denote the +vertex set of G. Suppose Q is a queue storing finite lists of elements of V with Q0 = ε. Let {oj}∞ +j=1 be an +infinite sequence of operations defined by induction according to the rules that: +(1) o1 = E(v0) for some v0 ∈ V . +(2) o2 = D. +(3) Whenever j ≥ 2 and oj = D is an operation that leads to the dequeuing of a vertex w, there is an +enumeration of the set E(w) of all vertices that are adjacent to w and have not already been enqueued +in operations with index less than j, E(w) = {w1,...,wk}, such that the next operations on the queue +are given by oj+i = E(wi) for i = 1,...,k and oj+k+1 = D. + +ZEBRA SURFACES +57 +Then, there are no dequeue errors, the queuing sequence {vi} of Q is an enumeration of V , and every vertex +is eventually dequeued. +Note that we have some freedom in our choice of operations that satisfy (1)-(3): We are free to choose +the order in which the vertices {w1,...,wk} are enqueued after each dequeue operation. +From a programming perspective, the reason this lemma holds is that we are constructing a spanning tree +formed inductively by including in the tree the edges from each dequeued vertex w to each of its adjacent +and not previously enqueued vertices (i.e., w1,...,wk), while simultaneously traversing the tree according in +the manner of a breadth-first search. We give an elementary formal proof: +Proof. The queuing sequence can be interpreted as a finite or infinite sequence {vi}m +i=0 for some m ∈ Z≥0 ∪ +{+∞}, being finite if there is a queuing error. We only enqueue vertices that have not already been enqueued, +so the map i ↦ vi is injective. We will show that every vertex will eventually be dequeued (before a dequeue +error occurs). Since there are infinitely many vertices, this guarantees that no no dequeue error can occur +and that {vi} is an infinite sequence enumerating the vertices. +Let w be any vertex. Since G is connected, there is a path of edges joining v0 to w, with vertices: +w0 = v0,w1,...,wk = w. +We will prove by induction that each wi is dequeued at some point. Since w0 = v0, we can see by inspection +that it is dequeued when o2 is performed. Now assume that wi was dequeued at some point. Then because +wi+1 is adjacent to wi, statement (3) ensures that wi+1 is enqueued shortly afterward if it hasn’t already +been enqueued. +Now observe that because queues remove from the front of the list and append to the +back, the vertex wi+1 will be removed after a finite number of dequeue operations (determined by its initial +position in the list). Since only finitely many enqueue operations occur between each dequeue operation, +wi+1 is dequeued in finite time. Furthermore, no dequeue errors can occur before wi+1 is dequeued, because +a dequeue error requires an empty list and wi+1 is in the list up to the point at which it is removed. By +induction, we see that wk = w is successfully dequeued. +□ +We will briefly outline the idea of the proof of Theorem 1.2. Let T be a leaf triangulation of a zebra +plane Z and let p ∈ Z. By a reduction, we’ll be able to assume that p is in the vertex set V of T . We’ll +let Q be a queue keeping track of finite lists of elements of V . The first operation we’ll perform on the +queue are o1 = E(p) and o2 = D which dequeues p. We’ll follow the restrictions on queue operations given in +Lemma 8.4. Thus, we will be defining a queue sequence {vi}∞ +i=0 enumerating V . But, recall that Lemma 8.4 +leaves us some freedom: the choice of the initial vertex, and the order in which vertices adjacent to the +most recently dequeued vertex are enqueued. Therefore, the sequence {vi} is defined inductively according +to our choices and initially we only know v0 = p. According to requirements of Lemma 8.4, the operations +immediately after o2 must enqueue the vertices adjacent to p, and we do so in arbitrary order. Then we are +required to dequeue one of the adjacent vertices: vertex v1 in our sequence. Proceeding inductively, once +vertex vi is determined, so are: +(23) +A(vi) = {w ∈ V ∶ w is adjacent to vi} +and +E(vi) = {w ∈ A(vi) ∶ we have ok ≠ E(w) for all k ≤ i}. +After dequeuing a vi ≠ p, we are required to enqueue the vertices in E(vi). As part of our induction, we +produce a proper consecutive subset A′(vi) ⊂ A(vi) for i ≥ 1 such that the vertices in the complement +A(vi) ∖ A′(vi) were already enqueued in previous steps of the induction, guaranteeing that E(vi) ⊂ A′(vi). +Recall that F(vi) denotes the flags with vertex vi. There is a natural bijection +fi ∶ A(vi) → F(vi); +w ↦ (vi,viw). +Using Lemma 8.3, we show that fi(A′(vi)) splits into three consecutive groups of flags: the counterclockwise +leaf flags, the leaf flags oriented away from vi, and the clockwise leaf flags. This allows us to enumerate the +collection A′(vi) as a list: +(a) First we list vertices w ∈ A′(vi) such that fi(w) is a counterclockwise transverse flag, in counter- +clockwise order. +(b) Second we list those w ∈ A′(vi) such that fi(w) is a clockwise transverse flag, in clockwise order. +(c) Last we list the w ∈ A′(vi) such that fi(w) is a leaf flag oriented away from vi, in counterclockwise +order. + +58 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Two examples of such enumerations are depicted by numbering vertices in Figure 24. +We enqueue the +subcollection E(vi) ⊂ A′(vi) in the order these vertices appear in the above list. Our inductive argument +will show that if we enqueue in this order, then we can continue to enqueue in this order. Whenever we +apply Lemma 8.3 after dequeuing a vertex vi, we also can conclude that all the triangles with vertex vi are +all contained in Up. Since our queue sequence enumerates V , we conclude that every triangle is contained +in Up and therefore Z ⊂ Up as desired. +Figure 24. Images related to the proof of Theorem 1.2. Left: A possible region X0. Middle +and right: Collections of all triangles with vertex vi (red dot), in cases when statements Ca +i +(middle) or Cb +i (right) are true. Dark gray regions illustrate Yi ⊂ Xi−1. Numbers are written +next to the vertices in A′(vi) = A(vi) ∖ Yi and indicate the order in which we enumerate +these vertices. +Proof of Theorem 1.2. Let Z be a zebra plane with a leaf triangulation T . Let p ∈ Z. We’ll show that +Z ⊂ Up, where Up as above denotes the union of all rays through p. +With no loss of generality, we may assume that p is a singularity and a vertex of a triangulation. To see +why there is no loss of generality, suppose p is not a singularity. We define a different triangulation ˆT of +Z. Observe that either p is in the interior of a triangle T, or p is in the interior of an edge e where two +triangles T1 and T2 meet. In the first case, partition T into three triangles along leaf segments from p to +the vertices of T to form ˆT . In the second case, cut T1 and T2 into two triangles each along leaf segments +from p to the vertices opposite e in T1 and T2. Both collections of leaf segments exist by Lemma 6.2. The +new triangulation ˆT is not a leaf triangulation, because p is not singular. But, if Z′ is the double cover of Z +branched only at p, then the preimages of edges of ˆT form a leaf triangulation T ′ for which the preimage p′ +of p is a singularity and a vertex. Our argument will then show that Z′ ⊂ Up′. Because the image of a ray +emanating from p′ under the covering map Z′ → Z is a ray emanating from p, we also would have Z ⊂ Up, +explaining this reduction. +Let V denote the collection of vertices of T , and now by assumption p ∈ V . Let Q be a queue keeping track +of finite lists of elements of V . The set V together with the edges of triangles forms an infinite connected +graph. We will perform operations as described in Lemma 8.4 to ensure that the queuing sequence is an +enumeration {vi}∞ +i=0 of V . For each i ≥ 0, let Xi denote the finite union of triangles: +(24) +Xi = ⋃{triangles T ∶ T has at least one vertex in {v0,...,vi}}. +Note that we are defining the vertices {vi} inductively, and we will be considering Xi to be defined as soon +as the vertices v0,...,vi are defined. +As in the outline, we have Q0 = ε, o1 = E(p) and o2 = D. Thus v0 = p. +It remains to specify how we enqueue vertices in a manner consistent with Lemma 8.4 after each dequeue. +The case of i = 0 is special. Vertex v0 = p is dequeued in operation o2. We list the collection of adjacent +vertices A(v0) = {v1,...,vk0} in arbitrary order. Then the next k0 + 1 operations will be +(25) +o3 = E(v1), +o4 = E(v2), +..., +ok0+2 = E(vk0), +and +ok0+3 = D. + +ZEBRA SURFACES +59 +We named these vertices so that their names correspond to their place in the queuing sequence. This also +determines unions of triangles X1,...,Xk0. +Consider the statements defined below for some i ≥ 1: +Ai : The vertices {v0,...,vi} have been determined and vertex vi is dequeued in some operation oj = D. +Bi : We have Xi−1 ⊂ Up. +Ci : Either statement Ca +i or statement Cb +i is true, where: +Ca +i : There is a fully transverse triangle in Xi−1 such that vi is the vertex opposite the edge through +which rays enter the triangle. +Cb +i : There is an edge e of the triangulation that is also a leaf of Rp oriented towards vi, and the two +triangles sharing this edge e are contained in Xi−1. +We will inductively prove that for every i ≥ 1, the statements Ai, Bi, and Ci are true. +We will first prove some base cases. First observe that A1 is true, because v1 will be dequeued in operation +ok0+3 by (25). Second we claim that B1 is true. To see this observe that X0 is the union of triangles with +vertex v0 = p, and Lemma 6.2 tells us that these triangles are foliated by leaves emanating from p, proving +B1. It also follows that statements Cb +1,...,Cb +k0 are all true, because for i = 1,...,k0, the edge pvi is a leaf +edge oriented towards vi, and the triangles sharing this edge are in X0 ⊂ Xi−1. Clearly Cb +i implies Ci, so we +have shown that +(26) +A1 ∧ B1 ∧ C1 ∧ C2 ∧ ... ∧ Ck0 +is true. +We will now prove the following implication for each i ≥ 1: +(27) +(Ai ∧ Bi ∧ Ci) �⇒ (Ai+1 ∧ Bi+1). +Suppose that Ai, Bi and Ci are true. Since Ai is true, {v0,...,vi} have been determined and vertex vi has +been dequeued. Verifying Ai+1 involves ensuring a vi+1 has been defined and dequeued, but before we can +dequeue vi+1 we must enqueue the vertices in E(vi), and so we also need to show we can enqueue these +vertices according to statements (a)-(c) of the proof outline. When combined with the truth of statement +Bi, statement Ca +i implies statement (a) of Lemma 8.3 is true with v = vi, and Cb +i implies statement (b) +of Lemma 8.3 is true. Using the conclusions of Lemma 8.3, we see that Bi and Ci together imply that all +triangles with vertex vi are contained in Up. That is, Bi+1 is true. To see Ai+1 is true, we need to show we +can enqueue the vertices in E(vi) and so we need to more carefully consider the vertices in A(vi). There are +two cases. If Ca +i is true, then there is a fully transverse triangle Ta satisfying Ca +i . Otherwise Cb +i must be +true and the pair of triangles {Tb,T ′ +b} sharing the edge e from the statement satisfy the statement. Define +Yi = Ta if Ca +i is true, and Yi = Tb ∪ T ′ +b if Cb +i is true. Then as a consequence of Ca +i or Cb +i (whichever is true) +we see that Yi ⊂ Xi−1. We define A′(vi) = A(vi) ∖ Yi, and so E(vi) ⊂ A′(vi) as in the outline. By (1)-(3) of +Lemma 8.3, with the possible exception of one leaf flag oriented towards vi which is necessarily contained +in Yi, the flags F(vi) fall into three consecutive groups: counterclockwise transverse flags, leaf flags oriented +away from vi, and clockwise transverse flags. Furthermore the order in which these edge types were just +listed matches the counterclockwise cyclic order on the intervals. Therefore, we can enumerate the vertices +in A′(vi) as described in statements (a)-(c) of the proof outline. (Note that fi(A′(vi)) excludes the leaf edge +oriented towards vi if it exists, as well as the first counterclockwise transverse flag in the counterclockwise +order, and excludes the first clockwise transverse flag in the clockwise order.) Because E(vi) ⊂ A′(vi), the +total ordering corresponding to this enumeration on A′(vi) restricts to a total ordering on E(vi) and we +enqueue these vertices in this order. Immediately after enqueuing all these vertices, we must carry out a +dequeue operation. Since we have been following the restrictions in Lemma 8.4, this cannot cause a dequeue +error so we must dequeue vertex vi+1. This proves that Ai+1 is true, and completes the proof of (27). +We now show the following implication holds for each i > k0: +(28) +(Ai ∧ (B1 ∧ B2 ∧ ... ∧ Bi) ∧ (C1 ∧ C2 ∧ ... ∧ Ci−1)) �⇒ Ci. +Assume the hypotheses. Since Ai is true, we know that {v0,...,vi} have been defined. Therefore vi was +enqueued at some point, and we must have vi ∈ E(vk) for some k < i. Since i > k0, we must have k ≥ 1. +Since 1 ≤ k < i, we know that Ck is true. We can then define the collection of triangles Yk as in the previous +paragraph. Since Yk ∩ E(vk) = ∅, we know vi /∈ Yk. We wish to prove Ci, and to do so we break into cases +depending on the type of the flag fk(vi) = (vk,vkvi), which was relevant to the order in which the vertices +E(vk) were enqueued. Recall that fk(E(vk)) consisted only of counterclockwise transverse flags, clockwise + +60 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +transverse flags, and leaf flags oriented away from vk. First suppose fk(vi) is a counterclockwise transverse +flag. Let u ∈ A(vk) be such that the flag fk(u) = fk(vi)c is next clockwise. Because vi /∈ Yk, it follows +from statements (1)-(3) of Lemma 8.3 (which apply because Bk and Ck are true as noted in the previous +paragraph) that fk(u) is also a counterclockwise transverse flag. (It might be helpful to look at Figure 24.) +Thus, rays enter the triangle △vkuvi through edge vku. We can then apply Lemma 8.2 and see that either +uvi is a transverse edge through which rays exit the triangle or uvi is a leaf edge oriented towards vi. In the +first case, △vkuvi is a fully transverse triangle in Xk ⊂ Xi−1 so Ca +i is true. In the second case, observe that +because of our choice of ordering, we have that u = vj for some j < i. (If u /∈ E(vk) then it was enqueued +earlier than elements of E(vk), and if u ∈ E(vk) it comes before vi in our enumeration of E(vk).) Therefore in +this second case, the two triangles sharing edge uvi are in Xj ⊂ Xi−1 and so statement Cb +i is true. The case +when fk(vi) is a clockwise transverse flag is handled by a mirror-symmetric argument. The final possibility +is that vkvi is a leaf edge oriented towards vi. In this case, the two triangles sharing this edge are contained +in Xk ⊂ Xi−1, so statement Cb +i is true here. We have shown that Ci is true in all possible cases. +Now observe that (26), (27), and (28) together imply that Ai, Bi and Ci are true for all i ≥ 1. Statement +Bi guarantees that Xi−1 ⊂ Up for all i. Since {vi} is an enumeration, we have Z = ⋃Xi−1 and so Z ⊂ Up as +desired. +□ +9. Closed Trails +9.1. Curves and deck transformations. Let S be a zebra surface. Recall that in Section 3.1 we defined +the PRU cover ˜S as the largest cover which is at most doubly branched over the poles. As in that section, +let Σ−1 = {p ∈ S ∶ α(p) = −1} be the set of poles. In Proposition 3.1 we proved that ˜S is a disk and a normal +cover of S. +The homotopy lifting property does not work when the range of the homotopy includes points in Σ−1 +because of the branching. Therefore, for this section, we will only consider paths and loops in γ ∶ [0,1] → S +where either: +(1) no poles lie in the interior of γ, i.e., γ((0,1)) ∩ Σ−1 = ∅. +(2) γ is an arc of a trail on S. +We allow γ to be a trail, because the lifting property works here in the sense that given a lift of the beginning +of γ (e.g., γ∣(0,ϵ) for some ϵ > 0), there is a unique way to continue the lift through the preimage of a point +in Σ−1 such that the lift is still a trail. (The preimage of a point in Σ−1 is non-singular, and the lift must go +straight through non-singular points.) +Two paths in S are pole-resolved (PR) homotopy equivalent rel endpoints if they have a lift to ˜S with the +same start and end points. Because the cover is normal, if γ1 is PR homotopy equivalent rel endpoints to +γ2, for any lift ˜γ1, there is a lift ˜γ2 with the same start and end points as ˜γ1. +Recall S+ = S ∖ Σ−1. Choose a basepoint p0 ∈ S+ and a preimage of this point ˜p0 ∈ ˜S. The pole-resolved +fundamental group is πPR +1 +(S,p0) is the collection of PR homotopy classes rel endpoints of loops starting and +ending at p0, with the operation of concatenation. This group is the same as π1(S+,p0)/N where N is as in +(8). In particular, if Σ−1 = ∅, then πPR +1 +(S,p0) = π1(S,p0). +If α and β are two parameterized curves such that the endpoint of α is the same as the starting point of +β, let α ● β denote their concatenation which follows α and then β. Two loops γ1,γ2 ∶ [0,1] → S+ will be +said to be pole-resolved (PR) free homotopy equivalent if there are paths η1,η2 ∶ [0,1] → S+ with ηi(0) = p0 +and ηi(1) = γi(0) for i ∈ {1,2} such that the concatenations η1 ● γ1 ● η−1 +1 +and η2 ● γ2 ● η−1 +2 +are PR homotopy +equivalent rel endpoints. Existence of such η1 and η2 is equivalent to the condition that for any curves βi +with βi(0) = p0 and βi(1) = γi(0), the elements in πPR +1 +(S,p0) given by [β1 ● γ1 ● β−1 +1 ] and [β2 ● γ2 ● β−1 +2 ] +are conjugate in πPR +1 +(S,p0). Thus, PR free homotopy classes of closed curves in S+ are in natural bijective +correspondence with conjugacy classes in πPR +1 +(S,p0). From the remarks above about trails, a closed trail +also determines such a conjugacy class. The collection of all closed curves determining a conjugacy class +in πPR +1 +(S,p0) is a PR free homotopy class of closed curves. The notion of PR free homotopy equivalence +coincides with the usual notion of free homotopy equivalence if S contains no poles. +From standard covering space theory, there is an isomorphism ∆ from the group πPR +1 +(S,p0) to the deck +group of the covering ˜S → S. To understand the associated deck group action, fix an element [γ] ∈ πPR +1 +(S,p0) +with representative γ and a point ˜q ∈ ˜S. Choose a path ˜η starting at ˜p0 and ending at ˜q such that the image + +ZEBRA SURFACES +61 +η in S is a path in the sense above. Let ̃ +γ ● η denote the lift of the concatenation γ ● η starting at ˜p0. Then +the image of ˜q under the deck transformation associated to γ, is the end point of ̃ +γ ● η denoted ∆γ(˜q). +Recall from Section 3.1 the notion of a polar loop in S+. We’ll call an element [γ] ∈ πPR +1 +(S,p0) polar if +curves in [γ] are PR free homotopy equivalent to a polar loop. +Proposition 9.1. Let [γ] ∈ πPR +1 +(S,p0). The following statements are equivalent: +● [γ] is polar. +● ∆γ is non-trivial and fixes a point in ˜S. +● [γ] is non-trivial and [γ]2 is the identity. +● ∆γ fixes a unique point in ˜S. +If these statements are false and [γ] is non-trivial, then [γ] is infinite order and all orbits of ∆γ are infinite. +Proof. First suppose [γ] is polar and let γ ∈ [γ]. We claim that ∆γ is non-trivial and fixes a point in ˜S. By +hypothesis, γ is PR homotopic rel endpoints to a concatenation η ● ℓ ● η−1, where η is a path in S+ starting +at p0 and ending in an open disk U ⊂ S with U ∩ Σ1 containing a single point q, and ℓ ⊂ U is a polar loop +enclosing q. Let β ⊂ U be a path joining the endpoint of η to q. Let ̃ +η ● β denote the lift of η ● β starting at +˜p0, and let ˜q denote the endpoint of ̃ +η ● β. We claim that ˜q is fixed by ∆γ. From our description of the deck +group action, ∆γ(˜q) is given by the endpoint of the lift of +(η ● ℓ ● η−1) ● (η ● β) +which is homotopic rel endpoints to +η ● ℓ ● β. +Thus ∆γ(˜q) = ˜q if and only if the lifts ̃ +η ● β and +̃ +η ● ℓ ● β terminate at the same point. We will explain that +this follows from the fact that both curves begin with η and are concatenated with paths that stay within U +terminating at the singularity q. Let ˜U be the connected component of the preimage of U in ˜S that contains +˜q. By Proposition 3.1, the restriction of the covering map is a map ˜U → U which is double branched over q. +Both ̃ +η ● β and +̃ +η ● ℓ ● β begin by following the same lift ˜η. By definition ̃ +η ● β ends at ˜q, so the endpoint of +˜η is contained in ˜U. Since ℓ ● β is contained entirely in U, the endpoint of the concatenation +̃ +η ● ℓ ● β must +be in ˜U. But, this endpoint must also be a lift of q, and the only lift of q that is contained in ˜U is ˜q, so +̃ +η ● ℓ ● β ends at ˜q as desired. This proves that ∆γ(˜q) = ˜q. To see ∆γ is non-trivial, let ˜r be the endpoint of +˜η. Then, ∆γ(˜r) is the endpoint of ̃ +η ● ℓ, which is distinct since the lift ˜ℓ of ℓ starting at ˜r does not lift as a +closed loop to ˜S because of the double branching. This completes the proof of our claim. +Now suppose ∆γ is non-trivial but has a fixed point ˜q. Since the restriction of the PRU covering map +to the preimage of S+ is a covering map, no point in the preimage of S+ can be fixed by a non-trivial deck +transformation. So the image q of ˜q must be a pole. Let U ⊂ S be a disk such that U ∩ Σ−1 = {q} as in +the previous paragraph and let ˜U be the connected component of the preimage containing ˜q. Then the deck +group of the restricted covering map cover ˜U → U must be order two and so ∆2 +γ is trivial. Thus [γ] is +non-trivial but [γ]2 is the identity in πPR +1 +(S,p0). +Now suppose [γ] is non-trivial but [γ]2 is the identity. The quotient ˜S/⟨∆γ⟩ must be an orientable surface +(intermediate between ˜S and S). If ∆γ has no fixed points then covering space theory guarantees that ˜S/⟨∆γ⟩ +is a surface with fundamental group isomorphic to Z/2Z, but all surfaces with finite fundamental group are +simply connected so such a quotient cannot exist. We conclude that ∆γ must have at least one fixed point. +Suppose it has two, ˜q and ˜q′. In ˜S/⟨∆γ⟩ construct a simple path α starting at the image of ˜q and ending at +the image ˜q′ whose interior does not contain lifts of points in Σ−1. Then α has two lifts ˜α1 and ˜α2, which are +both paths from ˜q to ˜q′. By construction, the union ˜α1 ∪ ˜α2 is a simple closed curve, which by the Jordan +Curve Theorem bounds a disk D ⊂ ˜S. Observe that ∆γ swaps ˜α1 and ˜α2, and therefore takes D to the +exterior of the curve ˜α1 ∪ ˜α2. Because ∆γ swaps the two curves, the union D ∪ ∆γ(D) is a 2-sphere. But, +this is impossible because this set is contained in ˜S which is a disk by Proposition 3.1. We conclude that +∆γ cannot have distinct fixed points. +To complete the equivalence of the four statements, suppose ∆γ has a unique fixed point ˜q ∈ ˜S. Let ˜ζ +be a simple loop through the basepoint in ˜S and contained in the preimage of S+ such that intersection of +the enclosed disk with Σ−1 is {˜q}. Let ζ ⊂ S+ be the image of ζ. Then from the first paragraph, ∆ζ fixes ˜q +but is non-trivial. Since ∆γ and ∆ζ fix ˜q and are non-trivial, they must agree in a neighborhood of ˜q (as +the covering is double branched at ˜q). Thus ∆γ = ∆ζ. From covering space theory, [γ] and [ζ] are equal in +πPR +1 +(S,p0), and so [γ] is polar. + +62 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Finally suppose the four statements are false for a non-trivial [γ] ∈ πPR +1 +(S,p0). Clearly if all orbits of ∆γ +are infinite, then [γ] is infinite order. So, it suffices to show that ∆γ has no periodic points. Suppose to +the contrary that ∆γ does have a periodic point. Then we can find a periodic point ˜q ∈ ˜S of minimal period +n ≥ 2. First, it could be that ∆n +γ is a trivial deck transformation. In this case there is a well-defined covering +˜S → ˜S/⟨∆γ⟩, and covering space theory tells us that ˜S/⟨∆γ⟩ has fundamental group Z/nZ. But, then ˜S/⟨∆γ⟩ +is a surface with a non-trivial finite fundamental group, which is impossible. If ∆n +γ is non-trivial, then the +four statements guarantee that ∆n +γ fixes a unique point. Let ˜q be this fixed point. But then each point in +the ∆γ orbit of ˜q (namely {˜q,∆γ(˜q),...,∆n−1 +γ +(˜q)}) is fixed by ∆n +γ, contradicting the uniqueness of ˜q (or that +n ≥ 2 is the minimal period). +□ +Corollary 9.2. If [γ] ∈ πPR +1 +(S,p0) is non-trivial and non-polar, then ˜S/⟨∆γ⟩ is a normal quotient of ˜S that +is homeomorphic to an annulus. +Proof. Clearly ˜S/⟨∆γ⟩ is an oriented surface because it is intermediate between ˜S and S. From Proposi- +tion 9.1 and covering space theory, ˜S/⟨∆γ⟩ has fundamental group isomorphic to Z. From the classification +of surfaces, ˜S/⟨∆γ⟩ must be homeomorphic to an annulus. +□ +9.2. Topological considerations for closed trails. Let (S,{Fm}) be a zebra surface with singular data +function α ∶ S → Z≥−1. A zebra automorphism of (S,{Fm}) is a homeomorphism δ ∶ S → S such that α○δ = α +and such that for each m ∈ ˆR, the pullback of Fm under δ is identical to Fm. +Let Z be a zebra plane and let δ ∶ Z → Z be a zebra automorphism with no fixed points. (Such auto- +morphisms naturally arise as deck transformations of PRU covers.) Let A = Z/⟨δ⟩ which is topologically an +open annulus with a zebra structure. +The fundamental group of an annulus is isomorphic to Z, and is isomorphic to H1(A;Z). We say that a +loop in the annulus is a core curve if its homology class generates H1(A;Z). We’ll say curve is essential if it +is not homotopic to a point. A basic topological fact about curves in the annulus is: +Proposition 9.3. Any essential simple closed curve in an annulus is a core curve. +Proof. First, identify A with C∗ = C ∖ {0}. It is a simple check that a simple parameterization of the circle +Cr of radius r centered at 0 is a core curve of C∗. The general statement follows from the Jordan curve +theorem. A simple curve γ in C∗ separates the sphere into two components. For the curve to be essential, +each must contain one of the two points removed. For r > 0 sufficiently small, Cr and γ must be disjoint and +so C∗ ∖ (Cr ∪ γ) must have three components. One component must be compact so the homology class of γ +is the same as that of Cr up to sign. So, γ is also a core curve. +□ +Proposition 9.4. A closed trail in A is homologically non-trivial. +Proof. If there were a homologically trivial closed trail, then it would lift to a closed trail in Z in violation +of Proposition 3.5. +□ +A cover of a closed curve is just a parameterization that wraps around the curve multiple times. A cover +of a closed trail is still a closed trail. Up to covers, closed trails in annuli are simple and distinct closed trails +do not intersect: +Proposition 9.5. Any closed trail in A covers a simple closed trail in A. If two closed trails in A intersect, +they cover the same simple closed trail in A (up to reparameterization). +Proof. Suppose γ1 and γ2 are homologous closed trails that intersect. Then we may parameterize the curves +by γi ∶ R/Z → A for i = 1,2 such that γ1(0) = γ2(0). Denote this common point by p = γ1(0). Consider the +curve η = γ1 ● γ−1 +2 , which starts at p, wraps around γ1 once, then wraps around γ2 backward once. Then η is +homologically trivial and so lifts to a closed curve ˜η = ˜γ1●˜γ−1 +2 , where ˜γ1 and ˜γ2 are trails in Z that descend to +γ1 and γ2 on A, respectively. Observe that Proposition 3.6 guarantees that ˜γ1 = ˜γ2 up to reparameterization +fixing the endpoints, so γ1 = γ2 up to reparameterization fixing 0. +We can use the fact we just proved to establish the proposition. First suppose γ ∶ R/Z → A is a closed +trail that is not simple. For the first assertion, it suffices to prove that γ is a non-trivial cover. (Since this +will decrease the absolute value of the homology class of γ, repeating the operation finitely many times must +lead to a simple curve.) Since γ is not simple there are distinct points t1 and t2 in the domain such that + +ZEBRA SURFACES +63 +γ(t1) = γ(t2). Let γ1 = γ ○ φ1 and γ2 = γ ○ φ2 be orientation-preserving reparameterizations of γ sending t1 +and t2 to zero, respectively. The curves γ1 and γ2 are homologous so the above paragraph applies and we get +that γ1 is a reparameterization of γ2, γ1 = γ2 ○ ψ, where ψ fixes zero. It then follows that γ = γ ○ φ2 ○ ψ ○ φ−1 +1 . +Since γ is locally one-to-one,this implies that φ2 ○ ψ ○ φ−1 +1 +is a finite-order homeomorphism of R/Z. The +combined reparameterization described by φ2 ○ ψ ○ φ−1 +1 +sends t1 to t2 and so is non-trivial, and γ covers the +quotient curve +(R/Z)/⟨φ2 ○ ψ ○ φ−1 +1 ⟩ → A, +which sends the ⟨φ2 ○ ψ ○ φ−1 +1 ⟩-orbit of x ∈ R/Z to γ(x). +Now suppose η1 and η2 are closed trails that intersect. From the previous paragraph, we know they cover +simple closed trails γ1 and γ2 respectively. Since η1 and η2 intersect, we know that γ1 and γ2 also must. +By Proposition 9.3, up to reversing the orientation of one of the curves, we may assume that γ1 and γ2 are +homologous. Then the first paragraph again gives that γ1 = γ2 up to reparameterization. +□ +The above focuses our attention on core curves when looking for closed trails. We will need the following +topological fact for later arguments: +Proposition 9.6. Let A be an open annulus and let q,r ∈ A be distinct points. Let γ be a simple closed +curve that is a core curve of A and passes through both q and r. Let A′ be the union of γ and one of the +components of A ∖ γ. Then any simple curve in A′ joining q to r is homotopic rel endpoints to one of the +arcs of γ joining q to r. Furthermore if β is a simple closed curve in A′ that passes through q and r and is +a core curve of A, then the two arcs of β from q to r are homotopic rel endpoints to the two arcs of γ from +q to r. +Proof. We can assume that A is the 2-sphere S2 with two points x,y ∈ S2 removed. Let γ be a simple core +curve of A, and let q and r be distinct points on γ. Let A′ be the union of γ and the component of A ∖ γ +containing x in its boundary. Now suppose that α ∶ [0,1] → A′ is a simple curve with α(0) = q and α(1) = r. +There is an ambient isotopy of A that fixes q and r but moves the rest of γ into the interior of A′. By +moving α under this isotopy, we see that we can assume that α ∩ γ = {q,r}. Thus, the union of α and either +of the arcs of γ from q to r forms a simple closed curve. By the Jordan curve theorem, each choice bounds +a closed disk contained in A′ ∪ {x}, and the union of the two disks is A′ ∪ {x}. These two disks intersect in +the curve α, so exactly one of the disks contains x. The disk that does not contain x can be used to define a +homotopy rel endpoints from α to the other boundary component of the disk, which is one of the two arcs +of γ joining q to r. +Now consider the last case where β is a simple core curve contained in A′ and passing through q and r. In +light of the previous paragraph, the arcs of β from q to r are each homotopic rel endpoints to one of the arcs +from γ. If the arcs of β were homotopic to the same arc of γ, then β would be contractible in A′. But this +is impossible because β is a core curve, so each homotopy class of arcs rel endpoints of γ must be attained +by an arc of β. +□ +9.3. Existence of closed trails. As in the previous section, let A = Z/⟨δ⟩ be an annular quotient of a zebra +plane. +Given a point ˜p ∈ Z, if there is an arc of a trail joining ˜p to δ(˜p), then we’ll denote it by ˜γ˜p. We use γp +to denote the image curve in A, which is a closed loop based at p that is also a core curve of the annulus. +Note that γp is independent of the choice of lift ˜p. The loop γp satisfies the angle condition for trails, except +possibly at the point p. So, in some sense it is close to being a closed trail. Our main result is that under +mild hypotheses it passes through a closed trail: +Theorem 9.7. Let p ∈ A be arbitrary and suppose that γp exists. If for all points q ∈ γp the curve γq exists, +then for some q ∈ γp, the curve γq is a simple closed trail. +Observe the following consequence: +Corollary 9.8. Suppose S is a zebra surface and [γ] ∈ πPR +1 +(S,p0) is non-trivial and non-polar. Let ˜S be +the PRU cover of S, and let A be the annulus ˜S/⟨∆γ⟩. Suppose there is a point p ∈ A such that γp exists. +Suppose further that there is a convex subset of ˜S that contains two consecutive periods of the preimage of +γp. Then, S contains a closed trail that is PR free homotopic to the curves in [γ]. + +64 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +We now turn our attention to proving the theorem. It will make our lives easier to assume that γp is a +simple curve. The following result allows us to do this: +Lemma 9.9 (Lollipop lemma). Every γp contains a simple γq as a subarc. Moreover, if γp is not simple, +then γp is (up to reparameterization) the concatenation of paths γp = α ● γq ● α−1 for some q ∈ γp such that +γq is a simple closed curve. Furthermore, α is a simple curve that is disjoint from γq except for the common +endpoint q. +We call this the Lollipop Lemma because it tells us that a non-simple γp traces the pattern of a lollipop: +γp begins by traveling up the stick (following a simple curve α), then travels around a circular candy (γq) +and returns to p by traveling back down the stick (α−1), and in addition both γq and α are simple and +disjoint except at their common endpoint. +Proof. Assume γp is not simple. Let X denote the set of pairs (a,b) ∈ [0,1] × [0,1] with a < b. Define +Y = {(a,b) ∈ X ∶ γp(a) = γp(b)}. +Then by continuity of γp, Y is a closed subset of X. Since γp is not simple, Y is non-empty. +There is a natural partial ordering on Y given by (a,b) ≤ (c,d) if [a,b] ⊂ [c,d]. We will apply Zorn’s +lemma to find a minimal element. (We remark that Zorn’s lemma is more than we need here, because our γp +is combinatorially fairly simple. However, Zorn’s lemma provides a framework enabling us to avoid thinking +about the combinatorial details.) To see that Zorn’s lemma applies, let {(ai,bi) ∈ Y ∶ i ∈ Λ} be a totally +ordered subset. We must find a lower bound. Let a = sup{ai} and b = inf{bi}. Then [a,b] = ⋂i∈Λ[ai,bi]. We +claim that a ≠ b. If this were not the case, we can choose a neighborhood of the common point γp(a) = γp(b) +that lifts to the zebra plane Z, and by continuity and the fact that the intervals nest down to this common +point, we can find a pair (ai,bi) such that γp([ai,bi]) is contained in this neighborhood. The lift of this +segment to Z violates the injectivity of trails on zebra planes. Thus, a ≠ b as claimed and because Y is +closed, we have (a,b) ∈ Y giving us our needed lower bound. +Zorn’s lemma guarantees the existence of a minimal (a,b) ∈ Y . The restriction γp∣[a,b] is a simple curve +in A. The curve γp∣[a,b] must be an essential simple closed curve, because otherwise its lift to Z violates +Proposition 3.5. Then Proposition 9.3 tells us that γp∣[a,b] is a core curve, and lifts of this restriction to Z +have endpoints differing by δ. It follows that the restriction γp∣[a,b] must coincide with γq or γ−1 +q , where q is +the common point, q = γp(a) = γp(b). We have shown that γp = α ● γ±1 +q ● β where α is a path from p to q and +β is a path from q back to p. +Figure 25. Illustration relevant to the case γp = α ● γ−1 +q ● β in the proof of Lemma 9.9. +We will first show that the power of γq must be positive. Suppose to the contrary that γp = α ● γ−1 +q +● β. +We will obtain a contradiction by considering lifts of these curves to Z; see Figure 25. Choose a lift ˜p ∈ Z +of p, and let ˜α denote the lift of α starting at ˜p. Then ˜α ends at some lift ˜q of q. Let ˜γq be the lift ending +at ˜q, so that following ˜γ−1 +q +ends at δ−1(˜q). Then define ˜β to be the lift of β starting at δ−1(˜q). This means +that the concatenation ˜α ● ˜γ−1 +q +● ˜β is a lift of γp, and ˜β must end at δ(˜p). Now observe that both ˜α ● ˜γ−1 +q +and δ−1(˜β−1 ● ˜γq) are arcs of trails running from ˜p to δ−1(˜q). Therefore, by Proposition 3.6, the two curves +coincide. Now observe that δ−2(˜q) is in δ−1(˜β−1 ● ˜γq) (as the endpoint of δ−1(˜β−1)) and therefore it must +also be in ˜α ● ˜γ−1 +q . Since ˜γ−1 +q +is the lift of a simple loop and joins ˜q to δ−1(˜q), it must be that δ−2(˜q) ∈ ˜α. +Thus by uniqueness of trails, we must have ˜α = δ−1(˜β−1) ● ϵ for some path ϵ from δ−2(˜q) to ˜q. Similarly, we + +ZEBRA SURFACES +65 +see that ˜q must be in δ−1(˜β−1 ● γq) but can’t be in δ−1(˜γq) and so must be in δ−1(˜β−1). We conclude by +uniqueness of trails that δ−1(˜β−1) = ˜α ● η for some path η joining ˜q to δ−2(˜q). But we have shown +(29) +˜α = δ−1(˜β−1) ● ϵ +and +δ−1(˜β−1) = ˜α ● η +from which it follows that ˜α = ˜α ● η ● ϵ, which is absurd: a compact path cannot be the concatenation of +itself with a non-trivial curve. +We have shown that γp = α●γq ●β. Again let ˜p be a lift of p and ˜α be a lift of α starting at ˜p. Define ˜q to +be the endpoint of ˜α, which is a lift of q. Let ˜γq be the lift of γq starting at ˜q. Then ˜γq ends at δ(˜q). Let ˜β +be the lift of β starting at δ(˜q). By definition of γp, we have that ˜β ends at δ(p). Now observe that both ˜α +and δ−1(˜β−1) run from ˜p to ˜q, so they coincide because they are trails. We conclude that α = β−1 as desired. +Now we know that γp = α ● γq ● α−1. With (a,b) ∈ Y minimal as above, we have γp(a) = γp(b) = q. We +claim the minimal pair (a,b) ∈ Y is unique. Suppose to the contrary that (c,d) ∈ Y is a distinct minimal +pair. Let q = γp(a) = γp(b) as above, and let r = γp(c) = γp(d). Let α = γp∣[0,a] as above and β = γp∣[0,c]. We +see that +γp = α ● γq ● α−1 = β ● γr ● β−1. +By definition of the partial order, we cannot have [a,b] ⊂ [c,d] or [c,d] ⊂ [a,b]. Up to swapping the minimal +pairs, we have two possible configurations of the points a,b,c,d ∈ [0,1], non-overlapping or overlapping: +a < b < c < d +or +a < c ≤ b < d. +First consider the non-overlapping possibility, a < b < c < d. In this case, +α−1 = γp∣[b,c] ● γr ● β−1 +and +β = α ● γp ● γp∣[b,c]. +Combining these rules creates an absurdity as in (29). In the overlapping case of a < c ≤ b < d, we have +β = α ● γp∣[a,c] +and +α−1 = γp∣[b,d] ● β−1, +which again leads to an absurdity, ruling out the possibility of multiple minimal pairs. +Now we claim that the curve α = γp∣[0,a] is simple. If not, then there is a pair (c,d) ∈ Y ∩ [0,a]2. Then +Y ∩ [0,a]2 is non-empty and we can apply Zorn’s lemma to produce a minimal (c′,d′) ∈ Y ∩ [0,a]2. Such a +minimal element would also be minimal in Y , in contradiction to the previous paragraph. +Finally suppose that α and γq share a point in common other than q. Then there is a c ∈ [0,a) and a +d ∈ (a,b) for which γp(c) = γp(d). Thus Y ∩ [c,d]2 is non-empty and Zorn’s lemma guarantees that there is +a minimal element (c′,d′) ∈ Y ∩ [c,d]2. This minimal element cannot be (a,b) because (a,b) /⊂ [c,d], so this +is also a contradiction to the uniqueness of the minimal element of Y . +□ +Corollary 9.10. If p and q are distinct points in the annulus A and γp = γq as sets, then γp is a simple +closed trail. +Proof. First assume γp = γq is not simple. Then this common curve must be a lollipop. Observe that a +lollipop is homeomorphic to a graph with two vertices: one of degree one and another of degree three. If the +lollipop is γp then p is the vertex of degree one. Thus γp = γq implies that p = q in the case of a lollipop. +Now consider the case when p ≠ q and γp = γq is a simple closed curve. Observe that γp satisfies the +angle condition to be a trail at every point other than p, and γq satisfies the angle condition at every point +other than q. So, p ≠ q and γp = γq implies that this simple closed curve satisfies the angle condition at all +points. +□ +Lemma 9.11. Suppose γp is a simple closed curve but is not a closed trail. If q ∈ γp and γq is defined, then +γp ∩ γq is connected. +To prove this, observe that since γp is a simple closed core curve of the annulus A, it splits A into two +sub-annuli. Since γp is not a closed trail, one of the angles made by γp at p must have measure smaller +than π. Let Ap denote the union of γp and the component of A ∖ γp containing the interior angle at p with +measure smaller than π. We have: +Proposition 9.12. Suppose γp exists and is a simple closed curve but is not a closed trail. Then for any +q ∈ Ap for which γq exists, we have γq ⊂ Ap. + +66 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Proof. Suppose to the contrary that γq /⊂ Ap. Then there is an arc α of γq ∖ Ap joining Ap to itself. Thus, α +has endpoints in γp = ∂Ap. Lift α to a curve ˜α in the PRU cover Z. Observe that ˜α is a trail and therefore +simple. The preimage of γp in Z is a bi-infinite path, so there is a polygonal disk ˜D bounded by ˜α and an +arc of the preimage of γp. But, there are only two possible points in ∂ ˜D where the interior angles are smaller +than π, namely the endpoints of ˜α. This violates Proposition 3.4, giving us our desired contradiction. +□ +Proof of Lemma 9.11. Clearly q ∈ γp ∩ γq, so to show γp ∩ γq is connected it suffices to show that any +r ∈ γp ∩γq ∖{q} can be joined to q by a segment in the intersection. Let r ∈ γp ∩γq ∖{q}. Let α ⊂ γp be an arc +from q to r within γp that does not contain p in its interior. (At least one of the two arcs from q to r in γp +must work.) Observe that γq ⊂ Ap by Proposition 9.12. If γq is a simple closed curve, then Proposition 9.6 +guarantees that there is an arc β of γq joining q to r that is homotopic rel endpoints to α. The same holds if +γp is a lollipop, but for continuity we postpone the argument to the following paragraph. So, either way we +get an arc β of γq that is homotopic to α. Since α and β are homotopic trails joining q to r, Proposition 3.6 +guarantees that α = β up to reparameterization. The common curve α = β is an interval in γp ∩ γq joining q +to r as desired. +Consider the case when γq is a lollipop and r ≠ q is another point of γp ∩ γq. We must show that two arcs +of γq from q to r are homotopic to the two arcs in γp from q to r. We will see that we can reduce to the case +when γq is a simple closed curve by performing a surgery on the annulus A. Viewing γq as a graph, let x +denote the vertex of γq that has degree three (i.e., the place where the stick meets the candy). The segment +qx is the stick of the lollipop. Let D be a closed topological disk such that γq ∩ D = {x} and such that D +is contained in the unbounded component of Ap ∖ γq. Let T be a triangle. Let A′ be the annulus formed +by cutting out qx ∪ D, and then gluing T in its place, where one vertex is sent to q, the two adjacent sides +are sent to the two arcs formed by cutting along qx, and the final edge is glued to the arc of ∂D joining x +to x. See Figure 26. Proposition 9.6 applies in A′, showing that the simple closed curve replacing γq has +arcs homotopic to the two curves in γp from q to r. There is a continuous surjective map φ ∶ A′ → A that +collapses T back to qx ∪ D, obtained by collapsing each leaf of a partial foliation of T (omitting a disk sent +to D). Post-composing the homotopies with φ gives the desired statement in A. +□ +Figure 26. Top left: A lollipop γq in an annulus A before surgery. Top right: The annulus +A′ formed from A by a surgery that makes γq a simple closed curve. Bottom: The triangle +T glued to A with qx ∪ D cut out to form A′. To reconstruct A from A′, we collapse leaves +in the orange partial foliation of T to points and send the unfoliated topological disk to D. +Lemma 9.13. Assume γp exists and is a simple closed curve, q ∈ γp ∖ {p} and γq is defined. Suppose +r ∈ γp ∩ γq and γr exists. Then γp ∩ γr ⊂ γp ∩ γq. +Proof. Assume the hypotheses described in the lemma. We break into several cases. +If γp is a closed trail, then γp = γq = γr so the conclusion is trivially true. Therefore, we may assume that +γp is a simple closed curve but not a closed trail. Let Ap be the subannulus bounded by γp such that the +interior angle at p is less than π as above. Proposition 9.12 guarantees that γq and γr are contained in Ap. +Now suppose that γq is a simple closed curve. If γq is a closed trail, then we’d have γr = γq and so the +conclusion is trivially true. So, assume that γq is not a closed trail. In this case we can define the region Aq + +ZEBRA SURFACES +67 +as above. Since q ∈ γp ∖ {p} and γp is a trail, the exterior angle of Ap at q is at least π. Since γq ⊂ Ap, the +annular region Aq must be on the same side of γq at q as Ap. Thus, Aq ⊂ Ap. Proposition 9.12 guarantees +that γr ⊂ Aq. To see γp ∩ γr ⊂ γp ∩ γq observe that if x ∈ γr ∩ γp, then x ∈ γr ∩ ∂Ap. Since γr ⊂ Aq, we have +x ∈ Aq ∩ ∂Ap. But since Aq ⊂ Ap, any point in both Aq and ∂Ap must lie in ∂Aq = γq. +It remains to handle the case when γq is not simple. By Lemma 9.9, γq is a lollipop. Then γq = α●γq′ ●α−1 +for some q′ ∈ γq. It is possible that r ∈ α. But in this case γr is the lollipop formed by removing the arc of +α from q to r, since this yields a trail. Thus in this case γr ⊂ γq and the conclusion is trivially true. So, we +may assume that r ∈ γq′. It is possible that γq′ is a closed trail, but in this case we have γr = γq′ ⊂ γq which +again trivially leads to the desired conclusion. Otherwise γq′ is a simple closed curve but not a closed trail, +and so Aq′ is well defined. Since γq is a trail, the side of γq′ where the angle at q′ appears with measure less +than π must not contain α. Thus, Aq′ ⊂ Ap. We have γr ⊂ Aq′ ⊂ Ap, so repeating the argument from the end +of the previous paragraph shows that γr ∩ γp ⊂ γq′ ∩ γp and we also have γq′ ⊂ γq. +□ +Proof of Theorem 9.7. Let p ∈ A. Assume γp exists and that γq exists for all q ∈ γp. We will show that some +γq is a closed trail. +First of all, we can assume without loss of generality that γp is a simple curve. (Otherwise replace γp by +the simple closed subarc guaranteed by Lemma 9.9, and observe that the hypotheses still hold.) We may +also assume that γp is not a closed trail, or else the conclusion is trivial. +For q ∈ γp, let Iq = γp ∩ γq, which is a closed connected subset of γp. Define +I = {Iq ∶ q ∈ γp}. +This collection is partially ordered by inclusion. Lemma 9.13 guarantees that r ∈ Iq implies that Ir ⊂ Iq, so +any nested family in I has a lower bound. Thus Zorn’s Lemma tells us that I has a minimal element Imin. +We break into several cases. First, it could be that Imin = γp. In this case choose a q ∈ γp ∖ {p}. We see +that Iq = Imin and thus γq = γp. Thus, γp is a closed trail by Corollary 9.10. +Now suppose that Imin is a non-degenerate interval. Choose distinct q and r from the interior of Imin. +Then both γq and γr contain Imin. Since γq and γr contain arcs in two directions leaving q and r, respectively, +these curves are not lollipops. Let q′ and r′ denote the endpoints of Imin. Each of γq and γr is the union +of two arcs joining q′ to r′, namely Imin and its complement. By Proposition 9.6, these two arcs that are +complements of Imin are homotopic rel endpoints. Thus by Proposition 3.6, they are equal and so we must +have γq = γr. This common curve must be a trail by Corollary 9.10. +The last possibility is that Imin consists of a single point, call it q. We claim that γq is a trail. Suppose +to the contrary that γq is not a closed trail. We break into subcases. +First, it could be that γq is a lollipop, so γq = α ● γr ● α−1. This case is illustrated on the left side of +Figure 27. Because γq ∩ γp = {q}, the curves γp and γr are disjoint and bound an annulus A′. We will apply +the Gauss-Bonnet Theorem to A′. The only interior angle whose measure is less than π occurs at p, and +the interior angle at r is at least 2π. So, the quantity on the left side of (5) is negative. But the Euler +characteristic of the annulus is zero, so this is a contradiction. It follows that γq could not have been a +lollipop after all. +Figure 27. Cases of the proof of Theorem 9.7 when Imin = {q}. + +68 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Otherwise γq is a simple closed curve, but not a closed trail. This case is illustrated on the right side of +Figure 27. This case is ruled out by another Gauss-Bonnet computation, this time involving the region R +between γp and γq. This region is a topological disk that touches itself at the point q, but it has a lift to +the zebra plane that does not touch itself so the Gauss-Bonnet Theorem applies. The right hand side of (5) +is therefore 2π. We will show the left side sums to at most π, giving us our desired contradiction. To get +this upper bound, we must sum π − θ where θ varies over the interior angles in the boundary of the region +R. Let θ1,...,θm denote the measures of angles over all points where γp bends, measured on the side of +γp containing R. Let η1,...,ηn denote the measures of angles over all points where γq bends, again taken +using the side of γq where R resides. Note that in making these definitions, we have intentionally included +angles at q that do not appear in the boundary of R, and omitted the angles at q that do appear. To fix +this, observe that γp and γq come together at q to form four angles. We will denote the measures of these +angles by a, b, c and d, as illustrated in Figure 27. If θ1 is the angle of γp based at q measured on the side +of γp containing R, then θ1 = a + b + c. Similarly, if η1 is the angle of γq based at q measured on the side +containing R, we have η1 = a + c + d. The interior angles of R based at q that appear in the boundary of R +are given by a and c. Therefore, the total contribution of the boundary to the left side of (5) is given by +(30) +⋆ = +m +∑ +i=1 +(π − θi) + +n +∑ +i=1 +(π − ηi) − (π − θ1) − (π − η1) + (π − a) + (π − c). +Suppose θ2 is the angle of γp measured at p on the side containing R. Since γp is simple but not a closed +trail, θ2 < π. Since γp is a trail, this is the only positive term in the sum ∑m +i=1(π − θi) and since this sum +adds to an integer multiple of π by Proposition 2.7, we see that ∑m +i=1(π −θi) ≤ 0. Since γq is simple but not a +closed trail, we have that b < π. (The angle b must be the one less than π, because the complement contains +d which has measure at least π because γp is a trail.) Therefore, η1 = a+c+d = (a+b+c+d)−b > a+b+c+d−π, +and π − η1 < 2π − a − b − c − d. The other ηi contribute non-positively to the sum, and since again the sum +must be an integer multiple of π, we have +n +∑ +i=1 +(π − ηi) ≤ π − a − b − c − d. +Also we can simplify the remaining terms in (30): +−(π − θ1) − (π − η1) + (π − a) + (π − c) = a + b + c + d. +Plugging all these quantities into (30), we see that +⋆ ≤ 0 + (π − a − b − c − d) + (a + b + c + d) = π. +This proves that the contribution of the boundary to the left side of (5) is at most π. The contribution of +any singularities in the interior of R is non-positive, so the left side of (5) is at most π, while the right side +is 2π as indicated above. This is a contradiction, so γq cannot be simple and not a closed trail. +□ +9.4. Uniqueness of closed trails. As in the previous two subsections, we let Z be a zebra plane, δ ∶ Z → Z +be a fixed-point free zebra automorphism, and A = Z/⟨δ⟩ be the annular quotient. +Theorem 9.14. If γ1 and γ2 are distinct closed trails that are core curves of A, then they are disjoint and +bound a compact annulus K ⊂ A such that there are no singularities in the interior of K and all interior +angles at singularities in ∂K = γ1 ∪ γ2 are of measure π. +Proof. That γ1 and γ2 are simple and disjoint follows from Proposition 9.5. Because they are simple disjoint +core curves, they bound a compact annulus K as described. We apply the Gauss-Bonnet Theorem. The +Euler characteristic of the annulus K is zero. All singularities in the interior of A contribute negatively to +the left side of (5), because there are no π-singularities. Also because γ1 and γ2 are trails, the boundary +contributes non-positively and positively unless all interior angles are π. We conclude that there can be no +singularities in the interior of K and all interior angles on the boundary have measure π. +□ +We have the following trivial consequence: +Corollary 9.15. If γ is a closed trail in a zebra surface S such that γ has at least one bending angle larger +than π on each side, then γ is the unique closed trail in the conjugacy class associated to γ in πPR +1 +(S,p0). + +ZEBRA SURFACES +69 +Proof. Let Z be the PRU cover of S. Then γ has a lift to a trail ˜γ ⊂ Z, which is bi-infinite by Theorem 3.19. +The deck transformation ∆γ preserves ˜γ, and since ˜γ is bi-infinite, ∆γ cannot be order two. It then follows +from results in Section 9.1 that A = Z/⟨∆γ⟩ is an annulus. If η were a distinct closed trail determining the +same conjugacy class, then we could lift η to a distinct closed trail on A and Theorem 9.14 would lead to a +contradiction to the hypothesized property about angles on each side of γ. +□ +9.5. Foliating quadrilaterals. We will describe a result that produces a foliation of a quadrilateral. We +do this to produce the foliations in cylinders. Concretely, consider an immersed trapezoid in a zebra surface, +where the restriction to the interior is an embedding and the two parallel sides have the same image. +Assuming the other sides do not meet each other, this results in an embedded annulus in the surface and +there is a natural homeomorphism from one parallel side to the other coming from the edge identification. +This homeomorphism determines a foliation, because of the following lemma. The statement is depicted on +the left side of Figure 28. +Lemma 9.16 (Edge homeomorphism foliation). Let P be the quadrilateral abcd in a zebra plane Z whose +interior angles add to 2π and whose interior angles are all less than or equal to π. Let h ∶ ab → dc be a +homeomorphism such that h(a) = d and h(b) = c. Then the collection of segments of leaves +{x h(x) ∶ x ∈ ab} +is pairwise disjoint, covers P, and foliates the interior of P. Moreover, the function µ ∶ P → ˆR that sends a +point p ∈ P to the slope of the segment x h(x) containing p is continuous. +Figure 28. A depiction of the statement of Lemma 9.16. +Proof. Proposition 3.3 guarantees that there are no singularities in the interior of P. We can find a Euclidean +polygon P ′ ⊂ R2 with sides of the same slopes. Choosing an appropriate homeomorphism ∂P → ∂P ′, we +may apply the Surgery Theorem to construct a new zebra plane Z′ = (R2 ∖ P ′) ∪ P. Observe that Z′ has no +singularities. For the proof we consider P to be a subset of Z′. This will simplify the logic below (but is not +essential for this argument), because P considered as a subset of Z could have singularities in ∂P. +First we will make some remarks about the geometry of P. Since all the interior angles are less than π, +P is convex by Theorem 7.3. Convexity guarantees that any two points in P can be joined by a trail, but +because Z′ does not contain singularities, any two points can be joined by segments of leaves. We will be +repeatedly use that a distinct pair of leaves can intersect in at most one point. +The above remarks show that for each x ∈ ab, the segment x h(x) exists. Furthermore, we claim that +distinct segments x h(x) and x′h(x′) cannot intersect. If they did intersect, then they intersect at exactly +one point transversely, because of the stellar structure at the point and Proposition 3.6. +Therefore, if +x h(x) ∩ x′h(x′) ≠ ∅, then the points x and h(x) would have to lie on opposite sides of x′h(x′), but that is +impossible from our definition of h. +The restriction of µ to ab is continuous as a consequence of Theorem 7.10. +Now we claim that the collection of segments x h(x) with x ∈ ab covers P. It is clear that every point +in ∂P is covered. Let q be a point in the interior of P. We will show q lies in one of the segments. Let +m0 denote the slope of aq and m1 denote the slope of bq. Assume without loss of generality that the slope +of ab = +∞. Then m0,m1 ∈ R and and m0 < m1 by Proposition 3.2. By considering the order of segments +emanating from a, we see that m0 < µ(a). Similar considerations at b tell us that µ(b) < m1. By Lemma 6.2, + +70 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +△abq is foliated by leaves of slope in [m0,m1] through q, and the function ν ∶ ab → [m0,m1] sending x to +the slope of xq is continuous. Since both ν and µ∣ab are continuous, take real values, and satisfy +ν(a) = m0 < µ(a) +and +µ(b) < ν(b) = m1, +there must be an x ∈ ab such that µ(x) = ν(x). But then the leaf x h(x) has slope ν(x) and so passes +through q. This completes the proof of our claim that P is covered. +Since the collection {x h(x)} is pairwise disjoint and covers P, there is a well defined function f1 ∶ P → ab +that sends p ∈ P to the x ∈ ab such that p ∈ x h(x). Choose a foliation F′ by segments of leaves joining ad to +bc. (If ab and cd are parallel, F′ can be taken to be a directional foliation, otherwise we can use Lemma 6.5 +to produce such a foliation.) Define f2 ∶ P → bc to send p ∈ P to the intersection of the leaf of F′ through p +with bc. We claim that +f1 × f2 ∶ P → ab × bc +is a homeomorphism. Note that the codomain is naturally homeomorphic to a rectangle in R2. The map +is one-to-one because each p ∈ P lies in exactly one x h(x) and exactly one leaf of F′. The map f1 × f2 is +surjective since every x h(x) intersects every leaf of F′. Since F′ is a foliation, we see that f2 is continuous. +To see that f1 is continuous observe that f −1 +1 (xy ∖ {x,y}) is the quadrilateral with vertices x, h(x), h(y) +and h(x) with the edges x h(x) and y h(y) removed. This set is open as a subset of P. We’ve shown that +f1 × f2 is a continuous bijection. Since the domain is compact and the codomain is Hausdorff, f1 × f2 is a +homeomorphism. Since f1 × f2 is a homeomorphism, the collection of segments of the form x h(x) = f −1 +1 (x) +foliate P. +□ +9.6. The closed trail theorem for annuli. We continue the notation from the previous subsection. The +following result guarantees that we can extend certain closed trails to open sets foliated by closed leaves. +Lemma 9.17. Let γ be a closed trail that is a core curve of an annulus A = Z/⟨δ⟩. Let A+ denote one of +the connected components of A ∖ γ. Suppose all interior angles in A+ at singularities along γ have measure +π. Then there is an open annulus U ⊂ A+ one of whose boundary components is γ that is foliated by closed +leaves (of possibly varying slope). Furthermore the function µ ∶ U ∪ γ → ˆR sending a point p to the slope of +the closed trail through p is continuous. +Proof. Let π ∶ Z → A denote the covering by the zebra plane. Choose a lift ˜γ ∶ [0,1] → Z, which is an arc of a +trail. Let ˜A+ = π−1(A+). Using Lemma 3.10, we can construct a rectangle ˜R ⊂ ˜A+ such that one edge is given +by ˜γ. Such a rectangle is convex by Theorem 7.3 and cannot contain singularities by Proposition 3.3. Let a +and b be the other vertices of ˜R so that the cyclically ordered vertices are a, ˜γ(0), ˜γ(1), and b. Because the +measure of the interior angle of A+ at the common point γ(0) = γ(1) is π, the image under the covering π of +the rays ���→ +˜γ(0)a and ���→ +˜γ(1)b must coincide. So, we can choose points a′ from the interior of ˜γ(0)a and b′ from +the interior of ˜γ(1)b such that the images of ���→ +˜γ(0)a′ and ���→ +˜γ(1)b′ coincide in A. This necessarily implies that +there are no singularities on the interiors of these paths. Also since they map to the same path in A, there is +a natural homeomorphism h ∶ ˜γ(0)a′ → ˜γ(1)b′ obtained by restricting a deck transformation. By convexity of +˜R, we can construct the arc of a trail a′b′. Let ˜Q denote the quadrilateral with vertices a′, ˜γ(0), ˜γ(1), and b′. +Let ˜Q− denote ˜Q with a′b′ removed. Then ˜Q−∖˜γ contains no singularities. We define U = π( ˜Q−)∖γ. Observe +that the restriction of π to ˜Q○ factors through the quotient ˜Q/h. Lemma 9.16 guarantees that we can foliate +˜R by leaves joining x ∈ ˜γ(0)a′ to h(x) ∈ ˜γ(1)b′ and that the slopes of these leaves vary continuously. For any +x ∈ ˜γ(0)a′ ∖ {˜γ(0)}, the segment x h(x) does not pass through any singularities and therefore projects to a +closed leaf in U. These closed leaves therefore foliate U. +□ +Lemma 9.18. If K ⊂ A is compact, then the union of all closed trails contained in K is closed. +Proof. Let T denote union of all closed trails contained in K. We need to show that T is closed, so let +p∞ ∈ T ∖ T. We will find a closed trail ℓ∞ ⊂ K containing p∞. +Since p∞ ∈ T, there is a sequence pn ∈ T converging to p∞. Let ℓn ⊂ T denote the closed trail through +pn. By Theorem 9.14, there can be at most two closed trails in A that are not closed leaves, so we may +assume that each ℓn is a closed leaf. Let ˜p∞ ∈ π−1(p∞) be a preimage. Then we likewise select preimages + +ZEBRA SURFACES +71 +˜pn ∈ π−1(pn) such that lim ˜pn = ˜p∞. Let ˜ℓn denote the portion of the bi-infinite leaf π−1(ℓn) running from +δ−1(˜pn) to δ(˜pn). Then ˜ℓn is a lift of two periods of ℓn. Assume that +(31) +⋃ +n +˜ℓn +is contained in a compact subset of Z. +(We will verify this assumption at the end of the proof.) Then Theorem 7.11 guarantees that there is a +limiting arc of a trail ˜ℓ∞ joining δ−1(˜p∞) with δ(˜p∞). Observe also that ˜p∞ ∈ ˜ℓ∞. (If ˜p∞ /∈ ˜ℓ∞ then there +is a compact neighborhood K of ˜p∞ that is disjoint from ℓ∞. Since ˜ℓn → ˜ℓ∞ in Cl(Z), we also must have +that ˜ℓn ∩ K = ∅ for n large enough. But this is impossible because ˜pn ∈ ˜ℓn converges to ˜p∞.) Thus ˜ℓ∞ is the +concatenation of trail arcs: +˜ℓ∞ = δ−1(˜p∞)˜p∞ ● ˜p∞δ(˜p∞) = δ−1(˜p∞)˜p∞ ● δ(δ−1(˜p∞)˜p∞). +It follows that the image ℓ∞ ⊂ A of ˜ℓ∞ is naturally a parameterized closed curve, with a parameterization +ϕ coming from the composition of a homeomorphism [0,1] → δ−1(˜p∞)p∞ and the covering map π ∶ Z → A. +The parameterized closed curve ϕ ∶ R/Z → ℓ∞ is a closed trail, because for every t ∈ R/Z there is an open +interval I containing t such that ϕ∣I lifts to an injective map I → ˜ℓ∞. Therefore, the closed curve ℓ∞ satisfies +the angle condition at all points ϕ(t) and so is a closed trail. +It remains to verify (31). Let π ∶ Z → A be the covering map. By compactness and using Corollary 3.11 +we construct a finite collection of generalized rectangles { ˜P1,..., ˜Pk} in Z such that the restriction of π to +each ˜Pi is injective and such that K ⊂ ⋃k +i=1 π(P ○ +i ). Define +˜P = {δm( ˜Pi) ∶ i ∈ {1,...,k} and m ∈ Z}. +Say that a polygonal chain is a finite sequence of elements ˜Q1, ˜Q2,..., ˜Qc ∈ ˜P such that for each j ∈ +{1,...,c − 1}, we have ˜Q○ +j ∩ ˜Q○ +j+1 ≠ ∅. We claim that for each n, ˜ℓn is covered by the union of interiors of +polygons in a polygonal chain of no more than 2k +1 polygons. Fix n. Let ˜γn ∶ R → Z be a parameterization +of the bi-infinite leaf π−1(ℓn) such that ˜γn(t + 1) = δ ○ ˜γn(t) for all t ∈ R. Observe that Proposition 3.7 +guarantees that ˜γ−1 +n ( ˜P ○ +i ) is either the empty set or an open interval. By the periodicity of ˜γn, we have +˜γ−1 +n (δm( ˜P ○ +i )) = m + ˜γ−1 +n ( ˜P ○ +i ) +for all m ∈ Z. +The images of these sets under π are all the same, so ℓn ∩ π( ˜P ○ +i ) is either empty or is homeomorphic to +an open interval. Choose a minimal subset C of P = {π( ˜P ○ +i ) ∶ +i = 1,...,k} that covers ℓn. Since ℓn is +homeomorphic to a circle and is being covered by open intervals, we can index the collection C as +C = {Q○ +j ∶ j ∈ Z/cZ} +with c = ∣C∣ +such that Q○ +j ∩ Q○ +j′ ∩ ℓn ≠ ∅ if and only if j − j′ ≡ ±1 (mod c). We can iteratively lift elements of C to cover +ℓn by a polygonal chain of preimages. Recalling that ˜ℓn is a lift of two periods of ℓn, we see that ˜ℓn can be +covered using the interiors of no more than 2c + 1 elements of ˜P. (It may be necessary to lift three copies of +the polygon whose interior covers the image in A of the endpoints of ˜ℓn, because there are three preimages +of this point in ˜ℓn.) We have c ≤ k, so this gives a covering of ˜ℓn as described above with at most 2k + 1 +polygons as desired. +We will use the above observation about polygonal chains to produce our compact set containing ⋃n ˜ℓn, +and thus verifying (31). We need a basic observation about the deck group {δm ∶ m ∈ Z} of the covering +Z → A. Observe that for every two compact sets L1,L2 ⊂ Z, the collection +(32) +{n ∈ Z ∶ δn(L1) ∩ L2 ≠ ∅} +is finite. +(To see this, note that there is a homeomorphism from A to R2 modulo a non-trivial translation. Then Z +can be identified with R2 and δ acts by translation, so (32) holds.) In our setting, it follows that for any +compact set K ⊂ Z, the collection +(33) +{ ˜P ∈ ˜P ∶ +˜P ∩ K ≠ ∅} +is finite, +because ˜P is a finite collection of compact sets and their images under the deck group. +Now let K0 = +{δ−1(˜pn)} ∪ {δ−1(˜p∞)}, which is compact because limδ−1(˜pn) = δ−1(˜p∞). Then (33) guarantees that the +collection ˜P0 ⊂ ˜P of all polygons in P intersecting K0 is finite. Then for j ≥ 0, inductively define +˜Pj+1 = { ˜Q ∈ ˜P ∶ there is a ˜P ∈ ˜Pj such that ˜Q ∩ ˜P ≠ ∅}. + +72 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +Then (33) tells us that if ˜Pj is finite then ˜Pj+1 is finite. So, by induction we conclude that ˜Pj is finite for all +j ≥ 0. Now observe that ˜P2k contains all polygonal chains that start by intersecting K0 and include at most +2k + 1 polygons. Therefore, from the claim in the previous paragraph, we see that the union of closures of +the polygon interiors in ˜P2k is a compact set containing ⋃ ˜ℓn, verifying (31) and completing the proof. +□ +Recall the definitions related to the standard cylinder C = [−1,1] × S1 given in the paragraph before the +theorem on closed trails, Theorem 1.3. We will state a version of this theorem that holds for annuli. +Theorem 9.19 (Closed trails in annuli). Let A = Z/⟨δ⟩ be an annulus as above. Then one of the following +mutually exclusive statements holds: +(NR) (Non-realization case) There is no closed trail in A. +(Cyl) (Cylinder case) There is an embedding ϵ ∶ C○ → A such that the closed leaves in A are precisely the +image under the embedding of the vertical closed leaves of C○. +(UT) (Unique trail case) There is a unique closed trail in A, and this closed trail has at least one bending +angle greater than π on each side. +Furthermore, +(1) If Z is convex, then case (NR) cannot occur. +(2) In case (Cyl), define σ to be the collection of signs s ∈ {±} such that ϵ(H○ +s) has compact closure. Set +¨C = C○ ∪ ⋃ +s∈σ +∂sC. +Then there is an embedding ¨ϵ ∶ ¨C → A whose restriction ¨ϵ∣C○ satisfies (Cyl) such that for each +s ∈ σ, the parameterized curve ¨ϵ∣∂sC is a closed trail in [γ] passing through a non-empty collection of +singularities, and every bending angle made when passing through such a singularity on the side of +¨ϵ(C○) has measure π. Furthermore, all closed trails in A are obtained as restrictions of ¨ϵ to vertical +circles in ¨C. +Proof. The three statements (NR), (Cyl) and (UT) are mutually exclusive, because they correspond to +different cardinalities of the set of closed trails in A. If there is a unique closed trail, it cannot have a side +where all bending angles are π by Lemma 9.17. So the second assertion in (UT) holds whenever the closed +trail is unique. Statement (1) is a consequence of Corollary 9.8. +It remains to show that if there is more than one closed trail in A, then statement (Cyl) applies and +that statement (2) holds. First suppose γ1 and γ2 are distinct closed trails. Then they bound a compact +subannulus K(γ1,γ2). The union of closed trails in K(γ1,γ2) is both open (by Lemma 9.17) and closed (by +Lemma 9.18) and is therefore all of K(γ1,γ2). Furthermore Theorem 9.14 guarantees that all closed trails in +K(γ1,γ2) are closed leaves except possibly for γ1 and γ2. It also follows from Lemma 9.17 that the union U +of all closed leaves in A is an open sub-annulus foliated by these closed leaves with leaf space homeomorphic +to (−1,1). Therefore, there is a homeomorphism ϵ ∶ C○ → U as described in statement (Cyl). +Now consider statement (2). +Fix a sign s and suppose ϵ(H○ +s) has compact closure. +Since H○ +s is not +compact and ϵ is an embedding, there is a point p ∈ ϵ(H○s) ∖ ϵ(H○ +s). Then Lemma 9.18 guarantees there is a +closed trail γs through p. Furthermore since γs is not a subset of U, we know that γs is not a closed leaf. +Then, Corollary 9.15 guarantees that bending angles are all π on one side of γs, and Lemma 9.17 guarantees +that there is an open set on this side with boundary γs as one boundary that is foliated by closed leaves. +Therefore, the foliation extends to include γs, and the leaf space in a neighborhood of γs is homeomorphic +to a half-open interval in R. Let ¨U = U ∪ ⋃s∈σ γs where σ is defined as in the statement. We see that ¨U is a +surface with boundary ∂ ¨U = ⋃s∈σ γs. The set ¨U is foliated by closed trails, where the boundary components +are some of the leaves. Thus ¨U is homeomorphic to the space ¨C, which is a trivial S1 over an interval. This +homeomorphism is ¨ϵ. We have ¨ϵ(∂sC) = γs, and the bending angles have already been discussed above. +Finally, to see all closed trails are images of vertical leaves in ¨C under ¨ϵ, suppose γ′ is any closed trail. +If it is a closed leaf, then it is contained in U and ¨ϵ(C○) contains U. Since distinct leaves are disjoint, γ′ +must be the image of a vertical leaf in C○. Otherwise γ′ passes through singularities. Again Corollary 9.15 +guarantees that bending angles are all π on one side of γ′ and Lemma 9.17 guarantees that there is an open +set on this side with boundary γ′ as one boundary that is foliated by closed leaves, so γ′ is contained in +the closure of U, and so must be a boundary component. This means that γ′ = γs for some sign s ∈ σ as +desired. +□ + +ZEBRA SURFACES +73 +9.7. The closed trail theorem. In this section, we prove Theorem 1.3. +Let (S,{Fm}) be a surface with a zebra structure, and fix a PR free homotopy class ⟦γ⟧ which is nontrivial, +non-polar, and not a power. Choose a non-singular basepoint p0 and let [γ] ∈ πPR +1 +(S,p0) be a representative +of the conjugacy class. Let ˜S be the PRU cover, and let ∆γ denote the deck transformation associated to +[γ]. Then A = ˜S/⟨∆γ⟩ is an annulus by Corollary 9.2. Observe that we have the sequence of covers +˜S +˜π�→ A +ˆπ�→ S. +Let ˆp0 ∈ A be a lift of the basepoint p0 ∈ S and let ˜p0 be a lift of ˆp0 to ˜S. +Proposition 9.20. Every closed trail τ representing ⟦γ⟧ has a lift to A. +Proof. Suppose that τ ∶ [0,1] → S is a parameterized closed trail representing ⟦γ⟧ with τ(0) non-singular. +Fix a path η starting at p0 and ending at τ(0). Then [η ●τ ●η−1] ∈ πPR +1 +(S,p0) lies in the conjugacy class ⟦γ⟧. +Since [γ] lies in the same conjugacy class, there is a [β0] ∈ πPR +1 +(S,p0) such that if β0 ∈ [β0] is a representative +then +(34) +γ = β0 ● η ● τ ● η−1 ● β−1 +0 +is in +[γ]. +This curve γ lifts to A and the portion of the lift corresponding to the subpath τ is the desired lift that is a +closed trail. +□ +This proposition enables us to prove part of Theorem 1.3. +Lemma 9.21. If A contains no closed trails then statement (NR) of Theorem 1.3 holds. If A contains only +one closed trail, then statement (UT) of Theorem 1.3 holds. +Proof. If there are no closed trails in A, then there can be no trails in ⟦γ⟧ in S. Also if there is a unique closed +trail in A, then the image of this trail in S must be the unique trail in S. The statement involving bending +angles of the (UT) case from Theorem 9.19 implies the bending angle statement for (UT) in Theorem 1.3. +□ +Recall from Section 9.1 that πPR +1 +(S,p0) acts on ˜S as the group of deck transformations of the cover ˜S → S. +For [β] ∈ πPR +1 +(S,p0), the corresponding deck transformation ∆β ∶ ˜S → ˜S descends to a well-defined deck +transformation ˆ∆β ∶ A → A of the cover ˆπ ∶ A → S if and only if [β] ∈ N(Γ) where +Γ = ⟨[γ]⟩ +and +N(Γ) = {g ∈ πPR +1 +(S,p0) ∶ gΓg−1 = Γ} +is the normalizer of Γ. +The group N(Γ) contains Γ as a normal subgroup, and β1,β2 ∈ N(Γ) induce the same deck transformation +of A if and only if they lie in the same coset of the quotient group N(Γ)/Γ. Thus the deck group of the +covering ˆπ is ˆ∆ ≅ N(Γ)/Γ. For background on this see [Bre13]. +After Lemma 9.21, it remains to consider the case when A contains a cylinder, case (Cyl) of Theorem 9.19. +In this case A contains closed leaves. +Proposition 9.22. If ˆℓ is a closed leaf in A, then either: +(1) The image ˆπ(ˆℓ) is a simple closed curve and the restriction of ˆπ to ˆℓ is a homeomorphism onto its +image. +(2) The image ˆπ(ˆℓ) is a saddle connection whose endpoints are distinct poles, and there is a deck trans- +formation ˆι ∶ A → A that is an involution and restricts to an orientation reversing homeomorphism +ˆℓ → ˆℓ that fixes the preimages of the poles and ˆπ∣ˆℓ descends to a homeomorphism ˆℓ/ˆι → ˆπ(ˆℓ). +Proof. First suppose ˆπ(ˆℓ) contains no poles. +Then ˆπ(ˆℓ) must be a closed leaf of one of the directional +foliations of S. Therefore, ˆπ(ˆℓ) is a simple closed curve. The restriction of ˆπ to ˆℓ gives a covering map to +ˆπ(ˆℓ). If this map were of degree d > 1, then the parameterized curve ˆπ ○ ˆℓ which lies in ⟦γ⟧ would be a d-fold +power of the parameterization of the image, making ⟦γ⟧ a power, a contradiction. Therefore φ∣ˆℓ is injective +and since it is clearly a local homeomorphism, it is a homeomorphism. +Now suppose that p ∈ ˆπ(ˆℓ) is a pole. Let ˆp ∈ ˆℓ be a preimage on ˆℓ. The preimage ˜ℓ = ˜π−1(ˆℓ) in ˜S is a +bi-infinite leaf, and the deck transformation ∆γ translates along ˜ℓ. Let ˜p ∈ ˜S be a preimage of ˆp. Since ˜p +projects to a pole on S, there is an involutive deck transformation ˜ι ∶ ˜S → ˜S of the cover ˜S → S that fixes +˜p. Observe that ˜ι reverses the orientation of ˜ℓ, and so conjugates ∆γ to its inverse. Thus from remarks +above, ˜ι descends to a deck transformation ˆι ∶ A → A of ˆπ. The transformation ˆι induces an orientation + +74 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +reversing homeomorphism of ˆℓ, and therefore must have exactly two fixed points: ˆp and some other point +ˆq ∈ ˆℓ. Then since ˆι is non-trivial and fixes the regular point ˆq, the image q = ˆπ(ˆq) must also be a pole. If there +were more poles on ˆπ(ˆℓ), then we’d get more involutive deck transformations reversing the orientation of ˆℓ, +and the composition of two would give a non-trivial deck transformation acting as an orientation preserving +homeomorphism of ˆℓ. But then again the map ˆπ∣ˆℓ ∶ ˆℓ → ˆπ(ˆℓ) would factor through a finite covering of the +circle making ⟦γ⟧ a power. Thus, p and q must be the only poles on ˆπ(ˆℓ), which must be a saddle connection +joining them. Again, the natural map from ˆℓ/ˆι to ˆπ(ˆℓ) is an injective local homeomorphism and so is a +homeomorphism. +□ +Proposition 9.23. If ˆℓ1 and ˆℓ2 are closed leaves in A, then the images ˆπ(ˆℓ1) and ˆπ(ˆℓ2) are either disjoint +or they coincide. If ˆπ(ˆℓ1) = ˆπ(ˆℓ2) then there is a deck transformation ˆδ ∈ ˆ∆ such that the restriction of ˆδ to +ˆℓ1 is a homeomorphism to ˆℓ2. +Proof. Suppose q ∈ ˆπ(ˆℓ1) ∩ ˆπ(ˆℓ2) is a point common to the images. +First, if ˆℓ1 and ˆℓ2 have distinct slopes then because ˆπ(ˆℓ1) and ˆπ(ˆℓ2) intersect transversely, they would +have to bound a bigon [FM11, Bigon Criterion]. This bigon lifts to ˜S in contradiction to Proposition 3.6. +Thus they must have the same slope. It also follows that ˆπ(ˆℓ1) = ˆπ(ˆℓ2) since both these are closed leaves or +saddle connections joining poles through a common point and of the same slope. +Parameterize ˆℓ1 and ˆℓ2 so that they start at preimages of q and satisfy +(35) +ˆπ ○ ˆℓ1(t) = ˆπ ○ ˆℓ2(t) +for all t. +For i ∈ {1,2}, let ˆηi be a curve in A from the basepoint ˆp0 to ˆℓi(0). Set ˆγi = ˆηi ● ˆℓi ● ˆη−1 +i +for i = 1,2. Because +of our choice of parameterizations for the ˆℓi, the two curves ˆπ(ˆγi) both represent [γ] or [γ−1]. Now consider +the closed curve β = ˆπ(ˆη2) ● ˆπ(ˆη−1 +1 ). From (35) it follows that β ● ˆπ(ˆγ1) ● β−1 is homotopic to ˆπ(ˆγ2). Since +each [ˆπ(ˆγi)] ∈ {[γ±1 +i ]}, the deck transformation ∆β descends to a deck transformation of A. Furthermore, +since β ● π(ˆη1) is homotopic to π(ˆη2), it carries ˆℓ1(0) to ˆℓ2(0). Thus, because the closed leaves ˆℓ1 and ˆℓ2 +have the same slope and this deck transformation of A sends a point on one to a point on the other, it must +send ˆℓ1 to ˆℓ2. +□ +Lemma 9.24. Assume A contains a closed leaf. Let ˆϵ ∶ C○ → A be the embedding guaranteed to exist from +statement (Cyl) of Theorem 9.19. The deck group ˆ∆ of the covering ˆπ ∶ A → S preserves ˆϵ(C○). The map +ϕ ∶ ˆϵ(C○)/ ˆ∆ → S; +[ˆp] ↦ ˆπ(ˆp) +is a homeomorphism onto its image. +Proof. By statement (Cyl) of Theorem 9.19, we know ˆϵ(C○) is the union of all closed leaves in A. Since ˆ∆ +consists of homeomorphisms of A that preserve the zebra structure, each deck transformation must preserve +the set ˆϵ(C○). +We now claim that ϕ is a local homeomorphism. Because ˆ∆ is the deck group of the branched covering +ˆπ, ϕ is a local homeomorphism except possibly at the preimages of poles. If p ∈ ˆϵ(C○) is such that ˆπ(p) is a +pole, then statement (2) of Proposition 9.22 guarantees that there is a non-trivial deck transformation in ˆ∆ +that fixes p. Thus, ϕ is also a local homeomorphism at preimages of poles and so is a local homeomorphism. +Then to prove that ϕ is a homeomorphism onto its image, it suffices to show that it is injective. Let ˆp1 and +ˆp2 be two points of ˆϵ(C○) such that ˆπ(ˆp1) = ˆπ(ˆp2). Let ˆℓi denote the closed leaf in A through ˆpi for i ∈ {1,2}. +Then there is a deck transformation ˆδ carrying ˆℓ1 to ˆℓ2 by homeomorphism by Proposition 9.23. If ˆπ(ˆℓ1) +is simple closed curve then restrictions of ˆπ to ˆℓ1 and ˆℓ2 are injective by statement (1) of Proposition 9.22. +Thus ˆδ(ˆp1) = ˆp2. If ˆπ(ˆℓ1) is a saddle connection joining poles then there is an involutive deck transformation +ˆι preserving ˆℓ1 and by statement (2) of Proposition 9.22, we either have ˆδ(ˆp1) = ˆp2 or ˆδ ○ ˆι(ˆp1) = ˆp2. +□ +Lemma 9.25. If A contains a closed leaf then either statement (TF) or statement (Cyl) of Theorem 1.3 +holds. +Proof. Suppose A contains a closed leaf. Then statement (Cyl) of Theorem 9.19 applies and we get an +embedding ϵ ∶ C○ → A that sends vertical closed leaves to closed leaves. The deck group ˆ∆ of the covering +ˆπ ∶ A → S acts on ϵ(C○) ⊂ A and sends closed leaves to closed leaves by Proposition 9.23. We will argue + +ZEBRA SURFACES +75 +that the conclusion of Lemma 9.24 implies one of the statements must hold. To do this we break into cases +depending on the nature of the action of ˆ∆. +First suppose the deck group ˆ∆ of the covering ˆπ ∶ A → S is trivial. Then we have ˆϵ(C○)/ ˆ∆ = ˆϵ(C○). The +composition ϕ ○ ˆϵ = ˆπ ○ ˆϵ ∶ C○ → S is the desired embedding satisfying (Cyl). It includes every closed leaf by +Proposition 9.20. +The action of ˆ∆ on ϵ(C○) induces an action on the space of leaves, which is homeomorphic to an open +interval. Let I denote this leaf space. +The next simplest case is when ˆ∆ = ⟨ˆι⟩ where ˆι acts on I as an orientation-reversing homeomorphism. In +this case ˆι must fix a unique point on I, and so there is a closed leaf ℓ ⊂ A that is fixed by ˆι. Furthermore ˆι +acts on ℓ as an orientation-reversing homeomorphism so there are two fixed points in ℓ. The images of these +fixed points must be poles, and ˆπ(ℓ) must be a saddle connection joining these poles. Let H denote one +of the two components of ϵ(C○) ∖ ℓ. Then we can define C′ = ϵ−1(H) which is a sub-cylinder of C○. Using +Lemma 9.24, we see that +ˆπ ○ ϵ∣C′ ∶ C′ → S +is an embedding of a cylinder satisfying statement (Cyl) of Theorem 1.3. In this case, one of the boundaries +must be the saddle connection ˆπ(ℓ). +Now suppose there is a non-trivial element ˆδ ∈ ˆ∆ that acts on I as an orientation preserving homeomor- +phism. We claim that the action of ˆδ on I is fixed-point free. If ˆδ has a fixed point in I, then because it is a +deck transformation, it would have to act on the corresponding closed leaf ˆℓ ⊂ A as an orientation-preserving +homeomorphism. Let ˜ℓ = ˜π−1(ˆℓ), which is a bi-infinite leaf, and let [δ] ∈ πPR +1 +(S,p0) denote a preimage of ˆδ. +Then the deck transformations associated to [γ] and [δ] both act as orientation-preserving homeomorphisms +of ˜ℓ, and the quotient ˜ℓ/⟨[γ],[δ]⟩ must be homeomorphic to a circle and so ⟨[γ],[δ]⟩ ≅ Z. But then, either +[γ] is a power (violating a hypothesis on [γ]), or [δ] ∈ Γ (violating that ˆδ was non-trivial). +Assuming it is non-trivial, the group of all ˆδ ∈ ˆ∆ that act on I as an orientation preserving homeomorphism +must be isomorphic to Z. Letting ˆδ be a generator, we see that ˆϵ(C○)/⟨ˆδ⟩ is homeomorphic to a torus. If +this is the full ˆ∆, then Lemma 9.24 tells us that statement (TF) holds. It could be that in addition there +is a ˆι ∈ ˆ∆ which acts on I as an orientation reversing homeomorphism. In this case, ˆι and ˆδ must generate +ˆ∆, since the composition of any two elements whose actions on I reverse orientation would have to lie in +the group ⟨ˆδ⟩. Again ˆι must have a unique fixed point in I and ˆι must act on the corresponding closed +leaf ˆℓ0 as an orientation-reversing homeomorphism with two fixed points which are mapped to poles under +ˆπ. The image ˆπ(ˆℓ0) is a saddle connection σ0 joining the poles. Recalling that ˆδ was a generator for the +orientation-preserving subgroup of ˆ∆, let ˆj = ˆδ○ˆι which also acts as an orientation reversing homeomorphism +on I, and so is an involution fixing some leaf ˆℓ1. Again ˆπ(ˆℓ1) is a saddle connection joining two poles of S. +Observe that ˆj sends ˆℓ0 to ˆδ(ℓ0) and so the fixed leaf ˆℓ1 must lie strictly between ˆℓ0 to ˆδ(ℓ0) . The region +from ˆℓ0 to ˆℓ1 forms a fundamental domain for the action of ⟨ˆδ⟩, and so S = ˆϵ(C○)/⟨ˆδ,ˆι⟩ is a sphere with four +poles. Let R be the region between ˆℓ0 and ˆℓ1, and let C′ = ˆϵ−1(R) be a subcylinder. Then the map +ϵ′ ∶ C′ → S; +ϵ′ = ˆπ ○ ˆϵ +satisfies statement (Cyl) of Theorem 1.3, and the boundary of ϵ′(C′) consists of the two saddle connections +σ0 and σ1. +□ +To complete the proof of Theorem 1.3, it remains to prove statements (1) and (2). Statement (1) follows +directly from Theorem 9.7, so (2) remains. We’ll prove (2) using the corresponding statement (2) from +Theorem 9.19. The difficulty is that it is easier for a subset of the smaller surface S to be compact than for +its preimage in A. The following result is the main tool we use for dealing with this difficulty. +Lemma 9.26. Let P ⊂ S be a generalized rectangle with n vertices. Let γ1,γ2 ⊂ S be two disjoint closed leaves +that bound an annulus C in S containing no singularities. Let #(P ∩ γi) denote the number of connected +components of P ∩ γi. Then, +∣#(P ∩ γ1) − #(P ∩ γ2)∣ ≤ n. +Proof. This has to do with the ways the annulus C can intersect P. Let C0 be a connected component of +C ∩ P. Say a boundary arc of C0 is a maximal arc of γ1 or γ2 contained in ∂C0. We claim that at least one +of the following statements holds for each C0. + +76 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +(1) The component C0 either has exactly two boundary arcs with one from γ1 and one from γ2. +(2) The component C0 has fewer than two boundary arcs and C0 contains a vertex of P. +The conclusion is immediate, from these statements, because components C0 with two boundary arcs con- +tribute equally to both #(P ∩ γ1) and #(P ∩ γ2), while there can be at most n components with fewer than +two boundary arcs. +Now fix C0. Suppose first that C0 has at least two boundary arcs from γ1. Then, there is a curve β ⊂ C0 +joining these two boundary arc from γ1. Since C is an annulus, C is isotopic within C to an arc β′ of γ1 +joining the two arcs. Lift P to a polygon ˜P ⊂ ˜S. Since β′ is isotopic to β, there is a lift ˜β′ of β′ which is +a segment of a leaf joining the two lifts of arcs of β1. But then ˜β′ must exit ˜P and later return to ˜P in +violation of Proposition 3.7. The same argument works of course when C0 has two boundary arcs from γ2, +so this proves that (1) must hold when C0 has two or more boundary arcs. +Now suppose C0 has at most one boundary arc. If it has zero boundary arcs then C0 must equal P, and so +(2) holds trivially. Now suppose C0 has one boundary arc and contains no vertices. Then the one boundary +arc must join an edge of P to itself. But this violates Proposition 3.6. Thus if C0 has only one boundary +arc, it must contain a vertex. +□ +Proof of Theorem 1.3. As discussed above, it remains to prove (2). Let ϵ ∶ C○ → S be as in statement (Cyl). +Let H○ +s ⊂ C○ be one of the two halves and suppose that ϵ(H○ +s) ⊂ K where K ⊂ S is compact. For t ∈ (−1,1) +let ℓt = ϵ({t} × S1) be the corresponding closed leaf. The set ϵ(H○s) ∖ ϵ(H○ +s) consists of accumulation points +of ℓt as t → s. To simplify notation, assume s = +1 so that H○ ++ = [0,1) × S1. The other case will work in a +symmetric way. +Let F = {P0,...,Pk} be a finite collection of generalized rectangles in S whose interiors cover K. Choose +a point q ∈ ϵ(H○+) ∖ ϵ(H○ ++). We may assume that q ∈ P ○ +0 . We claim that ℓt intersects P0 for t sufficiently +close to +1. Observe that q /∈ ℓ0 since ℓ0 ⊂ ϵ(H○ ++). Therefore, we can find a stellar neighborhood N of q with +N ⊂ P0 and N ∩ ℓ0 = ∅. Since q is an accumulation point of ℓt as t → 1, we must have that ℓt0 ∩ N ≠ ∅ for +some t0. Let γ be a segment in N of a leaf joining q to ℓt0 ∩ N, which is minimal in the sense that γ ∩ ℓt0 +consists only of the other endpoint of γ which we’ll denote by r. We claim that if t is between t0 and 1, then +ℓt must pass through γ. To see this consider the set B = ϵ([0,t]×S1). Observe that q /∈ B but r ∈ B, so there +must be a point in γ ∩ ∂B. Since γ ⊂ N and N ∩ ℓ0 = ∅, the point of γ ∩ ∂B must lie in γ ∩ ℓt. Therefore, ℓt +passes through N and hence also P0 as desired. +Let ˆϵ ∶ C○ → A be a lift of ϵ to A. This lift exists, because we can lift a closed leaf by Proposition 9.20 +and the fundamental group lifting criterion tells us that we can extend to the lift ˆϵ. (We remark that ˆϵ may +not be the same map as the map from (Cyl) of Theorem 9.19 because it may have a smaller image; see the +cases in the proof of Lemma 9.25.) Define ˆℓt = ˆϵ({t} × S1), which is a lift of ℓt. We can also lift r to a point +on ˆℓt0 and lift γ to a segment ˆγ. This determines a lift ˆq of q. +Now define ˆP to be the collection of all lifts of the Pi to A. Let ˆP0 be the lift of P0 to A such that the +lift carries q to ˆq. Then for t sufficiently close to 1, we have ˆℓt ∩ ˆP0 ≠ ∅. +We can think of ˆP as a graph G where the vertices are elements of ˆP and there is an edge between two lifts +whenever they intersect. Considering the covering ˜S → S, observe that for any two lifts of the generalized +polygons ˜Pi, ˜Pj ⊂ ˜S, the collection +{[δ] ∈ πPR +1 +(S,p0) ∶ ∆δ( ˜Pi) ∩ ˜Pj ≠ ∅} +is finite. This implies that all vertices of the graph G have finite degree. +Observe that by compactness, ℓ0 ⊂ S passes through finitely many of the polygons in F counting multi- +plicity. Then Lemma 9.26 gives us an N such that each ℓt with t ∈ (0,1) intersects at most N elements of F +counting multiplicity. +The union ˆU of all the element of ˆP that can be reached from ˆP0 by a path in G passing through at most +N edges is a finite union, and is therefore compact. Also observe that ˆU contains each ˆℓt for t sufficiently +close to 1. This is because for t large, ˆℓt passes through ˆP0. Then, it moves through at most N sequentially +overlapping elements of ˆP counting multiplicity before closing up. Thus ˆℓt ⊂ ˆU. Thus ˆϵ([t0,1) × S1) ⊂ ˆU and +so ˆϵ(H○ ++) ⊂ ˆU ∪ ˆϵ([0,t0] × S1), which is compact. +We’ve shown that if ϵ(H○ ++) has compact closure, then so does ˆϵ(H○ ++). The same works for negative signs. +As in statement (2), let σ denote the collection of signs so that ϵ(H○ +s) has compact closure, or equivalently + +ZEBRA SURFACES +77 +now ˆϵ(H○ +s) has compact closure. Then by statement (2) of Theorem 9.19, there is an embedding ¨ˆϵ ∶ ¨C → A +of the partial closure ¨C = C○ ∪ ⋃s∈σ ∂sC into A whose image is ˆϵ(C○) ∪ ⋃s∈σ ˆϵ(H○ +s). (Note that the image +of ˆϵ may not include all closed leaves of A, but is always a sub-cylinder, and so we can define ¨ˆϵ regardless.) +We define ¨ϵ = ˆπ ○ ¨ˆϵ ∶ ¨C → S, and observe that it satisfies statement (2), because it satisfies the corresponding +statement of Theorem 9.19. +□ +9.8. Non-compact translation surfaces. +Proof of Corollary 1.6. Let S be a non-compact translation surface whose universal cover ˜S is geodesically +convex. Fix a non-trivial deck transformation δ ∶ ˜S → ˜S. We will show δ is a hyperbolic isometry. Let p ∈ S +be a basepoint and ˜p ∈ ˜S be a preimage. From covering space theory, there is loop γ ⊂ S based at p such that +given any point ˜q ∈ ˜S, the point δ(˜q) is the endpoint of the lifted concatenation ̃ +γ ● α starting at ˜p, where +˜α is a path from ˜p to ˜q whose image in S is α. From our hypotheses and Theorem 1.3, there is a closed +geodesic g ∶ [0,1] → S in the free homotopy class [γ], and so we can replace γ by a concatenation of the +form β ● g ● β−1 where β is a path joining p to g(0). Let ˜q be any endpoint of the lift ̃ +β ● ¯g where ¯g starts at +g(0) and wraps around g any number of times, stopping anywhere on the curve. The set of all ˜q that are +attainable in this way is a lift of the universal cover of g and is therefore a geodesic. We have that δ(˜q) is +the endpoint of the lift +̃ +β ● g ● ¯g, so δ translates along this geodesic. By the equivalence between translating +along a geodesic and an isometry being hyperbolic described in Section 1.2.1, we see that δ is hyperbolic. +□ +10. Questions +Hopf tori give examples of zebra structures on the torus without singularities such that all closed leaves +lie in one homotopy class. Corollary 7.5 says that if there are two non-homotopic closed leaves then every +non-trivial free homotopy class of closed curves contains a closed leaf. But we are not sure of the following: +Question 10.1. Is there a zebra torus without singularities that has no closed leaves? +Every dilation structure on a closed surface has a cylinder [BGT21]. +Question 10.2. Does every zebra structure on a closed surface contain a cylinder? +Two zebra structures (S1,{F1 +m}) and (S2,{F2 +m}) should be considered isomorphic if there is an orientation- +preserving homeomorphism φ ∶ S1 → S2 such that F1 +m coincides with the pullback of F2 +m for each m ∈ ˆR. +The following question has been studied in the context of dilation surfaces (and more general complex affine +structures) in [ABW22] building off work in [Vee93]. +Question 10.3. Given a closed surface S and a singular data function α, is there a way to understand the +“stratum” of all compatible zebra structures on S up to isomorphism? Is there a natural topology? Is this +stratum naturally an orbifold modeled on a function space? +Question 10.4. Is there a notion of uniformization for zebra surfaces? That is, given a zebra surface, is +there a homeomorphism to a Riemann surface that respects angles? +We now make several definitions that we consider analogous to the affine automorphism group of a +translation or dilation surface, the derivative mapping from the affine automorphism group to GL(2,R), and +the Veech group of these surfaces. +Let S be an oriented topological surface. Let Z(S) denote the collection of all zebra structures on S. +Then each element of Z(S) is an indexed family of singular foliations {Fm ∶ +m ∈ ˆR} on S. Section 2.8 +described an action of Homeo+(ˆR) on Z(S), which we will denote by {Fm} ↦ ϕ({Fm}) for ϕ ∈ Homeo+(ˆR). +Also if h ∶ S → S is an orientation-preserving homeomorphism, then there is a natural action of h on Z(S) +that sends {Fm} to h({Fm}) = {h∗(Fm)}, where h∗(Fm) denotes the pushforward of the foliation under h. +We call h a stellar automorphism of (S,{Fm}) if there is a ϕ ∈ Homeo+(ˆR) such that h({Fm}) = ϕ({Fm}). +Let Aut☆ ++ (S,{Fm}) denote the group of stellar automorphisms of (S,{Fm}). Observing that ϕ is uniquely +determined by h, we see there is a natural group homomorphism +(36) +D ∶ Aut☆ ++ (S,{Fm}) → Homeo+(ˆR); +h ↦ ϕ, +that we call the stellar derivative. The stellar group of (S,{Fm}) is the image of D, which we denote by +G☆ ++ (S,{Fm}) ⊂ Homeo+(ˆR). The kernel of D is the zebra automorphism group Aut(S,{Fm}), the group of +zebra automorphisms as defined in Section 9.2. + +78 +W. PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS +The following question is answered in the context of closed dilation surfaces by [DFG19, Theorem 1]. +Question 10.5. When is G☆ ++ (S,{Fm}) a discrete subgroup of Homeo+(ˆR)? +We also wonder about discreteness of Aut☆ ++ (S,{Fm}). Note however that discreteness of both G☆ ++ (S,{Fm}) +and Aut(S,{Fm}) implies the discreteness of Aut☆ ++ (S,{Fm}) since the following sequence is exact: +1 → Aut(S,{Fm}) → Aut☆ ++ (S,{Fm}) +D +�→ G☆ ++ (S,{Fm}) → 1. +Note also that if the answer to Question 10.4 is affirmative, then Aut(S,{Fm}) is isomorphic to a subgroup +of a group of automorphisms of a Riemann surface and is therefore finite when S is a closed surface of genus +at least two. +The examples we understand of interesting subgroups of stellar groups all come from translation and +dilation surfaces and are therefore contained in SL(2,R) ⊂ Homeo+(ˆR). +Question 10.6. Is every G☆ ++ (S,{Fm}) conjugate within Homeo+(ˆR) to a subgroup of SL(2,R)? Is G☆ ++ (S,{Fm}) +always conjugate into a group of homeomorphism acting smoothly on the circle? +The map D of (36) gives an action of Aut☆ ++ (S,{Fm}) on the circle ˆR. +Question 10.7. Because we can deform the zebra structure, it is natural to wonder when the stellar deriv- +ative D of a structure (S,{Fm}) is locally rigid up to deformations of the zebra structure, meaning that +there is an open set of representations from Aut☆ ++ (S,{Fm}) to Homeo+(ˆR) containing D such that for any +representation D′ in the open set, there is another zebra surface (S′,{F′ +m}) and an injective homomor- +phism ψ ∶ Aut☆ ++ (S,{Fm}) → Aut☆ ++ (S′,{F′ +m}) such that D′ ○ ψ−1 is the restriction of the stellar derivative of +(S′,{F′ +m}) to the image of ψ. This is an instance of a collection of natural questions regarding the rigidity +of the action of Aut☆ ++ (S,{Fm}) on the circle. See [Man18] for background on rigidity questions for group +actions on the circle. +There is a natural group homomorphism sending elements of Aut☆ ++ (S,{Fm}) to the mapping class group +of S. +Question 10.8. Which mapping classes arise from elements of Aut☆ ++ (S,{Fm})? +In particular, are all +reducible mapping classes realizable as the stellar automorphism of a zebra surface? +All finite-order and pseudo-Anosov elements of the mapping class group are realized by affine automor- +phisms of half-translation surfaces. In contrast, the only reducible mapping classes that arise from affine +automorphisms of translation and dilation surfaces are multitwists [Wan21]. +The forgetful maps from half-translation and half-dilation structures to zebra structures leads to interest- +ing questions. +Question 10.9. Is there ever a homeomorphism φ ∶ S → S with S a closed half-translation surface that gives +a stellar automorphism of the induced zebra structure, but does not give an affine automorphism of S? What +about for half-dilation surfaces? +Question 10.10. Characterize the zebra structures that arise from half-translation (or half-dilation) struc- +tures. +There is a conjectural geometric characterization of dilation structures that arise from translation struc- +tures [BGT21, §1.5]. +Acknowledgments +We’d like to thank Jon Chaika who directed us in exploring the natural wonders in Utah, which partially +inspired this project. We’d also like to thank David Aulicino and Jane Wang for helpful conversations. W. P. +Hooper was supported by a grant from the Simons Foundation and by a PSC-CUNY Award, jointly funded +by The Professional Staff Congress and The City University of New York. F. Valdez would like to thank the +following grants: CONACYT Ciencia B´asica CB-2016 283960 and UNAM PAPIIT IN-101422. B. Weiss was +supported by grants BSF 2016256, ISF 2019/19, and ISF-NSFC 3739/21. + +ZEBRA SURFACES +79 +References +[ABW22] Paul Apisa, Matt Bainbridge, and Jane Wang, Moduli spaces of complex affine and dilation surfaces, arXiv, 2022. +[Bee93] Gerald Beer, Topologies on closed and closed convex sets, Vol. 268, Springer Science & Business Media, 1993. +[BGT21] Adrien Boulanger, Selim Ghazouani, and Guillaume Tahar, Closed geodesics in dilation surfaces, arXiv, 2021. +[BH13] Martin R Bridson and Andr´e Haefliger, Metric spaces of non-positive curvature, Vol. 319, Springer Science & Business +Media, 2013. +[BL18] Anja Bankovic and Christopher Leininger, Marked-length-spectral rigidity for flat metrics, Transactions of the Amer- +ican Mathematical Society 370 (2018), no. 3, 1867–1884. +[Bre13] Glen E Bredon, Topology and geometry, Vol. 139, Springer Science & Business Media, 2013. +[DFG19] Eduard Duryev, Charles Fougeron, and Selim Ghazouani, Dilation surfaces and their Veech groups, Journal of modern +dynamics 14 (2019), no. 1, 121. +[FLP21] Albert Fathi, Fran¸cois Laudenbach, and Valentin Po´enaru, Thurston’s work on surfaces (mn-48), Vol. 48, Princeton +University Press, 2021. +[FM11] Benson Farb and Dan Margalit, A primer on mapping class groups (pms-49), Princeton university press, 2011. +[Fra18] Ian Frankel, CAT(-1)-type properties for Teichm¨uller space, arXiv, 2018. +[HS06] Pascal Hubert and Thomas A Schmidt, An introduction to Veech surfaces, Vol. 1, Elsevier Amsterdam, 2006. +[Knu97] Donald Ervin Knuth, The art of computer programming, vol. 1: Fundamental algorithms, Addison-Wesley, 1997. +[Man18] K Mann, Rigidity and flexibility of group actions on the circle, Handbook of group actions IV, 2018, pp. 705–752. +[Nik01] Igor Nikolaev, Foliations on surfaces, Ergeb. 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+Dept. of Mathematics, City College of New York and CUNY Graduate Center, New York, NY, USA +Email address: whooper@ccny.cuny.edu +URL: http://wphooper.com +Centro de Ciencias Matem´aticas, UNAM Campus Morelia, M´exico +Email address: ferran@matmor.unam.mx +URL: https://www.matmor.unam.mx/~ferran/ +Dept. of Mathematics, Tel Aviv University, Tel Aviv, Israel +Email address: barakw@tauex.tau.ac.il +URL: http://www.math.tau.ac.il/~barakw/ + diff --git a/ztE2T4oBgHgl3EQfMwYs/content/tmp_files/load_file.txt b/ztE2T4oBgHgl3EQfMwYs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d335d82614053856383ede92d2ab3aedb9249e51 --- /dev/null +++ b/ztE2T4oBgHgl3EQfMwYs/content/tmp_files/load_file.txt @@ -0,0 +1,4455 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf,len=4454 +page_content='STELLAR FOLIATION STRUCTURES ON SURFACES W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We introduce the notion of a zebra structure on a surface, which is a more general geometric structure than a translation structure or a dilation structure that still gives a directional foliation of every slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We are concerned with the question of when a free homotopy class of loops (or a homotopy class of arcs relative to endpoints) has a canonical representative or family of representatives, either as closed leaves or chains of leaves joining singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We prove that such representations exist if the surface has a triangulation with edges joining singularities (in the zebra structure sense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Our results hold for both closed surfaces and non-compact surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Introduction A quadratic differential q on a Riemann surface X, possibly with simple poles, naturally endows that surface with a half-translation structure via coordinate charts obtained by integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Associated to q is its divisor, which may be interpreted as a function (1) α ∶ X → Z≥−1 = {−1,0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='} whose support is discrete and closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We have α(x) = −1 if x is a simple pole, and α(x) = k ≥ 1 if x is a zero of order k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The half-translation structure gives charts from a neighborhood of each x ∈ X to the α(x)+2-fold branched cover of C/⟨z ↦ −z⟩ branched over the origin, and transition maps are given by translations or 180○ rotations in local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Geometrically x ∈ X has the local structure of a Euclidean cone point with cone angle π(α(x)+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Thus the support Σ of α is the collection of singularities of the half-translation structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Throughout this paper, we allow our surfaces to be non-compact, though the closed surface case is of special interest and we prove new results in this setting as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Given a half-translation structure on an oriented topological surface S, for each slope m ∈ ˆR = R ∪ {∞}, the foliation of the plane by lines of slope m pulls back under charts to give a singular foliation Fm of the surface by leaves of slope m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Formally Fm is a foliation of S ∖ Σ whose local behavior near a point p is governed by the value α(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In particular, each Fm has α(p) + 2 prongs at each point p ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' More generally, a singular foliation F of a topological surface S is a foliation of S ∖ Σ, where Σ ⊂ S is a closed discrete subset and such that there is a singular data function α ∶ S → Z≥−1 whose support is Σ such that F is locally homeomorphic at p ∈ Σ to the horizontal foliation of a half-translation surface in a neighborhood of a cone point with cone angle π(α(p) + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Note that F determines both Σ and α: The subsurface S ∖ Σ is the union of leaves, and α can be determine by the number of prongs at a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Singular foliations need not come from quadratic differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Indeed, those that come from quadratic differentials carry the additional structure of a measured foliation, which appears because half-translation surfaces have a natural path metric obtained by pulling back the Euclidean metric on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In this paper, we investigate what happens when these additional structures are not required, but where we still have a family of singular foliations that fit together nicely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let S be an oriented topological surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Consider a family {Fm ∶ m ∈ ˆR} of singular foliations on S indexed by slope that determine the same singular set Σ and the same singular data function α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We say such a family is stellar if each p ∈ S has a neighborhood U such that U ∖ {p} is foliated by segments of leaves containing p in their closure in a manner homeomorphic to the standard half-translation model associated to α(p) as depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If {Fm} is stellar, we say it induces a stellar foliation structure or a zebra1 structure on (S,α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We call a pair (S,{Fm}m∈ˆR) where {Fm} is stellar a zebra surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Date: January 11, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 1The name zebra surface was inspired by the Zebra Slot Canyon in Grand Staircase-Escalante National Monument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='03727v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='GT] 10 Jan 2023 2 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The local structure of leaves through p, with α(p) ∈ {−1,0,1,2} from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Here the bold leaves are of slope zero and slopes cyclically increase in the counterclockwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' See Section 2 for a more detailed and formal definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Observe that half-translation structures induce zebra structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Another geometric structure on a surface that induces a zebra structure is a dilation structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This paper originated from our interest in dilation structures on surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' These structures have recently generated a lot of interest, see for example [ABW22,BGT21,DFG19,Wan21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We discuss some relevant results in Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='4, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='6, we show that there are zebra structures that do not arise from dilation or translation structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A broader discussion about foliations on surfaces can be found in [Nik01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' It is a fundamental observation in Teichm¨uller theory that the bundle Qg of quadratic differentials (without poles) over the moduli space of closed Riemann surfaces of genus g is naturally identified with the cotangent bundle of that space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The geodesic flow in the Teichm¨uller metric is conjugate via the identification of quadratic differentials with half-translation structures to the diagonal action of PSL(2,R) on the space of half-translation surfaces, acting affine-linearly by simultaneous post-composition with coordinate charts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This action of PSL(2,R) on Qg is an active area of research [Wri15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This action extends to an action of PSL(2,R) on zebra surfaces: In fact, the PSL(2,R)-action extends to a Homeo+(ˆR)-action given by reindexing the foliations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Our initial motivation when writing this paper was to extend fundamental facts about length-minimizing representatives of curves on translation surfaces to the more general context of dilation surfaces, where there is no natural notion of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' One motivation for studying this question is Thurston’s theory of simple closed curves and their relation with the classification of surface homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' When looking into this, we realized that surprisingly little is known about distinguished representatives of curves for related structures (such as dilation structures with cone-type singularities and translation structures on non-compact surfaces) and an answer can be given in the very general context of zebra surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Working in this more general context makes some things challenging, but the limited tools available lead to a natural and general approach to the problems under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We briefly introduce some important definitions so that we can state our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let (S,{Fm}) be a zebra surface, with S any oriented topological surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' As indicated above, this information determines a singular set Σ and a singular data function α ∶ S → Z≥−1 whose support is Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A leaf is a leaf of any of the foliations Fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Leaves are contained in S∖Σ and so do not contain singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A leaf is closed if it is homeomorphic to a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If a leaf is not closed, then it is homeomorphic to an open interval, and such a leaf can have singular endpoints in its closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A saddle connection is a leaf together with two singular endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A leaf triangulation of S is a triangulation of S whose edges are saddle connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We require that triangles to meet edge-to-edge and that the union of the triangles be all of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In such a triangulation, only finitely many triangles can meet at each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A trail is a maximal bi-infinite parameterized path that follows a sequence of leaves, transitioning between leaves only at singularities in such a way that the two angles made at the singular transitions are at least π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (Angles made between leaves meeting at a point can be measured using the stellar neighborhood of the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=') We call the angles made at singular transitions bending angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Each bending angle appears either on the right or left side of the trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We require trails to “bounce off” poles, returning along the leaf through ZEBRA SURFACES 3 which it arrived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (This “bouncing off” is not allowed at other singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=') A trail is closed if it can be reparameterized to be periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A zebra plane is a zebra structure (Z,{ ˜Fm}) where Z is an open disk and the singular data function ˜α is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (Examples of simply connected zebra surfaces which are not zebra planes can be found in [Pan09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=') If (S,{Fm}) is any connected zebra surface, and α is non-negative, then the structure lifts to the universal cover to give a zebra plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If S has poles, then there is a larger cover which is a zebra plane, where we require double branching over poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We call this the pole-resolved universal cover (the PRU cover), see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Note that because of the double branching, preimages of poles are non-singular points in the zebra plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The PRU cover coincides with the universal cover if S has no poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A basic question in the geometry of metric spaces is whether any two points can be joined by a geodesic, and if so, whether this geodesic is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' For example, the Cartan–Hadamard theorem guarantees that any two points can be connected by a unique geodesic segment in a complete non-positively curved simply- connected metric space [BH13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In our setting, we observe that distinct points on a zebra plane can be joined by at most one arc of a trail (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We say a subset of a zebra plane is convex if any two distinct points can be joined by an arc of a trail contained in that subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We prove: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If a zebra plane Z has a leaf triangulation, then Z is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The primary case of interest is when the zebra plane is the PRU cover of a closed surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' As a consequence if we can “triangulate” the closed surface in a manner that lifts to a leaf triangulation on the cover, then that cover is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Figure 2 shows a surface with no leaf triangulation whose PRU cover is not convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A dilation surface: edges are glued by dilations and translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The orange, pink, and blue points are dilation singularities, which are not singularities in the induced zebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The foliated region is a full zebra cylinder (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='4), and so no closed trail can cross this cylinder by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Of course, it is only possible for a zebra plane to have a leaf triangulation when α takes positive values, (that is, when there are singularities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We also provide a criterion for convexity of the PRU cover of a closed surface when α is non-positive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' see Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Some examples, like R2 which covers the square torus are convex, but others like the universal cover of a Hopf torus is not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We will need to extend the notion of homotopy of paths, to the case of zebra surfaces with poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' See Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='1 for details on this construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We call this extended notion of homotopy pole-resolved homotopy or PR homotopy, and it coincides with the usual notion if there are no poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The PR free homotopy classes of loops are in natural bijective correspondence with conjugacy classes in the deck group of the PRU covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We say a PR free homotopy class of closed curves is polar if there is a simple closed curve in the class that bounds a disk whose only interior singularity is a pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We say that a PR free homotopy class ⟦γ⟧ of closed curves is a power if there is another PR free homotopy class ⟦β⟧ and a k ≥ 2 such that ⟦γ⟧ is homotopic to the k-fold cover of ⟦β⟧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The closed standard cylinder is C = [−1,1] × S1 where S1 = R/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This cylinder comes equipped with its vertical foliation by fibers of the projection C → [−1,1], and has two boundary components ∂±C = {±1}×S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Its interior is the subset C○ = (−1,1) × S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We split C○ into two halves: H○ − = (−1,0] × S1 and H○ + = [0,1) × S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 4 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS We have: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3 (Closed trails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let (S,{Fm}) be a zebra surface, where S is any connected oriented topological surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Fix a PR free homotopy class of closed curves ⟦γ⟧ which is nontrivial, non-polar, and not a power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then one of the following mutually exclusive statements holds: (NR) (Non-realization case) There is no closed trail in ⟦γ⟧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (TF) (Toral foliation case) The surface S is the torus, the zebra structure has no singularities, and the collection of all closed trails in ⟦γ⟧ is a collection of simple closed leaves that foliate S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (Cyl) (Cylinder case) There is an embedding ϵ ∶ C○ → S such that the closed leaves of S in ⟦γ⟧ are precisely the images under the embedding of the vertical closed leaves of C○.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (UT) (Unique trail case) There is a unique closed trail in ⟦γ⟧ and this closed trail has at least one bending angle greater than π on each side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Furthermore, (1) If the PRU cover of S is convex, then case (NR) cannot occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (2) In case (Cyl), define σ to be the collection of signs s ∈ {±} such that ϵ(H○ s) has compact closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Set ¨C = C○ ∪ ⋃ s∈σ ∂sC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then there is a map ¨ϵ ∶ ¨C → S whose restriction ¨ϵ∣C○ satisfies (Cyl) such that for each s ∈ σ, the curve ¨ϵ∣∂sC is a closed trail in ⟦γ⟧ passing through a non-empty collection of singularities, and every bending angle made when passing through such a singularity on the side of ¨ϵ(C○) has measure π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Furthermore, all closed trails in ⟦γ⟧ are obtained as restrictions of ¨ϵ to vertical circles in ¨C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In order to prove this theorem, we prove a criterion for existence of a closed trail that does not require convexity of the PRU cover;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' see Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In the context of closed zebra surfaces with singularities and convex PRU covers, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3 specializes to the following: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose that S is a closed surface and {Fm} is a zebra structure on S with a non-empty singular set Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose also that the PRU cover of S is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then, if ⟦γ⟧ is a PR free homotopy class of closed curves that is nontrivial, non-polar, and not a power, then either there is a unique closed trail in ⟦γ⟧ as in case (UT) or there is a continuous map from the closed standard cylinder ¨ϵ ∶ C → S whose restriction to C○ is an embedding as in case (Cyl) and whose restriction to each boundary component is a closed trail as described in (2), and such that all closed trails in the homotopy class are given by restriction as in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' However, there certainly are some closed surfaces for which there is a ⟦γ⟧ that falls into case (NR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The PRU cover of such a surface is not convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' An example of this situation is depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The surface is constructed by starting with a polygonal annulus in R2 and making boundary identifications (which can be chosen to give the surface a dilation structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The homotopy class of a homotopically non-trivial loop traveling around the interior of this annulus gives an example of a homotopy class satisfying (NR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' See the caption of the figure for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Here we consider our main theorems in specific contexts moving roughly from more specific structures to more general structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The chart below depicts the various geometric structures on surfaces that we consider, together with arrows from one structure to another to indicate that a surface with the first structure is also a surface with the second structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=', a translation surface atlas is also a half-translation surface atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=') Translation Dilation with cone singularities Dilation with dilation singularities Euclidean cone Half-translation Half-dilation with cone singularities Half-dilation with dilation singulartites Zebra ZEBRA SURFACES 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Closed translation surfaces and cone surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A translation surface is an oriented surface with an atlas of charts to the plane whose transition functions are translations, where we allow cone points with cone angles that are integer multiples of 2π (so, in our notation, α takes even values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We briefly explain why our main results are true in the case of a closed translation surface and in a related case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A (Euclidean) cone surface is an oriented surface with an atlas of charts to the plane where transition functions are in the orientation-preserving isometry group, and where we allow cone singularities with any positive real cone angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Thus a translation surface is a special case of a cone surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If S is a closed translation surface, its universal cover is a Hadamard space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=', a complete metric space that is non-positively curved in the CAT(0) sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' More generally, we could consider a closed cone surface all of whose cone singularities have cone angles greater than 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The universal cover is again a Hadamard space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' For details see [BL18, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This also works for closed half-translation surfaces, but if the surface has poles, then we have to replace the universal cover with the PRU cover described in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' From local considerations, a curve on such a surface is a geodesic if and only if it satisfies our definition of a trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (Indeed, this is the motivation for our definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=') Hadamard spaces are well known to be geodesically convex, so this gives Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2 in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The strategy for deducing Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3 in this context is to use facts about isometries of Hadamard spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Isometries of metric spaces can categorized based on their attained translation lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Here the translation length of a point p ∈ X under an isometry ϕ ∶ X → X is TL(x) = d(x,ϕ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The isometry ϕ is elliptic if it has a fixed point, hyperbolic if TL attains a strictly positive minimum, and parabolic if the infimum of values of TL is not attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' An isometry of a locally CAT(0) space translates along some geodesic if and only if the isometry is hyperbolic [BH13, II, Thm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Given a PR homotopy class ⟦γ⟧ on a closed surface S as above, we get a deck transformation ∆γ ∶ ˜S → ˜S where ˜S is the Hadamard cover described above, which by hypothesis is not elliptic (because ⟦γ⟧ is nontrivial and non-polar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A cocompact group of isometries acting on a Hadamard space cannot contain parabolic isometries [BH13, II, Prop 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Therefore, ∆γ is hyperbolic and translates along a geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' It is not hard to move from this point to the description in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3 using elementary facts about Euclidean cone surfaces, though in the cone surface case cylinders are immersed rather than embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Also the closed trails in a cylinder in this case are parallel (globally in the translation surface case and locally in the cone surface case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Non-compact translation surfaces and cone surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We will briefly explain how the argument from Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='1 proving special cases of our main results breaks when we consider non-compact translation surfaces and Euclidean cone surfaces whose cone angles are larger than 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' There are two difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' First, there are non-compact translation surfaces, all of whose singularities are finite cone singularities, that admit leaf triangulations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=', triangulations by saddle connections joining singularities) but whose universal covers are not complete;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' see Figure 3 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Second, since the surface is not compact, it is unclear how to rule out parabolic isometries in the deck group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Part of an infinite triangulation of a connected open subset of the plane is depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let S be the smallest double cover of this disk with double branching over the vertices and no other branching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The surface S is naturally a translation surface with a leaf triangulation, but is not complete as a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3 apply to S, with the latter giving a closed trail in every nontrivial free homotopy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 6 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS It is interesting to note that the open unit disk in R2 has a complete metric for which geodesics are straight lines: The Klein disk model of the hyperbolic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We wonder: Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Which zebra surfaces have a complete CAT(0) metric whose geodesics are the trails?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Translation structures are special cases of zebra structures, so our results hold in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We have the following consequence: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If S is a non-compact translation surface whose universal cover ˜S is geodesically convex (or has a leaf triangulation), then every nontrivial deck transformation is a hyperbolic isometry of ˜S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We will explain that Figure 4 illustrates a failure of our conclusions to hold in the context of non-compact cone surfaces with all singularities having cone angles larger than 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Note that this surface is not a zebra surface, because it has cone singularities with cone angle strictly between 2π and 3π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The figure illustrates a Euclidean cone structure on the annulus, but with no geodesic core curve, because the annulus continues to get thinner as we move towards one boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This surface is depicted with a decomposition into quadrilaterals, which when cut along their diagonals gives a leaf triangulation, in the sense that edges are saddle connections and vertices are singularities with cone angle larger than 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' It follows that one of the following two implications must be incorrect in this context: A leaf triangulation implies convexity of the universal cover, or convexity of the cover implies the existence of a geodesic representative in every nontrivial free homotopy class of loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' However, we conjecture: Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Fix ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let S be non-compact Euclidean cone surface such that all cone singularities have angle at least 2π + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose that S admits a triangulation by saddle connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then the universal cover ˜S is convex and every nontrivial free homotopy class of closed curves in S contains a geodesic representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A Euclidean cone structure on the annulus built using infinitely many trapezoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Dilation surfaces with cone singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A half-dilation surface with cone singularities is a surface with an atlas of charts to the plane and finite covers of C/⟨z ↦ −z⟩ branched over the origin such that transition maps are in the group generated by translations, dilations, and rotations by π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A dilation surface with cone singularities is the same, only covers should be of the plane branched over the origin and the transition maps are in the group generated by translations and dilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The (dilational) holonomy around an oriented loop in such a surface is the ratio of lengths of a segment parallel translated around a loop and the original segment, measured in a fixed local coordinate chart and interpreted as an element of R+ [Wan21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Because our singularities are cone singularities, the holonomy around any contractible loop is trivial and the notion of holonomy around a loop gives rise to the holonomy homomorphism π1(S) → (R+,×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' ZEBRA SURFACES 7 Earlier we mentioned the Hopf tori, given by C∗/⟨z ↦ λz⟩ where C∗ = C ∖ {0} and λ is a positive real number, because these surfaces have non-convex universal covers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The fibers of the map arg ∶ C∗ → R/2πZ give a foliation of a Hopf torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' To see a Hopf torus has non-convex universal cover, observe that the torus has two closed leaves of every slope, and if p and q are points from distinct closed leaves of the same slope then there is no trail connecting p with q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' See [DFG19] for background on Hopf tori and related constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Consider an immersion of the cylinder C = [−1,1] × S1 into a Hopf torus T that sends vertical closed leaves of C to the foliation of T given by fibers of arg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The pullback of the dilation structure to C is called a dilation (or affine) cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The universal cover of C with this structure can be seen to be isomorphic to a sector in a branched cover of the plane, and we call the angle of this sector the angle of the dilation cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' It is not hard to see that a dilation surface (or half-dilation surface) containing an affine cylinder with angle π or more cannot be convex, and no closed curve crossing such a cylinder can have a closed trail representing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' See Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='We note that dilation surfaces can also contain flat cylinders, isomorphic to rotations of [0,w] × R/cZ for c and w positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We were unable to find a simple argument for proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3 in the context of dilation surfaces with cone singularities that bypasses the technique we use for zebra surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Dilation surfaces with dilation singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A dilation singularity is more general than a cone singu- larity: We allow a loop around a dilation singularity to have nontrivial dilational holonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The dilation singularities must locally look like certain natural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let U denote the closed upper half-plane, {z ∈ C ∶ Iz ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' One example of such a model is given by Mλ = U/ ∼ with λ > 0, where ∼ is the finest equivalence relation on U where for every positive x ∈ R, −x ∼ λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Here the singularity of Mλ is at the origin, and the dilational holonomy of a counterclockwise loop around the origin is λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In general, the model singularities are given by branched covers M n λ of Mλ of degree n ≥ 1 branched over the origin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The dilational holonomy around the singularity in M n λ is λn and the angle at the singularity is nπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' These models have natural local coordinate maps from M n λ ∖ {0} to C whose transition functions are in the group generated by translations, dilations, and rotations by π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If n is even, we can specify local coordinate maps to C where the transition functions are in the group generated by translations and dilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A half-dilation surface with dilation singularities is a surface together with an atlas of charts to the plane and the spaces M n λ , where the transition maps are in the group generated by translations, dilations, and rotations by π in local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A dilation surface with dilation singularities is the same, only the transition maps are in the group generated by translations and dilations, and the model singularities are of the form M 2n λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Note that the universal cover of a dilation surface with dilation type singularities has no natural metric, because now there is dilational holonomy around loops in the cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This makes metric methods to deduce convexity and existence of closed trails seem unlikely to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' It is conceivable there is a different metric worth considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' See Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Interestingly, some papers in the field of dilation surfaces only allow cone singularities, while others allow dilation singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Veech was probably the first to consider dilation surfaces, though he worked in the more general context of (singular) complex affine structures on surfaces [Vee93] [Vee97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Veech proved fundamental results on the moduli spaces of these structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This understanding was recently improved in [ABW22] which specifically considers moduli spaces of dilation surfaces allowing dilation singularities, and proving (among other results) that the moduli space of dilation surfaces with singular data fixed is an orbifold covering of the usual moduli space of the corresponding punctured surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The directional foliations on the spaces Mλ constructed above are isomorphic to the directional foliations of C∗/⟨z ↦ −z⟩, and the foliations on n-fold branched covers M n λ are isomorphic to those on the n-fold branched covers of C∗/⟨z ↦ −z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Thus, dilation surfaces with dilation singularities still induce zebra structures on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Veech proved that a dilation surface has a triangulation by saddle connections if and only if it contains no dilation cylinders with angle π or more [DFG19, Appendix].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Note however that dilation singularities with angle 2π are singularities in the dilation surface sense but are non-singular on the induced zebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Therefore a triangulation by saddle connections may not be a leaf triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Nonetheless, by combining Veech’s result with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='4 we obtain: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose S is a closed surface with a dilation structure and at least one singularity, but without dilation singularities with angle 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then, the following are equivalent: 8 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS S has a triangulation by saddle connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The universal cover ˜S is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Every nontrivial free homotopy class ⟦γ⟧ of closed curves is realized by either a unique closed trail or a cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' S contains no dilation cylinder with angle π or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We will observe however that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='8 does not hold if one allows dilation singularities with angle 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' For this, we need the notion of a zebra cylinder in a zebra surface S, which we take to be the union of all closed trails in a PR free homotopy class of closed curves ⟦γ⟧ that contains at least two such closed trails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3, these closed trails must have the structure of a cylinder, and each closed trail has constant slope (though these slopes vary with the closed trail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A zebra cylinder is full if every slope in ˆR is the slope of one of the closed trails in the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' When dilation singularities with angle 2π are allowed in a dilation surface, there can be multiple dilation and flat cylinders with homotopic core curves that join together to form one cylinder in the zebra sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (Dilation singularities with angle 2π are not considered to be singularities on the zebra surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=') An example of a full zebra cylinder made from two dilation cylinders and one flat cylinder is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The reason we cannot extend Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='8 to allow dilation singularities with angle 2π is the following: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose S is a zebra surface and that S has a PR free homotopy class of closed curves ⟦γ⟧ whose closed trails constitute a full zebra cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If ⟦β⟧ is any PR free homotopy class of closed curves whose geometric intersection number with ⟦γ⟧ is non-zero, then ⟦β⟧ contains no closed trails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Furthermore if such a ⟦β⟧ exists, then the PRU cover of S is not convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose ⟦γ⟧ and ⟦β⟧ are as stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let ¨ε ∶ ¨C → S be the full zebra cylinder containing the closed trails in ⟦γ⟧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose to the contrary that β is a closed trail in ⟦β⟧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Because of the intersection number condition, there must be an arc β′ ⊂ ¨ε( ¨C) of β that crosses every closed trail in the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since ¨ε(C○) contains no singularities, the slope of β′ must be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let m denote this slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let ℓ ⊂ ¨C be a vertical closed leaf whose image ¨ε(ℓ) is a closed trail whose constant slope is also m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We will derive a contradiction from the fact that β′ must both contain points in ¨ε(ℓ) and not in ¨ε(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This is clearly impossible when ℓ is contained in the interior of C○, because in this case both ¨ε(ℓ) and β′ are leaves of the same foliation of slope m restricted to the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If ℓ is one of the two boundary curves of C, then ¨ε(ℓ) is a trail of constant slope m and the bending angles along ¨ε(ℓ) on the side of ¨ε(C○) are all π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Because of these bending angles and the local structure at the singular points, no leaf of slope m emanating from a singularity on ¨ε(ℓ) enters ¨ε(C○), again contradicting the existence of β′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This completes the proof that ⟦β⟧ contains no closed trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The last statement follows directly from statement (1) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Zebra surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Based on Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='8, we state: Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose S is a closed surface with a zebra structure and at least one singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then, the following are equivalent: (a) S has a leaf triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (b) The PRU cover ˜S is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (c) Every PR free homotopy class ⟦γ⟧ of closed curves that is nontrivial and non-polar either contains a unique closed trail or contains closed leaves (as described in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In particular, case (NR) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3 does not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (d) S contains no full cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The implication (a) implies (b) is Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The implication (b) implies (c) is Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The implication (c) implies (d) follows directly from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' It remains to prove that (d) implies (a), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=', if S has no full cylinders, then it has a leaf triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Outline of paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We will now explain what is done in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Because the paper proves state- ments about zebra surfaces but some reader’s interests will only include translation surfaces or dilation surfaces, we try to point out what can be skipped for such a reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In Section 2, we carefully define what a zebra surface is and establish basic terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' All results here are well known for translation and dilation surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We prove that the Gauss-Bonnet theorem holds for zebra surfaces and subsurfaces with polygonal boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' ZEBRA SURFACES 9 In Section 3, we formally define the pole-resolved universal cover of a zebra surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This is also natural for half-translation surfaces with poles, but we have not seen it in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We consider basic geometric objects on zebra planes (such as PRU covers) such as polygons and trails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We prove basic results about these objects, which are all obvious when working with half-translation and half-dilation surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' For example, we construct rectangles, prove arcs of trails have maximal extensions as trails, and show trails on zebra planes are proper maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In Section 4, we define the notion of a zebra structure on a surface with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Our definition allows for a polygonal boundary, generalizing the natural idea of a dilation surface with piecewise-linear boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This is important for laying a rigorous foundation for the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In Section 5, we consider surgical constructions on zebra surfaces, building new zebra surfaces from subsurfaces of others with polygonal boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Because of the flexibility of the zebra structure, we allow gluing subsurfaces together by homeomorphism of edges in the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This is useful for simplifying several arguments appearing later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='6, we use surgery to show that there are zebra surfaces that do not arise from half-dilation structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Section 6 focuses on producing foliations of polygons in zebra planes by combining leaves from the foliations Fm with m varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' These foliations form the foundation of our later arguments, and some such foliations seem interesting even in the Euclidean plane (though proofs would be easier in this context where analytic methods are available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2, we show that a triangle in a zebra surface can be foliated by leaves emanating from a vertex, and that slopes of these leaves vary monotonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3, we prove that a polygon can be foliated by leaves passing through one edge, where the slopes of leaves passing through a given point on that edge are given by a monotone function (subject to obvious constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This second result has a slick proof using surgery on zebra surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In Section 7, we investigate the behavior of trail rays emanating from a point in a zebra plane Z, and also arcs of trails joining two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This work is fundamental for our convexity arguments later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Considering all the trail rays emanating from a point in Z leads to a foliation of an open subset of Z with a different singular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We use this structure to prove that polygonal regions in Z all of whose exterior angles are at least π are convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This is clear from CAT(0) arguments when the polygon is in a translation surface, but seems unclear for polygons in dilation surfaces with dilation-type singularities in the interior of the polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This result allows us to prove a criterion for convexity of PRU covers of closed zebra surfaces where α is non-positive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' see Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This statement applies to dilation tori, all of whose singularities are dilation-type with angle 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3, we prove a continuity statement for the map sending a pair of points to the arc of a trail between the two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In Section 8, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2, which says that a zebra plane Z with a leaf triangulation is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We choose an arbitrary point p ∈ Z and consider all trail rays emanating from p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We inductively show that these rays cover every triangle in our triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Given the results from the prior section, the proof is largely combinatorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This argument is likely of interest to anyone interested in non-compact translation surfaces or in dilation structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In Section 9, we consider the question of finding closed trails in a zebra surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We begin with a discussion of PR free homotopy classes of curves and a pole-resolved version of the fundamental group in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3, we prove a theorem that guarantees the existence of a closed trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This result, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='7, is of interest to those thinking about dilation surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Later subsections are concerned with developing the remainder of the structure described in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Arguments should be readable to experts interested in the contexts of translation or dilation surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We prove Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='6 in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Section 10 provides a list of open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We know very little about zebra surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Formal definitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Surfaces and structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' For us a surface is a second countable Hausdorff space that is locally homeomorphic to R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Throughout this paper, all surfaces are oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A closed surface is a compact connected surface without boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let X be a topological space and S be a surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' An atlas of charts from S to X is a collection of charts of the form φ ∶ U → X whose domains are open and cover S and such that each chart φ ∶ U → X has an open image φ(U), and is a homeomorphism from U to its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A transition map between two charts with intersecting domains φ1 ∶ U1 → X and φ2 ∶ U2 → X is the restriction of φ2 ○ φ−1 1 to φ1(U1 ∩ U2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 10 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS In general, geometric structures are specified by defining a pseudogroup of homeomorphisms between the open sets of X, and insisting that transition maps lie in the pseudogroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We call X the model space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We refer the uninitiated reader to Chapter 3 of [Thu14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Foliations can be considered to be a particular case of a geometric structure, see [Thu14, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Foliated surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The horizontal foliation H of R2 is the collection of all horizontal lines in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If U ⊂ R2 is open, we say that two points (x1,y1) and (x2,y2) are horizontally equivalent in U if y1 = y2 and the horizontal line segment between the points is contained in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The horizontal foliation of U is the collection H∣U of horizontal equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We call these equivalence classes leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The horizontal foliation pseudogroup of R2 is the collection of homeomorphisms h ∶ U → V between open subsets of R2 that induces a bijection from the leaves of H∣U to the leaves of H∣V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A foliation atlas on a surface S is an atlas of charts to R2 whose transition functions lie in the horizontal foliation pseudogroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A foliation atlas determines a foliation equivalence relation on S, namely the finest one such that given any chart φ ∶ U → R2, preimages of points in the same leaf of H∣φ(U) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A foliation of S is the collection of foliation equivalence classes obtained from a foliation atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We call the equivalence classes leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If S is a surface with a foliation F, and A ⊂ S is a subsurface (possibly with boundary), then the restricted foliation on A is the collection F∣A of connected components of intersections A ∩ ℓ, where ℓ varies over the leaves of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If A is an open set, then it is a surface and the restricted foliation is a foliation on A, because a foliation atlas for A can be obtained by restricting each chart φ ∶ U → R2 in the atlas for F to the function φ∣A∩U ∶ A ∩ U → R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='1 (Leaf topology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let S be a topological surface, perhaps with boundary, and let ℓ ⊂ S be a subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The leaf topology on ℓ is the coarsest topology such that for each open U ⊂ S, each connected component of U ∩ ℓ is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If ℓ ∈ F is a leaf of a foliated surface without boundary, then each point of ℓ has a neighborhood homeomorphic to an open interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This gives ℓ the structure of a connected 1-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A local homeomorphism f ∶ S0 → S1 is a map such that for every point p ∈ S0, there is an open neighborhood U of p such that f(U) is open in S1 and f∣U ∶ U → f(U) is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose S1 is a space with a foliation F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let S0 be another topological space and suppose f ∶ S0 → S1 is a local homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then there is a natural pullback equivalence relation, namely the finest equivalence relation on S0 such that the points p and q of S0 are equivalent when there is an open set U ⊂ S0 containing p and q such that f(U) is open, f∣U ∶ U → f(U) is a homeomorphism, and f(p) and f(q) lie on the same leaf of F1∣f(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The pullback foliation f ∗(F1) is the collection of equivalence classes of the pullback equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If (S0,F0) and (S1,F1) are two foliated spaces and f ∶ S0 → S1 is a homeomorphism such that F0 = f ∗(F1), then we say that f is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' That is, f must induce a bijection from F0 to F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Given a collection {(Si,Fi)} of foliated spaces as above, the disjoint union ⊔i Fi is a foliation on X = ⊔i Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In any of the foliated spaces (X,F) constructed as above, the foliation pseudogroup consists of all isomor- phisms between open subsets of X endowed with restricted foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' One can define a foliated surface (S,F) to be the geometric structure determined by a foliation atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' But, treating a foliation as its collection of leaves seems more natural, and from the collection of leaves derived from such an atlas we can recover an atlas defining the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The charts can be taken to be the collection of all φ ∶ U → R2 where U ⊂ S is open and φ is an isomorphism from (U,F∣U) to (f(U),H∣f(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Standard singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Note that the action of multiplication by −I on R2 preserves the horizontal foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let Π−1 denote R2/−I, which has a cone point with cone angle π at the image of the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We call this cone point the origin 0 ∈ Π−1 and write Π∗ −1 = Π−1 ∖{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Note that a collection of inverses of restrictions of the covering map R2 ∖ {0} → Π∗ −1 gives a foliation atlas on Π∗ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We call the foliation associated to this atlas the horizontal foliation H−1 of Π∗ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' For each integer n ≥ 0, we define Πn to be the branched cover of Π−1 of degree n + 2 branched over the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In all these spaces, we use 0 to denote the unique preimage of 0 ∈ Π−1, and call 0 the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Note that geometrically 0 ∈ Πn is a cone singularity with cone angle (n + 2)π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We define Π∗ n = Πn ∖ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The pullback of the horizontal foliation on Π∗ −1 under the covering map Π∗ n → Π∗ −1 is the horizontal foliation Hn of Π∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Note that Π0 is naturally homeomorphic to R2, and the foliation H0 of Π0 is carried by this homeomor- phism to the horizontal foliation of R2 ∖ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' ZEBRA SURFACES 11 We further define Π−2 to be the plane R2 equipped with the foliation of R2 ∖{0} by circles with center 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' For convenience we call this foliation the horizontal foliation of Π∗ −2 = Π−2 ∖ {0} and denote it by H−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This is locally homeomorphic to one of the straight-line foliations that arises near a double pole of a quadratic differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Singularities of this form show up in some of our arguments, but are not allowed in zebra surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A prong of Πn is a leaf of the horizontal foliation with 0 as an endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' There are n + 2 prongs in Πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Consider the model space (2) X = Π−1 ⊔ ⊔ n≥1 Πn, and let X∗ = Π∗ −1 ⊔ ⊔ n≥1 Π∗ n ⊂ X, which comes equipped with its horizontal foliation HX = ⊔Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let h ∶ U → V be an orientation-preserving homeomorphism between open subsets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We say h is in the horizontal pseudogroup (of X) if : (1) We have h(U ∩ X∗) = h(U) ∩ X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (2) The restriction h∣U∩X∗ is in the foliation pseudogroup of (X∗,HX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Observe that if Ui ⊂ U is a connected component then Ui ⊂ Πm for some m = m(i) and h(Ui) ⊂ Πn for some n = n(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Note that statement (1) implies that 0 ∈ Ui if and only if 0 ∈ h(Ui) and that h(0) = 0 if 0 ∈ Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A singular foliation atlas on a surface S is an atlas of charts to X whose transition functions lie in the horizontal pseudogroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We say a point p ∈ S is a singularity if there is a chart φ ∶ U → Πn such that φ(p) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Because elements of the pseudogroup send origins to origins, the notion of being a singularity is independent of the chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The singular set Σ ⊂ S is the collection of all singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Every point p has a neighborhood U such that U ∖ {p} contains no singularities, so Σ is a closed discrete subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Observe that a singular foliation atlas determines a foliation on S ∖ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We call this foliation a singular foliation on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The singular data of the atlas consists of Σ and the function α ∶ S → Z≥−1 whose support is Σ and which sends p ∈ Σ to the n such that there is a chart from a neighborhood of p to a neighborhood of 0 ∈ Πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Observe that α is well-defined, because a single chart tells you that there are n+2 prongs emanating from p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Here a prong emanating from a point p ∈ S is the germ of an injective path γ ∶ (0,1) → ℓ into a leaf ℓ with limt→0+ γ(t) = p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=', an equivalence class of such paths where two such paths γ1 ∶ (0,1) → ℓ and γ2 ∶ (0,1) → ℓ are equivalent if there are constants ϵ1,ϵ2 ∈ (0,1) such that the restrictions γ1∣(0,ϵ1) and γ2∣(0,ϵ2) are the same up to orientation-preserving reparameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (Our definition of prong is slightly non-standard in that typically prongs are only defined at singular points, but we define them at all points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=') If α(p) = −1, the point p is called a pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3 (Removable singularities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since allowing removable singularities would make statements of our main results more technical and since we don’t have much use for them in this paper, we have purposely not allowed removable singularities in our zebra surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' However, if the model space X is altered to include Π0 with its horizontal foliation, then a point mapping to 0 ∈ Π0 would be a removable singularity or marked point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A marked point in a singular foliation can be removed by replacing a chart to (Π0,H0) with a chart to (R2,H) or an isomorphic subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' So, removable singularities are much the same as regular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' However, treating a regular point as a singularity creates some problems, because including a regular point in Σ alters the space being foliated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Thus, this change alters the notion of what a leaf is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In particular, if we allow removable singularities, the statements of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3 or the definitions they depend on need to be suitable altered to make the results still true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Given a prong contained in a leaf ℓ emanating from a singular point p, a parameterization of the leaf (0,1) → ℓ can be extended to include p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In such a case we call the singularity an endpoint of the leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If a leaf has two endpoints, then we call the union of the leaf with its endpoints a saddle connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If the leaf has one endpoint, then we call this union a separatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A leaf with no endpoints is called bi-infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A leaf that is homeomorphic to a circle is said to be closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Sometimes we will allow double poles to appear in our singular foliations, though we do not allow these more general singular foliations in our zebra surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A generalized singular foliation atlas on a surface is defined as above but including Π−2 in the model space with its horizontal foliation F−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In this case α takes values in Z≥−2 and we call a point p where α(p) = −2 a double pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Note that a singular foliation is also a generalized singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 12 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='4 (Euler–Poincar´e Formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let F be a generalized singular foliation on a closed surface S with singular set Σ and singular data α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then, ∑ p∈Σ α(p) = −2χ(S), where χ(S) denotes the Euler characteristic of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This result follows from the Poincar´e-Hopf theorem, which gives the Euler characteristic in terms of the sums of indices of a zeros of a vector field with isolated zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' For a proof see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='1 of [FLP21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The main idea is to pass to a double cover, so the foliation becomes oriented and gives rise to a vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Note that here we allow α to take the values −2 and −1 while [FLP21] does not (though the proof goes through in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The extended real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let ˆR = R ∪ {∞}, which is homeomorphic to a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The usual increasing order on R extends to a cyclic order on ˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We use interval notation to denote subsets of ˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If m0 ≠ m1, we use interval notation such as (m0,m1) to denote the set of slopes m ∈ ˆR for which the triple (m0,m,m1) is in strictly increasing cyclic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then [m0,m1] will denote the associated closed interval, (m0,m1) ∪ {m0,m1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The stellar functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The stellar function on R2 ∖ {0} is the function which sends a point p to the slope of the line joining p to 0: (3) ρ ∶ R2 ∖ {0} → ˆR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (x,y) ↦ y x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Observe that the value of ρ is preserved by the action of −I, so ρ descends to a well-defined map ρ−1 ∶ Π∗ −1 → ˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then for n ≥ 0, we can obtain functions ρn ∶ Π∗ n → ˆR by composing the covering map Π∗ n → Π∗ −1 with ρ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We call ρn the stellar function on Π∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A ray in Πn of slope m is a connected component of ρ−1 n (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Definition of zebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let {Fm ∶ m ∈ ˆR} be a collection of singular foliations on a connected surface S with the same singular set Σ and the same singularity data function α ∶ S → Z≥−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A stellar neighborhood of a point p ∈ S is an open neighborhood U of p such that there is an integer n ≥ −1 and a homeomorphism h ∶ U → Πn such that h(p) = 0 and the following statements hold for each slope m ∈ ˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (1) For each ray r ⊂ Π∗ n of slope m, h−1(r) is contained in a leaf of Fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (2) For each prong of Fm emanating from p, there is a ray r ⊂ Π∗ n of slope m such that for any path γ ∶ (0,1) → r with limt→0+ γ(t) = 0, the preimage h−1 ○ γ represents the prong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The above two statements guarantee that for each m, h induces a bijection from prongs of Fm emanating from p and rays of slope m in Πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We call h a stellar homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' It follows by counting prongs and rays that n = α(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We say that the collection of singular foliations {Fm}m∈ˆR is a stellar foliation structure or a zebra structure on S if the foliations have the same singular sets, the same singular data, and every p ∈ S has a stellar neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We call a surface together with a zebra structure a zebra surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The action by homeomorphisms of the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Our foliations are parameterized by the topological circle ˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' There is a natural action of the group Homeo+(ˆR) of all orientation-preserving homeomorphisms of ˆR on zebra surfaces defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose {Fm} defines a zebra structure on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If ϕ ∈ Homeo+(ˆR), then for each m ∈ ˆR we can define F′ m = Fϕ−1(m), and {F′ m}m∈ˆR will define another zebra structure on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Note the original structure and the new structure have the same singular data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let Homeo−(ˆR) denote the collection of all orientation-reversing homeomorphisms of ˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then Homeo+(ˆR)∪ Homeo−(ˆR) forms the full homeomorphism group of ˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' For ϕ ∈ Homeo−(ˆR), we define ϕ(S,{Fm}) = ( ¯S,{Fϕ−1(m)}) where ¯S denotes S with its opposite orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' ZEBRA SURFACES 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Leaves on zebra surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' As in the introduction, a leaf of a zebra surface is a leaf of one of the foliations Fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The slope of a leaf on S is the m for which the leaf belongs to Fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The word horizontal means slope zero, and vertical means slope ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Terms like saddle connection, separatrix and bi-infinite leaf all make sense on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Observe: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='5 (Transversality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If leaves ℓ1 and ℓ2 have distinct slopes and intersect at a non-singular point, then they cross transversely in the sense that there is an open disk containing the intersection point such that there is only one intersection between ℓ1 and ℓ2 in this disk and the disk is cut in two by ℓ1 with points in ℓ2 in both halves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since we are on a zebra surface, the intersection point has a stellar neighborhood that gives the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Angles and Gauss-Bonnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Zebra surfaces have a natural notion of angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let pq and qr be two segments of leaves on a zebra surface, where we allow any of these three points to be singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let U be a stellar neighborhood of q and h ∶ U → Πα(q) be the corresponding stellar homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then ∡pqr indicates the counterclockwise angle measured at the origin of Πα(q) from h(pq ∩ U) to h(qr ∩ U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We normalize this measurement so (4) 0 ≤ ∡pqr < (α(q) + 2)π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose S′ is a subsurface of a zebra surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let q ∈ ∂S′, and suppose that in a neighborhood of q, ∂S′ = rq ∪ qp where rq and qp are segments of leaves and S′ is on the left as we move from r to q to p along this boundary curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then the interior angle of S′ at q is ∡pqr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='6 (The Gauss-Bonnet Theorem for zebra surfaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let K be a compact subsurface of a zebra surface with a boundary consisting of a union of disjoint simple closed curves that are piecewise given by segments of leaves (with finitely many pieces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let α be the singular data function on the surface containing K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' For a boundary point q, let θq denote the interior angle at q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then, (5) ∑ q∈∂K (π − θq) − ∑ p∈Σ∩K○ πα(p) = 2πχ(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Note that in (5), there are only finitely many points in each sum for which the contribution to the sum is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Consider K as equipped with a foliation of slope m which does not coincide with the slope of any of the finitely many leaves in the boundary of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Double K across its boundary to obtain a closed surface X to which we can apply the Euler–Poincar´e Formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The doubled surface satisfies χ(X) = 2χ(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The surface X inherits a foliation from the two copies of K, where we allow our leaves to pass between the copies of K through the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We will see that this foliation of slope m of K lifts to a generalized singular foliation of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We allow our leaves to pass between copies of K through the boundary, so this will be a foliation of the complement of the singular points in the interior of K and endpoints of boundary edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let ˜α denote the singularity data on X for the lifted foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (Checking that ˜α is well defined will prove that the lifted foliation to X is a generalized singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=') Each singular point p ∈ K○ has two lifts ˜p1, ˜p2 ∈ X, and we have ˜α(˜p1) = ˜α(˜p2) = α(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' For q ∈ ∂K an endpoint of a boundary edge, we have only one lift ˜q and by considering a stellar neighborhood of q we see that ˜α(˜q) = 2vq − 2 where vq is the number of prongs in K terminating at q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then by the Euler–Poincar´e Formula, we have ∑ q∈∂K (2 − 2vq) − ∑ p∈Σ∩K○ 2α(p) = 2χ(X) = 4χ(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Dividing by 2 and multiplying by π yields: (6) ∑ q∈∂K π(1 − vq) − ∑ p∈Σ∩K○ πα(p) = 2πχ(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Now consider a single boundary component γ of K whose vertices are qi for i ∈ Z/nZ written in increasing cyclic order as we travel around γ with the region K on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Given any i, choose a stellar homeomorphism hi ∶ Ui → Πα(qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' By possibly shrinking Ui, we can assume that h(K ∩ Ui) is a sector σi ⊂ Πα(qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then the angle of this sector coincides with the interior angle θqi, and the starting ray of the sector has the same 14 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS slope as qiqi+1 and the ending ray has the same slope as qi−1qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Consider the union of these sectors σi with the starting ray of σi glued to the ending ray of σi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Note that the glued rays have the same slopes, so this gluing of sectors produces a copy of Πk where ∑θqi = (k + 2)π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' It follows that the total number of prongs ∑vqi = k + 2 and so we have (7) n−1 ∑ i=0 θqi = n−1 ∑ i=0 πvqi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Substituting (7) into (6) (for each boundary component) yields (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ It is useful to observe the following result for future Gauss-Bonnet calculations: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let γ be a closed curve in a zebra surface that is piecewise given by segments of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let {θi ∶ i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=',n} denote the measures of angles at the transitions between the segments of leaves, measured uniformly on one side of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then the sum ∑n i=1 θi is an integer multiple of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let qi denote the endpoints of the segments of leaves making up γ, with the angle θi being measured at qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Orient γ, and consider a vector traveling around γ in the direction of the orientation and pointed along the curve on the interiors of the arcs making up γ, and turning at each qi between the arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then, the unit vector turns by a signed angle equivalent to π ± θi modulo πZ at qi, where the sign only depends on the side of γ where measurements were made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since when the vector travels completely around γ, it ends pointing in a direction with the same slope as its start, we have that ∑i(π ± θi) is an integer multiple of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since the signs are uniform, the conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Basic observations, definitions, and results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The pole-resolved universal cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let S, Σ, and α be as in Section 2, but add the hypothesis that S is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Define Σ−1 = α−1(−1) to be the set of poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The points in Σ−1 are our only source of “positive curvature.” The goal here is to define a variant of the universal cover but with no “positive curvature.” The cover described here was also considered in [Fra18, §3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let S+ = S ∖ Σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Similar to language used in the introduction, call a loop in S+ polar if it is freely homotopic in S+ to a simple loop bounding a disk in S containing exactly one point in Σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Choose a basepoint p0 ∈ S+ and define (8) N = ⟨γ2 ∶ γ is polar⟩ ⊂ π1(S+,p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then, N is a normal subgroup of π1(S+,p0), because being polar is a conjugacy invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' There is a largest branched cover ˜S of S which is at most doubly branched over each point in Σ−1 and is unbranched over other points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The restriction of the covering π ∶ ˜S → S to π−1(S+) → S+ is the normal cover of S+ associated to N ⊂ π1(S+,p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The surface ˜S is homeomorphic to a disk, and the covering π is doubly branched over every pole, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=', the covering map is locally 2 − 1 near any preimage of a pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We define ˜S to be the pole-resolved universal cover (PRU cover) of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' It follows from the result above that ˜S is a branched cover of the usual universal cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If Σ−1 = ∅, then the PRU cover coincides with the universal cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let ˜S+ denote the cover of S+ associated to the subgroup N, as in covering space theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We claim that ˜S+ is ˜S with the preimages of points in Σ−1 removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose ˜T is some other branched cover of S which is only branched over points in Σ−1 and at most doubly branched over these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Puncturing ˜T at preimages of Σ−1, we obtain a covering ˜T + of S+, which is associated to a subgroup G ⊂ π1(S+,p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' To show that ˜S+ covers ˜T +, it suffices to prove that N ⊂ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' To this end, let γ ∶ [0,1] → S+ be a polar loop with γ(0) = γ(1) = p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then there is a homotopy hs ∶ [0,1] → S+ such that h0 = γ and h1 is a loop in S bounding a disk enclosing exactly one point in Σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let η ∈ π1(S+,p0) be the loop which follows s ↦ hs(0) for s ∈ [0,1], then follows t ↦ h1(t) for t ∈ [0,1] and returns to the basepoint following s ↦ hs(0) parameterized backward from s = 1 to s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' It is not hard to show that η is homotopic rel endpoints to γ in S+, thus determining the same element of π1(S+,p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since ˜T is at most doubly branched over points in Σ−1, the square of the element of π1(S+,p0) associated to the common class of γ and η lies in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since γ was an arbitrary polar curve N ⊂ G as claimed, proving that ˜S+ is the largest ZEBRA SURFACES 15 such cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This argument also shows that ˜S+ is locally at most a double cover in neighborhoods of Σ−1, and we can fill in these points to form the branched cover ˜S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' It remains to show that the cover ˜S is actually doubly branched over points in Σ−1 and that ˜S is a topological disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' To see that ˜S → S is doubly branched over some point p ∈ Σ−1, it suffices to find a branched cover ˜T as above which is doubly branched over p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' To see that ˜S is a disk, it suffices to find a ˜T whose universal cover is a disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (If ˜T satisfies the double branching condition, the so does its universal cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=') If S has positive genus, this is clear since its universal cover is a disk and given any p ∈ Σ−1, we can find a linear map H1(S+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='Z/2Z) → Z/2Z sending a loop wrapping once around p to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Covering space theory associates this linear map to a double cover of S+ which is doubly branched over p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If S is a sphere, then by the Gauss-Bonnet Theorem there are at least four points in Σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Choosing four points including our favorite point p, we can puncture only at those four points and define a linear map as before such that the homology classes of the loops around each of these points are sent to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The associated double cover is a torus which is doubly branched over these four points as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since this torus has a disk as its universal cover, this also proves that ˜S is a disk in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ Now suppose {Fm}m∈ˆR is a family of foliations determining a zebra structure on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let ˜Fm be the singular foliation of ˜S whose leaves are lifts of leaves of Fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We call { ˜Fm}m∈ˆR the lifted family of foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' These foliations have common singularity data ˜α, where if ˜p ∈ ˜S projects to p ∈ S, we have ˜α(˜p) = α(p) if α(p) ≥ 0 and ˜α(˜p) = 0 if α(p) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Zebra planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A zebra plane is a zebra structure on the open topological disk such that the singular data function α is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' From the discussion above, the PRU cover of a zebra surface is always a zebra plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A polygon p0p1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='pn−1 in a zebra plane is a topological disk bounded by a simple closed curve of the form p0p1 ∪p1p2 ∪.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='∪pn−2pn−1 ∪pn−1p0, where each edge pipi+1 with i ∈ Z/nZ is a segment of a leaf from a directional foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The Jordan Curve Theorem guarantees that this curve bounds a topological disk, and we’ll use the counterclockwise ordering when describing polygons so that the polygon is on the left as we move from pi to pi+1 along pipi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We call the pi vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The interior angle of P at pi is ∡pi+1pipi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The external angle is ∡pi−1pipi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The sum of the interior and exterior angles at pi is α(pi)π + 2π, the total angle at pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We’ll call a vertex straight if the interior angle equals π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' As we are typically interested in the internal geometry of a polygon, we will typically ignore straight vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' So, a k-gon is a polygon with k vertices that are not straight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A triangle is a 3-gon, and we’ll use other similarly obvious terminology coming from plane figures to describe objects in ˜S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose S is a zebra surface and ˜S is its PRU cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let π ∶ ˜S → S denote the covering map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If ˜P is a polygon in ˜S such that the restriction π∣ ˜ P ∶ ˜P → S is injective on the interior of ˜P, then we’ll call the restriction π∣ ˜ P a polygon P in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' These are maps rather than subsets of S, because it gives the right notion of the interior of P (the image of the interior) and boundary (the further restriction to the boundary of ˜P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' These notions are confusing even when considering the square in the center of the usual square torus (in that the closed square is the whole torus, but you still want it to have boundary for instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The next proposition shows that interior angles and slopes of edges of a zebra triangle behave as they do in plane geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Triangles contain no singularities in their interiors and the sum of the interior angles of a triangle in a zebra plane is always π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let (m0,m1,m2) ∈ ˆR3 be a triple of slopes of edges of a triangle, listed in counterclockwise order as we travel around the boundary of the triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then the triple of slopes is distinct and appear in decreasing cyclic order on ˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let θi for i = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=',2 be the interior angles of a triangle T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since the curve is simple, we have θi > 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' By the Gauss-Bonnet Theorem, we have θ0 + θ1 + θ2 = π − ∑ p∈T + πα(p) > 0, where the sum is taken over all interior singularities of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since α(p) ≥ 1 at all singularities in a zebra plane, there must be no singularities in T + and thus θ0 + θ1 + θ2 = π as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' It follows the slopes of the sides of 16 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS a triangle must be the same as the slopes of the sides of a Euclidean triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Therefore, the slopes appear in decreasing cyclic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The sum of the interior angles of a quadrilateral Q in a zebra plane is either π or 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If the sum is 2π, then Q contains no singularities in its interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If the sum is π, then Q contains a singularity q in its interior with α(q) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' From the Gauss-Bonnet Theorem, the interior angles satisfy θ0 + θ1 + θ2 + θ3 = 2π − ∑ p∈T + πα(p) > 0, immediately giving the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ We also have the following more general result: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Any closed n-gon P in a zebra plane must have at least three interior angles whose measure is less than π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Again by the Gauss-Bonnet Theorem, ∑ q∈∂P (π − θq) = 2π + ∑ p∈K○ πα(p) ≥ 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since for each q, we have π − θq < π, there must be at least three q ∈ ∂P for which π − θq > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Arcs of trails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let Z be a zebra plane and let { ˜Fm} denote the associated foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let p ∈ Z be a singularity and pq and pr be segments of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We say these arcs satisfy the angle condition at p if ∡qpr ≥ π and ∡rpq ≥ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A parameterized arc of a trail on Z is a parameterized curve γ ∶ I → Z, where I ⊂ R is a non-degenerate interval, such that if t is any point in the interior of I then the following statements are satisfied: If γ(t) is not singular, then in a neighborhood of t, γ(t) moves injectively along a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If γ(t) is singular, then the two arcs made at γ(t) formed by increasing and decreasing t satisfy the angle condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' That is, the angle made by γ at γ(t) is at least π on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose γ ∶ I → Z and η ∶ J → Z are parameterized arcs of trails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We say that γ is a subarc of η if there is an orientation-preserving continuous injective map φ ∶ I → J such that γ = η ○ φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We say γ is a proper subarc of η if the map φ is not surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The condition that two arcs of trails are each subarcs of the other (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=', reparameterizations of one another) is an equivalence relation, and we’ll call an equivalence class an arc of a trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The notion of subarc induces a well-defined partial ordering on arcs of trails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' An arc of a trail on S is the image of an arc of a trail on the PRU cover ˜S under the covering ˜S → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We likewise use the cover to define the other notions above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='5 (No monogons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If τ ∶ I → Z is a parameterized arc of a trail in a zebra plane, then τ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose τ is not injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then we can assume without loss of generality that I = [a,b] and τ(a) = τ(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Observe that because I is closed and bounded and τ is continuous and locally injective, the set J of all t ∈ [a,b] for which there is a t′ ∈ [a,t) such that τ(t) = τ(t′) is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let x = inf J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then there is an x′ ∈ [a,x) such that τ(x) = τ(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Restricting τ to [x′,x] yields a simple closed curve, which by the Jordan Curve Theorem bounds a disk D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Observe that D must be a polygon, and since τ is an arc of a trail, all its interior angles are larger than π except possibly at τ(x′) = τ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This violates Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='6 (No bigons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose τ1 and τ2 are arcs of trails in a zebra plane Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then τ1 ∩ τ2 is the empty set, is a single point, or is a common subarc (possibly with different induced orientations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Assume τ1 and τ2 intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Consider each τi to be parameterized by functions with the same name, τi ∶ Ii → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The statement can be observed to be true as long as τ −1 1 (τ2) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If τ −1 1 (τ2) is disconnected, we can let J1 ⊂ I1 ∖ τ −1 1 (τ2) be a bounded open interval such that both boundary points lie in τ −1 1 (τ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since τ2 is injective, there is also a unique interval J2 ⊂ I2 such that τ2(∂J2) = τ1(∂J1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' By construction τ1(J1)∪τ2(J2) forms a simple closed curve, which again by the Jordan Curve Theorem bounds ZEBRA SURFACES 17 a disk D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This time the only possible interior angles less than π are the two points in τ1(∂J1), again violating Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let P be a polygon in a zebra plane all of whose exterior angles are at least π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then there is no parameterized arc of a trail τ ∶ [0,1] → Z such that τ(0),τ(1) ∈ ∂P and τ((0,1)) ∩ P = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If this were the case, the union of τ and an arc of P bound a polygon Q whose interior is in the complement of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Points in ∂Q ∩ τ have interior angles at least π since τ is an arc of a trail, and points on ∂Q ∩ P that are not endpoints of τ have interior angles for Q which are the same as the exterior angles for P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Thus, Q can have at most two interior angles less than π (namely, the endpoints of τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Again this violates Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Trapezoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We speak of two segments of leaves in a zebra plane Z as being parallel if they are segments of leaves coming from the same foliation ˜Fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A trapezoid in Z is a 4-gon with a pair of opposite edges that are parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A parallelogram is a 4-gon such that both opposite pairs of edges are parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let T be a trapezoid in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then the angles of T add to 2π and there are no singularities in the interior of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose our trapezoid is pqrs, with vertices ordered counterclockwise and with pq parallel to rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then ∡rqp + ∡srq = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The other pair of angles add to π as well, so the sum of all the angles is 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='3 tells us that T has no singular points in its interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ We say a trail has constant slope if there is an m such that every segment of a leaf contained in the trail has slope m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let pq,rs ⊂ Z be arcs of trails of the same constant slope m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose that qr and sp are disjoint segments of leaves whose slopes are not m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then the curve pq ∪ qr ∪ rs ∪ sp is a 4-gon (trapezoid) whose vertices are p, q, r and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In particular, all interior angles at singularities in the interior of segments pq and rs are π, and the interior angles at p, q, r and s are each less than π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The curve pq ∪ qr ∪ rs ∪ sp is a closed curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If we can show it is simple, then by the Jordan curve theorem it bounds a disk, which is our polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose it has n sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' As in Euclidean geometry, the Gauss-Bonnet Theorem guarantees that the sum of the interior angles is (n − 2)π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Because pq and rs are parallel, we have ∡q + ∡r = aπ and ∡s + ∡p = bπ for some integers a,b ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=',tn−4 are the singularities in the interiors of pq and rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then ∡ti = kiπ for some integer ki ≥ 1, because both these arcs of trains have constant slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Thus the sum of the interior angles is (a+b+∑n−4 i=1 ki)π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The smallest this sum can be is (n − 2)π and so we must have a = b = k1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' = kn−4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Now we will argue that pq ∪ qr ∪ rs ∪ sp is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We know that the arcs pq, qr, rs, and sp are simple by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Using the slope conditions, we see that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='6 guarantees that the intersection of adjacent edges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=', pq ∩ qr) consists only of the common vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Therefore, the only way that the curve can fail to be simple is if opposite edges intersect in their interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' By hypothesis rq ∩ ps = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We claim that pq ∩ rs = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose to the contrary that pq ∩ rs ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='6, they intersect in either a single point or a common compact subarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let x ∈ pq be the point closest to q in pq ∩ rs (where “closest” is measured in a parameterization of pq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then the path γ = xq ∪ qr ∪ rx is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Observe that γ has at most two points at which there are angles whose measure is less than π (on either side of the curve), namely the points q and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (It could be that x coincides with either q or r, but otherwise because the arcs of γ on both sides of x are parallel, the angle at x must be at least π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=') This contradicts Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='4, proving our claim and completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='10 (Trapezoid construction lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let pq ⊂ Z be an arc of a trail, where the angle measured on the left side as we move from p to q at any singularities in the interior of pq is π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let U ⊂ Z be an open set containing pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let ps and qr be additional segments such that 0 < ∡qps < π and 0 < ∡rqp < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then there exist s′ ∈ ps ∖ {p} and r′ ∈ qr ∖ {q} and a segment s′r′ parallel to pq forming a trapezoid pqr′s′ contained in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Furthermore, we can construct the trapezoid in such a way so that leaves parallel to pq passing through the interior of the trapezoid pass through the interiors of ps′ and qr′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' 18 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Depiction of the trapezoid produced by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' See Figure 5 for an illustration of the trapezoid construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We briefly discuss the idea of the proof before giving a detailed proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' By compactness, we can produce a finite covering of pq by foliation charts for the foliation parallel to pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We can find stellar neighborhoods of p and q contained in the charts containing p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The angle conditions at p and q and the fact that there are only finitely many charts can be used to guarantee that we can find a leaf parallel to pq that joins a point of ps in the stellar neighborhood of p to a point of qr in the stellar neighborhood of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A technical point is that the charts might contain a singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Assume without loss of generality that pq is horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' For each point x ∈ pq there is a foliation chart from a neighborhood of x to (−1,1) × (−1,1) (or if x is singular, a singular chart to Πn carrying x to 0) for the horizontal foliation that intersects pq in an interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' By possibly shrinking the chart, we can assume that this neighborhood of x is contained in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' For this proof, all our foliation charts lie in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We can normalize these charts so that the portion of pq in the domain maps to a subset of (−1,1) × {0} and so that the portion of the chart to the left of pq when moving from p to q is mapped into (−1,1) × (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In case x is singular, since the angle made on the left side of pq at x is π, the portion of U in the chart to the left of pq is mapped into a half-plane of Πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We can put coordinates on this closed half-plane of the form R × [0,+∞), and like in the non-singular case we can restrict the chart and rescale it so that the portion of pq in the domain can be mapped to (−1,1) × {0} and the portion to the left of pq in the domain can be mapped to (−1,1) × (0,1) in these coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Using compactness of pq we can produce a minimal finite subcovering of pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Restrict these charts to the points whose images have non-negative y-coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We can order this collection of restricted charts φi ∶ Bi → (−1,1) × [0,1) for i = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=',n such that Bi ∩ pq gives a sequence of open subintervals of pq such that p ∈ B0, q ∈ Bn and pq ∩ Bi ∩ Bj ≠ ∅ if and only if ∣i − j∣ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We claim that we can define intervals Ji ⊂ [0,1) which are open as subsets of [0,1) together with continuous strictly increasing functions ψi ∶ Ji → [0,1) such that for all y ∈ Ji, the leaf φ−1 0 ((−1,1)×{y}) can be continued across B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' , Bi as (9) i ⋃ k=0 φ−1 k ((−1,1) × {ψk(y)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We do this by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Define J0 = [0,1) and φ0 ∶ J0 → [0,1) to be the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Now assuming Ji and φi are defined, we can let Ui be the connected component of Bi ∩ Bi+1 containing pq ∩ Bi ∩ Bi+1 and there is a continuous strictly increasing function hi ∶ πy ○ φi(Ui) → πy ○ φi+1(Ui), where πy(x,y) = y coming from the transition between the charts such that φ−1 i ((−1,1)×{y}) continues across Ui as φ−1 i+1((−1,1)× {hi(y)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then by defining Ji+1 = Ji ∩ ψ−1 i (Ui) and ψi+1 = hi ○ ψi we see that (9) is satisfied, completing the induction and giving definitions for Jn and ψ0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=',ψn satisfying (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let N be a stellar neighborhood at p and h ∶ N → Πn′ be the stellar homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then the connected component of N ∩ pq containing p maps under h to the closure of a horizontal ray r0 ⊂ Πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let H ⊂ Πn be the closed half-space consisting of 0 and all rays r ⊂ Πn′ such that the counterclockwise angle from r0 to r lies in [0,π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Observe that p ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' By restricting to a smaller neighborhood N and rescaling the stellar homeomorphism h, we can assume that h−1(H) ⊂ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let ℓ be the connected component of (ps ∖ {s}) ∩ N ZEBRA SURFACES 19 containing p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then h(ℓ) is a ray contained in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since h−1(H) ⊂ B0, we see that ℓ ⊂ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since ℓ is not horizontal and contains p, the horizontal leaves in B0 meet ℓ transversely and so πy ○φ0(ℓ) = [0,b0) for some b0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Similarly, there is a half-open arc ℓ′ ⊂ qr ∩ Bn containing q such that interval πy ○ φn(ℓ′) = [0,bn) for some bn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since Jn is an open subset of [0,1) containing 0 and ψn is strictly increasing and preserves zero, we can find y ∈ (0,b0) so that ψn(y) ∈ (0,bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then we have a segment of a leaf that cuts across B0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' , Bn as described in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let s′ be the place this leaf crosses ps and r′ be the place this leaf crosses qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We conclude that pqr′s′ is a trapezoid using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Furthermore, if 0 < y′ < y, then the leaf as constructed in (9) (with y′ replacing y) cuts across this trapezoid passing through the interiors of edges ps′ and qr′ as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='11 (Generalized rectangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let p ∈ Z and let U ⊂ Z be an open set containing p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then there is a 2α(p) + 4-gon P ⊂ U with alternating horizontal and vertical sides and all interior angles of π 2 such that p ∈ P ○.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Furthermore, there is a bijection between the horizontal prongs at p and the vertical edges of P such that each horizontal prong has a realization as a horizontal path joining the corresponding edge to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' These α(p) + 2 paths cut P into α(p) + 2 rectangles, each of which has p in the interior of an edge formed by two of the prong realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Every horizontal leaf that enters the interior of P either crosses through the interior of one of the rectangles joining opposite vertical sides of the rectangle, or follows one of the α(p)+2 horizontal paths and terminates at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We call the polygon P a generalized rectangle because of the alternating horizontal and vertical sides and angles of π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' An example is depicted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A generalized rectangle surrounding a point p where α(p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Construct horizontal segments of leaves with endpoints at p realizing every prong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' By possibly shortening them, we can assume they are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' There are n = α(p) + 2 such arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Denote them {βi ∶ i ∈ Z/nZ} and order them counterclockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then the counterclockwise angle from βi to βj at p is π if and only if j = i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let ei denote the path βi ∪ {p} ∪ βi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='10, for each i we can produce a rectangle Ri one of whose edges is ei such that the counterclockwise angle from βi to βi+1 at p is interior to Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Define Q = (⊔Ri)/ ∼ where ∼ identifies the corresponding points in the common subarcs {p}∪βi ⊂ Ri−1∩Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Observe that Q is homeomorphic to a closed disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let π ∶ Q → Z be the natural map induced by the inclusions of Ri into Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Observe that π restricted to each Ri is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' By considering horizontal foliation charts at p and at points of each βi, we can see that π is locally injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If we knew π were globally injective, then P = π(Q) will be a polygon satisfying the statements in the corollary, with the statement about the horizontal leaves following from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Now assume π ∶ Q → Z is not injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We will alter the construction to produce a smaller Q′ ⊂ Q with the same properties such that π restricted to Q′ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This will complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We will actually construct a sequence of subsets Qn playing the role of Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' For each i, construct a sequence of rectangles Rn i ⊂ ˜Ri such that are nested ( ˜Rn+1 i ⊂ ˜Rn i for all n) and satisfy ⋂∞ n=0 Rn i = ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' These Ri n can be produced from Ri by cutting along a horizontal leaf through the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We define Qn ⊂ Q to be the union over i of the Rn i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If the restriction of π to Qn is injective, then we can define P = π(Qn) to be our generalized rectangle, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Now suppose to the contrary that the restriction of π to Qn is not injective for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then for each n, we can find distinct points xn,yn ∈ Qn such that π(xn) = π(yn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' By passing to a 20 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' PATRICK HOOPER, FERR´AN VALDEZ, AND BARAK WEISS subsequence, we can assume that limxn = x and limyn = y both exist in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then π(x) = π(y) by continuity of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Observe that ⋂n Qn = {p} ∪ ⋃i βi, and each Qn is closed, so we have x,y ∈ {p} ∪ ⋃i βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' But π restricted to {p} ∪ ⋃i βi is injective because the βi were constructed to be pairwise disjoint realizations of prongs of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Thus we must actually have x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' But then the facts that xn ≠ yn, π(xn) = π(yn) and limxn = limyn violates the local injectivity of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Properness of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let ℓ be a separatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then it has a parameterization of the form ℓ ∶ [0,+∞) → Z where ℓ(0) is its singular endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Bi-infinite leaves have a parameterizations of the form ℓ ∶ R → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Recall that a proper map between topological spaces is one for which preimages of compact subsets are compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A separatrix or bi-infinite leaf in a zebra plane that is parameterized as above is a proper map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Singular foliations of the disk where α is allowed to take the value −1 can have separatrices and bi-infinite leaves that fail to be proper in the above sense even if there are only finitely many singularities [Ros83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This phenomenon also occurs with periodically arranged singularities where α = −1 [Pan09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Avoid- ing this phenomenon is a reason for requiring α to be non-negative in a zebra plane and for our definition of the PRU cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let ℓ be as above, and assume without loss of generality that its slope is horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If ℓ is not proper, there is a compact set K ⊂ Z whose preimage is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since its preimage is necessarily closed, the preimage must not be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' So, we can find a sequence tn in the domain of ℓ such that each ℓ(tn) ∈ K and ∣tn∣ → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then by passing to a subsequence, we can assume that ℓ(tn) converges to some point p ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Using Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='11, we can produce a generalized rectangle P containing p in its interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Therefore there are infinitely many ℓ(tn) ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Fix such an n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since ℓ is horizontal and doesn’t approach p as ∣t∣ → +∞, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='11 guarantees that the portion of the horizontal leaf through ℓ(tn) must enter, cut across one of the rectangles making up P and then exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' In particular the connected component of ℓ−1(P) containing tn is a closed and bounded interval I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' But then by hypothesis there is a tm in our sequence such that ℓ(tm) ∈ P and tm /∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This contradicts Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='7, which tells us that ℓ cannot later return to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let T = pqrs be a trapezoid with pq and rs parallel and of slope m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then the restriction of Fm to the interior of T consists of segments of leaves joining ps to qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Consider a parameterized leaf ℓ ∶ (a−,a+) → Z of slope m that intersects the interior T ○.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We claim that ℓ(t) must exit T or approach a point in ∂T as t approaches either endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If s is a sign and limt→as ℓ(t) is a singularity, this follows from the fact that T ○ has no singularities by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' On the other hand, if limt→as ℓ(t) is not a singularity, then ℓ must be a bi-infinite leaf or must be a separatrix with the limit to the other endpoint limt→a−s ℓ(t) a singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='12 guarantees that this parameterization can be made proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Setting ts to be the closest element of (a−,a+) to as such that ℓ(ts) ∈ T, we see ℓ exits T at ℓ(ts) and never returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Now consider a maximal segment of a leaf of Fm in T ○.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' From the previous paragraph, traversing such a maximal segment in either direction approaches a point in ∂T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The segment can’t approach a point in pq or in rs because these are trails of slope m with interior angles of π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' This means there are no prongs of slope m approaching points in pq or rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Now observe that the two points approached in the two different directions can’t lie on the same edge by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Thus each maximal segment must join ps to qr as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Trails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' A trail is an arc of a trail which is maximal with respect to the subarc partial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The following result tells us that trails exist, and every arc of a trail can be extended to a trail: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If γ is an arc of a trail in a zebra surface, then γ is a subarc of a trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let I ⊂ R be an interval, ¯I ⊂ R∪{±∞} be its closure, and γ ∶ I → Z be a parameterized arc of a trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let ℓ be a leaf which, since Z is a zebra surface, is not closed and thus is homeomorphic to an open interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If γ(a) is in ℓ then ℓ∖γ(a) has two connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We’ll say that γ finishes a leaf ℓ in the positive direction if there are a ∈ I and b ∈ ¯I with a < b such that γ(a) ∈ ℓ and ℓ∖γ([a,b)) has one connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' That is, there is a c ∈ (a,b] such that γ((a,c)) is one of the connected components of ℓ ∖ γ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then, any further ZEBRA SURFACES 21 extension of γ in the positive direction will require adding points not in ℓ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=', a singularity and a portion of a new leaf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We make a similar definition of finishing a leaf in the negative direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The following is the main ingredient in the proof of this theorem: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let Z be a zebra plane, let I ⊂ R be an interval with endpoints −1 and 1, and let γ ∶ I → Z be a parameterized arc of a trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (1) (Right limit) If limt→1− γ(t) exists, then there is a parameterized arc of a trail η ∶ I ∪ [1,2) → Z extending γ such that η finishes a leaf in the positive direction that γ does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (2) (Left limit) If limt→−1+ γ(t) exists, then γ can be similarly extended to left as an arc of a trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Conversely if neither limit exists, then γ is a trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We will prove statement (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Statement (2) will follow by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let γ(1) denote the limit limt→1− γ(t) (regardless of whether 1 is formally in the domain of γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' If γ(1) is not singular, let m be the slope of γ at γ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then we can extend γ by following the leaf of Fm through γ(1) until it finishes the leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Now suppose γ(1) is singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let L be a leaf with an endpoint at γ(1) which satisfies the angle condition at γ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' (One can see by inspection of the angle condition that such a leaf always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=') Then γ can be extended to finish L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' The final statement can be proved by showing the contrapositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Suppose γ ∶ I → Z is a proper subarc of an arc of a trail η ∶ J → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' By a change of coordinates, we may assume that I has endpoints −1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let φ ∶ I → J be the continuous orientation-preserving map satisfying γ = η ○ φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Since φ is not surjective, we may assume without loss of generality that the limit limt→1− φ(t) exists in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Then we have lim t→1− γ(t) = lim t→1− η ○ φ(t) = η( lim t→1− φ(t)) by continuity of η, so the limit in statement (1) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' □ Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Observe that it suffices to prove the statement for zebra planes, because arcs of trails on zebra surfaces are defined to be images of arcs of trails on their PRU cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' So, throughout this proof, we will only consider trails in a zebra plane Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' Let γ1 ∶ J1 → Z be an arc of a trail where J1 ⊂ R is a bounded interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfMwYs/content/2301.03727v1.pdf'} +page_content=' We produce k ∈ N ∪ {+∞} and a finite or infinite sequence of parameterized arcs of trails {γi ∶ Ji → Z}1≤i